Research ArticleA Class of Transformation Matrices and Its Applications

Wenhui Liu1 Feiqi Deng1 Jiarong Liang2 and Haijun Liu3

1 College of Automation Science and Engineering South China University of Technology Guangzhou 510641 China2 School of Computer Electronics and Information Guangxi University Nanning 530004 China3 School of Mathematics and Statistics Zhengzhou University Zhengzhou 450001 China

Correspondence should be addressed to Feiqi Deng aufqdengscuteducn

Received 8 November 2013 Revised 13 February 2014 Accepted 13 February 2014 Published 1 April 2014

Academic Editor Turgut Ozis

Copyright copy 2014 Wenhui Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies a class of transformation matrices and its applications Firstly we introduce a class of transformation matricesbetween two different vector operators and give some important properties of it Secondly we consider its two applications Thefirst one is to improve Qian Jilingrsquos formula And the second one is to deal with the observability of discrete-time stochastic linearsystems with Markovian jump and multiplicative noises A new necessary and sufficient condition for the weak observability willbe given in the second application

1 Introduction

The vector operator is an important concept in matrix analy-sis and an effective mathematical tool in many applications[1] For the symmetric case we will consider a new vectoroperator We find that there exists a class of transformationmatrices between the two vector operators and they can beused to solve many problems In order to demonstrate theimportance of these transformation matrices this paper willconsider their two applications in control theory

Qian Jilingrsquos formula is an important and interestingformula to compute the Lyapunov functions [2] Our paperwill improve Qian Jilingrsquos formula by the help of thesetransformation matrices

Observability and detectability are two basic concepts incontrol theory [3] These concepts of deterministic systemshad been extended to the stochastic systems over the past fewdecades such as [4ndash11] and references therein Particularlythis paper will focus on the observabilitydetectability fordiscrete-time stochastic linear systems The definition ofthe uniform observabilitydetectability [4] for determin-istic discrete-time time-varying linear systems had beenextended to stochastic linear systems in [5] Actually theweakobservabilitydetectability in [5] and the uniform observabil-itydetectability in [4] are consistent Reference [5] showedthat the weak detectability was weaker than the mean square

detectability Still the weak detectability plays the samerole in the discussion of algebraic Lyapunov and Riccatiequations Under the same framework [6] investigated theobservability (ie the weak observability in [5]) problemsfor a class of discrete-time stochastic linear systems subjectto Markovian jump and multiplicative noises and got somenecessary and sufficient conditions Reference [7] studiedthe equivalence between two different definitions of theobservabilitydetectability for discrete-time stochastic linearsystems These are good works about the observability anddetectability of discrete-time stochastic linear systems How-ever our paper will further study the weak observability fordiscrete-time stochastic linear systems with the help of thesetransformation matrices The system in [6] is more compli-cated than ours but we get some more profound conclusionswhich are not included in [6] Although the systems in [5 7]are two special cases of our system our results are not theirparallel results We need these transformation matrices inorder to get these new results

The outline of this paper is as follows In Section 2we define a class of transformation matrices which existsindependently and study its important properties Section 3considers its applications We use it to improve Qian Jilingrsquosformula in Section 31 We use it to deal with the weakobservability of discrete-time stochastic linear systems withMarkovian jump and multiplicative noises in Section 32

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 742098 11 pageshttpdxdoiorg1011552014742098

2 Abstract and Applied Analysis

A new necessary and sufficient condition for the weakobservability will be givenNotations 119877119899 is 119899-dimensional real Euclidean space 119862119899 is 119899-dimensional complex space 119860119879 |119860| and tr(119860) respectivelydenote the transpose determinant and trace of 119860 119860 gt

(ge)0 means that 119860 is a symmetric positive (semipositive)definite matrix E(sdot) denotes the mathematical expectationof a random variable 119885+ ≜ 0 1 2 119878 ≜ 1 2 119904where 119904 is a positive integer 119877119899times119898 denotes 119899 times 119898 real matrixspace with the inner product ⟨119860 119861⟩ ≜ tr(119860119879119861) 119877119899times119898

119904≜

(1198801 119880

119904) | 119880

119894isin 119877119899times119898

119894 isin 119878 with the inner product⟨119880 119881⟩ ≜ sum

119904

119894=1tr(119880119879119894119881119894) 1119860(119909) ≜

1 119909isinA0 119909notin119860

2 Transformation Matrix and Its Properties

21 Transformation Matrix between Two Vector Operators

Definition 1 (see [1]) 1198711 119862119899times119898

rarr 119862119899119898times1 is called a

vector operator by column if 1198711(119883) = (119909

11 11990921 119909

1198991 11990912

11990922 119909

1198992 119909

1119898 1199092119898 119909

119899119898)119879 for any119883 = (119909

119894119895)119899times119898

Definition 2 1198712 119882119899times119899

rarr 119862(119899(119899+1)2)times1 is called a half

vector operator by column if 1198712(119883) = (119909

11 11990921 119909

1198991

11990922 119909

1198992 119909

119899119899)119879 for any 119883 = (119909

119894119895)119899times119899

where 119882119899times119899 ≜119883 | 119883

119879

= 119883 isin 119862119899times119899

Definition 3 (see [1]) For matrices 119860 = (119886119894119895)119899times119898

and 119861 =

(119887119894119895)119904times119905 one calls matrix 119860 otimes 119861 = (119886

119894119895119861)119899times119898

the Kroneckerproduct of matrices 119860 and 119861

It is easy to find that 1198711and 119871

2have many similar

properties such as the following

(a) 1198711and 119871

2are all linear operators

(b) 119883 = 119884 hArr 1198711(119883) = 119871

1(119884)

(c) 119883 = 119884 hArr 1198712(119883) = 119871

2(119884) where119883119884 isin 119882

119899times119899

For convenience we will adopt the following notations119864119899119894119895

denotes a 119899 times 119899 matrix its (119894 119895)-element is 1 and itis zero elsewhere

119867(119899) denotes a 1198992times(119899(119899+1)2)matrix Its column vectorsare denoted by 119867

11

11986721

1198671198991

11986722

1198671198992

119867119899119899

from left to right and row vectors are denoted by11986711 11986721 119867

1198991 119867

1119899 1198672119899 119867

119899119899from top to bottom

where

119867119894119895

≜

1198711(119864119899119894119895

) 119894 = 119895

1198711(119864119899119894119895

+ 119864119899119895119894

) 119894 gt 119895

(1)

It is easy to prove

119867119894119895=

119871119879

2(119864119899119895119894

) 119894 lt 119895

119871119879

2(119864119899119894119895

) 119894 ge 119895

(2)

We will denote119867 = 119867119899≜ 119867(119899)

119867minus

(119899) denotes a (119899(119899 + 1)2) times 1198992 matrix Its

column vectors are denoted by 119867minus11

119867minus21

119867minus1198991

119867minus1119899

119867minus2119899

119867minus119899119899 from left to right and row vectors are

denoted by 119867minus

11 119867minus

21 119867

minus

1198991 119867minus

22 119867

minus

1198992 119867

minus

119899119899from

top to bottom where 119867minus

119894119895≜ 119871119879

1(119864119899119894119895

) We will denote119867minus

= 119867minus

119899≜ 119867minus

(119899)For example

119867(1) = [1] 119867minus

(1) = [1]

119867 (2) =[[[

[

1 0 0

0 1 0

0 1 0

0 0 1

]]]

]

119867minus

(2) = [

[

1 0 0 0

0 1 0 0

0 0 0 1

]

]

(3)

and so forthConsidering Property 3 we generally call 119867 and 119867

minus

transformation matrices between 1198711and 119871

2

22 Properties of Transformation Matrix

Property 1 Consider

119867minus

119899119867119899= 119868119899(119899+1)2

(4)

Proof For 119896 = 119897

119867minus

119894119895119867119896119897

= 119871119879

1(119864119899119894119895

) 1198711(119864119899119896119896

) = 1 119894 = 119895 = 119896

0 others(5)

For 119896 gt 119897

119867minus

119894119895119867119896119897

= 119871119879

1(119864119899119894119895

) 1198711(119864119899119896119897

+ 119864119899119897119896

) = 1 119894 = 119896 119895 = 119897

0 others(6)

Thus119867minus119899119867119899= 119868119899(119899+1)2

Remark 4 Generally119867119899119867minus

119899= 1198681198992 such as

1198672119867minus

2=[[[

[

1 0 0 0

0 1 0 0

0 1 0 0

0 0 0 1

]]]

]

(7)

Property 2 Consider119867119867minus

119867 = 119867119867minus119867119867minus = 119867minus (119867minus119867)119879 = 119867minus119867

Remark 5 Generally (119867119867minus)119879 =119867119867minus such as119867

2119867minus

2which is

not a symmetric matrixThis indicates that119867minus is the 1 2 3-generalized inverse matrix of 119867 but not the 1 2 3 4-generalized inverse matrix of119867 (see [1])

Property 3 If 119883119879 = 119883 ≜ (119909119894119895)119899times119899

then 1198671198712(119883) = 119871

1(119883)

119867minus

1198711(119883) = 119871

2(119883)

Abstract and Applied Analysis 3

Proof Consider

∵ 1198671198941198951198712(119883) = 119871

119879

2(119864119899119894119895

) 1198712(119883) = 119909

119894119895(119894 ge 119895)

1198671198941198951198712(119883) = 119871

119879

2(119864119899119895119894

) 1198712(119883) = 119909

119895119894= 119909119894119895

(119894 lt 119895)

there4 1198671198712(119883) = 119871

1(119883)

1198712(119883) = 119867

minus

1198711(119883) (by Property 1)

(8)

Property 4 For matrices 119860119899times119899 119861119903times119899

and 119883119879119899times119899

= 119883 we havethe following

(a) 1198711(119860119883 + 119883119860

119879

) = (119868 otimes 119860 + 119860 otimes 119868)1198711(119883)

(b) 1198711(119861119883119861

119879

) = (119861 otimes 119861)1198711(119883)

(c) 1198712(119860119883 + 119883119860

119879

) = 119867minus

(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)

(d) 1198712(119861119883119861

119879

) = 119867minus

(119861 otimes 119861)1198671198712(119883)

Proof We can get (a) and (b) in [1] so we only need to prove(c) and (d)

Obviously119860119883+119883119860119879 and 119861119883119861119879 are symmetricalThere-fore

1198712(119860119883 + 119883119860

119879

) = 119867minus

1198711(119860119883 + 119883119860

119879

)

= 119867minus

(119868 otimes 119860 + 119860 otimes 119868) 1198711(119883)

= 119867minus

(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)

1198712(119861119883119861

119879

) = 119867minus

1198711(119861119883119861

119879

) = 119867minus

(119861 otimes 119861)1198671198712(119883)

(9)

Remark 6 If 1198711and 119871

2are defined by row that is

1198711(119883) = (119909

11 11990912 119909

1119898 11990921 11990922

1199092119898 119909

1198991 1199091198992 119909

119899119898)119879 for 119883 isin 119862

119899times119898

1198712(119883) = (119909

11 11990912 119909

1119899 11990922

1199092119899 119909

119899119899)119879 for 119883 isin 119882

119899times119899

(10)

we find that Properties 3 and 4 are still true

Property 5 For all 119860 isin 119862119899times119899 then

(a) 120590(119860 otimes 119860) = 120590(119867minus

(119860 otimes 119860)119867) = 120590(119867minus

(119860119879

otimes 119860119879

)119867)

(b) 120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868+119868otimes119860)119867) = 120590(119867minus(119860119879otimes119868 + 119868 otimes 119860

119879

)119867)

where 120590(sdot) denotes the set of all eigenvalues of matrix

Proof (a) Let

1205901(119860) ≜ 120582 isin 119862 | 119860119883119860

119879

= 120582119883119883 = 0 119883 isin 119862119899times119899

1205902(119860) ≜ 120582 isin 119862 | 119860119883119860

119879

= 120582119883119883119879

= 119883 = 0119883 isin 119862119899times119899

∵ 119860119883119860119879

= 120582119883 lArrrArr (119860 otimes 119860) 1198711(119883) = 120582119871

1(119883)

there4 1205901(119860) = 120590 (119860 otimes 119860)

(11)

If 119883119879 = 119883 then 119860119883119860119879 = 120582119883 hArr 119867minus

(119860 otimes 119860)1198671198712(119883) =

1205821198712(119883) Thus 120590

2(119860) = 120590(119867

minus

(119860 otimes 119860)119867)Next we prove 120590

1(119860) = 120590

2(119860) Obviously 120590

2(119860) sube 120590

1(119860)

so we only need to prove 1205901(119860) sube 120590

2(119860)

For any 120582 isin 1205901(119860) exist isin 120590(119860) such that 120582 = 119909 =

119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862

119899 (see [1]) Let119883 ≜ 119909119910

119879

119885 ≜ 119883 + 119883119879 then

120582119885 = 119909119910119879

+ 119910119909119879

= 119860119909119910119879

119860119879

+ 119860119910119909119879

119860119879

= 119860119885119860119879

(12)

If 119885 = 0 then 120582 isin 1205902(119860) Actually we always have 119885 = 0

Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie

119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let

119909119894= 0 119910119895= 0 then 119910

119894= 0 119909

119895= 0 119911

119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0

This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119860) = 120590

1(119860) = 120590

2(119860) = 120590(119867

minus

(119860 otimes 119860)119867)And because 120590(119860 otimes 119860) = 120590

1(119860119879

otimes 119860119879

) = 120590(119867minus

(119860119879

otimes 119860119879

)119867)therefore 120590(119860 otimes 119860) = 120590(119867minus(119860 otimes 119860)119867) = 120590(119867minus(119860119879 otimes 119860119879)119867)

(b) Let

1205903(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860

119879

= 120582119883119883 = 0 119883 isin 119862119899times119899

1205904(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860

119879

= 120582119883119883119879

= 119883 = 0119883 isin 119862119899times119899

∵ 119860119883 + 119883119860119879

= 120582119883 lArrrArr (119860 otimes 119868 + 119868 otimes 119860) 1198711(119883) = 120582119871

1(119883)

there4 1205903(119860) = 120590 (119860 otimes 119868 + 119868 otimes 119860)

(13)

If 119883119879 = 119883 then 119860119883 + 119883119860119879

= 120582119883 hArr 119867minus

(119860 otimes 119868 + 119868 otimes

119860)1198671198712(119883) = 120582119871

2(119883)

Thus 1205904(119860) = 120590(119867

minus

(119860 otimes 119868 + 119868 otimes 119860)119867)Next we prove 120590

3(119860) = 120590

4(119860) Obviously 120590

4(119860) sube 120590

3(119860)

so we only need to prove 1205903(119860) sube 120590

4(119860)

For any 120582 isin 1205903(119860) exist isin 120590(119860) such that 120582 = +

119909 = 119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862

119899 (see [1])Let119883 ≜ 119909119910

119879

119885 ≜ 119883 + 119883119879 then

120582119885 = 119909119910119879

+ 119909119910119879

+ 119910119909119879

+ 119910119909119879

= 119860119909119910119879

+ 119909119910119879

119860119879

+ 119910119909119879

119860119879

+ 119860119910119909119879

= 119860119885 + 119885119860119879

(14)

If 119885 = 0 then 120582 isin 1205904(119860) Actually we always have 119885 = 0

Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie

119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let

4 Abstract and Applied Analysis

119909119894= 0 119910119895= 0 then 119910

119894= 0 119909

119895= 0 119911

119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0

This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

3(119860) = 120590

4(119860) = 120590(119867

minus

(119860 otimes 119868 +

119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

1(119860119879

otimes 119868 + 119868 otimes 119860119879

) =

120590(119867minus

(119860119879

otimes119868+119868otimes119860119879

)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867

minus

(119860119879

otimes 119868 + 119868 otimes 119860119879

)119867)

3 Applications of Transformation Matrix

31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices

Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877

119899times119899 is a Hurwitz matrix) thenone has the following

(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique

solution 119884 isin 119877119899times119899

(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909

119879

119884119909 such that 119860119879119884 + 119884119860 =

minus119863 and the expression is

119881 =1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

(15)

where

Δ = 119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)119867

119883 = (1198831 1198832 119883

119899)

1198831= (1199092

1 211990911199092 2119909

1119909119899)

1198832= (1199092

2 211990921199093 2119909

2119909119899)

119883119899= (1199092

119899)

(16)

Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int

infin

0

119890119905119860119879

119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz

matrix so 119884 gt 0 and

119860119879

119884 + 119884119860 = 119860119879

int

infin

0

119890119905119860119879

119863119890119905119860d119905 + int

infin

0

119890119905119860119879

119863119890119905119860d119905119860

= int

infin

0

d (119890119905119860119879

119863119890119905119860

) = minus119863

(17)

By (a) the quadratic Lyapunov function 119881 = 119909119879

119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique

Next we prove that 119881 can be expressed as (15) Note that119860119879

119884 + 119884119860 = minus119863 then

119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)1198671198712(119884) = minus119871

2(119863) (18)

By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and

119910119894119895=

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|(1 le 119895 le 119894 le 119899) (19)

where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ

and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors

are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)

By expanding the determinant we have

1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

=

119899

sum

119894=1

1199092

119894

10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816

|Δ|

+ sum

1le119895lt119894le119899

2119909119895119909119894

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|

=

119899

sum

119894119895=1

119909119894119910119894119895119909119895= 119881

(20)

For the uniqueness of 119881 (15) is the desired expression

Remark 8 The dimensions of Δ and [0 119883

1198712(119863) Δ

] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus

32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices

321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) + 119861120579(119896)

119906 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(21)

for 119896 isin 119885+ where 119909(119896) isin 119877

119899 119906(119896) isin 119877119898 and 119910(119896) isin

119877119903 respectively denote the system state control input and

measured output 119860119894119895isin 119877119899times119899 119861

119895isin 119877119899times119898 and 119862

119894119895isin 119877119903times119899

120579(119896) 119896 isin 119885+

is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is

Abstract and Applied Analysis 5

119875 = (119901119894119895)119904times119904

and initial distribution is 1205830

= 1205830119894

≜

119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908

119873(119896) isin 119877 are wide sense

stationary processes and independent of each other such that

E (119908119894(119896)) = 0 E (119908

119894(119896) 119908119895(119896)) = 120575

119894119895≜

1 119894 = 119895

0 119894 = 119895(22)

and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain

120579(119896) 119896 isin 119885+

For the stochastic system (21) its solution and output

processes with 119909(0) = 1199090and 120579(0) sim 120583

0are respectively

denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909

0 1205830) We will simply

denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909

0 1205830) 119910(119896) =

119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909

0 1205830)

If 119906(119896) equiv 0 the stochastic system (21) becomes

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(23)

for 119896 isin 119885+For convenience we will use the following notations

119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119894)

119883 (119896) ≜ (1198831(119896) 119883

119904(119896))

119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))

119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1

120579(119896)=119894)

(119896) ≜ (1198841(119896) 119884

119904(119896))

119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))

W0≜ 119883 (0) | 119909

0isin 119877119899

1205830isin Θ119883

119894(0) ≜ 119909

0119909119879

01205830119894 119894 isin 119878

(24)

It is easy to get 1198831(119896) + sdot sdot sdot + 119883

119904(119896) = 119883(119896) 119884

1(119896) +

sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =

sum119904

119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877

119899times119899

119904

Specifically E|119909(0)|2 = |1199090|2

For the system (23) we define several operators in119877119899times119899119904

asfollows

E119894(119880) =

119904

sum

119895=1

119901119894119895119880119895

F119894(119880) =

119873

sum

119897=0

119860119879

119897119894E119894(119880)119860

119897119894=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894119880119895119860119897119894

Flowast

119894(119880) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895119880119895119860119879

119897119895

E (119880) = (E1(119880) E

119904(119880))

F (119880) = (F1(119880) F

119904(119880))

Flowast

(119880) = (Flowast

1(119880) F

lowast

119904(119880))

F119896

(119880)

= F (F119896minus1

(119880)) (119896 = 1 2 ) where F0

(119880) = 119880

Flowast119896

(119880)

= Flowast

(Flowast119896minus1

(119880)) (119896 = 1 2 ) where Flowast0

(119880) = 119880

(25)

Flowast and F are called dual operators for each other (seeLemma 15)

Also we define the operator O119894(119897) = 119862

119894+ F119894(O(119897 minus

1)) for 119894 isin 119878 119897 = 1 2 where

119862119894=

119873

sum

119897=0

119862119879

119897119894119862119897119894

O (0) = 0

O (119897) = (O1(119897) O

119904(119897))

(26)

Then

O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862

119904))

= 119862 +F (119862) +F2

(119862) + sdot sdot sdot +F119897minus1

(119862)

119897 = 1 2

(27)

322 Some Preliminaries and Auxiliary Results Let

119882119897

119894(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119904

sum

119894=1

119882119897

119894(119883 (119905))

for 119905 isin 119885+ 119897 = 1 2

(28)

Lemma 9 For the system (23) one has O119894(119897) ge O

119894(119897 minus 1) for

all 119894 isin 119878 119897 = 1 2

Proof Firstly it is easy to get O119894(1) = 119862

119894= sum119873

119897=0119862119879

119897119894119862119897119894ge 0 =

O119894(0) for all 119894 isin 119878 Secondly we assume thatO

119894(119897) ge O

119894(119897minus1)for

6 Abstract and Applied Analysis

all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave

O119894(119898 + 1) minus O

119894(119898)

= F119894(O (119898)) minusF

119894(O (119898 minus 1))

=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894(O119895(119898) minus O

119895(119898 minus 1))119860

119897119894

ge 0

(29)

By mathematical induction it is right

Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =

F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1

Proof It is easy to get the result by mathematical inductionso we will omit the details

Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

(ie for all 119883(0) = 1198830isinW0) one has

119883 (119896 + 1) = Flowast

(119883 (119896)) for 119896 isin 119885+

119883 (119896 + 119905) = Flowast119905

(119883 (119896)) for 119896 119905 isin 119885+(30)

Proof For all 119894 isin 119878 we have

Flowast

119894(119883 (119896)) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119895)119860119879

119897119895

=

119873

sum

119897=0

119904

sum

119895=1

119901119895119894E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896)=119895

)

=

119873

sum

119897=0

E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896+1)=119894

)

=E

[

[

1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

times [

[

1198600120579(119896)

119909 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896+1)=119894

=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894

= 119883119894(119896 + 1) 119896 isin 119885

+

(31)

then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin

119885+

Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

for 119905 isin 119885+ 119897 = 1 2

(32)

Proof Because

119884119894(119896) = E

[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

times[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896)=119894

=

119873

sum

119897=0

119862119897119894119883119894(119896) 119862119879

119897119894

(33)

therefore

⟨119883119894(119896) 119862

119894⟩ = tr[119883

119894(119896)(

119873

sum

119897=0

119862119879

119897119894119862119897119894)] = tr (119884

119894(119896))

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩ =

119897minus1

sum

119896=0

tr (119884119894(119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

⟨119883 (119896) 119862⟩ =

119904

sum

119894=1

tr (119883119894(119896) 119862119894) = tr (119884 (119896))

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119897minus1

sum

119896=0

tr (119884 (119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

(34)

Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897

119894(119883(119905))

Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2

Proof It is easy to get the result by mathematical inductionso we will omit the details

Abstract and Applied Analysis 7

Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119898 isin 119885+

119905 isin 119885+ one has

⟨119883 (119905) F119897

(O (119898))⟩ = ⟨Flowast119897

(119883 (119905)) O (119898)⟩ 119897 isin 119885+

(35)

Proof Firstly it is easy to get

⟨119883 (119905) F0

(O (119898))⟩

= ⟨119883 (119905) O (119898)⟩

= ⟨Flowast0

(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+

(36)

Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have

⟨119883 (119905) F119902+1

(O (119898))⟩ = ⟨119883 (119905) F119902

(O (119898 + 1))⟩

minus ⟨119883 (119905) F119902

(119862)⟩

= ⟨Flowast119902

(119883 (119905)) O (119898 + 1)⟩

minus ⟨Flowast119902

(119883 (119905)) 119862⟩

= ⟨119883 (119905 + 119902) O (119898 + 1)⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= 119882119898+1

(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩

=

119898

sum

119896=0

⟨119883 (119905 + 119902 + 119896) 119862⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= ⟨119883 (119905 + 119902 + 1) O (119898)⟩

= ⟨Flowast119902+1

(119883 (119905)) O (119898)⟩

(37)

By mathematical induction it is right

Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ 119905 isin 119885+ one has

⟨119883 (119905) F119897

(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)

Proof Consider

⟨119883 (119905) F119897

(119862)⟩ = ⟨119883 (119905) F119897

(O (1))⟩

= ⟨Flowast119897

(119883 (119905)) O (1)⟩

= ⟨119883 (119905 + 119897) 119862⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]

(39)

In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1

Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one

has

119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)

Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

lim119896rarrinfin

E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin

E [119909119879 (119896) 119909 (119896)] = 0(41)

Proof This proof is omitted

Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899

119903 ≜ rank(119883) then thereexist nonzero vectors 119909

1 119909

119903isin 119877119899 such that 119883 = 119909

1119909119879

1+

sdot sdot sdot + 119909119903119909119879

119903

323 A Useful Formula Define two operators L1(119880) =

[

1198711(1198801)

1198711(119880119904)

] for 119880119894isin 119877119899times119899 andL

2(119880) = [

1198712(1198801)

1198712(119880119904)

] for 119880119894isin 119883 |

119883119879

= 119883 isin 119877119899times119899

in 119877119899times119899119904

For convenience we naturally think 119884 isin 119883 | 119883

119879

= 119883 isin

119877119899times119899

when we use 1198712(119884) in this subsection

Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899

119904 one has

L1(F (119880)) = AL

1(119880)

L2(F (119880)) = BL

2(119880)

⟨119880 119881⟩ = L119879

1(119880)L

1(119881)

⟨119880 119881⟩ = L119879

1(119880) (119868 otimes 119867)L

2(119881)

(42)

where

A ≜

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971 119860

119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

B ≜

119873

sum

119897=0

diag (119867minus (1198601198791198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

(43)

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

A new necessary and sufficient condition for the weakobservability will be givenNotations 119877119899 is 119899-dimensional real Euclidean space 119862119899 is 119899-dimensional complex space 119860119879 |119860| and tr(119860) respectivelydenote the transpose determinant and trace of 119860 119860 gt

(ge)0 means that 119860 is a symmetric positive (semipositive)definite matrix E(sdot) denotes the mathematical expectationof a random variable 119885+ ≜ 0 1 2 119878 ≜ 1 2 119904where 119904 is a positive integer 119877119899times119898 denotes 119899 times 119898 real matrixspace with the inner product ⟨119860 119861⟩ ≜ tr(119860119879119861) 119877119899times119898

119904≜

(1198801 119880

119904) | 119880

119894isin 119877119899times119898

119894 isin 119878 with the inner product⟨119880 119881⟩ ≜ sum

119904

119894=1tr(119880119879119894119881119894) 1119860(119909) ≜

1 119909isinA0 119909notin119860

2 Transformation Matrix and Its Properties

21 Transformation Matrix between Two Vector Operators

Definition 1 (see [1]) 1198711 119862119899times119898

rarr 119862119899119898times1 is called a

vector operator by column if 1198711(119883) = (119909

11 11990921 119909

1198991 11990912

11990922 119909

1198992 119909

1119898 1199092119898 119909

119899119898)119879 for any119883 = (119909

119894119895)119899times119898

Definition 2 1198712 119882119899times119899

rarr 119862(119899(119899+1)2)times1 is called a half

vector operator by column if 1198712(119883) = (119909

11 11990921 119909

1198991

11990922 119909

1198992 119909

119899119899)119879 for any 119883 = (119909

119894119895)119899times119899

where 119882119899times119899 ≜119883 | 119883

119879

= 119883 isin 119862119899times119899

Definition 3 (see [1]) For matrices 119860 = (119886119894119895)119899times119898

and 119861 =

(119887119894119895)119904times119905 one calls matrix 119860 otimes 119861 = (119886

119894119895119861)119899times119898

the Kroneckerproduct of matrices 119860 and 119861

It is easy to find that 1198711and 119871

2have many similar

properties such as the following

(a) 1198711and 119871

2are all linear operators

(b) 119883 = 119884 hArr 1198711(119883) = 119871

1(119884)

(c) 119883 = 119884 hArr 1198712(119883) = 119871

2(119884) where119883119884 isin 119882

119899times119899

For convenience we will adopt the following notations119864119899119894119895

denotes a 119899 times 119899 matrix its (119894 119895)-element is 1 and itis zero elsewhere

119867(119899) denotes a 1198992times(119899(119899+1)2)matrix Its column vectorsare denoted by 119867

11

11986721

1198671198991

11986722

1198671198992

119867119899119899

from left to right and row vectors are denoted by11986711 11986721 119867

1198991 119867

1119899 1198672119899 119867

119899119899from top to bottom

where

119867119894119895

≜

1198711(119864119899119894119895

) 119894 = 119895

1198711(119864119899119894119895

+ 119864119899119895119894

) 119894 gt 119895

(1)

It is easy to prove

119867119894119895=

119871119879

2(119864119899119895119894

) 119894 lt 119895

119871119879

2(119864119899119894119895

) 119894 ge 119895

(2)

We will denote119867 = 119867119899≜ 119867(119899)

119867minus

(119899) denotes a (119899(119899 + 1)2) times 1198992 matrix Its

column vectors are denoted by 119867minus11

119867minus21

119867minus1198991

119867minus1119899

119867minus2119899

119867minus119899119899 from left to right and row vectors are

denoted by 119867minus

11 119867minus

21 119867

minus

1198991 119867minus

22 119867

minus

1198992 119867

minus

119899119899from

top to bottom where 119867minus

119894119895≜ 119871119879

1(119864119899119894119895

) We will denote119867minus

= 119867minus

119899≜ 119867minus

(119899)For example

119867(1) = [1] 119867minus

(1) = [1]

119867 (2) =[[[

[

1 0 0

0 1 0

0 1 0

0 0 1

]]]

]

119867minus

(2) = [

[

1 0 0 0

0 1 0 0

0 0 0 1

]

]

(3)

and so forthConsidering Property 3 we generally call 119867 and 119867

minus

transformation matrices between 1198711and 119871

2

22 Properties of Transformation Matrix

Property 1 Consider

119867minus

119899119867119899= 119868119899(119899+1)2

(4)

Proof For 119896 = 119897

119867minus

119894119895119867119896119897

= 119871119879

1(119864119899119894119895

) 1198711(119864119899119896119896

) = 1 119894 = 119895 = 119896

0 others(5)

For 119896 gt 119897

119867minus

119894119895119867119896119897

= 119871119879

1(119864119899119894119895

) 1198711(119864119899119896119897

+ 119864119899119897119896

) = 1 119894 = 119896 119895 = 119897

0 others(6)

Thus119867minus119899119867119899= 119868119899(119899+1)2

Remark 4 Generally119867119899119867minus

119899= 1198681198992 such as

1198672119867minus

2=[[[

[

1 0 0 0

0 1 0 0

0 1 0 0

0 0 0 1

]]]

]

(7)

Property 2 Consider119867119867minus

119867 = 119867119867minus119867119867minus = 119867minus (119867minus119867)119879 = 119867minus119867

Remark 5 Generally (119867119867minus)119879 =119867119867minus such as119867

2119867minus

2which is

not a symmetric matrixThis indicates that119867minus is the 1 2 3-generalized inverse matrix of 119867 but not the 1 2 3 4-generalized inverse matrix of119867 (see [1])

Property 3 If 119883119879 = 119883 ≜ (119909119894119895)119899times119899

then 1198671198712(119883) = 119871

1(119883)

119867minus

1198711(119883) = 119871

2(119883)

Abstract and Applied Analysis 3

Proof Consider

∵ 1198671198941198951198712(119883) = 119871

119879

2(119864119899119894119895

) 1198712(119883) = 119909

119894119895(119894 ge 119895)

1198671198941198951198712(119883) = 119871

119879

2(119864119899119895119894

) 1198712(119883) = 119909

119895119894= 119909119894119895

(119894 lt 119895)

there4 1198671198712(119883) = 119871

1(119883)

1198712(119883) = 119867

minus

1198711(119883) (by Property 1)

(8)

Property 4 For matrices 119860119899times119899 119861119903times119899

and 119883119879119899times119899

= 119883 we havethe following

(a) 1198711(119860119883 + 119883119860

119879

) = (119868 otimes 119860 + 119860 otimes 119868)1198711(119883)

(b) 1198711(119861119883119861

119879

) = (119861 otimes 119861)1198711(119883)

(c) 1198712(119860119883 + 119883119860

119879

) = 119867minus

(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)

(d) 1198712(119861119883119861

119879

) = 119867minus

(119861 otimes 119861)1198671198712(119883)

Proof We can get (a) and (b) in [1] so we only need to prove(c) and (d)

Obviously119860119883+119883119860119879 and 119861119883119861119879 are symmetricalThere-fore

1198712(119860119883 + 119883119860

119879

) = 119867minus

1198711(119860119883 + 119883119860

119879

)

= 119867minus

(119868 otimes 119860 + 119860 otimes 119868) 1198711(119883)

= 119867minus

(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)

1198712(119861119883119861

119879

) = 119867minus

1198711(119861119883119861

119879

) = 119867minus

(119861 otimes 119861)1198671198712(119883)

(9)

Remark 6 If 1198711and 119871

2are defined by row that is

1198711(119883) = (119909

11 11990912 119909

1119898 11990921 11990922

1199092119898 119909

1198991 1199091198992 119909

119899119898)119879 for 119883 isin 119862

119899times119898

1198712(119883) = (119909

11 11990912 119909

1119899 11990922

1199092119899 119909

119899119899)119879 for 119883 isin 119882

119899times119899

(10)

we find that Properties 3 and 4 are still true

Property 5 For all 119860 isin 119862119899times119899 then

(a) 120590(119860 otimes 119860) = 120590(119867minus

(119860 otimes 119860)119867) = 120590(119867minus

(119860119879

otimes 119860119879

)119867)

(b) 120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868+119868otimes119860)119867) = 120590(119867minus(119860119879otimes119868 + 119868 otimes 119860

119879

)119867)

where 120590(sdot) denotes the set of all eigenvalues of matrix

Proof (a) Let

1205901(119860) ≜ 120582 isin 119862 | 119860119883119860

119879

= 120582119883119883 = 0 119883 isin 119862119899times119899

1205902(119860) ≜ 120582 isin 119862 | 119860119883119860

119879

= 120582119883119883119879

= 119883 = 0119883 isin 119862119899times119899

∵ 119860119883119860119879

= 120582119883 lArrrArr (119860 otimes 119860) 1198711(119883) = 120582119871

1(119883)

there4 1205901(119860) = 120590 (119860 otimes 119860)

(11)

If 119883119879 = 119883 then 119860119883119860119879 = 120582119883 hArr 119867minus

(119860 otimes 119860)1198671198712(119883) =

1205821198712(119883) Thus 120590

2(119860) = 120590(119867

minus

(119860 otimes 119860)119867)Next we prove 120590

1(119860) = 120590

2(119860) Obviously 120590

2(119860) sube 120590

1(119860)

so we only need to prove 1205901(119860) sube 120590

2(119860)

For any 120582 isin 1205901(119860) exist isin 120590(119860) such that 120582 = 119909 =

119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862

119899 (see [1]) Let119883 ≜ 119909119910

119879

119885 ≜ 119883 + 119883119879 then

120582119885 = 119909119910119879

+ 119910119909119879

= 119860119909119910119879

119860119879

+ 119860119910119909119879

119860119879

= 119860119885119860119879

(12)

If 119885 = 0 then 120582 isin 1205902(119860) Actually we always have 119885 = 0

Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie

119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let

119909119894= 0 119910119895= 0 then 119910

119894= 0 119909

119895= 0 119911

119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0

This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119860) = 120590

1(119860) = 120590

2(119860) = 120590(119867

minus

(119860 otimes 119860)119867)And because 120590(119860 otimes 119860) = 120590

1(119860119879

otimes 119860119879

) = 120590(119867minus

(119860119879

otimes 119860119879

)119867)therefore 120590(119860 otimes 119860) = 120590(119867minus(119860 otimes 119860)119867) = 120590(119867minus(119860119879 otimes 119860119879)119867)

(b) Let

1205903(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860

119879

= 120582119883119883 = 0 119883 isin 119862119899times119899

1205904(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860

119879

= 120582119883119883119879

= 119883 = 0119883 isin 119862119899times119899

∵ 119860119883 + 119883119860119879

= 120582119883 lArrrArr (119860 otimes 119868 + 119868 otimes 119860) 1198711(119883) = 120582119871

1(119883)

there4 1205903(119860) = 120590 (119860 otimes 119868 + 119868 otimes 119860)

(13)

If 119883119879 = 119883 then 119860119883 + 119883119860119879

= 120582119883 hArr 119867minus

(119860 otimes 119868 + 119868 otimes

119860)1198671198712(119883) = 120582119871

2(119883)

Thus 1205904(119860) = 120590(119867

minus

(119860 otimes 119868 + 119868 otimes 119860)119867)Next we prove 120590

3(119860) = 120590

4(119860) Obviously 120590

4(119860) sube 120590

3(119860)

so we only need to prove 1205903(119860) sube 120590

4(119860)

For any 120582 isin 1205903(119860) exist isin 120590(119860) such that 120582 = +

119909 = 119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862

119899 (see [1])Let119883 ≜ 119909119910

119879

119885 ≜ 119883 + 119883119879 then

120582119885 = 119909119910119879

+ 119909119910119879

+ 119910119909119879

+ 119910119909119879

= 119860119909119910119879

+ 119909119910119879

119860119879

+ 119910119909119879

119860119879

+ 119860119910119909119879

= 119860119885 + 119885119860119879

(14)

If 119885 = 0 then 120582 isin 1205904(119860) Actually we always have 119885 = 0

Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie

119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let

4 Abstract and Applied Analysis

119909119894= 0 119910119895= 0 then 119910

119894= 0 119909

119895= 0 119911

119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0

This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

3(119860) = 120590

4(119860) = 120590(119867

minus

(119860 otimes 119868 +

119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

1(119860119879

otimes 119868 + 119868 otimes 119860119879

) =

120590(119867minus

(119860119879

otimes119868+119868otimes119860119879

)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867

minus

(119860119879

otimes 119868 + 119868 otimes 119860119879

)119867)

3 Applications of Transformation Matrix

31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices

Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877

119899times119899 is a Hurwitz matrix) thenone has the following

(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique

solution 119884 isin 119877119899times119899

(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909

119879

119884119909 such that 119860119879119884 + 119884119860 =

minus119863 and the expression is

119881 =1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

(15)

where

Δ = 119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)119867

119883 = (1198831 1198832 119883

119899)

1198831= (1199092

1 211990911199092 2119909

1119909119899)

1198832= (1199092

2 211990921199093 2119909

2119909119899)

119883119899= (1199092

119899)

(16)

Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int

infin

0

119890119905119860119879

119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz

matrix so 119884 gt 0 and

119860119879

119884 + 119884119860 = 119860119879

int

infin

0

119890119905119860119879

119863119890119905119860d119905 + int

infin

0

119890119905119860119879

119863119890119905119860d119905119860

= int

infin

0

d (119890119905119860119879

119863119890119905119860

) = minus119863

(17)

By (a) the quadratic Lyapunov function 119881 = 119909119879

119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique

Next we prove that 119881 can be expressed as (15) Note that119860119879

119884 + 119884119860 = minus119863 then

119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)1198671198712(119884) = minus119871

2(119863) (18)

By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and

119910119894119895=

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|(1 le 119895 le 119894 le 119899) (19)

where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ

and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors

are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)

By expanding the determinant we have

1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

=

119899

sum

119894=1

1199092

119894

10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816

|Δ|

+ sum

1le119895lt119894le119899

2119909119895119909119894

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|

=

119899

sum

119894119895=1

119909119894119910119894119895119909119895= 119881

(20)

For the uniqueness of 119881 (15) is the desired expression

Remark 8 The dimensions of Δ and [0 119883

1198712(119863) Δ

] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus

32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices

321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) + 119861120579(119896)

119906 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(21)

for 119896 isin 119885+ where 119909(119896) isin 119877

119899 119906(119896) isin 119877119898 and 119910(119896) isin

119877119903 respectively denote the system state control input and

measured output 119860119894119895isin 119877119899times119899 119861

119895isin 119877119899times119898 and 119862

119894119895isin 119877119903times119899

120579(119896) 119896 isin 119885+

is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is

Abstract and Applied Analysis 5

119875 = (119901119894119895)119904times119904

and initial distribution is 1205830

= 1205830119894

≜

119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908

119873(119896) isin 119877 are wide sense

stationary processes and independent of each other such that

E (119908119894(119896)) = 0 E (119908

119894(119896) 119908119895(119896)) = 120575

119894119895≜

1 119894 = 119895

0 119894 = 119895(22)

and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain

120579(119896) 119896 isin 119885+

For the stochastic system (21) its solution and output

processes with 119909(0) = 1199090and 120579(0) sim 120583

0are respectively

denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909

0 1205830) We will simply

denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909

0 1205830) 119910(119896) =

119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909

0 1205830)

If 119906(119896) equiv 0 the stochastic system (21) becomes

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(23)

for 119896 isin 119885+For convenience we will use the following notations

119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119894)

119883 (119896) ≜ (1198831(119896) 119883

119904(119896))

119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))

119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1

120579(119896)=119894)

(119896) ≜ (1198841(119896) 119884

119904(119896))

119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))

W0≜ 119883 (0) | 119909

0isin 119877119899

1205830isin Θ119883

119894(0) ≜ 119909

0119909119879

01205830119894 119894 isin 119878

(24)

It is easy to get 1198831(119896) + sdot sdot sdot + 119883

119904(119896) = 119883(119896) 119884

1(119896) +

sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =

sum119904

119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877

119899times119899

119904

Specifically E|119909(0)|2 = |1199090|2

For the system (23) we define several operators in119877119899times119899119904

asfollows

E119894(119880) =

119904

sum

119895=1

119901119894119895119880119895

F119894(119880) =

119873

sum

119897=0

119860119879

119897119894E119894(119880)119860

119897119894=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894119880119895119860119897119894

Flowast

119894(119880) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895119880119895119860119879

119897119895

E (119880) = (E1(119880) E

119904(119880))

F (119880) = (F1(119880) F

119904(119880))

Flowast

(119880) = (Flowast

1(119880) F

lowast

119904(119880))

F119896

(119880)

= F (F119896minus1

(119880)) (119896 = 1 2 ) where F0

(119880) = 119880

Flowast119896

(119880)

= Flowast

(Flowast119896minus1

(119880)) (119896 = 1 2 ) where Flowast0

(119880) = 119880

(25)

Flowast and F are called dual operators for each other (seeLemma 15)

Also we define the operator O119894(119897) = 119862

119894+ F119894(O(119897 minus

1)) for 119894 isin 119878 119897 = 1 2 where

119862119894=

119873

sum

119897=0

119862119879

119897119894119862119897119894

O (0) = 0

O (119897) = (O1(119897) O

119904(119897))

(26)

Then

O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862

119904))

= 119862 +F (119862) +F2

(119862) + sdot sdot sdot +F119897minus1

(119862)

119897 = 1 2

(27)

322 Some Preliminaries and Auxiliary Results Let

119882119897

119894(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119904

sum

119894=1

119882119897

119894(119883 (119905))

for 119905 isin 119885+ 119897 = 1 2

(28)

Lemma 9 For the system (23) one has O119894(119897) ge O

119894(119897 minus 1) for

all 119894 isin 119878 119897 = 1 2

Proof Firstly it is easy to get O119894(1) = 119862

119894= sum119873

119897=0119862119879

119897119894119862119897119894ge 0 =

O119894(0) for all 119894 isin 119878 Secondly we assume thatO

119894(119897) ge O

119894(119897minus1)for

6 Abstract and Applied Analysis

all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave

O119894(119898 + 1) minus O

119894(119898)

= F119894(O (119898)) minusF

119894(O (119898 minus 1))

=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894(O119895(119898) minus O

119895(119898 minus 1))119860

119897119894

ge 0

(29)

By mathematical induction it is right

Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =

F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1

Proof It is easy to get the result by mathematical inductionso we will omit the details

Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

(ie for all 119883(0) = 1198830isinW0) one has

119883 (119896 + 1) = Flowast

(119883 (119896)) for 119896 isin 119885+

119883 (119896 + 119905) = Flowast119905

(119883 (119896)) for 119896 119905 isin 119885+(30)

Proof For all 119894 isin 119878 we have

Flowast

119894(119883 (119896)) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119895)119860119879

119897119895

=

119873

sum

119897=0

119904

sum

119895=1

119901119895119894E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896)=119895

)

=

119873

sum

119897=0

E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896+1)=119894

)

=E

[

[

1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

times [

[

1198600120579(119896)

119909 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896+1)=119894

=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894

= 119883119894(119896 + 1) 119896 isin 119885

+

(31)

then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin

119885+

Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

for 119905 isin 119885+ 119897 = 1 2

(32)

Proof Because

119884119894(119896) = E

[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

times[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896)=119894

=

119873

sum

119897=0

119862119897119894119883119894(119896) 119862119879

119897119894

(33)

therefore

⟨119883119894(119896) 119862

119894⟩ = tr[119883

119894(119896)(

119873

sum

119897=0

119862119879

119897119894119862119897119894)] = tr (119884

119894(119896))

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩ =

119897minus1

sum

119896=0

tr (119884119894(119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

⟨119883 (119896) 119862⟩ =

119904

sum

119894=1

tr (119883119894(119896) 119862119894) = tr (119884 (119896))

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119897minus1

sum

119896=0

tr (119884 (119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

(34)

Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897

119894(119883(119905))

Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2

Proof It is easy to get the result by mathematical inductionso we will omit the details

Abstract and Applied Analysis 7

Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119898 isin 119885+

119905 isin 119885+ one has

⟨119883 (119905) F119897

(O (119898))⟩ = ⟨Flowast119897

(119883 (119905)) O (119898)⟩ 119897 isin 119885+

(35)

Proof Firstly it is easy to get

⟨119883 (119905) F0

(O (119898))⟩

= ⟨119883 (119905) O (119898)⟩

= ⟨Flowast0

(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+

(36)

Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have

⟨119883 (119905) F119902+1

(O (119898))⟩ = ⟨119883 (119905) F119902

(O (119898 + 1))⟩

minus ⟨119883 (119905) F119902

(119862)⟩

= ⟨Flowast119902

(119883 (119905)) O (119898 + 1)⟩

minus ⟨Flowast119902

(119883 (119905)) 119862⟩

= ⟨119883 (119905 + 119902) O (119898 + 1)⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= 119882119898+1

(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩

=

119898

sum

119896=0

⟨119883 (119905 + 119902 + 119896) 119862⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= ⟨119883 (119905 + 119902 + 1) O (119898)⟩

= ⟨Flowast119902+1

(119883 (119905)) O (119898)⟩

(37)

By mathematical induction it is right

Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ 119905 isin 119885+ one has

⟨119883 (119905) F119897

(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)

Proof Consider

⟨119883 (119905) F119897

(119862)⟩ = ⟨119883 (119905) F119897

(O (1))⟩

= ⟨Flowast119897

(119883 (119905)) O (1)⟩

= ⟨119883 (119905 + 119897) 119862⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]

(39)

In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1

Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one

has

119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)

Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

lim119896rarrinfin

E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin

E [119909119879 (119896) 119909 (119896)] = 0(41)

Proof This proof is omitted

Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899

119903 ≜ rank(119883) then thereexist nonzero vectors 119909

1 119909

119903isin 119877119899 such that 119883 = 119909

1119909119879

1+

sdot sdot sdot + 119909119903119909119879

119903

323 A Useful Formula Define two operators L1(119880) =

[

1198711(1198801)

1198711(119880119904)

] for 119880119894isin 119877119899times119899 andL

2(119880) = [

1198712(1198801)

1198712(119880119904)

] for 119880119894isin 119883 |

119883119879

= 119883 isin 119877119899times119899

in 119877119899times119899119904

For convenience we naturally think 119884 isin 119883 | 119883

119879

= 119883 isin

119877119899times119899

when we use 1198712(119884) in this subsection

Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899

119904 one has

L1(F (119880)) = AL

1(119880)

L2(F (119880)) = BL

2(119880)

⟨119880 119881⟩ = L119879

1(119880)L

1(119881)

⟨119880 119881⟩ = L119879

1(119880) (119868 otimes 119867)L

2(119881)

(42)

where

A ≜

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971 119860

119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

B ≜

119873

sum

119897=0

diag (119867minus (1198601198791198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

(43)

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Proof Consider

∵ 1198671198941198951198712(119883) = 119871

119879

2(119864119899119894119895

) 1198712(119883) = 119909

119894119895(119894 ge 119895)

1198671198941198951198712(119883) = 119871

119879

2(119864119899119895119894

) 1198712(119883) = 119909

119895119894= 119909119894119895

(119894 lt 119895)

there4 1198671198712(119883) = 119871

1(119883)

1198712(119883) = 119867

minus

1198711(119883) (by Property 1)

(8)

Property 4 For matrices 119860119899times119899 119861119903times119899

and 119883119879119899times119899

= 119883 we havethe following

(a) 1198711(119860119883 + 119883119860

119879

) = (119868 otimes 119860 + 119860 otimes 119868)1198711(119883)

(b) 1198711(119861119883119861

119879

) = (119861 otimes 119861)1198711(119883)

(c) 1198712(119860119883 + 119883119860

119879

) = 119867minus

(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)

(d) 1198712(119861119883119861

119879

) = 119867minus

(119861 otimes 119861)1198671198712(119883)

Proof We can get (a) and (b) in [1] so we only need to prove(c) and (d)

Obviously119860119883+119883119860119879 and 119861119883119861119879 are symmetricalThere-fore

1198712(119860119883 + 119883119860

119879

) = 119867minus

1198711(119860119883 + 119883119860

119879

)

= 119867minus

(119868 otimes 119860 + 119860 otimes 119868) 1198711(119883)

= 119867minus

(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)

1198712(119861119883119861

119879

) = 119867minus

1198711(119861119883119861

119879

) = 119867minus

(119861 otimes 119861)1198671198712(119883)

(9)

Remark 6 If 1198711and 119871

2are defined by row that is

1198711(119883) = (119909

11 11990912 119909

1119898 11990921 11990922

1199092119898 119909

1198991 1199091198992 119909

119899119898)119879 for 119883 isin 119862

119899times119898

1198712(119883) = (119909

11 11990912 119909

1119899 11990922

1199092119899 119909

119899119899)119879 for 119883 isin 119882

119899times119899

(10)

we find that Properties 3 and 4 are still true

Property 5 For all 119860 isin 119862119899times119899 then

(a) 120590(119860 otimes 119860) = 120590(119867minus

(119860 otimes 119860)119867) = 120590(119867minus

(119860119879

otimes 119860119879

)119867)

(b) 120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868+119868otimes119860)119867) = 120590(119867minus(119860119879otimes119868 + 119868 otimes 119860

119879

)119867)

where 120590(sdot) denotes the set of all eigenvalues of matrix

Proof (a) Let

1205901(119860) ≜ 120582 isin 119862 | 119860119883119860

119879

= 120582119883119883 = 0 119883 isin 119862119899times119899

1205902(119860) ≜ 120582 isin 119862 | 119860119883119860

119879

= 120582119883119883119879

= 119883 = 0119883 isin 119862119899times119899

∵ 119860119883119860119879

= 120582119883 lArrrArr (119860 otimes 119860) 1198711(119883) = 120582119871

1(119883)

there4 1205901(119860) = 120590 (119860 otimes 119860)

(11)

If 119883119879 = 119883 then 119860119883119860119879 = 120582119883 hArr 119867minus

(119860 otimes 119860)1198671198712(119883) =

1205821198712(119883) Thus 120590

2(119860) = 120590(119867

minus

(119860 otimes 119860)119867)Next we prove 120590

1(119860) = 120590

2(119860) Obviously 120590

2(119860) sube 120590

1(119860)

so we only need to prove 1205901(119860) sube 120590

2(119860)

For any 120582 isin 1205901(119860) exist isin 120590(119860) such that 120582 = 119909 =

119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862

119899 (see [1]) Let119883 ≜ 119909119910

119879

119885 ≜ 119883 + 119883119879 then

120582119885 = 119909119910119879

+ 119910119909119879

= 119860119909119910119879

119860119879

+ 119860119910119909119879

119860119879

= 119860119885119860119879

(12)

If 119885 = 0 then 120582 isin 1205902(119860) Actually we always have 119885 = 0

Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie

119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let

119909119894= 0 119910119895= 0 then 119910

119894= 0 119909

119895= 0 119911

119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0

This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119860) = 120590

1(119860) = 120590

2(119860) = 120590(119867

minus

(119860 otimes 119860)119867)And because 120590(119860 otimes 119860) = 120590

1(119860119879

otimes 119860119879

) = 120590(119867minus

(119860119879

otimes 119860119879

)119867)therefore 120590(119860 otimes 119860) = 120590(119867minus(119860 otimes 119860)119867) = 120590(119867minus(119860119879 otimes 119860119879)119867)

(b) Let

1205903(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860

119879

= 120582119883119883 = 0 119883 isin 119862119899times119899

1205904(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860

119879

= 120582119883119883119879

= 119883 = 0119883 isin 119862119899times119899

∵ 119860119883 + 119883119860119879

= 120582119883 lArrrArr (119860 otimes 119868 + 119868 otimes 119860) 1198711(119883) = 120582119871

1(119883)

there4 1205903(119860) = 120590 (119860 otimes 119868 + 119868 otimes 119860)

(13)

If 119883119879 = 119883 then 119860119883 + 119883119860119879

= 120582119883 hArr 119867minus

(119860 otimes 119868 + 119868 otimes

119860)1198671198712(119883) = 120582119871

2(119883)

Thus 1205904(119860) = 120590(119867

minus

(119860 otimes 119868 + 119868 otimes 119860)119867)Next we prove 120590

3(119860) = 120590

4(119860) Obviously 120590

4(119860) sube 120590

3(119860)

so we only need to prove 1205903(119860) sube 120590

4(119860)

For any 120582 isin 1205903(119860) exist isin 120590(119860) such that 120582 = +

119909 = 119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862

119899 (see [1])Let119883 ≜ 119909119910

119879

119885 ≜ 119883 + 119883119879 then

120582119885 = 119909119910119879

+ 119909119910119879

+ 119910119909119879

+ 119910119909119879

= 119860119909119910119879

+ 119909119910119879

119860119879

+ 119910119909119879

119860119879

+ 119860119910119909119879

= 119860119885 + 119885119860119879

(14)

If 119885 = 0 then 120582 isin 1205904(119860) Actually we always have 119885 = 0

Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie

119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let

4 Abstract and Applied Analysis

119909119894= 0 119910119895= 0 then 119910

119894= 0 119909

119895= 0 119911

119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0

This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

3(119860) = 120590

4(119860) = 120590(119867

minus

(119860 otimes 119868 +

119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

1(119860119879

otimes 119868 + 119868 otimes 119860119879

) =

120590(119867minus

(119860119879

otimes119868+119868otimes119860119879

)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867

minus

(119860119879

otimes 119868 + 119868 otimes 119860119879

)119867)

3 Applications of Transformation Matrix

31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices

Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877

119899times119899 is a Hurwitz matrix) thenone has the following

(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique

solution 119884 isin 119877119899times119899

(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909

119879

119884119909 such that 119860119879119884 + 119884119860 =

minus119863 and the expression is

119881 =1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

(15)

where

Δ = 119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)119867

119883 = (1198831 1198832 119883

119899)

1198831= (1199092

1 211990911199092 2119909

1119909119899)

1198832= (1199092

2 211990921199093 2119909

2119909119899)

119883119899= (1199092

119899)

(16)

Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int

infin

0

119890119905119860119879

119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz

matrix so 119884 gt 0 and

119860119879

119884 + 119884119860 = 119860119879

int

infin

0

119890119905119860119879

119863119890119905119860d119905 + int

infin

0

119890119905119860119879

119863119890119905119860d119905119860

= int

infin

0

d (119890119905119860119879

119863119890119905119860

) = minus119863

(17)

By (a) the quadratic Lyapunov function 119881 = 119909119879

119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique

Next we prove that 119881 can be expressed as (15) Note that119860119879

119884 + 119884119860 = minus119863 then

119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)1198671198712(119884) = minus119871

2(119863) (18)

By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and

119910119894119895=

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|(1 le 119895 le 119894 le 119899) (19)

where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ

and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors

are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)

By expanding the determinant we have

1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

=

119899

sum

119894=1

1199092

119894

10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816

|Δ|

+ sum

1le119895lt119894le119899

2119909119895119909119894

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|

=

119899

sum

119894119895=1

119909119894119910119894119895119909119895= 119881

(20)

For the uniqueness of 119881 (15) is the desired expression

Remark 8 The dimensions of Δ and [0 119883

1198712(119863) Δ

] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus

32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices

321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) + 119861120579(119896)

119906 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(21)

for 119896 isin 119885+ where 119909(119896) isin 119877

119899 119906(119896) isin 119877119898 and 119910(119896) isin

119877119903 respectively denote the system state control input and

measured output 119860119894119895isin 119877119899times119899 119861

119895isin 119877119899times119898 and 119862

119894119895isin 119877119903times119899

120579(119896) 119896 isin 119885+

is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is

Abstract and Applied Analysis 5

119875 = (119901119894119895)119904times119904

and initial distribution is 1205830

= 1205830119894

≜

119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908

119873(119896) isin 119877 are wide sense

stationary processes and independent of each other such that

E (119908119894(119896)) = 0 E (119908

119894(119896) 119908119895(119896)) = 120575

119894119895≜

1 119894 = 119895

0 119894 = 119895(22)

and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain

120579(119896) 119896 isin 119885+

For the stochastic system (21) its solution and output

processes with 119909(0) = 1199090and 120579(0) sim 120583

0are respectively

denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909

0 1205830) We will simply

denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909

0 1205830) 119910(119896) =

119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909

0 1205830)

If 119906(119896) equiv 0 the stochastic system (21) becomes

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(23)

for 119896 isin 119885+For convenience we will use the following notations

119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119894)

119883 (119896) ≜ (1198831(119896) 119883

119904(119896))

119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))

119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1

120579(119896)=119894)

(119896) ≜ (1198841(119896) 119884

119904(119896))

119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))

W0≜ 119883 (0) | 119909

0isin 119877119899

1205830isin Θ119883

119894(0) ≜ 119909

0119909119879

01205830119894 119894 isin 119878

(24)

It is easy to get 1198831(119896) + sdot sdot sdot + 119883

119904(119896) = 119883(119896) 119884

1(119896) +

sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =

sum119904

119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877

119899times119899

119904

Specifically E|119909(0)|2 = |1199090|2

For the system (23) we define several operators in119877119899times119899119904

asfollows

E119894(119880) =

119904

sum

119895=1

119901119894119895119880119895

F119894(119880) =

119873

sum

119897=0

119860119879

119897119894E119894(119880)119860

119897119894=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894119880119895119860119897119894

Flowast

119894(119880) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895119880119895119860119879

119897119895

E (119880) = (E1(119880) E

119904(119880))

F (119880) = (F1(119880) F

119904(119880))

Flowast

(119880) = (Flowast

1(119880) F

lowast

119904(119880))

F119896

(119880)

= F (F119896minus1

(119880)) (119896 = 1 2 ) where F0

(119880) = 119880

Flowast119896

(119880)

= Flowast

(Flowast119896minus1

(119880)) (119896 = 1 2 ) where Flowast0

(119880) = 119880

(25)

Flowast and F are called dual operators for each other (seeLemma 15)

Also we define the operator O119894(119897) = 119862

119894+ F119894(O(119897 minus

1)) for 119894 isin 119878 119897 = 1 2 where

119862119894=

119873

sum

119897=0

119862119879

119897119894119862119897119894

O (0) = 0

O (119897) = (O1(119897) O

119904(119897))

(26)

Then

O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862

119904))

= 119862 +F (119862) +F2

(119862) + sdot sdot sdot +F119897minus1

(119862)

119897 = 1 2

(27)

322 Some Preliminaries and Auxiliary Results Let

119882119897

119894(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119904

sum

119894=1

119882119897

119894(119883 (119905))

for 119905 isin 119885+ 119897 = 1 2

(28)

Lemma 9 For the system (23) one has O119894(119897) ge O

119894(119897 minus 1) for

all 119894 isin 119878 119897 = 1 2

Proof Firstly it is easy to get O119894(1) = 119862

119894= sum119873

119897=0119862119879

119897119894119862119897119894ge 0 =

O119894(0) for all 119894 isin 119878 Secondly we assume thatO

119894(119897) ge O

119894(119897minus1)for

6 Abstract and Applied Analysis

all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave

O119894(119898 + 1) minus O

119894(119898)

= F119894(O (119898)) minusF

119894(O (119898 minus 1))

=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894(O119895(119898) minus O

119895(119898 minus 1))119860

119897119894

ge 0

(29)

By mathematical induction it is right

Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =

F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1

Proof It is easy to get the result by mathematical inductionso we will omit the details

Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

(ie for all 119883(0) = 1198830isinW0) one has

119883 (119896 + 1) = Flowast

(119883 (119896)) for 119896 isin 119885+

119883 (119896 + 119905) = Flowast119905

(119883 (119896)) for 119896 119905 isin 119885+(30)

Proof For all 119894 isin 119878 we have

Flowast

119894(119883 (119896)) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119895)119860119879

119897119895

=

119873

sum

119897=0

119904

sum

119895=1

119901119895119894E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896)=119895

)

=

119873

sum

119897=0

E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896+1)=119894

)

=E

[

[

1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

times [

[

1198600120579(119896)

119909 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896+1)=119894

=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894

= 119883119894(119896 + 1) 119896 isin 119885

+

(31)

then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin

119885+

Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

for 119905 isin 119885+ 119897 = 1 2

(32)

Proof Because

119884119894(119896) = E

[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

times[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896)=119894

=

119873

sum

119897=0

119862119897119894119883119894(119896) 119862119879

119897119894

(33)

therefore

⟨119883119894(119896) 119862

119894⟩ = tr[119883

119894(119896)(

119873

sum

119897=0

119862119879

119897119894119862119897119894)] = tr (119884

119894(119896))

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩ =

119897minus1

sum

119896=0

tr (119884119894(119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

⟨119883 (119896) 119862⟩ =

119904

sum

119894=1

tr (119883119894(119896) 119862119894) = tr (119884 (119896))

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119897minus1

sum

119896=0

tr (119884 (119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

(34)

Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897

119894(119883(119905))

Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2

Proof It is easy to get the result by mathematical inductionso we will omit the details

Abstract and Applied Analysis 7

Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119898 isin 119885+

119905 isin 119885+ one has

⟨119883 (119905) F119897

(O (119898))⟩ = ⟨Flowast119897

(119883 (119905)) O (119898)⟩ 119897 isin 119885+

(35)

Proof Firstly it is easy to get

⟨119883 (119905) F0

(O (119898))⟩

= ⟨119883 (119905) O (119898)⟩

= ⟨Flowast0

(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+

(36)

Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have

⟨119883 (119905) F119902+1

(O (119898))⟩ = ⟨119883 (119905) F119902

(O (119898 + 1))⟩

minus ⟨119883 (119905) F119902

(119862)⟩

= ⟨Flowast119902

(119883 (119905)) O (119898 + 1)⟩

minus ⟨Flowast119902

(119883 (119905)) 119862⟩

= ⟨119883 (119905 + 119902) O (119898 + 1)⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= 119882119898+1

(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩

=

119898

sum

119896=0

⟨119883 (119905 + 119902 + 119896) 119862⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= ⟨119883 (119905 + 119902 + 1) O (119898)⟩

= ⟨Flowast119902+1

(119883 (119905)) O (119898)⟩

(37)

By mathematical induction it is right

Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ 119905 isin 119885+ one has

⟨119883 (119905) F119897

(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)

Proof Consider

⟨119883 (119905) F119897

(119862)⟩ = ⟨119883 (119905) F119897

(O (1))⟩

= ⟨Flowast119897

(119883 (119905)) O (1)⟩

= ⟨119883 (119905 + 119897) 119862⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]

(39)

In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1

Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one

has

119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)

Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

lim119896rarrinfin

E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin

E [119909119879 (119896) 119909 (119896)] = 0(41)

Proof This proof is omitted

Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899

119903 ≜ rank(119883) then thereexist nonzero vectors 119909

1 119909

119903isin 119877119899 such that 119883 = 119909

1119909119879

1+

sdot sdot sdot + 119909119903119909119879

119903

323 A Useful Formula Define two operators L1(119880) =

[

1198711(1198801)

1198711(119880119904)

] for 119880119894isin 119877119899times119899 andL

2(119880) = [

1198712(1198801)

1198712(119880119904)

] for 119880119894isin 119883 |

119883119879

= 119883 isin 119877119899times119899

in 119877119899times119899119904

For convenience we naturally think 119884 isin 119883 | 119883

119879

= 119883 isin

119877119899times119899

when we use 1198712(119884) in this subsection

Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899

119904 one has

L1(F (119880)) = AL

1(119880)

L2(F (119880)) = BL

2(119880)

⟨119880 119881⟩ = L119879

1(119880)L

1(119881)

⟨119880 119881⟩ = L119879

1(119880) (119868 otimes 119867)L

2(119881)

(42)

where

A ≜

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971 119860

119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

B ≜

119873

sum

119897=0

diag (119867minus (1198601198791198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

(43)

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

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Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

119909119894= 0 119910119895= 0 then 119910

119894= 0 119909

119895= 0 119911

119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0

This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

3(119860) = 120590

4(119860) = 120590(119867

minus

(119860 otimes 119868 +

119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590

1(119860119879

otimes 119868 + 119868 otimes 119860119879

) =

120590(119867minus

(119860119879

otimes119868+119868otimes119860119879

)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867

minus

(119860119879

otimes 119868 + 119868 otimes 119860119879

)119867)

3 Applications of Transformation Matrix

31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices

Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877

119899times119899 is a Hurwitz matrix) thenone has the following

(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique

solution 119884 isin 119877119899times119899

(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909

119879

119884119909 such that 119860119879119884 + 119884119860 =

minus119863 and the expression is

119881 =1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

(15)

where

Δ = 119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)119867

119883 = (1198831 1198832 119883

119899)

1198831= (1199092

1 211990911199092 2119909

1119909119899)

1198832= (1199092

2 211990921199093 2119909

2119909119899)

119883119899= (1199092

119899)

(16)

Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int

infin

0

119890119905119860119879

119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz

matrix so 119884 gt 0 and

119860119879

119884 + 119884119860 = 119860119879

int

infin

0

119890119905119860119879

119863119890119905119860d119905 + int

infin

0

119890119905119860119879

119863119890119905119860d119905119860

= int

infin

0

d (119890119905119860119879

119863119890119905119860

) = minus119863

(17)

By (a) the quadratic Lyapunov function 119881 = 119909119879

119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique

Next we prove that 119881 can be expressed as (15) Note that119860119879

119884 + 119884119860 = minus119863 then

119867minus

(119868 otimes 119860119879

+ 119860119879

otimes 119868)1198671198712(119884) = minus119871

2(119863) (18)

By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and

119910119894119895=

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|(1 le 119895 le 119894 le 119899) (19)

where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ

and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors

are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)

By expanding the determinant we have

1

|Δ|

10038161003816100381610038161003816100381610038161003816

0 119883

1198712(119863) Δ

10038161003816100381610038161003816100381610038161003816

=

119899

sum

119894=1

1199092

119894

10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816

|Δ|

+ sum

1le119895lt119894le119899

2119909119895119909119894

10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816

|Δ|

=

119899

sum

119894119895=1

119909119894119910119894119895119909119895= 119881

(20)

For the uniqueness of 119881 (15) is the desired expression

Remark 8 The dimensions of Δ and [0 119883

1198712(119863) Δ

] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus

32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices

321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) + 119861120579(119896)

119906 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(21)

for 119896 isin 119885+ where 119909(119896) isin 119877

119899 119906(119896) isin 119877119898 and 119910(119896) isin

119877119903 respectively denote the system state control input and

measured output 119860119894119895isin 119877119899times119899 119861

119895isin 119877119899times119898 and 119862

119894119895isin 119877119903times119899

120579(119896) 119896 isin 119885+

is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is

Abstract and Applied Analysis 5

119875 = (119901119894119895)119904times119904

and initial distribution is 1205830

= 1205830119894

≜

119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908

119873(119896) isin 119877 are wide sense

stationary processes and independent of each other such that

E (119908119894(119896)) = 0 E (119908

119894(119896) 119908119895(119896)) = 120575

119894119895≜

1 119894 = 119895

0 119894 = 119895(22)

and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain

120579(119896) 119896 isin 119885+

For the stochastic system (21) its solution and output

processes with 119909(0) = 1199090and 120579(0) sim 120583

0are respectively

denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909

0 1205830) We will simply

denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909

0 1205830) 119910(119896) =

119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909

0 1205830)

If 119906(119896) equiv 0 the stochastic system (21) becomes

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(23)

for 119896 isin 119885+For convenience we will use the following notations

119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119894)

119883 (119896) ≜ (1198831(119896) 119883

119904(119896))

119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))

119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1

120579(119896)=119894)

(119896) ≜ (1198841(119896) 119884

119904(119896))

119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))

W0≜ 119883 (0) | 119909

0isin 119877119899

1205830isin Θ119883

119894(0) ≜ 119909

0119909119879

01205830119894 119894 isin 119878

(24)

It is easy to get 1198831(119896) + sdot sdot sdot + 119883

119904(119896) = 119883(119896) 119884

1(119896) +

sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =

sum119904

119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877

119899times119899

119904

Specifically E|119909(0)|2 = |1199090|2

For the system (23) we define several operators in119877119899times119899119904

asfollows

E119894(119880) =

119904

sum

119895=1

119901119894119895119880119895

F119894(119880) =

119873

sum

119897=0

119860119879

119897119894E119894(119880)119860

119897119894=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894119880119895119860119897119894

Flowast

119894(119880) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895119880119895119860119879

119897119895

E (119880) = (E1(119880) E

119904(119880))

F (119880) = (F1(119880) F

119904(119880))

Flowast

(119880) = (Flowast

1(119880) F

lowast

119904(119880))

F119896

(119880)

= F (F119896minus1

(119880)) (119896 = 1 2 ) where F0

(119880) = 119880

Flowast119896

(119880)

= Flowast

(Flowast119896minus1

(119880)) (119896 = 1 2 ) where Flowast0

(119880) = 119880

(25)

Flowast and F are called dual operators for each other (seeLemma 15)

Also we define the operator O119894(119897) = 119862

119894+ F119894(O(119897 minus

1)) for 119894 isin 119878 119897 = 1 2 where

119862119894=

119873

sum

119897=0

119862119879

119897119894119862119897119894

O (0) = 0

O (119897) = (O1(119897) O

119904(119897))

(26)

Then

O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862

119904))

= 119862 +F (119862) +F2

(119862) + sdot sdot sdot +F119897minus1

(119862)

119897 = 1 2

(27)

322 Some Preliminaries and Auxiliary Results Let

119882119897

119894(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119904

sum

119894=1

119882119897

119894(119883 (119905))

for 119905 isin 119885+ 119897 = 1 2

(28)

Lemma 9 For the system (23) one has O119894(119897) ge O

119894(119897 minus 1) for

all 119894 isin 119878 119897 = 1 2

Proof Firstly it is easy to get O119894(1) = 119862

119894= sum119873

119897=0119862119879

119897119894119862119897119894ge 0 =

O119894(0) for all 119894 isin 119878 Secondly we assume thatO

119894(119897) ge O

119894(119897minus1)for

6 Abstract and Applied Analysis

all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave

O119894(119898 + 1) minus O

119894(119898)

= F119894(O (119898)) minusF

119894(O (119898 minus 1))

=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894(O119895(119898) minus O

119895(119898 minus 1))119860

119897119894

ge 0

(29)

By mathematical induction it is right

Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =

F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1

Proof It is easy to get the result by mathematical inductionso we will omit the details

Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

(ie for all 119883(0) = 1198830isinW0) one has

119883 (119896 + 1) = Flowast

(119883 (119896)) for 119896 isin 119885+

119883 (119896 + 119905) = Flowast119905

(119883 (119896)) for 119896 119905 isin 119885+(30)

Proof For all 119894 isin 119878 we have

Flowast

119894(119883 (119896)) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119895)119860119879

119897119895

=

119873

sum

119897=0

119904

sum

119895=1

119901119895119894E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896)=119895

)

=

119873

sum

119897=0

E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896+1)=119894

)

=E

[

[

1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

times [

[

1198600120579(119896)

119909 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896+1)=119894

=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894

= 119883119894(119896 + 1) 119896 isin 119885

+

(31)

then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin

119885+

Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

for 119905 isin 119885+ 119897 = 1 2

(32)

Proof Because

119884119894(119896) = E

[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

times[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896)=119894

=

119873

sum

119897=0

119862119897119894119883119894(119896) 119862119879

119897119894

(33)

therefore

⟨119883119894(119896) 119862

119894⟩ = tr[119883

119894(119896)(

119873

sum

119897=0

119862119879

119897119894119862119897119894)] = tr (119884

119894(119896))

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩ =

119897minus1

sum

119896=0

tr (119884119894(119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

⟨119883 (119896) 119862⟩ =

119904

sum

119894=1

tr (119883119894(119896) 119862119894) = tr (119884 (119896))

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119897minus1

sum

119896=0

tr (119884 (119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

(34)

Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897

119894(119883(119905))

Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2

Proof It is easy to get the result by mathematical inductionso we will omit the details

Abstract and Applied Analysis 7

Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119898 isin 119885+

119905 isin 119885+ one has

⟨119883 (119905) F119897

(O (119898))⟩ = ⟨Flowast119897

(119883 (119905)) O (119898)⟩ 119897 isin 119885+

(35)

Proof Firstly it is easy to get

⟨119883 (119905) F0

(O (119898))⟩

= ⟨119883 (119905) O (119898)⟩

= ⟨Flowast0

(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+

(36)

Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have

⟨119883 (119905) F119902+1

(O (119898))⟩ = ⟨119883 (119905) F119902

(O (119898 + 1))⟩

minus ⟨119883 (119905) F119902

(119862)⟩

= ⟨Flowast119902

(119883 (119905)) O (119898 + 1)⟩

minus ⟨Flowast119902

(119883 (119905)) 119862⟩

= ⟨119883 (119905 + 119902) O (119898 + 1)⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= 119882119898+1

(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩

=

119898

sum

119896=0

⟨119883 (119905 + 119902 + 119896) 119862⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= ⟨119883 (119905 + 119902 + 1) O (119898)⟩

= ⟨Flowast119902+1

(119883 (119905)) O (119898)⟩

(37)

By mathematical induction it is right

Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ 119905 isin 119885+ one has

⟨119883 (119905) F119897

(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)

Proof Consider

⟨119883 (119905) F119897

(119862)⟩ = ⟨119883 (119905) F119897

(O (1))⟩

= ⟨Flowast119897

(119883 (119905)) O (1)⟩

= ⟨119883 (119905 + 119897) 119862⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]

(39)

In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1

Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one

has

119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)

Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

lim119896rarrinfin

E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin

E [119909119879 (119896) 119909 (119896)] = 0(41)

Proof This proof is omitted

Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899

119903 ≜ rank(119883) then thereexist nonzero vectors 119909

1 119909

119903isin 119877119899 such that 119883 = 119909

1119909119879

1+

sdot sdot sdot + 119909119903119909119879

119903

323 A Useful Formula Define two operators L1(119880) =

[

1198711(1198801)

1198711(119880119904)

] for 119880119894isin 119877119899times119899 andL

2(119880) = [

1198712(1198801)

1198712(119880119904)

] for 119880119894isin 119883 |

119883119879

= 119883 isin 119877119899times119899

in 119877119899times119899119904

For convenience we naturally think 119884 isin 119883 | 119883

119879

= 119883 isin

119877119899times119899

when we use 1198712(119884) in this subsection

Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899

119904 one has

L1(F (119880)) = AL

1(119880)

L2(F (119880)) = BL

2(119880)

⟨119880 119881⟩ = L119879

1(119880)L

1(119881)

⟨119880 119881⟩ = L119879

1(119880) (119868 otimes 119867)L

2(119881)

(42)

where

A ≜

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971 119860

119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

B ≜

119873

sum

119897=0

diag (119867minus (1198601198791198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

(43)

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

119875 = (119901119894119895)119904times119904

and initial distribution is 1205830

= 1205830119894

≜

119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908

119873(119896) isin 119877 are wide sense

stationary processes and independent of each other such that

E (119908119894(119896)) = 0 E (119908

119894(119896) 119908119895(119896)) = 120575

119894119895≜

1 119894 = 119895

0 119894 = 119895(22)

and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain

120579(119896) 119896 isin 119885+

For the stochastic system (21) its solution and output

processes with 119909(0) = 1199090and 120579(0) sim 120583

0are respectively

denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909

0 1205830) We will simply

denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909

0 1205830) 119910(119896) =

119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909

0 1205830)

If 119906(119896) equiv 0 the stochastic system (21) becomes

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)

119910 (119896) = 1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)

(23)

for 119896 isin 119885+For convenience we will use the following notations

119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119894)

119883 (119896) ≜ (1198831(119896) 119883

119904(119896))

119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))

119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1

120579(119896)=119894)

(119896) ≜ (1198841(119896) 119884

119904(119896))

119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))

W0≜ 119883 (0) | 119909

0isin 119877119899

1205830isin Θ119883

119894(0) ≜ 119909

0119909119879

01205830119894 119894 isin 119878

(24)

It is easy to get 1198831(119896) + sdot sdot sdot + 119883

119904(119896) = 119883(119896) 119884

1(119896) +

sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =

sum119904

119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877

119899times119899

119904

Specifically E|119909(0)|2 = |1199090|2

For the system (23) we define several operators in119877119899times119899119904

asfollows

E119894(119880) =

119904

sum

119895=1

119901119894119895119880119895

F119894(119880) =

119873

sum

119897=0

119860119879

119897119894E119894(119880)119860

119897119894=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894119880119895119860119897119894

Flowast

119894(119880) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895119880119895119860119879

119897119895

E (119880) = (E1(119880) E

119904(119880))

F (119880) = (F1(119880) F

119904(119880))

Flowast

(119880) = (Flowast

1(119880) F

lowast

119904(119880))

F119896

(119880)

= F (F119896minus1

(119880)) (119896 = 1 2 ) where F0

(119880) = 119880

Flowast119896

(119880)

= Flowast

(Flowast119896minus1

(119880)) (119896 = 1 2 ) where Flowast0

(119880) = 119880

(25)

Flowast and F are called dual operators for each other (seeLemma 15)

Also we define the operator O119894(119897) = 119862

119894+ F119894(O(119897 minus

1)) for 119894 isin 119878 119897 = 1 2 where

119862119894=

119873

sum

119897=0

119862119879

119897119894119862119897119894

O (0) = 0

O (119897) = (O1(119897) O

119904(119897))

(26)

Then

O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862

119904))

= 119862 +F (119862) +F2

(119862) + sdot sdot sdot +F119897minus1

(119862)

119897 = 1 2

(27)

322 Some Preliminaries and Auxiliary Results Let

119882119897

119894(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) ≜

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119904

sum

119894=1

119882119897

119894(119883 (119905))

for 119905 isin 119885+ 119897 = 1 2

(28)

Lemma 9 For the system (23) one has O119894(119897) ge O

119894(119897 minus 1) for

all 119894 isin 119878 119897 = 1 2

Proof Firstly it is easy to get O119894(1) = 119862

119894= sum119873

119897=0119862119879

119897119894119862119897119894ge 0 =

O119894(0) for all 119894 isin 119878 Secondly we assume thatO

119894(119897) ge O

119894(119897minus1)for

6 Abstract and Applied Analysis

all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave

O119894(119898 + 1) minus O

119894(119898)

= F119894(O (119898)) minusF

119894(O (119898 minus 1))

=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894(O119895(119898) minus O

119895(119898 minus 1))119860

119897119894

ge 0

(29)

By mathematical induction it is right

Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =

F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1

Proof It is easy to get the result by mathematical inductionso we will omit the details

Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

(ie for all 119883(0) = 1198830isinW0) one has

119883 (119896 + 1) = Flowast

(119883 (119896)) for 119896 isin 119885+

119883 (119896 + 119905) = Flowast119905

(119883 (119896)) for 119896 119905 isin 119885+(30)

Proof For all 119894 isin 119878 we have

Flowast

119894(119883 (119896)) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119895)119860119879

119897119895

=

119873

sum

119897=0

119904

sum

119895=1

119901119895119894E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896)=119895

)

=

119873

sum

119897=0

E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896+1)=119894

)

=E

[

[

1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

times [

[

1198600120579(119896)

119909 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896+1)=119894

=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894

= 119883119894(119896 + 1) 119896 isin 119885

+

(31)

then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin

119885+

Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

for 119905 isin 119885+ 119897 = 1 2

(32)

Proof Because

119884119894(119896) = E

[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

times[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896)=119894

=

119873

sum

119897=0

119862119897119894119883119894(119896) 119862119879

119897119894

(33)

therefore

⟨119883119894(119896) 119862

119894⟩ = tr[119883

119894(119896)(

119873

sum

119897=0

119862119879

119897119894119862119897119894)] = tr (119884

119894(119896))

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩ =

119897minus1

sum

119896=0

tr (119884119894(119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

⟨119883 (119896) 119862⟩ =

119904

sum

119894=1

tr (119883119894(119896) 119862119894) = tr (119884 (119896))

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119897minus1

sum

119896=0

tr (119884 (119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

(34)

Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897

119894(119883(119905))

Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2

Proof It is easy to get the result by mathematical inductionso we will omit the details

Abstract and Applied Analysis 7

Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119898 isin 119885+

119905 isin 119885+ one has

⟨119883 (119905) F119897

(O (119898))⟩ = ⟨Flowast119897

(119883 (119905)) O (119898)⟩ 119897 isin 119885+

(35)

Proof Firstly it is easy to get

⟨119883 (119905) F0

(O (119898))⟩

= ⟨119883 (119905) O (119898)⟩

= ⟨Flowast0

(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+

(36)

Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have

⟨119883 (119905) F119902+1

(O (119898))⟩ = ⟨119883 (119905) F119902

(O (119898 + 1))⟩

minus ⟨119883 (119905) F119902

(119862)⟩

= ⟨Flowast119902

(119883 (119905)) O (119898 + 1)⟩

minus ⟨Flowast119902

(119883 (119905)) 119862⟩

= ⟨119883 (119905 + 119902) O (119898 + 1)⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= 119882119898+1

(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩

=

119898

sum

119896=0

⟨119883 (119905 + 119902 + 119896) 119862⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= ⟨119883 (119905 + 119902 + 1) O (119898)⟩

= ⟨Flowast119902+1

(119883 (119905)) O (119898)⟩

(37)

By mathematical induction it is right

Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ 119905 isin 119885+ one has

⟨119883 (119905) F119897

(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)

Proof Consider

⟨119883 (119905) F119897

(119862)⟩ = ⟨119883 (119905) F119897

(O (1))⟩

= ⟨Flowast119897

(119883 (119905)) O (1)⟩

= ⟨119883 (119905 + 119897) 119862⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]

(39)

In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1

Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one

has

119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)

Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

lim119896rarrinfin

E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin

E [119909119879 (119896) 119909 (119896)] = 0(41)

Proof This proof is omitted

Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899

119903 ≜ rank(119883) then thereexist nonzero vectors 119909

1 119909

119903isin 119877119899 such that 119883 = 119909

1119909119879

1+

sdot sdot sdot + 119909119903119909119879

119903

323 A Useful Formula Define two operators L1(119880) =

[

1198711(1198801)

1198711(119880119904)

] for 119880119894isin 119877119899times119899 andL

2(119880) = [

1198712(1198801)

1198712(119880119904)

] for 119880119894isin 119883 |

119883119879

= 119883 isin 119877119899times119899

in 119877119899times119899119904

For convenience we naturally think 119884 isin 119883 | 119883

119879

= 119883 isin

119877119899times119899

when we use 1198712(119884) in this subsection

Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899

119904 one has

L1(F (119880)) = AL

1(119880)

L2(F (119880)) = BL

2(119880)

⟨119880 119881⟩ = L119879

1(119880)L

1(119881)

⟨119880 119881⟩ = L119879

1(119880) (119868 otimes 119867)L

2(119881)

(42)

where

A ≜

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971 119860

119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

B ≜

119873

sum

119897=0

diag (119867minus (1198601198791198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

(43)

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave

O119894(119898 + 1) minus O

119894(119898)

= F119894(O (119898)) minusF

119894(O (119898 minus 1))

=

119873

sum

119897=0

119904

sum

119895=1

119901119894119895119860119879

119897119894(O119895(119898) minus O

119895(119898 minus 1))119860

119897119894

ge 0

(29)

By mathematical induction it is right

Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =

F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1

Proof It is easy to get the result by mathematical inductionso we will omit the details

Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

(ie for all 119883(0) = 1198830isinW0) one has

119883 (119896 + 1) = Flowast

(119883 (119896)) for 119896 isin 119885+

119883 (119896 + 119905) = Flowast119905

(119883 (119896)) for 119896 119905 isin 119885+(30)

Proof For all 119894 isin 119878 we have

Flowast

119894(119883 (119896)) =

119873

sum

119897=0

119904

sum

119895=1

119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1

120579(119896)=119895)119860119879

119897119895

=

119873

sum

119897=0

119904

sum

119895=1

119901119895119894E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896)=119895

)

=

119873

sum

119897=0

E (119860119897120579(119896)

119909 (119896) 119909119879

(119896) 119860119879

119897120579(119896)1120579(119896+1)=119894

)

=E

[

[

1198600120579(119896)

119909 (119896) +

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

times [

[

1198600120579(119896)

119909 (119896)

+

119873

sum

119895=1

119860119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896+1)=119894

=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894

= 119883119894(119896 + 1) 119896 isin 119885

+

(31)

then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin

119885+

Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

for 119905 isin 119885+ 119897 = 1 2

(32)

Proof Because

119884119894(119896) = E

[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

times[

[

1198620120579(119896)

119909 (119896) +

119873

sum

119895=1

119862119895120579(119896)

119909 (119896)119908119895(119896)]

]

119879

1120579(119896)=119894

=

119873

sum

119897=0

119862119897119894119883119894(119896) 119862119879

119897119894

(33)

therefore

⟨119883119894(119896) 119862

119894⟩ = tr[119883

119894(119896)(

119873

sum

119897=0

119862119879

119897119894119862119897119894)] = tr (119884

119894(119896))

119882119897

119894(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883119894(119905 + 119896) 119862

119894⟩ =

119897minus1

sum

119896=0

tr (119884119894(119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894

]

⟨119883 (119896) 119862⟩ =

119904

sum

119894=1

tr (119883119894(119896) 119862119894) = tr (119884 (119896))

119882119897

(119883 (119905)) =

119897minus1

sum

119896=0

⟨119883 (119905 + 119896) 119862⟩ =

119897minus1

sum

119896=0

tr (119884 (119905 + 119896))

=

119897minus1

sum

119896=0

E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]

(34)

Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897

119894(119883(119905))

Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2

Proof It is easy to get the result by mathematical inductionso we will omit the details

Abstract and Applied Analysis 7

Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119898 isin 119885+

119905 isin 119885+ one has

⟨119883 (119905) F119897

(O (119898))⟩ = ⟨Flowast119897

(119883 (119905)) O (119898)⟩ 119897 isin 119885+

(35)

Proof Firstly it is easy to get

⟨119883 (119905) F0

(O (119898))⟩

= ⟨119883 (119905) O (119898)⟩

= ⟨Flowast0

(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+

(36)

Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have

⟨119883 (119905) F119902+1

(O (119898))⟩ = ⟨119883 (119905) F119902

(O (119898 + 1))⟩

minus ⟨119883 (119905) F119902

(119862)⟩

= ⟨Flowast119902

(119883 (119905)) O (119898 + 1)⟩

minus ⟨Flowast119902

(119883 (119905)) 119862⟩

= ⟨119883 (119905 + 119902) O (119898 + 1)⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= 119882119898+1

(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩

=

119898

sum

119896=0

⟨119883 (119905 + 119902 + 119896) 119862⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= ⟨119883 (119905 + 119902 + 1) O (119898)⟩

= ⟨Flowast119902+1

(119883 (119905)) O (119898)⟩

(37)

By mathematical induction it is right

Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ 119905 isin 119885+ one has

⟨119883 (119905) F119897

(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)

Proof Consider

⟨119883 (119905) F119897

(119862)⟩ = ⟨119883 (119905) F119897

(O (1))⟩

= ⟨Flowast119897

(119883 (119905)) O (1)⟩

= ⟨119883 (119905 + 119897) 119862⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]

(39)

In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1

Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one

has

119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)

Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

lim119896rarrinfin

E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin

E [119909119879 (119896) 119909 (119896)] = 0(41)

Proof This proof is omitted

Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899

119903 ≜ rank(119883) then thereexist nonzero vectors 119909

1 119909

119903isin 119877119899 such that 119883 = 119909

1119909119879

1+

sdot sdot sdot + 119909119903119909119879

119903

323 A Useful Formula Define two operators L1(119880) =

[

1198711(1198801)

1198711(119880119904)

] for 119880119894isin 119877119899times119899 andL

2(119880) = [

1198712(1198801)

1198712(119880119904)

] for 119880119894isin 119883 |

119883119879

= 119883 isin 119877119899times119899

in 119877119899times119899119904

For convenience we naturally think 119884 isin 119883 | 119883

119879

= 119883 isin

119877119899times119899

when we use 1198712(119884) in this subsection

Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899

119904 one has

L1(F (119880)) = AL

1(119880)

L2(F (119880)) = BL

2(119880)

⟨119880 119881⟩ = L119879

1(119880)L

1(119881)

⟨119880 119881⟩ = L119879

1(119880) (119868 otimes 119867)L

2(119881)

(42)

where

A ≜

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971 119860

119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

B ≜

119873

sum

119897=0

diag (119867minus (1198601198791198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

(43)

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

Submit your manuscripts athttpwwwhindawicom

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119898 isin 119885+

119905 isin 119885+ one has

⟨119883 (119905) F119897

(O (119898))⟩ = ⟨Flowast119897

(119883 (119905)) O (119898)⟩ 119897 isin 119885+

(35)

Proof Firstly it is easy to get

⟨119883 (119905) F0

(O (119898))⟩

= ⟨119883 (119905) O (119898)⟩

= ⟨Flowast0

(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+

(36)

Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have

⟨119883 (119905) F119902+1

(O (119898))⟩ = ⟨119883 (119905) F119902

(O (119898 + 1))⟩

minus ⟨119883 (119905) F119902

(119862)⟩

= ⟨Flowast119902

(119883 (119905)) O (119898 + 1)⟩

minus ⟨Flowast119902

(119883 (119905)) 119862⟩

= ⟨119883 (119905 + 119902) O (119898 + 1)⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= 119882119898+1

(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩

=

119898

sum

119896=0

⟨119883 (119905 + 119902 + 119896) 119862⟩

minus ⟨119883 (119905 + 119902) 119862⟩

= ⟨119883 (119905 + 119902 + 1) O (119898)⟩

= ⟨Flowast119902+1

(119883 (119905)) O (119898)⟩

(37)

By mathematical induction it is right

Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ 119905 isin 119885+ one has

⟨119883 (119905) F119897

(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)

Proof Consider

⟨119883 (119905) F119897

(119862)⟩ = ⟨119883 (119905) F119897

(O (1))⟩

= ⟨Flowast119897

(119883 (119905)) O (1)⟩

= ⟨119883 (119905 + 119897) 119862⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]

(39)

In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1

Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one

has

119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)

Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

one has

lim119896rarrinfin

E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin

E [119909119879 (119896) 119909 (119896)] = 0(41)

Proof This proof is omitted

Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899

119903 ≜ rank(119883) then thereexist nonzero vectors 119909

1 119909

119903isin 119877119899 such that 119883 = 119909

1119909119879

1+

sdot sdot sdot + 119909119903119909119879

119903

323 A Useful Formula Define two operators L1(119880) =

[

1198711(1198801)

1198711(119880119904)

] for 119880119894isin 119877119899times119899 andL

2(119880) = [

1198712(1198801)

1198712(119880119904)

] for 119880119894isin 119883 |

119883119879

= 119883 isin 119877119899times119899

in 119877119899times119899119904

For convenience we naturally think 119884 isin 119883 | 119883

119879

= 119883 isin

119877119899times119899

when we use 1198712(119884) in this subsection

Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899

119904 one has

L1(F (119880)) = AL

1(119880)

L2(F (119880)) = BL

2(119880)

⟨119880 119881⟩ = L119879

1(119880)L

1(119881)

⟨119880 119881⟩ = L119879

1(119880) (119868 otimes 119867)L

2(119881)

(42)

where

A ≜

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971 119860

119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

B ≜

119873

sum

119897=0

diag (119867minus (1198601198791198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

(43)

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

Submit your manuscripts athttpwwwhindawicom

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Proof When 119880119879119894= 119880119894for 119894 isin 119878 then

L2(F (119880)) =

[[[[[[[[[

[

1198712(

119873

sum

119897=0

119904

sum

119895=1

1199011119895119860119879

11989711198801198951198601198971)

1198712(

119873

sum

119897=0

119904

sum

119895=1

119901119904119895119860119879

119897119904119880119895119860119897119904)

]]]]]]]]]

]

=

119873

sum

119897=0

[[[[[[[[

[

119867minus

(119860119879

1198971otimes 119860119879

1198971)119867

119904

sum

119895=1

11990111198951198712(119880119895)

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867

119904

sum

119895=1

1199011199041198951198712(119880119895)

]]]]]]]]

]

= Σ119873

119897=0diag (119867minus (119860119879

1198971otimes 119860119879

1198971)119867

119867minus

(119860119879

119897119904otimes 119860119879

119897119904)119867) (119875 otimes 119868)L

2(119880)

=BL2(119880)

(44)

By the same way it is easy to get

L1(F (119880)) =

119873

sum

119897=0

diag (1198601198791198971otimes 119860119879

1198971

119860119879

119897119904otimes 119860119879

119897119904) (119875 otimes 119868)L

1(119880)

= AL1(119880)

(45)

Consider

⟨119880 119881⟩ =

119904

sum

119894=1

tr (119880119879119894119881119894) =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=L119879

1(119880)L

1(119881)

⟨119880 119881⟩ =

119904

sum

119894=1

119871119879

1(119880119894) 1198711(119881119894)

=

119904

sum

119894=1

119871119879

1(119880119894)1198671198712(119881119894)

=L119879

1(119880) (119868 otimes 119867)L

2(119881)

(46)

Let 119889 ≜ 119904(119899(119899 + 1)2)

Remark 21 For the symmetric case L2(sdot) is more suit-

able than L1(sdot) in applications (such as Lemma 22 and

Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool

Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin Θ

119905 isin 119885+ one has

119882d(119883 (119905)) = 0 lArrrArr 119882

119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(47)

Proof By Lemma 12 it is easy to get

119882119897

(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)

lArrrArr 119864(119910119879

(119896) 119910 (119896)) = 0 (119896 ge 119905)

(48)

Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889

(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882

119897

(119883(119896)) =

0 (119897 ge 1 119896 ge 119905)

∵L2(F (119880)) =BL

2(119880) (by Theorem 20)

O (119897) = 119862 +F (O (119897 minus 1))

there4L2(O (119897)) =L

2(119862) +BL

2(O (119897 minus 1))

= 119902 +B [119902 +BL2(O (119897 minus 2))]

where 119902 ≜L2(119862)

= 119902 +B119902 + sdot sdot sdot +B119897minus1

119902 (119897 = 1 2 )

∵ ⟨119883 (119905) F119897

(119862)⟩

= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)

there4 ⟨119883 (119905) F119897

(119862)⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)L

2(F119897

(119862)) (by Theorem 20)

=L119879

1(119883 (119905)) (119868 otimes 119867)BL

2(F119897minus1

(119862))

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119897

119902 ge 0 (119897 isin 119885+

)

(49)

If119882119889(119883(119905)) = 0 then

⟨119883 (119905) O (119889)⟩ =L119879

1(119883 (119905)) (119868 otimes 119867)L

2(O (119889))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (119902 +B119902 + sdot sdot sdot +B119889minus1

119902)

= 0

(50)

ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1

For B isin 119877119889times119889 there exist 119886

0(119897) 1198861(119897) 119886

119889minus1(119897) isin 119877 by

the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that

B119897

= 1198860(119897)B0

+ 1198861(119897)B1

+ sdot sdot sdot + 119886119889minus1

(119897)B119889minus1

(119897 isin 119885+

)

(51)

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

For 119897 ge 1 119896 ge 119905 we have

119882119897

(119883 (119896)) = ⟨119883 (119905) F119896minus119905

(O (119897))⟩

=L119879

1(119883 (119905)) (119868 otimes 119867)B

119896minus119905

L2(O (119897))

=L119879

1(119883 (119905)) (119868 otimes 119867)

times (B119896minus119905

+B119896minus119905+1

+ sdot sdot sdot +B119896minus119905+119897minus1

) 119902

=L119879

1(119883 (119905)) (119868 otimes 119867)

times [1198870(])B0 + 119887

1(])B1

+ sdot sdot sdot + 119887119889minus1

(])B119889minus1] 119902

= 1198870(])L119879

1(119883 (119905)) (119868 otimes 119867)B

0

119902

+ 1198871(])L119879

1(119883 (119905)) (119868 otimes 119867)B

1

119902

+ sdot sdot sdot + 119887119889minus1

(])L1198791(119883 (119905)) (119868 otimes 119867)B

119889minus1

119902

= 0

(52)

where 1198870(]) 1198871(]) 119887

119889minus1(]) isin 119877

Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583

0isin

Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)

324 Main Results for the Weak Observability

Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583

0isin Θ one has

119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)

Remark 25 By Lemma 22 (53) is equivalent to

119882119889

(1198830) = 0 997904rArr 119909

0= 0 (54)

Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878

Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)

by Lemmas 14 and 22 we have

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

tr (119883119894(0)O119894(119889))

=

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090= 0

(55)

Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909

0= 0

That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable

then for all 1205830isin Θ we have

1199090= 0 997904rArr exist119896

0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896

0) = 0) = 1

(56)

By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for

119896 isin 119885+ Thus we can consistently select 0 le 119896

0le 119889 for above

1199090 such that

119882119889+1

(1198830) =

119889

sum

119896=0

E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0

(57)

Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly

semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =

0) Select 1205830isin Θ such that 120583

0119895= 0 (119895 = 119894) Let 119883

0=

(VV1198791205831199001 VV119879120583

119900119904) then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119889) V = 0 (58)

By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883

0) gt 0

Thus O119894(119889) gt 0 for 119894 isin 119878

Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt

0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878

Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)

is strictly semipositive definite (ie exist0 = V isin 119877119899 such that

V119879O119894(119896 + 1)V = 0) Select 120583

0isin Θ to satisfy 120583

0119895= 0 (119895 = 119894)

Let1198830= (VV119879120583

1199001 VV119879120583

119900119904) then

119882119896+1

(1198830) = ⟨119883

0O (119896 + 1)⟩ =

119904

sum

119895=1

1205830119895V119879O119895(119896 + 1) V = 0

(59)

It contradicts119882119896+1(1198830) ge 119882

119896

(1198830) = ⟨119883

0O(119896)⟩ gt 0 Thus

O119894(119896 + 1) gt 0 for 119894 isin 119878

Theorem28 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873

119889= 119889 then

119882119889

(1198830) = ⟨119883

0O (119889)⟩ =

119904

sum

119894=1

1205830119894119909119879

0O119894(119889) 1199090

ge

119904

sum

119894=1

[1205830119894120582min (O119894 (119889))]

100381610038161003816100381611990901003816100381610038161003816

2

ge

119904

sum

119894=1

[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816

2

= 120574100381610038161003816100381611990901003816100381610038161003816

2

(60)

where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583

0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then

119882119897

(1198830) =

119897minus1

sum

119896=0

E (119910119879 (119896) 119910 (119896)) = 0 (61)

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909

0|2 so 119909

0= 0

That is the system (23) is119882-observable

By Theorem 28 and [6] it is easy to get the followingtheorem

Theorem29 For the system (23) the following two statementsare equivalent

(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O

119894(119896) gt 0 for all 119894 isin 119878

Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability

If we do not consider the noises that is 119860119895119894= 119862119895119894= 0

for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system

119909 (119896 + 1) = 1198600120579(119896)

119909 (119896)

119910 (119896) = 1198620120579(119896)

119909 (119896)

(62)

If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system

119909 (119896 + 1) = 1198600119909 (119896) +

119873

sum

119895=1

119860119895119909 (119896)119908

119895(119896)

119910 (119896) = 1198620119909 (119896) +

119873

sum

119895=1

119862119895119909 (119896)119908

119895(119896)

(63)

where 119860119895≜ 1198601198951 119862119895≜ 1198621198951

for 119895 = 0 1 119873For the above two special situations it is easy to get the

following theorems

Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent

(a) the Markovian jump linear system (62) is 119882-observable

(b) exist119873119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 and 120583

0isin Θ one has

119882119873119889(1198830) ge 120574|119909

0|2

(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878

(d) O119894(119889) gt 0 for all 119894 isin 119878

Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent

(a) the stochastic linear system (63) is119882-observable(b) exist119873

119889ge 1 120574 gt 0 for all 119909

0isin 119877119899 one has119882119873119889(119909

0119909119879

0) ge

120574|1199090|2

(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0

Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899

2) Furthermore O119894(119889)119894isin119878

are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))

4 Conclusions

This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01

References

[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010

[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987

[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002

[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981

[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001

[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006

[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004

[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007

[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008

[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009

[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of