Research Article Parameter and State Estimator for State Space …downloads.hindawi.com/journals/tswj/2014/106505.pdf · 2019-07-31 · Research Article Parameter and State Estimator

Research ArticleParameter and State Estimator for State Space Models

Ruifeng Ding12 and Linfan Zhuang12

1 Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education) Jiangnan University Wuxi 214122 China2 School of Internet of Things Engineering Jiangnan University Wuxi 214122 China

Correspondence should be addressed to Ruifeng Ding rfding12126com

Received 30 August 2013 Accepted 31 December 2013 Published 2 March 2014

Academic Editors M Hajarian and C Saravanan

Copyright copy 2014 R Ding and L Zhuang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper proposes a parameter and state estimator for canonical state space systems from measured input-output data The keyis to solve the system state from the state equation and to substitute it into the output equation eliminating the state variablesand the resulting equation contains only the system inputs and outputs and to derive a least squares parameter identificationalgorithm Furthermore the system states are computed from the estimated parameters and the input-output data Convergenceanalysis using the martingale convergence theorem indicates that the parameter estimates converge to their true values Finally anillustrative example is provided to show that the proposed algorithm is effective

1 Introduction

Parameter estimation and identification have had importantapplications in systemmodelling system control and systemanalysis [1ndash5] and thus have receivedmuch research attentionin recent decades [6ndash11] Several identification methods havebeen developed for state space models for example thesubspace identification methods [12] Gibson and Ninnesspresented a robust maximum-likelihood estimation for fullyparameterized linear time-invariant (LTI) state space mod-els the idea is to use the expectation maximization (EM)algorithm to estimate maximum-likelihood degrees [13]Raghavan et al studied the EM-based state space modelidentification problems with irregular output sampling [14]

The state space model includes not only the unknownparameter matricesvectors but also the unknown noiseterms in the formation vector and unmeasurable state vectorMany algorithms can estimate the system states assumingthat the system parameter matricesvectors are available butsuch state estimation algorithm cannot work if the systemparameters are unknown [15] Recently Ding presenteda combined state and least squares parameter estimationalgorithm for dynamic systems [16]

In the area of state space model identification Dingand Chen proposed a hierarchical identification estimationalgorithm for estimating the system parameters and states

[17] Li et al assumed that the system states were availableand used the measurable states and input-output data toestimate the parameters of lifted state space models forgeneral dual-rate systems [18] Recently some identificationmethods have been developed for example the least squaresmethods [19 20] the gradient-based methods [21 22] thebias compensation methods [23 24] and the maximumlikelihood methods [25ndash30] The objective of this paper is topresent a new parameter and state estimation-based residualalgorithm from the given input-output data and further toanalyze the convergence of the proposed algorithm

The convergence analysis of identification algorithms hasalways been one of the important projects in the field ofcontrol By using the stochastic martingale theory Ding etal studied the properties of stochastic gradient identifica-tion algorithms under weak conditions [31] Ding and Liudiscussed the gradient-based identification approach andconvergence for multivariable systems with output measure-ment noise [32] Other identification methods for linearor nonlinear systems [33ndash42] include the auxiliary modelidentification methods [43ndash57] the hierarchical identifi-cation methods [58ndash73] and the two-stage or multistageidentification methods [74ndash78]

This paper is organized as follows Section 2 introducesthe system description and its identification model paperSection 3 derives a basic parameter identification algorithm

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 106505 10 pageshttpdxdoiorg1011552014106505

2 The Scientific World Journal

for canonical state space systems and analyzes the perfor-mance of the proposed algorithm Section 4 gives a stateestimation algorithm Section 5 provides an example for theproposed algorithm Finally concluding remarks are given inSection 6

2 System Description andIdentification Model

Let us introduce some notation [15] ldquo119860 = 119883rdquo or ldquo119883 = 119860rdquostands for ldquo119860 is defined as 119883rdquo the symbol I(I

119899) stands for an

identity matrix of appropriate size (119899 times 119899) the superscript Tdenotes the matrix transpose |X| = det[X] represents thedeterminant of a square matrix X the norm of a matrix Xis defined by X2 = tr[XXT

] 1119899

= 1119899times1

represents an119899 times 1 vector whose elements are all 1 120582min[X] represents theminimum eigenvalues of X for 119892(119905) ⩾ 0 we write 119891(119905) =

119874(119892(119905)) if there exists a positive constant 1205751such that |119891(119905)| ⩽

1205751119892(119905)In order to study the convergence of the algorithm

proposed in [15] here we simply give that algorithm in[15] Consider a linear system described by the followingobservability canonical state space model [15]

x (119905 + 1) = Ax (119905) + b119906 (119905)

119910 (119905) = cx (119905) + V (119905) (1)

A =

[[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 d

d 0

0 0 sdot sdot sdot 0 1

minus119886119899

minus119886119899minus1


sdot sdot sdot minus1198861

]]]]]]]

]

isin R119899times119899

b =

[[[[

[

1198871

1198872

119887119899

]]]]

]

isin R119899

c = [1 0 0 0] isin R1times119899

(2)

where x(119905) isin R119899 is the state vector 119906(119905) isin R is the systeminput119910(119905) isin R is the system output and V(119905) isin R is a randomnoise with zero mean Assume that the order 119899 is known and119906(119905) = 0 119910(119905) = 0 and V(119905) = 0 for 119905 ⩽ 0

The system in (1) is an observability canonical form andits observability matrixQ

119900is an identity matrix that is

Q119900=

[[[[

[

ccA

cA119899minus1

]]]]

]

= I119899 (3)

For the system in (1) the objective of this paper isto develop a new algorithm to estimate the parametermatrixvector A and b (ie the parameters 119886

119894and 119887

119894) and

the system state vector x(119905) from the available measurementinput-output data 119906(119905) 119910(119905)

Since the available measurement input-output data119906(119905) 119910(119905) are known but the state vector x(119905) is unknownit is required to eliminate the state vector from (1) and obtaina new expression which only involves the input and outputin order to obtain the estimates of the parameters in (1)The following derives the identification model based on themethod in [15]

Define some vectorsmatrix

120593119910(119905) = [119910 (119905 minus 119899) 119910 (119905 minus 119899 + 1) 119910 (119905 minus 1)]

Tisin R119899


Tisin R119899

120593V (119905) = [V (119905 minus 119899) V (119905 minus 119899 + 1) V (119905 minus 1)]Tisin R119899

M =

[[[[[[[

[


cb 0 sdot sdot sdot 0 0

cAb cb d

d 0 0

cA119899minus2b cA119899minus3b sdot sdot sdot cb 0

]]]]]]]

]


(4)

From (1) we have

119910 (119905) = cx (119905) + V (119905) (5)

119910 (119905 + 1) = cx (119905 + 1) + V (119905 + 1)

= c [Ax (119905) + b119906 (119905)] + V (119905 + 1)

= cAx (119905) + cb119906 (119905) + V (119905 + 1)

(6)

119910 (119905 + 2) = cAx (119905 + 1) + cb119906 (119905 + 1) + V (119905 + 2)

= cA [Ax (119905) + b119906 (119905)] + cb119906 (119905 + 1) + V (119905 + 2)

= cA2x (119905) + cAb119906 (119905) + cb119906 (119905 + 1) + V (119905 + 2)

(7)

119910 (119905 + 119899 minus 1) = cA119899minus1x (119905) + cA119899minus2b119906 (119905) + cA119899minus3b119906 (119905 minus 1)

+ sdot sdot sdot + cb119906 (119905 + 119899 minus 2) + V (119905 + 119899 minus 1)

(8)

119910 (119905 + 119899) = cA119899x (119905) + cA119899minus1b119906 (119905) + cA119899minus2b119906 (119905 minus 1)

+ sdot sdot sdot + cb119906 (119905 + 119899 minus 1) + V (119905 + 119899)

(9)

Combining (5) with (8) gives

120593119910(119905 + 119899) = Q

119900x (119905) +M120593

119906(119905 + 119899) + 120593V (119905 + 119899)

= x (119905) +M120593119906(119905 + 119899) + 120593V (119905 + 119899)

(10)

or

x (119905) = 120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899) (11)

The Scientific World Journal 3

Define the parameter vector 120579 and the information vector120593(119905) as

120579 = [120579119886

120579119887

] isin R2119899

120579119886= [cA119899]T isin R

119899

120579119887= [minuscA119899M + [cA119899minus1b cA119899minus1b cb]]

Tisin R119899

120593 (119905 + 119899) = [120593T119910(119905 + 119899) minus 120593

TV (119905 + 119899) 120593

T119906(119905 + 119899)]

Tisin R2119899

(12)

Substituting (11) into (9) gives

119910 (119905 + 119899)

= cA119899 [120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899)]

+ cA119899minus1b119906 (119905) + cA119899minus2b119906 (119905 minus 1)


= cA119899 [120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899)]

+ [cA119899minus1b cA119899minus2b cb][[[[

[

119906 (119905)

119906 (119905 minus 1)

119906 (119905 + 119899 minus 1)

]]]]

]

+ V (119905 + 119899)

= cA119899 [120593119910(119905 + 119899) minus 120593V (119905 + 119899)] minus cA119899M120593

119906(119905 + 119899)

+ [cA119899minus1b cA119899minus2b cb]120593119906(119905 + 119899) + V (119905 + 119899)

= [120593T119910(119905 + 119899) minus 120593

TV (119905 + 119899) 120593

T119906(119905 + 119899)] [

120579119886

120579119887

] + V (119905 + 119899)

= 120593T(119905 + 119899) 120579 + V (119905 + 119899)

(13)

Replacing 119905 in (13) with 119905 minus 119899 yields

119910 (119905) = 120593T(119905) 120579 + V (119905) (14)

which is called the identification model or identificationexpression of the state-space model

3 The Parameter EstimationAlgorithm and Its Convergence

The recursive least squares algorithm for estimating 120579 isexpressed as

(119905) = (119905 minus 1) + P (119905) (119905) [119910 (119905) minus T(119905) (119905 minus 1)] (15)

Pminus1 (119905) = Pminus1 (119905 minus 1) + (119905) T(119905) P (0) = 119901

0I (16)

V (119905) = 119910 (119905) minus T(119905) (119905) (17)

(119905) = [119910 (119905 minus 119899) minus V (119905 minus 119899)

119910 (119905 minus 119899 + 1) minus V (119905 minus 119899 + 1)

119910 (119905 minus 1) minus V (119905 minus 1) 119906 (119905 minus 119899)

119906 (119905 minus 119899 + 1) 119906 (119905 minus 1)]T

(18)

This algorithm is commonly used for convergence analysisTo avoid computing the matrix inversion this algorithm isequivalently expressed as

(119905) = (119905 minus 1) + L (119905) [119910 (119905) minus T(119905) (119905 minus 1)] (19)

L (119905) = P (119905) (119905) =P (119905 minus 1) (119905)

1 + T(119905)P (119905 minus 1) (119905)

(20)

P (119905) = [I minus L (119905) T(119905)]P (119905 minus 1) P (0) = 119901

0I (21)

V (119905) = 119910 (119905) minus T(119905) (119905) (22)

(119905) = [T(119905 minus 119899) (119905 minus 119899)

T(119905 minus 119899 + 1) (119905 minus 119899 + 1)

T(119905 minus 1) (119905 minus 1)

119906 (119905 minus 119899) 119906 (119905 minus 119899 + 1) 119906 (119905 minus 1)]T

(23)

where L(119905) isin R2119899 is the gain vectorDefine the parameter estimation error vector (119905) =

(119905)minus120579 and the nonnegative function119879(119905) = T(119905)Pminus1(119905)(119905)

Theorem 1 For the system in (1) and algorithm in (15)ndash(18) assume that V(119905)F

119905 is a martingale difference sequence

defined on a probability space ΩF 119875 where F119905 is the

120590 algebra sequence generated by the observations up to andincluding time 119905The noise sequence V(119905) satisfies the followingassumptions

(A1) E [V(119905) | F119905minus1

] = 0 as(A2) E [V2(119905) | F

119905minus1] ⩽ 1205902

lt infin as(A3) 1198601015840(119911) = 119860

minus1

(119911) minus 12 is strictly positive real

Then the following inequality holds

E [119879 (119905) + 119878 (119905) | F119905minus1

]

⩽ 119879 (119905 minus 1) + 119878 (119905 minus 1) + 2T(119905)P (119905) (119905) 120590

2

(24)


where

119878 (119905) = 2

119905

sum

119894=1

(119894) 119910 (119894) ⩾ 0 (25)

(119905) = minusT(119905) (119905) (26)

119910 (119905) =1

2

T(119905) (119905) + [119910 (119905) minus

T(119905) (119905) minus V (119905)] (27)

Proof Define the innovation vector 119890(119905) = 119910(119905) minus T(119905)(119905 minus

1) Using (17) it follows that

V (119905) = [1 minus T(119905)P (119905) (119905)] 119890 (119905)

=119890 (119905)

1 + T(119905)P (119905 minus 1) (119905)

(28)

Subtracting 120579 from both sides of (15) and using (14) we have

(119905) = (119905) minus 120579 = (119905 minus 1) + P (119905) (119905) 119890 (119905)

= (119905 minus 1) + P (119905 minus 1) (119905) V (119905) (29)

According to the definition of 119879(119905) and using (16) and (29)we have

119879 (119905) = 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905) minus T (119905)P (119905) (119905)

times [1 minus T(119905)P (119905) (119905)] 119890

2

(119905)

⩽ 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905)

= 119879 (119905 minus 1) + 2T(119905) (119905)

times [1

2T(119905) (119905) + (V (119905) minus V (119905))] + 2

T(119905) (119905) V (119905)

(30)

Using (26) (27) and (29) and 0 ⩽ T(119905)P(119905)(119905) ⩽ 1 we have

119879 (119905) ⩽ 119879 (119905 minus 1) minus 2 (119905) 119910 (119905) + 2T(119905)

times [ (119905 minus 1) + P (119905) (119905) 119890 (119905)] V (119905)

= 119879 (119905 minus 1) minus 2 (119905) 119910 (119905) + 2T(119905) (119905 minus 1) V (119905)

+ 2T(119905)P (119905) (119905) [119890 (119905) minus V (119905)] V (119905) + V2 (119905)

(31)

Since T(119905)(119905minus1) 119890(119905)minus V(119905) T(119905)P(119905)(119905) are uncorrelatedwith V(119905) and are F

119905minus1-measurable taking the conditional

expectation with respect toF119905minus1

and using (A1)-(A2) give

E [119879 (119905) | F119905minus1

] ⩽ 119879 (119905 minus 1) minus 2E [ (119905) 119910 (119905) | F119905minus1

]

+ 2T(119905)P (119905) (119905) 120590

2

as(32)

The state spacemodel in (1) can be transformed into an input-output representation

119910 (119905) = c(119911I minus A)minus1b119906 (119905) + V (119905)

=c adj [119911I minus A] bdet [119911I minus A]

119906 (119905) + V (119905)

=119861 (119911)

119860 (119911)119906 (119905) + V (119905)

(33)

where adj[119911I minus A] is the adjoint matrix of [119911I minus A] 119860(119911)and 119861(119911) are polynomials in a unit backward shift operator119911minus1

[119911minus1

119910(119905) = 119910(119905 minus 1)] and

119860 (119911) = 119911minus119899 det [119911I minus A]

119861 (119911) = 119911minus119899c adj [119911I minus A] b

(34)

Referring to the proof of Lemma 3 in [43] using (33) we have

119860 (119911) [V (119905) minus V (119905)] = 119860 (119911) V (119905) minus 119860 (119911) 119910 (119905) + 119861 (119911) 119906 (119905)

= minus119860 (119911) T(119905) (119905) + 119861 (119911) 119906 (119905)

= minusT(119905) (119905) = (119905)

(35)

Using (17) (26) and (35) from (27) we get

119910 (119905) =1

2T(119905) (119905) + [V (119905) minus V (119905)]

= [119860minus1

(119911) minus1

2] (119905)

(36)

Since 1198601015840

(119911) is a strictly positive real function referring toAppendix C in [79] we can obtain the conclusion 119878(119905) ⩾ 0Adding both sides of (32) by 119878(119905) gives the conclusion ofTheorem 1

Theorem 2 For the system in (1) and the algorithm in (15)ndash(18) assume that (A1)ndash(A3) hold and that 119860(119911) is stable thatis all zeros of119860(119911) are inside the unit circle then the parameterestimation error satisfies

10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119903 (119905)]

119888

120582min [Pminus1 (119905)]) 119886119904 119891119900119903 119886119899119910 119888 gt 1

(37)

Proof Using the formula 120582min[Q]x2 ⩽ xTQx ⩽

120582max[Q]x2 and from the definition of 119879(119905) we have

10038171003817100381710038171003817 (119905)

10038171003817100381710038171003817

2

⩽T(119905)Pminus1 (119905) (119905)120582min [Pminus1 (119905)]

=119879 (119905)

120582min [Pminus1 (119905)] (38)

Let

119882(119905) =119879 (119905) + 119878 (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888 119888 gt 1 (39)


Since ln |Pminus1(119905)| is nondecreasing usingTheorem 1 yields

E [119882 (119905) | F119905minus1

] ⩽119879 (119905 minus 1) + 119878 (119905 minus 1)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

+2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

⩽ 119881 (119905 minus 1) +2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

as

(40)

Referring to the proof of Theorem 2 in [43] we have

10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874(

[ln 10038161003816100381610038161003816Pminus1 (119905)10038161003816100381610038161003816]

119888

120582min [Pminus1 (119905)])

= 119874([ln 119903 (119905)]

119888

120582min [Pminus1 (119905)]) as for any 119888 gt 1

(41)

Assume that there exist positive constants 120574 1198881 1198882 and

1199050such that the following generalized persistent excitation

condition (unbounded condition number) holds

1198881I ⩽ 1

119905

119905

sum

119895=1

120593 (119895)120593T(119895) ⩽ 119888

2119905120574I as for 119905 ⩾ 119905

0 (42)

Then for any 119888 gt 1 we have

10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119905]119888

119905) 997888rarr 0 as for any 119888 gt 1 (43)

4 The State Estimation Algorithm

Referring to the method in [15] the state estimate x(119905) of thestate vector x(119905) can be expressed as

x (119905 minus 119899) = 120593119910(119905) minus M (119905)120593

119906(119905) minus V (119905) (44)


T (45)


T (46)

V (119905) = [V (119905 minus 119899) V (119905 minus 119899 + 1) V (119905 minus 1)]T (47)

M (119905) =

[[[[[[[[

[


1(119905) 0 sdot sdot sdot 0 0

2(119905)

1(119905) d

d 0 0

119899minus1

(119905) 119899minus2

(119905) sdot sdot sdot 1(119905) 0

]]]]]]]]

]

(48)

[[[[[[[

[

1(119905)

2(119905)

119899minus1

(119905)

119899(119905)

]]]]]]]

]

=

[[[[[[

[

119886119899minus1

(119905) 119886119899minus2

(119905) sdot sdot sdot 1198861(119905) 1

119886119899minus2

(119905) 119886119899minus3

(119905) sdot sdot sdot 1 0

1198861(119905) 1 sdot sdot sdot 0 0


]]]]]]

]

minus1

119887(119905)

(49)

(119905) = [119886(119905)

119887(119905)

] (50)

119886(119905) = [minus119886

119899(119905) minus119886

119899minus1(119905) minus119886

1(119905)]

T (51)

To summarize we list the steps involved in the algorithm in(19)ndash(23) and (44)ndash(51) to compute the parameter estimate(119905) and the state estimate x(119905 minus 119899)

(1) Let 119905 = 1 set the initial values (119894) = 11198991199010 P(0) =

1199010I 119906(119894) = 0 119910(119894) = 0 V(119894) = 0 or V(119894) = 1119901

0for

119894 ⩽ 0 1199010= 106 Give a small positive number 120576

(2) Collect the input-output data 119906(119905) and 119910(119905) form (119905)using (23) 120593

119910(119905) using (45) and 120593

119906(119905) using (46)

(3) Compute the gain vector L(119905) using (20) and thecovariance matrix P(119905) using (21)

(4) Update the parameter estimation vector (119905) using(19)

(5) Compute V(119905) using (22) and form V(119905) using (47)

(6) Determine 119886119894(119905) using (51) and compute

119894(119905) using

(49) then form M(119905) using (48)(7) Compute the state estimate x(119905 minus 119899) using (44)

(8) If they are sufficiently close if (119905) minus (119905 minus 1) ⩽ 120576then terminate the procedure and obtain the estimate(119905) otherwise increase 119905 by 1 and go to step 2

5 Example

Consider the following single-input single-output second-order system in canonical form

x (119905 + 1) = [0 1

minus070 135] x (119905) + [

1

1] 119906 (119905)

119910 (119905) = [1 0] x (119905) + V (119905)

(52)

The simulation conditions are the same as in [15] Thatis the input 119906(119905) is taken as an independent persistentexcitation signal sequence with zero mean and unit variancesand V(119905) as a white noise sequence with zero mean andvariances 120590

2

= 0202 and 120590

2

= 1002 respectively Apply

the proposed parameter and state estimation algorithm in(19)ndash(23) and (44)ndash(51) to estimate the parameters and statesof this example system the parameter estimates and theirestimation errors are shown in Tables 1 and 2 the parameterestimation errors 120575 versus 119905 are shown in Figure 1 the states119909119894(119905) and their estimates 119909

119894(119905) versus 119905 are shown in Figures 2

and 3 where 120575 = (119905)minus120579120579 (x2 = xTx) is the parameterestimation error

From the simulation results of Tables 1 and 2 and Figures1ndash3 we can draw the following conclusions

(1) A lower noise level leads to a faster rate of convergenceof the parameter estimates to the true parameters

(2) The parameter estimation errors 120575 become smaller(in general) as the data length 119905 increases see


Table 1 The parameter estimates and errors (1205902 = 0202)

119905 1205791

1205792

1205793

1205794

120575 ()

100 minus070568 135339 minus039169 101645 244445200 minus071515 138650 minus041330 100994 406207500 minus070678 136377 minus038556 099926 2089971000 minus070624 136483 minus037257 100253 1501802000 minus070481 135720 minus035450 099836 0533953000 minus070098 135107 minus034969 099994 008005True values minus070000 135000 minus035000 100000


119905 1205791

1205792

1205793

1205794

120575 ()


0 500 1000 1500 2000 2500 30000

010203040506070809

120575 1205902= 100

2

1205902= 020

2

t

Figure 1 The parameter estimation errors 120575 versus 119905 (1205902 = 0202

and 1205902

= 1002)

Tables 1 and 2 and Figure 1 In other words increasingdata length generally results in smaller parameterestimation errors

(3) The state estimates are close to their true values with 119905

increasing see Figures 2 and 3These indicate that theproposed parameter and state estimation algorithmare effective

6 Conclusions

In this paper the identification problems for linear systemsbased on the canonical state space models with unknownparameters and states are studied A new parameter and stateestimation algorithm has been presented directly from input-output data The analysis using the martingale convergence

0 20 40 60 80 100 120 140 160 180 200

Stat

e est

imat

es

t (s)

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

Figure 2 The state estimation errors 120575 versus 119905 (1205902 = 0202) Solid

line the true 1199091(119905) dots the estimated 119909

1(119905)

0 20 40 60 80 100 120 140 160 180 200

Stat

e esti

mat

es

t (s)

3

2

1

0

minus1

minus2

minus3



2(119905)


theorem indicates that the proposed algorithms can giveconsistent parameter estimationThe simulation results showthat the proposed algorithms are effective The method inthis paper can combine the multiinnovation identificationmethods [80ndash92] the iterative identification methods [93ndash100] and other identification methods [101ndash111] to presentnew identification algorithms or to study adaptive controlproblems for linear or nonlinear single-rate or dual-ratescalar or multivariable systems [112ndash117]

Conflict of Interests

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

Acknowledgments

This work was supported by the PAPD of Jiangsu HigherEducation Institutions and the 111 Project (B12018)

References

[1] F Ding System IdentificationmdashNew Theory and Methods Sci-ence Press Beijing China 2013

[2] F Ding System IdentificationmdashPerformances Analysis for Iden-tification Methods Science Press Beijing China 2014

[3] Y B Hu ldquoIterative and recursive least squares estimationalgorithms for moving average systemsrdquo Simulation ModellingPractice andTheory vol 34 pp 12ndash19 2013

[4] Z Y Wang Y X Shen Z C Ji and F Ding ldquoFiltering basedrecursive least squares algorithm for Hammerstein FIR-MAsystemsrdquo Nonlinear Dynamics vol 73 no 1-2 pp 1045ndash10542013

[5] V Singh ldquoStability of discrete-time systems joined with asaturation operator on the statespace generalized form of Liu-Michelrsquos criterionrdquo Automatica vol 47 no 3 pp 634ndash637 2011

[6] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2012 Article ID 684074 14pages 2012

[7] Y Shi and H Fang ldquoKalman filter-based identification forsystems with randomly missing measurements in a networkenvironmentrdquo International Journal of Control vol 83 no 3 pp538ndash551 2010

[8] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009

[9] Y Shi and B Yu ldquoRobust mixed 1198672119867infin

control of networkedcontrol systems with random time delays in both forward andbackward communication linksrdquo Automatica vol 47 no 4 pp754ndash760 2011

[10] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics vol 73 no 1-2 pp 583ndash592 2013

[11] P P Hu F Ding and J Sheng ldquoAuxiliary model based leastsquares parameter estimation algorithm for feedback nonlinearsystems using the hierarchical identification principlerdquo Journal

of the Franklin InstitutemdashEngineering and AppliedMathematicsvol 350 no 10 pp 3248ndash3259 2013

[12] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[13] S Gibson and B Ninness ldquoRobust maximum-likelihood esti-mation of multivariable dynamic systemsrdquo Automatica vol 41no 10 pp 1667ndash1682 2005

[14] H Raghavan A K Tangirala R B Gopaluni and S L ShahldquoIdentification of chemical processes with irregular outputsamplingrdquo Control Engineering Practice vol 14 no 5 pp 467ndash480 2006

[15] L Zhuang F Pan and F Ding ldquoParameter and state estimationalgorithm for single-input single-output linear systems usingthe canonical state space modelsrdquo Applied Mathematical Mod-elling vol 36 no 8 pp 3454ndash3463 2012

[16] F Ding ldquoCombined state and least squares parameter estima-tion algorithms for dynamic systemsrdquo Applied MathematicalModelling vol 38 no 1 pp 403ndash412 2014

[17] F Ding and T Chen ldquoHierarchical identification of lifted state-space models for general dual-rate systemsrdquo IEEE Transactionson Circuits and Systems I Regular Papers vol 52 no 6 pp 1179ndash1187 2005

[18] D Li S L Shah and T Chen ldquoIdentification of fast-rate modelsfrom multirate datardquo International Journal of Control vol 74no 7 pp 680ndash689 2001

[19] L Ljung System Identification Theory for the User Prentice-Hall Englewood Cliffs NJ USA 2nd edition 1999

[20] Y Xiao F Ding Y Zhou M Li and J Dai ldquoOn consistency ofrecursive least squares identification algorithms for controlledauto-regression modelsrdquo Applied Mathematical Modelling vol32 no 11 pp 2207ndash2215 2008

[21] F Ding X M Liu H B Chen and G Y Yao ldquoHierarchicalgradient based and hierarchical least squares based iterativeparameter identification for CARARMA systemsrdquo Signal Pro-cessing vol 97 pp 31ndash39 2014

[22] FDing andTChen ldquoHierarchical gradient-based identificationof multivariable discrete-time systemsrdquo Automatica vol 41 no2 pp 315ndash325 2005

[23] Y Zhang ldquoUnbiased identification of a class of multi-inputsingle-output systems with correlated disturbances using biascompensation methodsrdquo Mathematical and Computer Mod-elling vol 53 no 9-10 pp 1810ndash1819 2011

[24] Y Zhang and G Cui ldquoBias compensation methods for stochas-tic systems with colored noiserdquo Applied Mathematical Mod-elling vol 35 no 4 pp 1709ndash1716 2011

[25] WWang J Li and R F Ding ldquoMaximum likelihood parameterestimation algorithm for controlled autoregressive autoregres-sive modelsrdquo International Journal of Computer Mathematicsvol 88 no 16 pp 3458ndash3467 2011

[26] W Wang F Ding and J Dai ldquoMaximum likelihood leastsquares identification for systems with autoregressive movingaverage noiserdquo Applied Mathematical Modelling vol 36 no 5pp 1842ndash1853 2012

[27] J Li and F Ding ldquoMaximum likelihood stochastic gradientestimation for Hammerstein systems with colored noise basedon the key term separation techniquerdquo Computers and Mathe-matics with Applications vol 62 no 11 pp 4170ndash4177 2011

[28] J Li F Ding and G W Yang ldquoMaximum likelihood leastsquares identificationmethod for input nonlinear finite impulseresponsemoving average systemsrdquoMathematical andComputerModelling vol 55 no 3-4 pp 442ndash450 2012


[29] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013

[30] J H Li F Ding and L Hua ldquoMaximum likelihood Newtonrecursive and the Newton iterative estimation algorithms forHammerstein CARAR systemsrdquo Nonlinear Dynamics vol 75no 1-2 pp 234ndash245 2014

[31] F Ding H Yang and F Liu ldquoPerformance analysis of stochasticgradient algorithms under weak conditionsrdquo Science in China Fvol 51 no 9 pp 1269ndash1280 2008

[32] F Ding and X-P Liu ldquoAuxiliary model-based stochastic gra-dient algorithm for multivariable output error systemsrdquo ActaAutomatica Sinica vol 36 no 7 pp 993ndash998 2010

[33] D Q Wang and F Ding ldquoLeast squares based and gradientbased iterative identification for Wiener nonlinear systemsrdquoSignal Processing vol 91 no 5 pp 1182ndash1189 2011

[34] D Q Wang F Ding and Y Y Chu ldquoData filtering basedrecursive least squares algorithm for Hammerstein systemsusing the key-term separation principlerdquo Information Sciencesvol 222 pp 203ndash212 2013

[35] D Q Wang and F Ding ldquoHierarchical least squares estimationalgorithm for Hammerstein-Wiener systemsrdquo IEEE Signal Pro-cessing Letters vol 19 no 12 pp 825ndash828 2012

[36] F Ding and T Chen ldquoIdentification of Hammerstein nonlinearARMAX systemsrdquo Automatica vol 41 no 9 pp 1479ndash14892005

[37] F Ding Y Shi and T Chen ldquoGradient-based identificationmethods for hammerstein nonlinear ARMAXmodelsrdquoNonlin-ear Dynamics vol 45 no 1-2 pp 31ndash43 2006

[38] D QWang and F Ding ldquoExtended stochastic gradient identifi-cation algorithms for Hammerstein-Wiener ARMAX systemsrdquoComputers and Mathematics with Applications vol 56 no 12pp 3157ndash3164 2008

[39] DQWang F Ding andXM Liu ldquoLeast squares algorithm foran input nonlinear system with a dynamic subspace state spacemodelrdquo Nonlinear Dynamics vol 75 no 1-2 pp 49ndash61 2014

[40] D Q Wang T Shan and R Ding ldquoData filtering basedstochastic gradient algorithms for multivariable CARAR-likesystemsrdquo Mathematical Modelling and Analysis vol 18 no 3pp 374ndash385 2013

[41] D Q Wang F Ding and D Q Zhu ldquoData filtering based leastsquares algorithms for multivariable CARAR-like systemsrdquoInternational Journal of Control Automation and Systems vol11 no 4 pp 711ndash717 2013

[42] F Ding X P Liu and G Liu ldquoIdentification methods forHammerstein nonlinear systemsrdquo Digital Signal Processing vol21 no 2 pp 215ndash238 2011

[43] F Ding and T Chen ldquoCombined parameter and output estima-tion of dual-rate systems using an auxiliarymodelrdquoAutomaticavol 40 no 10 pp 1739ndash1748 2004

[44] F Ding and T Chen ldquoParameter estimation of dual-ratestochastic systems by using an output error methodrdquo IEEETransactions on Automatic Control vol 50 no 9 pp 1436ndash14412005

[45] F Ding Y Shi and T Chen ldquoAuxiliary model-based least-squares identification methods for Hammerstein output-errorsystemsrdquo Systems and Control Letters vol 56 no 5 pp 373ndash3802007

[46] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[47] F Ding and T Chen ldquoIdentification of dual-rate systems basedon finite impulse response modelsrdquo International Journal ofAdaptive Control and Signal Processing vol 18 no 7 pp 589ndash598 2004

[48] F Ding and Y Gu ldquoPerformance analysis of the auxiliary modelbased least squares identification algorithm for one-step statedelay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012

[49] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013

[50] D Q Wang Y Chu G W Yang and F Ding ldquoAuxiliary modelbased recursive generalized least squares parameter estimationfor Hammerstein OEAR systemsrdquoMathematical and ComputerModelling vol 52 no 1-2 pp 309ndash317 2010

[51] D Q Wang Y Chu and F Ding ldquoAuxiliary model-based RELSand MI-ELS algorithm for Hammerstein OEMA systemsrdquoComputers andMathematics with Applications vol 59 no 9 pp3092ndash3098 2010

[52] L L Han J Sheng F Ding and Y Shi ldquoAuxiliary modelidentification method for multirate multi-input systems basedon least squaresrdquo Mathematical and Computer Modelling vol50 no 7-8 pp 1100ndash1106 2009

[53] L L Han F Wu J Sheng and F Ding ldquoTwo recursiveleast squares parameter estimation algorithms for multiratemultiple-input systems by using the auxiliary modelrdquo Mathe-matics and Computers in Simulation vol 82 no 5 pp 777ndash7892012

[54] Y Gu and F Ding ldquoAuxiliary model based least squaresidentification method for a state space model with a unit time-delayrdquoAppliedMathematicalModelling vol 36 no 12 pp 5773ndash5779 2012

[55] J Chen and F Ding ldquoLeast squares and stochastic gradientparameter estimation for multivariable nonlinear Box-Jenkinsmodels based on the auxiliary model and the multi-innovationidentification theoryrdquo Engineering Computations vol 29 no 8pp 907ndash921 2012

[56] J Chen Y Zhang and R F Ding ldquoGradient-based parameterestimation for input nonlinear systems with ARMA noisesbased on the auxiliary modelrdquo Nonlinear Dynamics vol 72 no4 pp 865ndash871 2013

[57] J Chen Y Zhang and R F Ding ldquoAuxiliarymodel basedmulti-innovation algorithms for multivariable nonlinear systemsrdquoMathematical and Computer Modelling vol 52 no 9-10 pp1428ndash1434 2010

[58] F Ding and T Chen ldquoHierarchical least squares identificationmethods for multivariable systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 3 pp 397ndash402 2005

[59] F Ding L Qiu and T Chen ldquoReconstruction of continuous-time systems from their non-uniformly sampled discrete-timesystemsrdquo Automatica vol 45 no 2 pp 324ndash332 2009

[60] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011

[61] LWang F Ding and P X Liu ldquoConvergence ofHLS estimationalgorithms for multivariable ARX-like systemsrdquo Applied Math-ematics and Computation vol 190 no 2 pp 1081ndash1093 2007

[62] H Han L Xie F Ding and X Liu ldquoHierarchical least-squaresbased iterative identification for multivariable systems with


moving average noisesrdquoMathematical and ComputerModellingvol 51 no 9-10 pp 1213ndash1220 2010

[63] Y J Liu F Ding and Y Shi ldquoLeast squares estimation for a classof non-uniformly sampled systems based on the hierarchicalidentification principlerdquo Circuits Systems and Signal Processingvol 31 no 6 pp 1985ndash2000 2012

[64] Z Zhang F Ding and X Liu ldquoHierarchical gradient basediterative parameter estimation algorithm for multivariable out-put errormoving average systemsrdquoComputers andMathematicswith Applications vol 61 no 3 pp 672ndash682 2011

[65] D Q Wang R Ding and X Z Dong ldquoIterative parameterestimation for a class of multivariable systems based on thehierarchical identification principle and the gradient searchrdquoCircuits Systems and Signal Processing vol 31 no 6 pp 2167ndash2177 2012

[66] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006

[67] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008

[68] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005

[69] F Ding and T Chen ldquoIterative least-squares solutions of cou-pled Sylvester matrix equationsrdquo Systems and Control Lettersvol 54 no 2 pp 95ndash107 2005

[70] F Ding ldquoTransformations between some special matricesrdquoComputers andMathematics with Applications vol 59 no 8 pp2676ndash2695 2010

[71] L Xie J Ding and F Ding ldquoGradient based iterative solutionsfor general linear matrix equationsrdquo Computers and Mathemat-ics with Applications vol 58 no 7 pp 1441ndash1448 2009

[72] J Ding Y J Liu and F Ding ldquoIterative solutions to matrixequations of the form119860

119894119883119861119894= 119865119894rdquoComputers andMathematics

with Applications vol 59 no 11 pp 3500ndash3507 2010[73] L Xie Y J Liu and H Yang ldquoGradient based and least squares

based iterative algorithms for matrix equations119860119883119861+119862119883119879

119863 =

119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010

[74] F Ding ldquoTwo-stage least squares based iterative estimationalgorithm for CARARMA system modelingrdquo Applied Mathe-matical Modelling vol 37 no 7 pp 4798ndash4808 2013

[75] F Ding and H H Duan ldquoTwo-stage parameter estimationalgorithms for Box-Jenkins systemsrdquo IET Signal Processing vol7 no 8 pp 646ndash654 2013

[76] H Duan J Jia and R F Ding ldquoTwo-stage recursive leastsquares parameter estimation algorithm for output error mod-elsrdquoMathematical and Computer Modelling vol 55 no 3-4 pp1151ndash1159 2012

[77] G Yao and R F Ding ldquoTwo-stage least squares based iterativeidentification algorithm for controlled autoregressive movingaverage (CARMA) systemsrdquo Computers and Mathematics withApplications vol 63 no 5 pp 975ndash984 2012

[78] S J Wang and R Ding ldquoThree-stage recursive least squaresparameter estimation for controlled autoregressive autoregres-sive systemsrdquoAppliedMathematical Modelling vol 37 no 12-13pp 7489ndash7497 2013

[79] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984

[80] F Ding and T Chen ldquoPerformance analysis ofmulti-innovationgradient type identification methodsrdquo Automatica vol 43 no1 pp 1ndash14 2007

[81] F Ding P X Liu and G Liu ldquoMultiinnovation least-squaresidentification for system modelingrdquo IEEE Transactions onSystems Man and Cybernetics B Cybernetics vol 40 no 3 pp767ndash778 2010

[82] F Ding P X Liu and G Liu ldquoAuxiliary model based multi-innovation extended stochastic gradient parameter estimationwith coloredmeasurement noisesrdquo Signal Processing vol 89 no10 pp 1883ndash1890 2009

[83] F Ding ldquoSeveral multi-innovation identification methodsrdquoDigital Signal Processing vol 20 no 4 pp 1027ndash1039 2010

[84] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013

[85] F Ding H B Chen and M Li ldquoMulti-innovation least squaresidentification methods based on the auxiliary model for MISOsystemsrdquo Applied Mathematics and Computation vol 187 no 2pp 658ndash668 2007

[86] L L Han and F Ding ldquoMulti-innovation stochastic gradientalgorithms formulti-inputmulti-output systemsrdquoDigital SignalProcessing vol 19 no 4 pp 545ndash554 2009

[87] D Q Wang and F Ding ldquoPerformance analysis of the auxiliarymodels based multi-innovation stochastic gradient estimationalgorithm for output error systemsrdquo Digital Signal Processingvol 20 no 3 pp 750ndash762 2010

[88] L Xie Y J Liu H Z Yang and F Ding ldquoModelling and identi-fication for non-uniformly periodically sampled-data systemsrdquoIET Control Theory and Applications vol 4 no 5 pp 784ndash7942010

[89] Y J Liu L Yu and F Ding ldquoMulti-innovation extendedstochastic gradient algorithm and its performance analysisrdquoCircuits Systems and Signal Processing vol 29 no 4 pp 649ndash667 2010

[90] Y J Liu Y Xiao and X Zhao ldquoMulti-innovation stochastic gra-dient algorithm for multiple-input single-output systems usingthe auxiliary modelrdquo Applied Mathematics and Computationvol 215 no 4 pp 1477ndash1483 2009

[91] L L Han and F Ding ldquoParameter estimation for multi-rate multi-input systems using auxiliary model and multi-innovationrdquo Journal of Systems Engineering and Electronics vol21 no 6 pp 1079ndash1083 2010

[92] L L Han and F Ding ldquoIdentification for multirate multi-input systems using themulti-innovation identification theoryrdquoComputers andMathematics with Applications vol 57 no 9 pp1438ndash1449 2009

[93] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013

[94] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 226 no 1 pp 43ndash55 2012

[95] F Ding P X Liu and G Liu ldquoGradient based and least-squares based iterative identification methods for OE andOEMA systemsrdquo Digital Signal Processing vol 20 no 3 pp664ndash677 2010


[96] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013

[97] Y J Liu D Q Wang and F Ding ldquoLeast squares basediterative algorithms for identifying Box-Jenkins models withfinite measurement datardquo Digital Signal Processing vol 20 no5 pp 1458ndash1467 2010

[98] H Y Hu and F Ding ldquoAn iterative least squares estimationalgorithm for controlled moving average systems based onmatrix decompositionrdquoAppliedMathematics Letters vol 25 no12 pp 2332ndash2338 2012

[99] D Q Wang G W Yang and R F Ding ldquoGradient-based itera-tive parameter estimation for Box-Jenkins systemsrdquo ComputersandMathematics with Applications vol 60 no 5 pp 1200ndash12082010

[100] D Q Wang ldquoLeast squares-based recursive and iterative esti-mation for output error moving average systems using datafilteringrdquo IET ControlTheory and Applications vol 5 no 14 pp1648ndash1657 2011

[101] F Ding G Liu and X P Liu ldquoPartially coupled stochasticgradient identification methods for non-uniformly sampledsystemsrdquo IEEE Transactions on Automatic Control vol 55 no8 pp 1976ndash1981 2010

[102] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013

[103] F Ding and T Chen ldquoPerformance bounds of forgetting factorleast-squares algorithms for time-varying systems with finitemeasurement datardquo IEEE Transactions on Circuits and SystemsI Regular Papers vol 52 no 3 pp 555ndash566 2005

[104] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers and Mathematics with Applications vol 59no 8 pp 2615ndash2627 2010

[105] F Ding G Liu and X P Liu ldquoParameter estimation with scarcemeasurementsrdquo Automatica vol 47 no 8 pp 1646ndash1655 2011

[106] Y J Liu L Xie and F Ding ldquoAn auxiliary model based ona recursive least-squares parameter estimation algorithm fornon-uniformly sampled multirate systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part I Journal of Systemsand Control Engineering vol 223 no 4 pp 445ndash454 2009

[107] J Ding L L Han and X Chen ldquoTime series ARmodeling withmissing observations based on the polynomial transformationrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 527ndash536 2010

[108] F Ding T Chen and L Qiu ldquoBias compensation based recur-sive least-squares identification algorithm for MISO systemsrdquoIEEE Transactions on Circuits and Systems II Express Briefs vol53 no 5 pp 349ndash353 2006

[109] F Ding P X Liu and H Yang ldquoParameter identificationand intersample output estimation for dual-rate systemsrdquo IEEETransactions on Systems Man and Cybernetics A Systems andHumans vol 38 no 4 pp 966ndash975 2008

[110] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011

[111] J Ding Y Shi H Wang and F Ding ldquoA modified stochasticgradient based parameter estimation algorithm for dual-ratesampled-data systemsrdquo Digital Signal Processing vol 20 no 4pp 1238ndash1247 2010

[112] F Ding and T Chen ldquoLeast squares based self-tuning controlof dual-rate systemsrdquo International Journal of Adaptive Controland Signal Processing vol 18 no 8 pp 697ndash714 2004

[113] F Ding and T Chen ldquoA gradient based adaptive controlalgorithm for dual-rate systemsrdquo Asian Journal of Control vol8 no 4 pp 314ndash323 2006

[114] F Ding T Chen and Z Iwai ldquoAdaptive digital control ofHammerstein nonlinear systems with limited output samplingrdquoSIAM Journal on Control and Optimization vol 45 no 6 pp2257ndash2276 2007

[115] J Zhang F Ding and Y Shi ldquoSelf-tuning control based onmulti-innovation stochastic gradient parameter estimationrdquoSystems and Control Letters vol 58 no 1 pp 69ndash75 2009

[116] Y J Liu F Ding and Y Shi ldquoAn efficient hierarchical identi-fication method for general dual-rate sampled-data systemsrdquoAutomatica 2014

[117] F Ding ldquoHierarchical parameter estimation algorithms formultivariable systems using measurement informationrdquo Infor-mation Sciences 2014

International Journal of

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Submit your manuscripts athttpwwwhindawicom

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



for canonical state space systems and analyzes the perfor-mance of the proposed algorithm Section 4 gives a stateestimation algorithm Section 5 provides an example for theproposed algorithm Finally concluding remarks are given inSection 6

2 System Description andIdentification Model

Let us introduce some notation [15] ldquo119860 = 119883rdquo or ldquo119883 = 119860rdquostands for ldquo119860 is defined as 119883rdquo the symbol I(I

119899) stands for an

identity matrix of appropriate size (119899 times 119899) the superscript Tdenotes the matrix transpose |X| = det[X] represents thedeterminant of a square matrix X the norm of a matrix Xis defined by X2 = tr[XXT

] 1119899

= 1119899times1

represents an119899 times 1 vector whose elements are all 1 120582min[X] represents theminimum eigenvalues of X for 119892(119905) ⩾ 0 we write 119891(119905) =

119874(119892(119905)) if there exists a positive constant 1205751such that |119891(119905)| ⩽

1205751119892(119905)In order to study the convergence of the algorithm

proposed in [15] here we simply give that algorithm in[15] Consider a linear system described by the followingobservability canonical state space model [15]

x (119905 + 1) = Ax (119905) + b119906 (119905)

119910 (119905) = cx (119905) + V (119905) (1)

A =

[[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 d

d 0


minus119886119899



sdot sdot sdot minus1198861

]]]]]]]

]


b =

[[[[

[

1198871

1198872

119887119899

]]]]

]

isin R119899

c = [1 0 0 0] isin R1times119899

(2)

where x(119905) isin R119899 is the state vector 119906(119905) isin R is the systeminput119910(119905) isin R is the system output and V(119905) isin R is a randomnoise with zero mean Assume that the order 119899 is known and119906(119905) = 0 119910(119905) = 0 and V(119905) = 0 for 119905 ⩽ 0

The system in (1) is an observability canonical form andits observability matrixQ

119900is an identity matrix that is

Q119900=

[[[[

[

ccA

cA119899minus1

]]]]

]

= I119899 (3)

For the system in (1) the objective of this paper isto develop a new algorithm to estimate the parametermatrixvector A and b (ie the parameters 119886

119894and 119887

119894) and

the system state vector x(119905) from the available measurementinput-output data 119906(119905) 119910(119905)

Since the available measurement input-output data119906(119905) 119910(119905) are known but the state vector x(119905) is unknownit is required to eliminate the state vector from (1) and obtaina new expression which only involves the input and outputin order to obtain the estimates of the parameters in (1)The following derives the identification model based on themethod in [15]

Define some vectorsmatrix


Tisin R119899


Tisin R119899

120593V (119905) = [V (119905 minus 119899) V (119905 minus 119899 + 1) V (119905 minus 1)]Tisin R119899

M =

[[[[[[[

[


cb 0 sdot sdot sdot 0 0

cAb cb d

d 0 0

cA119899minus2b cA119899minus3b sdot sdot sdot cb 0

]]]]]]]

]


(4)

From (1) we have

119910 (119905) = cx (119905) + V (119905) (5)

119910 (119905 + 1) = cx (119905 + 1) + V (119905 + 1)

= c [Ax (119905) + b119906 (119905)] + V (119905 + 1)

= cAx (119905) + cb119906 (119905) + V (119905 + 1)

(6)

119910 (119905 + 2) = cAx (119905 + 1) + cb119906 (119905 + 1) + V (119905 + 2)

= cA [Ax (119905) + b119906 (119905)] + cb119906 (119905 + 1) + V (119905 + 2)

= cA2x (119905) + cAb119906 (119905) + cb119906 (119905 + 1) + V (119905 + 2)

(7)

119910 (119905 + 119899 minus 1) = cA119899minus1x (119905) + cA119899minus2b119906 (119905) + cA119899minus3b119906 (119905 minus 1)

+ sdot sdot sdot + cb119906 (119905 + 119899 minus 2) + V (119905 + 119899 minus 1)

(8)

119910 (119905 + 119899) = cA119899x (119905) + cA119899minus1b119906 (119905) + cA119899minus2b119906 (119905 minus 1)


(9)

Combining (5) with (8) gives

120593119910(119905 + 119899) = Q

119900x (119905) +M120593

119906(119905 + 119899) + 120593V (119905 + 119899)

= x (119905) +M120593119906(119905 + 119899) + 120593V (119905 + 119899)

(10)

or

x (119905) = 120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899) (11)



120579 = [120579119886

120579119887

] isin R2119899

120579119886= [cA119899]T isin R

119899


Tisin R119899

120593 (119905 + 119899) = [120593T119910(119905 + 119899) minus 120593

TV (119905 + 119899) 120593

T119906(119905 + 119899)]

Tisin R2119899

(12)


119910 (119905 + 119899)

= cA119899 [120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899)]



= cA119899 [120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899)]


[

119906 (119905)

119906 (119905 minus 1)

119906 (119905 + 119899 minus 1)

]]]]

]

+ V (119905 + 119899)


119906(119905 + 119899)


= [120593T119910(119905 + 119899) minus 120593

TV (119905 + 119899) 120593

T119906(119905 + 119899)] [

120579119886

120579119887

] + V (119905 + 119899)

= 120593T(119905 + 119899) 120579 + V (119905 + 119899)

(13)


119910 (119905) = 120593T(119905) 120579 + V (119905) (14)






0I (16)

V (119905) = 119910 (119905) minus T(119905) (119905) (17)




119906 (119905 minus 119899 + 1) 119906 (119905 minus 1)]T

(18)



L (119905) = P (119905) (119905) =P (119905 minus 1) (119905)

1 + T(119905)P (119905 minus 1) (119905)

(20)


0I (21)

V (119905) = 119910 (119905) minus T(119905) (119905) (22)

(119905) = [T(119905 minus 119899) (119905 minus 119899)

T(119905 minus 119899 + 1) (119905 minus 119899 + 1)

T(119905 minus 1) (119905 minus 1)


(23)







(A1) E [V(119905) | F119905minus1

] = 0 as(A2) E [V2(119905) | F

119905minus1] ⩽ 1205902

lt infin as(A3) 1198601015840(119911) = 119860

minus1



E [119879 (119905) + 119878 (119905) | F119905minus1

]

⩽ 119879 (119905 minus 1) + 119878 (119905 minus 1) + 2T(119905)P (119905) (119905) 120590

2

(24)


where

119878 (119905) = 2

119905

sum

119894=1

(119894) 119910 (119894) ⩾ 0 (25)

(119905) = minusT(119905) (119905) (26)

119910 (119905) =1

2

T(119905) (119905) + [119910 (119905) minus

T(119905) (119905) minus V (119905)] (27)



V (119905) = [1 minus T(119905)P (119905) (119905)] 119890 (119905)

=119890 (119905)

1 + T(119905)P (119905 minus 1) (119905)

(28)


(119905) = (119905) minus 120579 = (119905 minus 1) + P (119905) (119905) 119890 (119905)

= (119905 minus 1) + P (119905 minus 1) (119905) V (119905) (29)


119879 (119905) = 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905) minus T (119905)P (119905) (119905)

times [1 minus T(119905)P (119905) (119905)] 119890

2

(119905)

⩽ 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905)

= 119879 (119905 minus 1) + 2T(119905) (119905)

times [1

2T(119905) (119905) + (V (119905) minus V (119905))] + 2

T(119905) (119905) V (119905)

(30)


119879 (119905) ⩽ 119879 (119905 minus 1) minus 2 (119905) 119910 (119905) + 2T(119905)

times [ (119905 minus 1) + P (119905) (119905) 119890 (119905)] V (119905)


+ 2T(119905)P (119905) (119905) [119890 (119905) minus V (119905)] V (119905) + V2 (119905)

(31)





E [119879 (119905) | F119905minus1


]

+ 2T(119905)P (119905) (119905) 120590

2

as(32)




119906 (119905) + V (119905)

=119861 (119911)

119860 (119911)119906 (119905) + V (119905)

(33)


[119911minus1

119910(119905) = 119910(119905 minus 1)] and



(34)


119860 (119911) [V (119905) minus V (119905)] = 119860 (119911) V (119905) minus 119860 (119911) 119910 (119905) + 119861 (119911) 119906 (119905)

= minus119860 (119911) T(119905) (119905) + 119861 (119911) 119906 (119905)

= minusT(119905) (119905) = (119905)

(35)


119910 (119905) =1

2T(119905) (119905) + [V (119905) minus V (119905)]

= [119860minus1

(119911) minus1

2] (119905)

(36)

Since 1198601015840



10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119903 (119905)]

119888

120582min [Pminus1 (119905)]) 119886119904 119891119900119903 119886119899119910 119888 gt 1

(37)



10038171003817100381710038171003817 (119905)

10038171003817100381710038171003817

2


=119879 (119905)

120582min [Pminus1 (119905)] (38)

Let

119882(119905) =119879 (119905) + 119878 (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888 119888 gt 1 (39)



E [119882 (119905) | F119905minus1

] ⩽119879 (119905 minus 1) + 119878 (119905 minus 1)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

+2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

⩽ 119881 (119905 minus 1) +2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

as

(40)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874(

[ln 10038161003816100381610038161003816Pminus1 (119905)10038161003816100381610038161003816]

119888

120582min [Pminus1 (119905)])

= 119874([ln 119903 (119905)]

119888


(41)




1198881I ⩽ 1

119905

119905

sum

119895=1

120593 (119895)120593T(119895) ⩽ 119888

2119905120574I as for 119905 ⩾ 119905

0 (42)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119905]119888

119905) 997888rarr 0 as for any 119888 gt 1 (43)



x (119905 minus 119899) = 120593119910(119905) minus M (119905)120593

119906(119905) minus V (119905) (44)


T (45)


T (46)


M (119905) =

[[[[[[[[

[



2(119905)

1(119905) d

d 0 0

119899minus1

(119905) 119899minus2

(119905) sdot sdot sdot 1(119905) 0

]]]]]]]]

]

(48)

[[[[[[[

[

1(119905)

2(119905)

119899minus1

(119905)

119899(119905)

]]]]]]]

]

=

[[[[[[

[

119886119899minus1

(119905) 119886119899minus2

(119905) sdot sdot sdot 1198861(119905) 1

119886119899minus2

(119905) 119886119899minus3


1198861(119905) 1 sdot sdot sdot 0 0


]]]]]]

]

minus1

119887(119905)

(49)

(119905) = [119886(119905)

119887(119905)

] (50)

119886(119905) = [minus119886

119899(119905) minus119886

119899minus1(119905) minus119886

1(119905)]

T (51)



1199010I 119906(119894) = 0 119910(119894) = 0 V(119894) = 0 or V(119894) = 1119901

0for



119910(119905) using (45) and 120593

119906(119905) using (46)





119894(119905) using



5 Example


x (119905 + 1) = [0 1

minus070 135] x (119905) + [

1

1] 119906 (119905)

119910 (119905) = [1 0] x (119905) + V (119905)

(52)


2

= 0202 and 120590

2










119905 1205791

1205792

1205793

1205794

120575 ()



119905 1205791

1205792

1205793

1205794

120575 ()


0 500 1000 1500 2000 2500 30000

010203040506070809

120575 1205902= 100

2

1205902= 020

2

t


and 1205902

= 1002)




6 Conclusions


0 20 40 60 80 100 120 140 160 180 200

Stat

e est

imat

es

t (s)

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5



1(119905)

0 20 40 60 80 100 120 140 160 180 200

Stat

e esti

mat

es

t (s)

3

2

1

0

minus1

minus2

minus3



2(119905)





Acknowledgments


References

















































































119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120579 = [120579119886

120579119887

] isin R2119899

120579119886= [cA119899]T isin R

119899


Tisin R119899

120593 (119905 + 119899) = [120593T119910(119905 + 119899) minus 120593

TV (119905 + 119899) 120593

T119906(119905 + 119899)]

Tisin R2119899

(12)


119910 (119905 + 119899)

= cA119899 [120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899)]



= cA119899 [120593119910(119905 + 119899) minusM120593

119906(119905 + 119899) minus 120593V (119905 + 119899)]


[

119906 (119905)

119906 (119905 minus 1)

119906 (119905 + 119899 minus 1)

]]]]

]

+ V (119905 + 119899)


119906(119905 + 119899)


= [120593T119910(119905 + 119899) minus 120593

TV (119905 + 119899) 120593

T119906(119905 + 119899)] [

120579119886

120579119887

] + V (119905 + 119899)

= 120593T(119905 + 119899) 120579 + V (119905 + 119899)

(13)


119910 (119905) = 120593T(119905) 120579 + V (119905) (14)






0I (16)

V (119905) = 119910 (119905) minus T(119905) (119905) (17)




119906 (119905 minus 119899 + 1) 119906 (119905 minus 1)]T

(18)



L (119905) = P (119905) (119905) =P (119905 minus 1) (119905)

1 + T(119905)P (119905 minus 1) (119905)

(20)


0I (21)

V (119905) = 119910 (119905) minus T(119905) (119905) (22)

(119905) = [T(119905 minus 119899) (119905 minus 119899)

T(119905 minus 119899 + 1) (119905 minus 119899 + 1)

T(119905 minus 1) (119905 minus 1)


(23)







(A1) E [V(119905) | F119905minus1

] = 0 as(A2) E [V2(119905) | F

119905minus1] ⩽ 1205902

lt infin as(A3) 1198601015840(119911) = 119860

minus1



E [119879 (119905) + 119878 (119905) | F119905minus1

]

⩽ 119879 (119905 minus 1) + 119878 (119905 minus 1) + 2T(119905)P (119905) (119905) 120590

2

(24)


where

119878 (119905) = 2

119905

sum

119894=1

(119894) 119910 (119894) ⩾ 0 (25)

(119905) = minusT(119905) (119905) (26)

119910 (119905) =1

2

T(119905) (119905) + [119910 (119905) minus

T(119905) (119905) minus V (119905)] (27)



V (119905) = [1 minus T(119905)P (119905) (119905)] 119890 (119905)

=119890 (119905)

1 + T(119905)P (119905 minus 1) (119905)

(28)


(119905) = (119905) minus 120579 = (119905 minus 1) + P (119905) (119905) 119890 (119905)

= (119905 minus 1) + P (119905 minus 1) (119905) V (119905) (29)


119879 (119905) = 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905) minus T (119905)P (119905) (119905)

times [1 minus T(119905)P (119905) (119905)] 119890

2

(119905)

⩽ 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905)

= 119879 (119905 minus 1) + 2T(119905) (119905)

times [1

2T(119905) (119905) + (V (119905) minus V (119905))] + 2

T(119905) (119905) V (119905)

(30)


119879 (119905) ⩽ 119879 (119905 minus 1) minus 2 (119905) 119910 (119905) + 2T(119905)

times [ (119905 minus 1) + P (119905) (119905) 119890 (119905)] V (119905)


+ 2T(119905)P (119905) (119905) [119890 (119905) minus V (119905)] V (119905) + V2 (119905)

(31)





E [119879 (119905) | F119905minus1


]

+ 2T(119905)P (119905) (119905) 120590

2

as(32)




119906 (119905) + V (119905)

=119861 (119911)

119860 (119911)119906 (119905) + V (119905)

(33)


[119911minus1

119910(119905) = 119910(119905 minus 1)] and



(34)


119860 (119911) [V (119905) minus V (119905)] = 119860 (119911) V (119905) minus 119860 (119911) 119910 (119905) + 119861 (119911) 119906 (119905)

= minus119860 (119911) T(119905) (119905) + 119861 (119911) 119906 (119905)

= minusT(119905) (119905) = (119905)

(35)


119910 (119905) =1

2T(119905) (119905) + [V (119905) minus V (119905)]

= [119860minus1

(119911) minus1

2] (119905)

(36)

Since 1198601015840



10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119903 (119905)]

119888

120582min [Pminus1 (119905)]) 119886119904 119891119900119903 119886119899119910 119888 gt 1

(37)



10038171003817100381710038171003817 (119905)

10038171003817100381710038171003817

2


=119879 (119905)

120582min [Pminus1 (119905)] (38)

Let

119882(119905) =119879 (119905) + 119878 (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888 119888 gt 1 (39)



E [119882 (119905) | F119905minus1

] ⩽119879 (119905 minus 1) + 119878 (119905 minus 1)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

+2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

⩽ 119881 (119905 minus 1) +2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

as

(40)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874(

[ln 10038161003816100381610038161003816Pminus1 (119905)10038161003816100381610038161003816]

119888

120582min [Pminus1 (119905)])

= 119874([ln 119903 (119905)]

119888


(41)




1198881I ⩽ 1

119905

119905

sum

119895=1

120593 (119895)120593T(119895) ⩽ 119888

2119905120574I as for 119905 ⩾ 119905

0 (42)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119905]119888

119905) 997888rarr 0 as for any 119888 gt 1 (43)



x (119905 minus 119899) = 120593119910(119905) minus M (119905)120593

119906(119905) minus V (119905) (44)


T (45)


T (46)


M (119905) =

[[[[[[[[

[



2(119905)

1(119905) d

d 0 0

119899minus1

(119905) 119899minus2

(119905) sdot sdot sdot 1(119905) 0

]]]]]]]]

]

(48)

[[[[[[[

[

1(119905)

2(119905)

119899minus1

(119905)

119899(119905)

]]]]]]]

]

=

[[[[[[

[

119886119899minus1

(119905) 119886119899minus2

(119905) sdot sdot sdot 1198861(119905) 1

119886119899minus2

(119905) 119886119899minus3


1198861(119905) 1 sdot sdot sdot 0 0


]]]]]]

]

minus1

119887(119905)

(49)

(119905) = [119886(119905)

119887(119905)

] (50)

119886(119905) = [minus119886

119899(119905) minus119886

119899minus1(119905) minus119886

1(119905)]

T (51)



1199010I 119906(119894) = 0 119910(119894) = 0 V(119894) = 0 or V(119894) = 1119901

0for



119910(119905) using (45) and 120593

119906(119905) using (46)





119894(119905) using



5 Example


x (119905 + 1) = [0 1

minus070 135] x (119905) + [

1

1] 119906 (119905)

119910 (119905) = [1 0] x (119905) + V (119905)

(52)


2

= 0202 and 120590

2










119905 1205791

1205792

1205793

1205794

120575 ()



119905 1205791

1205792

1205793

1205794

120575 ()


0 500 1000 1500 2000 2500 30000

010203040506070809

120575 1205902= 100

2

1205902= 020

2

t


and 1205902

= 1002)




6 Conclusions


0 20 40 60 80 100 120 140 160 180 200

Stat

e est

imat

es

t (s)

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5



1(119905)

0 20 40 60 80 100 120 140 160 180 200

Stat

e esti

mat

es

t (s)

3

2

1

0

minus1

minus2

minus3



2(119905)





Acknowledgments


References

















































































119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

119878 (119905) = 2

119905

sum

119894=1

(119894) 119910 (119894) ⩾ 0 (25)

(119905) = minusT(119905) (119905) (26)

119910 (119905) =1

2

T(119905) (119905) + [119910 (119905) minus

T(119905) (119905) minus V (119905)] (27)



V (119905) = [1 minus T(119905)P (119905) (119905)] 119890 (119905)

=119890 (119905)

1 + T(119905)P (119905 minus 1) (119905)

(28)


(119905) = (119905) minus 120579 = (119905 minus 1) + P (119905) (119905) 119890 (119905)

= (119905 minus 1) + P (119905 minus 1) (119905) V (119905) (29)


119879 (119905) = 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905) minus T (119905)P (119905) (119905)

times [1 minus T(119905)P (119905) (119905)] 119890

2

(119905)

⩽ 119879 (119905 minus 1) + T(119905) (119905)

T(119905) (119905)

+ 2T(119905) (119905) V (119905)

= 119879 (119905 minus 1) + 2T(119905) (119905)

times [1

2T(119905) (119905) + (V (119905) minus V (119905))] + 2

T(119905) (119905) V (119905)

(30)


119879 (119905) ⩽ 119879 (119905 minus 1) minus 2 (119905) 119910 (119905) + 2T(119905)

times [ (119905 minus 1) + P (119905) (119905) 119890 (119905)] V (119905)


+ 2T(119905)P (119905) (119905) [119890 (119905) minus V (119905)] V (119905) + V2 (119905)

(31)





E [119879 (119905) | F119905minus1


]

+ 2T(119905)P (119905) (119905) 120590

2

as(32)




119906 (119905) + V (119905)

=119861 (119911)

119860 (119911)119906 (119905) + V (119905)

(33)


[119911minus1

119910(119905) = 119910(119905 minus 1)] and



(34)


119860 (119911) [V (119905) minus V (119905)] = 119860 (119911) V (119905) minus 119860 (119911) 119910 (119905) + 119861 (119911) 119906 (119905)

= minus119860 (119911) T(119905) (119905) + 119861 (119911) 119906 (119905)

= minusT(119905) (119905) = (119905)

(35)


119910 (119905) =1

2T(119905) (119905) + [V (119905) minus V (119905)]

= [119860minus1

(119911) minus1

2] (119905)

(36)

Since 1198601015840



10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119903 (119905)]

119888

120582min [Pminus1 (119905)]) 119886119904 119891119900119903 119886119899119910 119888 gt 1

(37)



10038171003817100381710038171003817 (119905)

10038171003817100381710038171003817

2


=119879 (119905)

120582min [Pminus1 (119905)] (38)

Let

119882(119905) =119879 (119905) + 119878 (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888 119888 gt 1 (39)



E [119882 (119905) | F119905minus1

] ⩽119879 (119905 minus 1) + 119878 (119905 minus 1)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

+2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

⩽ 119881 (119905 minus 1) +2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

as

(40)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874(

[ln 10038161003816100381610038161003816Pminus1 (119905)10038161003816100381610038161003816]

119888

120582min [Pminus1 (119905)])

= 119874([ln 119903 (119905)]

119888


(41)




1198881I ⩽ 1

119905

119905

sum

119895=1

120593 (119895)120593T(119895) ⩽ 119888

2119905120574I as for 119905 ⩾ 119905

0 (42)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119905]119888

119905) 997888rarr 0 as for any 119888 gt 1 (43)



x (119905 minus 119899) = 120593119910(119905) minus M (119905)120593

119906(119905) minus V (119905) (44)


T (45)


T (46)


M (119905) =

[[[[[[[[

[



2(119905)

1(119905) d

d 0 0

119899minus1

(119905) 119899minus2

(119905) sdot sdot sdot 1(119905) 0

]]]]]]]]

]

(48)

[[[[[[[

[

1(119905)

2(119905)

119899minus1

(119905)

119899(119905)

]]]]]]]

]

=

[[[[[[

[

119886119899minus1

(119905) 119886119899minus2

(119905) sdot sdot sdot 1198861(119905) 1

119886119899minus2

(119905) 119886119899minus3


1198861(119905) 1 sdot sdot sdot 0 0


]]]]]]

]

minus1

119887(119905)

(49)

(119905) = [119886(119905)

119887(119905)

] (50)

119886(119905) = [minus119886

119899(119905) minus119886

119899minus1(119905) minus119886

1(119905)]

T (51)



1199010I 119906(119894) = 0 119910(119894) = 0 V(119894) = 0 or V(119894) = 1119901

0for



119910(119905) using (45) and 120593

119906(119905) using (46)





119894(119905) using



5 Example


x (119905 + 1) = [0 1

minus070 135] x (119905) + [

1

1] 119906 (119905)

119910 (119905) = [1 0] x (119905) + V (119905)

(52)


2

= 0202 and 120590

2










119905 1205791

1205792

1205793

1205794

120575 ()



119905 1205791

1205792

1205793

1205794

120575 ()


0 500 1000 1500 2000 2500 30000

010203040506070809

120575 1205902= 100

2

1205902= 020

2

t


and 1205902

= 1002)




6 Conclusions


0 20 40 60 80 100 120 140 160 180 200

Stat

e est

imat

es

t (s)

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5



1(119905)

0 20 40 60 80 100 120 140 160 180 200

Stat

e esti

mat

es

t (s)

3

2

1

0

minus1

minus2

minus3



2(119905)





Acknowledgments


References

















































































119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











E [119882 (119905) | F119905minus1

] ⩽119879 (119905 minus 1) + 119878 (119905 minus 1)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

+2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

⩽ 119881 (119905 minus 1) +2

T(119905)P (119905) (119905)

[ln 1003816100381610038161003816Pminus1 (119905)1003816100381610038161003816]119888

1205902

as

(40)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874(

[ln 10038161003816100381610038161003816Pminus1 (119905)10038161003816100381610038161003816]

119888

120582min [Pminus1 (119905)])

= 119874([ln 119903 (119905)]

119888


(41)




1198881I ⩽ 1

119905

119905

sum

119895=1

120593 (119895)120593T(119895) ⩽ 119888

2119905120574I as for 119905 ⩾ 119905

0 (42)


10038171003817100381710038171003817 (119905) minus 120579

10038171003817100381710038171003817

2

= 119874([ln 119905]119888

119905) 997888rarr 0 as for any 119888 gt 1 (43)



x (119905 minus 119899) = 120593119910(119905) minus M (119905)120593

119906(119905) minus V (119905) (44)


T (45)


T (46)


M (119905) =

[[[[[[[[

[



2(119905)

1(119905) d

d 0 0

119899minus1

(119905) 119899minus2

(119905) sdot sdot sdot 1(119905) 0

]]]]]]]]

]

(48)

[[[[[[[

[

1(119905)

2(119905)

119899minus1

(119905)

119899(119905)

]]]]]]]

]

=

[[[[[[

[

119886119899minus1

(119905) 119886119899minus2

(119905) sdot sdot sdot 1198861(119905) 1

119886119899minus2

(119905) 119886119899minus3


1198861(119905) 1 sdot sdot sdot 0 0


]]]]]]

]

minus1

119887(119905)

(49)

(119905) = [119886(119905)

119887(119905)

] (50)

119886(119905) = [minus119886

119899(119905) minus119886

119899minus1(119905) minus119886

1(119905)]

T (51)



1199010I 119906(119894) = 0 119910(119894) = 0 V(119894) = 0 or V(119894) = 1119901

0for



119910(119905) using (45) and 120593

119906(119905) using (46)





119894(119905) using



5 Example


x (119905 + 1) = [0 1

minus070 135] x (119905) + [

1

1] 119906 (119905)

119910 (119905) = [1 0] x (119905) + V (119905)

(52)


2

= 0202 and 120590

2










119905 1205791

1205792

1205793

1205794

120575 ()



119905 1205791

1205792

1205793

1205794

120575 ()


0 500 1000 1500 2000 2500 30000

010203040506070809

120575 1205902= 100

2

1205902= 020

2

t


and 1205902

= 1002)




6 Conclusions


0 20 40 60 80 100 120 140 160 180 200

Stat

e est

imat

es

t (s)

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5



1(119905)

0 20 40 60 80 100 120 140 160 180 200

Stat

e esti

mat

es

t (s)

3

2

1

0

minus1

minus2

minus3



2(119905)





Acknowledgments


References

















































































119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119905 1205791

1205792

1205793

1205794

120575 ()



119905 1205791

1205792

1205793

1205794

120575 ()


0 500 1000 1500 2000 2500 30000

010203040506070809

120575 1205902= 100

2

1205902= 020

2

t


and 1205902

= 1002)




6 Conclusions


0 20 40 60 80 100 120 140 160 180 200

Stat

e est

imat

es

t (s)

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5



1(119905)

0 20 40 60 80 100 120 140 160 180 200

Stat

e esti

mat

es

t (s)

3

2

1

0

minus1

minus2

minus3



2(119905)





Acknowledgments


References

















































































119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References

















































































119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























































119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























119863 =

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Parameter and State Estimator for State Space …downloads.hindawi.com/journals/tswj/2014/106505.pdf · 2019-07-31 · Research Article Parameter and State Estimator