Research ArticleAn Improved Iterative Method for Solving the Discrete AlgebraicRiccati Equation

Li Wang

School of Mathematics and Computational Science Hunan University of Science and Technology Hunan 411201 China

Correspondence should be addressed to Li Wang wanglileigh163com

Received 24 March 2020 Accepted 21 April 2020 Published 20 May 2020

Guest Editor Hou-Sheng Su

Copyright copy 2020 Li Wang +is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+e discrete algebraic Riccati equation has wide applications especially in networked systems and optimal control systems In thispaper according to the damped Newton method two iterative algorithms with a stepsize parameter is proposed to solve thediscrete algebraic Riccati equation one of which is an extension of Algorithm (41) in Dai and Bai (2011) A numerical exampledemonstrates the convergence effect of the presented algorithm

1 Introduction and Preliminaries

+e discrete algebraic Riccati equation plays an impor-tant part in engineering such as optimal control systems[1] modified filtering [2 3] and networked systems[4ndash7] Consider the following discrete-time linearsystem

x(k + 1) Ax(k) + Bu(k) (1)

where x(k) isin Rn is the state variable u(k) isin Rr is the inputvariable B isin Rntimesr is the input matrix and A isin Rntimesn is thesystem matrix and is always invertible [8] +e optimal statefeedback controller of (1) is

ulowast(k) minus G + B

TPB1113872 1113873

minus 1B

TPAx(k) (2)

which minimizes the quadratic performance index of (1) andis closely related to the discrete algebraic Riccati equation(DARE)

P ATPA minus A

TPB G + B

TPB1113872 1113873

minus 1B

TPA + Q (3)

where Q isin Rntimesn is semipositive definite G isin Rrtimesr is positivedefinite and P isin Rntimesn is the positive definite solution of theDARE (3) Let R BGminus 1BT ge 0 According to the matrixidentity

Xminus 1

+ YZ1113872 1113873minus 1

X minus XY(I + ZXY)minus 1

ZX (4)

equation (3) can be transformed into

P AT

Pminus 1

+ R1113872 1113873minus 1

A + Q (5)

Due to the wide applications of the DARE many workshave been proposed to discuss the DARE Various boundsand solutions about the DARE have been provided such asupper and lower solution bounds [9ndash14] bounds about sumand product of eigenvalues [15 16] determinant of thesolution [17] and the existence of the solution [18ndash21]However in an optimal control system we often need tocompute the solution of the DARE to find the optimal statefeedback controller which minimizes the quadratic perfor-mance index It is very difficult to solve the DARE especiallywhen the dimensions of the coefficient matrices are high Somany researchers provide a lot of iterative methods to solvethis equation Komaroff present a fixed-point iterative al-gorithm that needs to compute twice matrix inversion ateach step [22] By Newtonrsquos method Guo derived themaximal symmetric solution of the DARE in [23] +estructure-preserving doubling algorithms are discussed in[24ndash27] +e Schur method is adopted to solve algebraicRiccati equations [28] Recently Dai and Bai propose aniterative algorithm that partially avoids computing thematrix inversions by making use of the Schulz iteration [29]

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 3283157 6 pageshttpsdoiorg10115520203283157

In Section 2 we propose two iterative algorithms with astepsize parameter to solve the DARE by the dampedNewton method One of the iterative algorithms is an ex-tension of Algorithm 41 in [29] Numerical example is givenin Section 3 to demonstrate the convergence effect of ouralgorithms

We first introduce some symbol conventions R denotesthe real number field Rntimesm denotes the set of n times m realmatrices For X (xij) isin Rntimesn let XT Xminus 1 X andλmin(X) denote the transpose inverse spectral norm andthe minimal eigenvalue of the matrix X respectively +einequality Xgt (ge )0 meansX is a symmetric positive (semi-)definite matrix and the inequality Xgt (ge )Y means X minus Y isa symmetric positive (semi-) definite matrix +e identitymatrix with appropriate dimensions is represented by I

Lemma 1 (see [30]) If A B isin Rntimesn are symmetric positivedefinite matrices then

AgeB if and only if Bminus 1 geA

minus 1 (6)

Lemma 2 (see [31]) Let C and P be Hermitian matrices ofthe same order and let Pgt 0 (en

CPC + Pminus 1 ge 2C (7)

Lemma 3 (see [32]) Let S andT be symmetric positivedefinite matrices (en

Sge T if SgeTge 0 (8)

2 Improved Iterative Algorithms for Solvingthe DARE

To find the positive definite solution of the DARE (5) Daiand Bai in [29] proposed an algorithm that partially avoidscomputing the matrix inversion as follows

Algorithm 1 (see [29]) Take Y0 (Qminus 1 + R)minus 1 Fork 0 1 2 middot middot middot compute

Pk+1 ATYkA + Q

Yk+1 Yk 2I minus Pminus1k+1 + R( 1113857Yk( 1113857

⎧⎨

⎩ (9)

In this section we propose two iterative algorithms tosolve the DARE (5) which are motivated by the dampedNewton method [33] and the methods in [34 35] Let usrecall the damped Newton method to find the root ofF(Z) Zminus 1 minus B

Zk+1 Zk + tZk I minus BZk( 1113857 (1 + t)Zk minus tZkBZk (10)

where tgt 0 is a stepsize parameter If the initial matrix is nearthe solution of the problem the unit stepsize t 1 can beaccepted in the local Newton method However it is notsuitable to choose t 1 if the initial matrix is far from thesolution of the problem [33]

+e DARE (5) can be translated into F(P) 0 where

F(P) Aminus T

(P minus Q)Aminus 1

1113960 1113961minus 1

minus Pminus 1

+ R1113872 1113873 (11)

Let Z Aminus T(P minus Q)Aminus 1 and B Pminus 1 + R +en to findthe root of F(P) is equivalent to find the root of F(Z) we cansolve the DARE (5) by constructing an iterative schemeAccording to (10) we present the following iterative algo-rithms for the DARE (5)

Algorithm 2

Step 1 set P0 Q Y0 (Qminus 1 + R)minus 1 and tgt 0Step 2 compute

Yk+1 (1 minus t)Yk + t 2Yk minus Yk Pminus1k + R( 1113857Yk1113858 1113859

Pk+1 ATYk+1A + Q k 0 1 2 middot middot middot 1113896 (12)

Algorithm 3

Step 1 set Y0 (Qminus 1 + R)minus 1 P0 Q and tgt 0Step 2 compute

Pk+1 ATYkA + Q

Yk+1 (1 minus t)Yk + t 2Yk minus Yk Pminus1k + R( 1113857Yk1113858 1113859 k 0 1 2 middot middot middot

⎧⎨

⎩ (13)

About Algorithms 2 and 3 we have the following results

Theorem 1 Let Pminus be the positive definite solution of theDARE (5) and Qgt 0 (e iterative sequences Pk1113864 1113865 and Yk1113864 1113865

are generated by Algorithm 2 with t isin (0 1] then

P0 leP1 leP2 le middot middot middot limk⟶infin

Pk Pminus

Y0 leY1 leY2 le middot middot middot limk⟶infin

Yk Pminus1minus + R1113872 1113873

minus 1

(14)

Proof We first prove Pk and Yk are monotone increasing byinduction Since Pminus is positive definite solution of DARE (5)then

Pminus AT

Pminus1minus + R1113872 1113873

minus 1A + Q (15)

+us Pminus geQ

(i) Since

2 Mathematical Problems in Engineering

Y1 (1 minus t)Y0 + t 2Y0 minus Y0 Pminus10 + R1113872 1113873Y01113960 1113961

(1 minus t)Y0 + t 2Y0 minus Y0 Qminus 1

+ R1113872 1113873Y01113960 1113961

(1 minus t)Y0 + t 2Y0 minus Y01113858 1113859

Y0

(16)

then by Lemma 1 we obtain

P1 ATY1A + Q A

TY0A + Q

AT

Qminus 1

+ R1113872 1113873minus 1

A + Q(17)

leAT

Pminus1minus + R1113872 1113873

minus 1A + Q

Pminus(18)

From (17) we also obtain P1 geQ P0 then

Pminus geP1 geP0 (19)

Y0 Y1 Qminus 1

+ R1113872 1113873minus 1le P

minus11 + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(20)

By Lemma 2 and (20) we have

Y2 (1 minus t)Y1 + t 2Y1 minus Y1 Pminus11 + R1113872 1113873Y11113960 1113961

le (1 minus t)Y1 + t Pminus11 + R1113872 1113873

minus 1

le (1 minus t) Pminus11 + R1113872 1113873

minus 1+ t P

minus11 + R1113872 1113873

minus 1

Pminus11 + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(21)

By (20) and Lemma 1 we obtain

Y2 (1 minus t)Y1 + t 2Y1 minus Y1 Pminus11 + R1113872 1113873Y11113960 1113961

ge (1 minus t)Y1 + t 2Y1 minus Y1 Qminus 1

+ R1113872 1113873Y11113960 1113961

(1 minus t)Y1 + t 2Y1 minus Y11113858 1113859

Y1

(22)

thereby

P2 ATY2A + Q

geATY1A + Q P1

(23)

By (21) we obtain

P2 ATY2A + Q

leAT

Pminus11 + R1113872 1113873

minus 1A + Q

leAT

Pminus1minus + R1113872 1113873

minus 1A + Q Pminus

(24)

+us from the above-mentioned proof we have

P0 leP1 leP2 lePminus Y0 Y1

leY2 le Pminus11 + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(25)

(ii) Assume that

Piminus1 lePi le middot middot middot lePminus Yiminus1 leYi le Pminus1iminus 1 + R1113872 1113873

minus 1

le Pminus1minus + R1113872 1113873

minus 1 i 1 2 middot middot middot k

(26)

From (26) we get Yk le (Pminus1kminus 1 + R)minus 1 le (Pminus1

k + R)minus 1then

Yminus1k geP

minus1k + R (27)

+us

Yk+1 (1 minus t)Yk + t 2Yk minus Yk Pminus1k + R1113872 1113873Yk1113960 1113961

ge (1 minus t)Yk + t 2Yk minus YkYminus1k Yk1113960 1113961

ge (1 minus t)Yk + tYk Yk

(28)

Yk+1 (1 minus t)Yk + t 2Yk minus Yk Pminus1k + R1113872 1113873Yk1113960 1113961

le (1 minus t)Yk + t Pminus1k + R1113872 1113873

minus 1

le (1 minus t) Pminus1k + R1113872 1113873

minus 1+ t P

minus1k + R1113872 1113873

minus 1

le Pminus1k + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(29)

By (28) and (29) we have

Pk+1 ATYk+1A + Q

geATYkA + Q Pk

(30)

Pk+1 ATYk+1A + Q

leAT

Pminus1minus + R1113872 1113873

minus 1A + Q Pminus

(31)

So we obtain

Pk lePk+1 lePminus Yk leYk+1 le Pminus1k + R1113872 1113873

minus 1

le Pminus1minus + R1113872 1113873

minus 1 k 0 1 2 middot middot middot

(32)

+us the proof of induction is completed Moreover asPk and Yk are monotone increasing and they are boundedthen lim

k⟶infinPk and lim

k⟶infinYk exist Taking limits in Algo-

rithm 2 gives limk⟶infin

Yk (Pminus1minus + R)minus 1 and

limk⟶infin

Pk Pminus

Mathematical Problems in Engineering 3

Theorem 2 Let Pminus be the positive definite solution of theDARE (5) After k steps of iteration for Algorithm 2 we haveI minus Yk(Pminus1

k + R)lt ε then

AT

Pminus1k + R1113872 1113873

minus 1A + Q minus Pk

le ε Pminus minus Q

(33)

Proof According to (15) we have

AT

Pminus1minus + R1113872 1113873

minus 1A Pminus minus Q (34)

+en by Algorithm 2 Lemma 3 and (34) we obtain

AT

Pminus1k + R1113872 1113873

minus 1A + Q minus Pk

AT

Pminus1k + R1113872 1113873

minus 1A minus A

TYkA

AT

Pminus1k + R1113872 1113873

minus 1minus Yk1113876 1113877A

AT

I minus Yk Pminus1k + R1113872 11138731113960 1113961 P

minus1k + R1113872 1113873

minus 1A

le I minus Yk Pminus1k + R1113872 1113873

middot AT

Pminus1k + R1113872 1113873

minus 1A

le I minus Yk Pminus1k + R1113872 1113873

middot AT

Pminus1minus + R1113872 1113873

minus 1A

I minus Yk Pminus1k + R1113872 1113873

middot Pminus minus Q

le ε Pminus minus Q

(35)

because of Pk lePminusAs the proof method is similar to +eorem 1 we list the

monotonicity and convergence of Algorithm 3 withoutproof

Theorem 3 Let Pminus be the positive definite solution of theDARE (5) and Qgt 0 (e iterative sequences Pk1113864 1113865 and Yk1113864 1113865

are generated by Algorithm 3 with t isin 0 1] and start fromY0 (Qminus 1 + R)minus 1 and P0 Q then Pk is monotone in-creasing and converges to Pminus and Yk is monotone increasingand converges to (Pminus1

minus + R)minus 1

Remark 1 For Algorithms 2 and 3 we find the steps ofiteration for Algorithm 2 are less than Algorithm 3 and theconvergence speed of the Algorithm 2 is faster than Algo-rithm 3 from the numerical examples +erefore in thefollowing example we only discuss the superiority and ef-fectiveness of Algorithm 2

3 Numerical Examples

In this section we present the following numerical exampleto show the effectiveness of our results We also discuss theperformance of Algorithm 2 with different t values +ewhole process is carried out on Matlab 71 and the precisionis 10minus 8

Example 1 Consider the discrete system (1) with

A

227 013 012 01

minus013 234 012 005

011 minus017 19 003

001 007 002 11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B

115 0 001 0

0 08 0 0

0 004 09 0

002 0 0 18

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Q

012 0 01 0

0 22 0 0

01 0 14 0

0 0 0 07

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

G I

(36)

In [29] Dai and Bai choose the starting matrixY0 (Qminus 1 + R)minus 1 After 17 steps of iteration the requiredprecision is derived and the residual AT(Pminus 1 + R)minus 1A +

Q minus P is 20754e minus 009For Algorithm 2 we choose P0 Q Y0 (Qminus 1 + R)minus 1

and give the steps of iteration and the residual as Table 1 with

4 Mathematical Problems in Engineering

a different parameter twhen the process is stopped under therequired precision When t is near 1 we find that the steps ofiteration are less than [29] Especially when t 12 it onlyneeds 10 steps for Algorithm 2 to converge to the iterativesolution

P10

33299 minus03120 05202 01433

minus03120 96394 minus01292 01904

05202 minus01292 49731 00820

01433 01904 00820 09962

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(37)

with the residual AT(Pminus 1 + R)minus 1A + Q minus P 56438eminus

009 and Algorithm 2 has faster convergence speed thanAlgorithm 1 from Figure 1 Moreover from Table 1 we seethat Algorithm 2 is more efficient when tgt 1 Although weonly prove the convergence of Algorithm 2 when t isin 0 1] inthis paper Algorithm 2 works well in practical computationwhen tgt 1

Data Availability

All data generated or analyzed during this study are includedwithin this article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e work was supported in part by the National NaturalScience Foundation for Youths of China (11801164) Na-tional Natural Science Foundation of China (11971413) KeyProject of National Natural Science Foundation of China(91430213) General Project of Hunan Provincial NaturalScience Foundation of China (2015JJ2134) and the GeneralProject of Hunan Provincial Education Department ofChina (15C1320)

References

[1] H Kwakernaak and R Sivan Linear Optimal Control SystemsWiley-Interscience New York NY USA 1972

[2] L Wu and D W C Ho ldquoFuzzy filter design for Ito stochasticsystems with application to sensor fault detectionrdquo IEEETransactions on Fuzzy Systems vol 17 no 1 pp 233ndash2422009

[3] X Su P Shi L Wu and Y D Song ldquoA novel approach tofilter design for TCS fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20no 6 pp 1114ndash1129 2012

[4] W N Anderson T D Morley and G E Trapp ldquoLaddernetworks fixpoints and the geometric meanrdquo Circuits Sys-tems and Signal Processing vol 2 no 3 pp 259ndash268 1983

[5] M Z Q Chen L Zhang H Su and G Chen ldquoStabilizingsolution and parameter dependence of modified algebraicRiccati equation with application to discrete-time networksynchronizationrdquo IEEE Transactions on Automatic Controlvol 61 no 1 pp 228ndash233 2016

[6] H S Su H Wu and J Lam ldquoPositive edge-consensus fornodal networks via output feedbackrdquo IEEE Transactions onAutomatic Control vol 64 no 3 pp 1224ndash1249 2019

[7] M Lan and S Chand ldquoSolving linear quadratic discrete-timeoptimal controls usingneural networksrdquo in Proceedings of theIEEE Conference on Decision amp Control IEEE Honolulu HIUSA December 1990

[8] R A Kennedy ldquoLinear system theoryrdquo Automatica vol 30no 11 pp 1811ndash1813 1994

[9] C-H Lee ldquoMatrix bounds of the solutions of the continuousand discrete Riccati equationsmdasha unified approachrdquo Inter-national Journal of Control vol 76 no 6 pp 635ndash642 2003

[10] H H Choi ldquoUpper matrix bounds for the discrete algebraicRiccati matrix equationrdquo IEEE Transactions on AutomaticControl vol 46 no 3 pp 504ndash508 2001

[11] R Davies P Shi and R Wiltshire ldquoNew upper solutionbounds of the discrete algebraic Riccati matrix equationrdquoJournal of Computational and Applied Mathematics vol 213no 2 pp 307ndash315 2008

[12] J Liu L Wang and J Zhang ldquo+e solution bounds and fixedpoint iterative algorithm for the discrete coupled algebraicRiccati equation applied to automatic controlrdquo IMA Journalof Mathematical Control and Information pp 1ndash22 2016

[13] J Liu L Wang and J Zhang ldquoNew matrix bounds and it-erative algorithms for the discrete coupled algebraic Riccatiequationrdquo International Journal of Control vol 90 no 11pp 2326ndash2337 2017

[14] N Komaroff ldquoUpper bounds for the solution of the discreteRiccati equationrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1370ndash1373 1992

[15] J Zhang J Liu and Y Zha ldquo+e improved eigenvalue boundsfor the solution of the discrete algebraic Riccati equationrdquo

Table 1 Numerical results

t Iterations Residual t Iterations Residual06 35 90186e ndash 009 12 10 56438eminus 00908 24 54562eminus 009 13 12 92094eminus 00909 20 47874eminus 009 15 18 94721eminus 0091 17 20754eminus 009 18 40 86619eminus 00911 14 10160eminus 009 20 104 91141eminus 009

10minus10

10minus8

10minus6

10minus4

10minus2

100

102

Resid

ual

Algorithm 1Algorithm 2

2 4 6 8 10 12 14 16 180Iteration steps

Figure 1 +e relationship between iteration step and residual

Mathematical Problems in Engineering 5

IMA Journal of Mathematical Control and Informationpp 1ndash20 2016

[16] N Komaroff and B Shahian ldquoLower summation bounds forthe discrete Riccati and Lyapunov equationsrdquo IEEE Trans-actions on Automatic Control vol 37 no 7 pp 1078ndash10801992

[17] M T Tran and M E Sawan ldquoOn the discrete Riccati matrixequationrdquo SIAM Journal on Algebraic Discrete Methods vol 6no 1 pp 107-108 1985

[18] J Liu and J Zhang ldquo+e existence uniqueness and the fixediterative algorithm of the solution for the discrete coupledalgebraic Riccati equationrdquo International Journal of Controlvol 84 no 8 pp 1430ndash1441 2011

[19] R Huang J Z Liu and L Zhu ldquoAccurate solutions of di-agonally dominant tridiagonal linear systemsrdquo BIT NumericalMathematics vol 54 no 3 pp 711ndash727 2014

[20] Q H Liu X X Li and J Yan ldquoOn the large time behaviour ofsolutions for a class of time-dependent Hamilton-Jacobiequationsrdquo Science China Mathematics vol 59 no 5pp 875ndash8890 2016

[21] Z-H He ldquoSome new results on a system of Sylvester-typequaternion matrix equationsrdquo Linear and Multilinear Alge-bra pp 1ndash23 2019

[22] N Komaroff ldquoIterative matrix bounds and computationalsolutions to the discrete algebraic Riccati equationrdquo IEEETransactions on Automatic Control vol 39 no 8 pp 1676ndash1678 1994

[23] C-H Guo ldquoNewtonrsquos method for discrete algebraic Riccatiequations when the closed-loop matrix has eigenvalues on theunit circlerdquo SIAM Journal on Matrix Analysis and Applica-tions vol 20 no 2 pp 279ndash294 1998

[24] W-W Lin and S-F Xu ldquoConvergence analysis of structure-preserving doubling algorithms for riccati-type matrixequationsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 28 no 1 pp 26ndash39 2006

[25] E K-W Chu H-Y Fan W-W Lin and C-S WangldquoStructure-preserving algorithms for periodic discrete-timealgebraic Riccati equationsrdquo International Journal of Controlvol 77 no 8 pp 767ndash788 2004

[26] L-Z Lu W-W Lin and C E M Pearce ldquoAn efficient al-gorithm for the discrete-time algebraic Riccati equationrdquoIEEE Transactions on Automatic Control vol 44 no 6pp 1216ndash1220 1999

[27] T-M Hwang E K-W Chu and W-W Lin ldquoA generalizedstructure-preserving doubling algorithm for generalizeddiscrete-time algebraic Riccati equationsrdquo InternationalJournal of Control vol 78 no 14 pp 1063ndash1075 2005

[28] A Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24no 6 pp 913ndash921 1979

[29] H Dai and Z Z Bai ldquoOn eigenvalue bounds and iterationmethods for discrete algebraic Riccati equationsrdquo Journal ofComputational Mathematics vol 29 no 3 pp 341ndash366 2011

[30] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 2012

[31] X Zhan ldquoComputing the extremal positive definite solutionsof a matrix equationrdquo SIAM Journal on Scientific Computingvol 17 no 5 pp 1167ndash1174 1996

[32] A W Marshall and I Olkin Inequalities (eory of Majori-zation and its Applications Academic Press New York NYUSA 1979

[33] W Sun and Y Yuan Optimization (eory and MethodsSpringer Science and Business Media LLC New York NYUSA 2006

[34] M Monsalve and M Raydan ldquoA new inversion-free methodfor a rational matrix equationrdquo Linear Algebra and Its Ap-plications vol 433 no 1 pp 64ndash71 2010

[35] G Schulz ldquoIterative Berechung der reziproken MatrixrdquoZAMMmdashZeitschrift fur Angewandte Mathematik und Mech-anik vol 13 no 1 pp 57ndash59 1933

6 Mathematical Problems in Engineering

Y1 (1 minus t)Y0 + t 2Y0 minus Y0 Pminus10 + R1113872 1113873Y01113960 1113961

(1 minus t)Y0 + t 2Y0 minus Y0 Qminus 1

+ R1113872 1113873Y01113960 1113961

(1 minus t)Y0 + t 2Y0 minus Y01113858 1113859

Y0

(16)

then by Lemma 1 we obtain

P1 ATY1A + Q A

TY0A + Q

AT

Qminus 1

+ R1113872 1113873minus 1

A + Q(17)

leAT

Pminus1minus + R1113872 1113873

minus 1A + Q

Pminus(18)

From (17) we also obtain P1 geQ P0 then

Pminus geP1 geP0 (19)

Y0 Y1 Qminus 1

+ R1113872 1113873minus 1le P

minus11 + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(20)

By Lemma 2 and (20) we have

Y2 (1 minus t)Y1 + t 2Y1 minus Y1 Pminus11 + R1113872 1113873Y11113960 1113961

le (1 minus t)Y1 + t Pminus11 + R1113872 1113873

minus 1

le (1 minus t) Pminus11 + R1113872 1113873

minus 1+ t P

minus11 + R1113872 1113873

minus 1

Pminus11 + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(21)

By (20) and Lemma 1 we obtain

Y2 (1 minus t)Y1 + t 2Y1 minus Y1 Pminus11 + R1113872 1113873Y11113960 1113961

ge (1 minus t)Y1 + t 2Y1 minus Y1 Qminus 1

+ R1113872 1113873Y11113960 1113961

(1 minus t)Y1 + t 2Y1 minus Y11113858 1113859

Y1

(22)

thereby

P2 ATY2A + Q

geATY1A + Q P1

(23)

By (21) we obtain

P2 ATY2A + Q

leAT

Pminus11 + R1113872 1113873

minus 1A + Q

leAT

Pminus1minus + R1113872 1113873

minus 1A + Q Pminus

(24)

+us from the above-mentioned proof we have

P0 leP1 leP2 lePminus Y0 Y1

leY2 le Pminus11 + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(25)

(ii) Assume that

Piminus1 lePi le middot middot middot lePminus Yiminus1 leYi le Pminus1iminus 1 + R1113872 1113873

minus 1

le Pminus1minus + R1113872 1113873

minus 1 i 1 2 middot middot middot k

(26)

From (26) we get Yk le (Pminus1kminus 1 + R)minus 1 le (Pminus1

k + R)minus 1then

Yminus1k geP

minus1k + R (27)

+us

Yk+1 (1 minus t)Yk + t 2Yk minus Yk Pminus1k + R1113872 1113873Yk1113960 1113961

ge (1 minus t)Yk + t 2Yk minus YkYminus1k Yk1113960 1113961

ge (1 minus t)Yk + tYk Yk

(28)

Yk+1 (1 minus t)Yk + t 2Yk minus Yk Pminus1k + R1113872 1113873Yk1113960 1113961

le (1 minus t)Yk + t Pminus1k + R1113872 1113873

minus 1

le (1 minus t) Pminus1k + R1113872 1113873

minus 1+ t P

minus1k + R1113872 1113873

minus 1

le Pminus1k + R1113872 1113873

minus 1le P

minus1minus + R1113872 1113873

minus 1

(29)

By (28) and (29) we have

Pk+1 ATYk+1A + Q

geATYkA + Q Pk

(30)

Pk+1 ATYk+1A + Q

leAT

Pminus1minus + R1113872 1113873

minus 1A + Q Pminus

(31)

So we obtain

Pk lePk+1 lePminus Yk leYk+1 le Pminus1k + R1113872 1113873

minus 1

le Pminus1minus + R1113872 1113873

minus 1 k 0 1 2 middot middot middot

(32)

+us the proof of induction is completed Moreover asPk and Yk are monotone increasing and they are boundedthen lim

k⟶infinPk and lim

k⟶infinYk exist Taking limits in Algo-

rithm 2 gives limk⟶infin

Yk (Pminus1minus + R)minus 1 and

limk⟶infin

Pk Pminus

Mathematical Problems in Engineering 3

Theorem 2 Let Pminus be the positive definite solution of theDARE (5) After k steps of iteration for Algorithm 2 we haveI minus Yk(Pminus1

k + R)lt ε then

AT

Pminus1k + R1113872 1113873

minus 1A + Q minus Pk

le ε Pminus minus Q

(33)

Proof According to (15) we have

AT

Pminus1minus + R1113872 1113873

minus 1A Pminus minus Q (34)

+en by Algorithm 2 Lemma 3 and (34) we obtain

AT

Pminus1k + R1113872 1113873

minus 1A + Q minus Pk

AT

Pminus1k + R1113872 1113873

minus 1A minus A

TYkA

AT

Pminus1k + R1113872 1113873

minus 1minus Yk1113876 1113877A

AT

I minus Yk Pminus1k + R1113872 11138731113960 1113961 P

minus1k + R1113872 1113873

minus 1A

le I minus Yk Pminus1k + R1113872 1113873

middot AT

Pminus1k + R1113872 1113873

minus 1A

le I minus Yk Pminus1k + R1113872 1113873

middot AT

Pminus1minus + R1113872 1113873

minus 1A

I minus Yk Pminus1k + R1113872 1113873

middot Pminus minus Q

le ε Pminus minus Q

(35)

because of Pk lePminusAs the proof method is similar to +eorem 1 we list the

monotonicity and convergence of Algorithm 3 withoutproof

Theorem 3 Let Pminus be the positive definite solution of theDARE (5) and Qgt 0 (e iterative sequences Pk1113864 1113865 and Yk1113864 1113865

are generated by Algorithm 3 with t isin 0 1] and start fromY0 (Qminus 1 + R)minus 1 and P0 Q then Pk is monotone in-creasing and converges to Pminus and Yk is monotone increasingand converges to (Pminus1

minus + R)minus 1

Remark 1 For Algorithms 2 and 3 we find the steps ofiteration for Algorithm 2 are less than Algorithm 3 and theconvergence speed of the Algorithm 2 is faster than Algo-rithm 3 from the numerical examples +erefore in thefollowing example we only discuss the superiority and ef-fectiveness of Algorithm 2

3 Numerical Examples

In this section we present the following numerical exampleto show the effectiveness of our results We also discuss theperformance of Algorithm 2 with different t values +ewhole process is carried out on Matlab 71 and the precisionis 10minus 8

Example 1 Consider the discrete system (1) with

A

227 013 012 01

minus013 234 012 005

011 minus017 19 003

001 007 002 11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B

115 0 001 0

0 08 0 0

0 004 09 0

002 0 0 18

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Q

012 0 01 0

0 22 0 0

01 0 14 0

0 0 0 07

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

G I

(36)

In [29] Dai and Bai choose the starting matrixY0 (Qminus 1 + R)minus 1 After 17 steps of iteration the requiredprecision is derived and the residual AT(Pminus 1 + R)minus 1A +

Q minus P is 20754e minus 009For Algorithm 2 we choose P0 Q Y0 (Qminus 1 + R)minus 1

and give the steps of iteration and the residual as Table 1 with

4 Mathematical Problems in Engineering

a different parameter twhen the process is stopped under therequired precision When t is near 1 we find that the steps ofiteration are less than [29] Especially when t 12 it onlyneeds 10 steps for Algorithm 2 to converge to the iterativesolution

P10

33299 minus03120 05202 01433

minus03120 96394 minus01292 01904

05202 minus01292 49731 00820

01433 01904 00820 09962

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(37)

with the residual AT(Pminus 1 + R)minus 1A + Q minus P 56438eminus

009 and Algorithm 2 has faster convergence speed thanAlgorithm 1 from Figure 1 Moreover from Table 1 we seethat Algorithm 2 is more efficient when tgt 1 Although weonly prove the convergence of Algorithm 2 when t isin 0 1] inthis paper Algorithm 2 works well in practical computationwhen tgt 1

Data Availability

All data generated or analyzed during this study are includedwithin this article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e work was supported in part by the National NaturalScience Foundation for Youths of China (11801164) Na-tional Natural Science Foundation of China (11971413) KeyProject of National Natural Science Foundation of China(91430213) General Project of Hunan Provincial NaturalScience Foundation of China (2015JJ2134) and the GeneralProject of Hunan Provincial Education Department ofChina (15C1320)

References

[1] H Kwakernaak and R Sivan Linear Optimal Control SystemsWiley-Interscience New York NY USA 1972

[2] L Wu and D W C Ho ldquoFuzzy filter design for Ito stochasticsystems with application to sensor fault detectionrdquo IEEETransactions on Fuzzy Systems vol 17 no 1 pp 233ndash2422009

[3] X Su P Shi L Wu and Y D Song ldquoA novel approach tofilter design for TCS fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20no 6 pp 1114ndash1129 2012

[4] W N Anderson T D Morley and G E Trapp ldquoLaddernetworks fixpoints and the geometric meanrdquo Circuits Sys-tems and Signal Processing vol 2 no 3 pp 259ndash268 1983

[5] M Z Q Chen L Zhang H Su and G Chen ldquoStabilizingsolution and parameter dependence of modified algebraicRiccati equation with application to discrete-time networksynchronizationrdquo IEEE Transactions on Automatic Controlvol 61 no 1 pp 228ndash233 2016

[6] H S Su H Wu and J Lam ldquoPositive edge-consensus fornodal networks via output feedbackrdquo IEEE Transactions onAutomatic Control vol 64 no 3 pp 1224ndash1249 2019

[7] M Lan and S Chand ldquoSolving linear quadratic discrete-timeoptimal controls usingneural networksrdquo in Proceedings of theIEEE Conference on Decision amp Control IEEE Honolulu HIUSA December 1990

[8] R A Kennedy ldquoLinear system theoryrdquo Automatica vol 30no 11 pp 1811ndash1813 1994

[9] C-H Lee ldquoMatrix bounds of the solutions of the continuousand discrete Riccati equationsmdasha unified approachrdquo Inter-national Journal of Control vol 76 no 6 pp 635ndash642 2003

[10] H H Choi ldquoUpper matrix bounds for the discrete algebraicRiccati matrix equationrdquo IEEE Transactions on AutomaticControl vol 46 no 3 pp 504ndash508 2001

[11] R Davies P Shi and R Wiltshire ldquoNew upper solutionbounds of the discrete algebraic Riccati matrix equationrdquoJournal of Computational and Applied Mathematics vol 213no 2 pp 307ndash315 2008

[12] J Liu L Wang and J Zhang ldquo+e solution bounds and fixedpoint iterative algorithm for the discrete coupled algebraicRiccati equation applied to automatic controlrdquo IMA Journalof Mathematical Control and Information pp 1ndash22 2016

[13] J Liu L Wang and J Zhang ldquoNew matrix bounds and it-erative algorithms for the discrete coupled algebraic Riccatiequationrdquo International Journal of Control vol 90 no 11pp 2326ndash2337 2017

[14] N Komaroff ldquoUpper bounds for the solution of the discreteRiccati equationrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1370ndash1373 1992

[15] J Zhang J Liu and Y Zha ldquo+e improved eigenvalue boundsfor the solution of the discrete algebraic Riccati equationrdquo

Table 1 Numerical results

t Iterations Residual t Iterations Residual06 35 90186e ndash 009 12 10 56438eminus 00908 24 54562eminus 009 13 12 92094eminus 00909 20 47874eminus 009 15 18 94721eminus 0091 17 20754eminus 009 18 40 86619eminus 00911 14 10160eminus 009 20 104 91141eminus 009

10minus10

10minus8

10minus6

10minus4

10minus2

100

102

Resid

ual

Algorithm 1Algorithm 2

2 4 6 8 10 12 14 16 180Iteration steps

Figure 1 +e relationship between iteration step and residual

Mathematical Problems in Engineering 5

IMA Journal of Mathematical Control and Informationpp 1ndash20 2016

[16] N Komaroff and B Shahian ldquoLower summation bounds forthe discrete Riccati and Lyapunov equationsrdquo IEEE Trans-actions on Automatic Control vol 37 no 7 pp 1078ndash10801992

[17] M T Tran and M E Sawan ldquoOn the discrete Riccati matrixequationrdquo SIAM Journal on Algebraic Discrete Methods vol 6no 1 pp 107-108 1985

[18] J Liu and J Zhang ldquo+e existence uniqueness and the fixediterative algorithm of the solution for the discrete coupledalgebraic Riccati equationrdquo International Journal of Controlvol 84 no 8 pp 1430ndash1441 2011

[19] R Huang J Z Liu and L Zhu ldquoAccurate solutions of di-agonally dominant tridiagonal linear systemsrdquo BIT NumericalMathematics vol 54 no 3 pp 711ndash727 2014

[20] Q H Liu X X Li and J Yan ldquoOn the large time behaviour ofsolutions for a class of time-dependent Hamilton-Jacobiequationsrdquo Science China Mathematics vol 59 no 5pp 875ndash8890 2016

[21] Z-H He ldquoSome new results on a system of Sylvester-typequaternion matrix equationsrdquo Linear and Multilinear Alge-bra pp 1ndash23 2019

[22] N Komaroff ldquoIterative matrix bounds and computationalsolutions to the discrete algebraic Riccati equationrdquo IEEETransactions on Automatic Control vol 39 no 8 pp 1676ndash1678 1994

[23] C-H Guo ldquoNewtonrsquos method for discrete algebraic Riccatiequations when the closed-loop matrix has eigenvalues on theunit circlerdquo SIAM Journal on Matrix Analysis and Applica-tions vol 20 no 2 pp 279ndash294 1998

[24] W-W Lin and S-F Xu ldquoConvergence analysis of structure-preserving doubling algorithms for riccati-type matrixequationsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 28 no 1 pp 26ndash39 2006

[25] E K-W Chu H-Y Fan W-W Lin and C-S WangldquoStructure-preserving algorithms for periodic discrete-timealgebraic Riccati equationsrdquo International Journal of Controlvol 77 no 8 pp 767ndash788 2004

[26] L-Z Lu W-W Lin and C E M Pearce ldquoAn efficient al-gorithm for the discrete-time algebraic Riccati equationrdquoIEEE Transactions on Automatic Control vol 44 no 6pp 1216ndash1220 1999

[27] T-M Hwang E K-W Chu and W-W Lin ldquoA generalizedstructure-preserving doubling algorithm for generalizeddiscrete-time algebraic Riccati equationsrdquo InternationalJournal of Control vol 78 no 14 pp 1063ndash1075 2005

[28] A Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24no 6 pp 913ndash921 1979

[29] H Dai and Z Z Bai ldquoOn eigenvalue bounds and iterationmethods for discrete algebraic Riccati equationsrdquo Journal ofComputational Mathematics vol 29 no 3 pp 341ndash366 2011

[30] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 2012

[31] X Zhan ldquoComputing the extremal positive definite solutionsof a matrix equationrdquo SIAM Journal on Scientific Computingvol 17 no 5 pp 1167ndash1174 1996

[32] A W Marshall and I Olkin Inequalities (eory of Majori-zation and its Applications Academic Press New York NYUSA 1979

[33] W Sun and Y Yuan Optimization (eory and MethodsSpringer Science and Business Media LLC New York NYUSA 2006

[34] M Monsalve and M Raydan ldquoA new inversion-free methodfor a rational matrix equationrdquo Linear Algebra and Its Ap-plications vol 433 no 1 pp 64ndash71 2010

[35] G Schulz ldquoIterative Berechung der reziproken MatrixrdquoZAMMmdashZeitschrift fur Angewandte Mathematik und Mech-anik vol 13 no 1 pp 57ndash59 1933

6 Mathematical Problems in Engineering

Theorem 2 Let Pminus be the positive definite solution of theDARE (5) After k steps of iteration for Algorithm 2 we haveI minus Yk(Pminus1

k + R)lt ε then

AT

Pminus1k + R1113872 1113873

minus 1A + Q minus Pk

le ε Pminus minus Q

(33)

Proof According to (15) we have

AT

Pminus1minus + R1113872 1113873

minus 1A Pminus minus Q (34)

+en by Algorithm 2 Lemma 3 and (34) we obtain

AT

Pminus1k + R1113872 1113873

minus 1A + Q minus Pk

AT

Pminus1k + R1113872 1113873

minus 1A minus A

TYkA

AT

Pminus1k + R1113872 1113873

minus 1minus Yk1113876 1113877A

AT

I minus Yk Pminus1k + R1113872 11138731113960 1113961 P

minus1k + R1113872 1113873

minus 1A

le I minus Yk Pminus1k + R1113872 1113873

middot AT

Pminus1k + R1113872 1113873

minus 1A

le I minus Yk Pminus1k + R1113872 1113873

middot AT

Pminus1minus + R1113872 1113873

minus 1A

I minus Yk Pminus1k + R1113872 1113873

middot Pminus minus Q

le ε Pminus minus Q

(35)

because of Pk lePminusAs the proof method is similar to +eorem 1 we list the

monotonicity and convergence of Algorithm 3 withoutproof

Theorem 3 Let Pminus be the positive definite solution of theDARE (5) and Qgt 0 (e iterative sequences Pk1113864 1113865 and Yk1113864 1113865

are generated by Algorithm 3 with t isin 0 1] and start fromY0 (Qminus 1 + R)minus 1 and P0 Q then Pk is monotone in-creasing and converges to Pminus and Yk is monotone increasingand converges to (Pminus1

minus + R)minus 1

Remark 1 For Algorithms 2 and 3 we find the steps ofiteration for Algorithm 2 are less than Algorithm 3 and theconvergence speed of the Algorithm 2 is faster than Algo-rithm 3 from the numerical examples +erefore in thefollowing example we only discuss the superiority and ef-fectiveness of Algorithm 2

3 Numerical Examples

In this section we present the following numerical exampleto show the effectiveness of our results We also discuss theperformance of Algorithm 2 with different t values +ewhole process is carried out on Matlab 71 and the precisionis 10minus 8

Example 1 Consider the discrete system (1) with

A

227 013 012 01

minus013 234 012 005

011 minus017 19 003

001 007 002 11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B

115 0 001 0

0 08 0 0

0 004 09 0

002 0 0 18

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Q

012 0 01 0

0 22 0 0

01 0 14 0

0 0 0 07

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

G I

(36)

In [29] Dai and Bai choose the starting matrixY0 (Qminus 1 + R)minus 1 After 17 steps of iteration the requiredprecision is derived and the residual AT(Pminus 1 + R)minus 1A +

Q minus P is 20754e minus 009For Algorithm 2 we choose P0 Q Y0 (Qminus 1 + R)minus 1

and give the steps of iteration and the residual as Table 1 with

4 Mathematical Problems in Engineering

a different parameter twhen the process is stopped under therequired precision When t is near 1 we find that the steps ofiteration are less than [29] Especially when t 12 it onlyneeds 10 steps for Algorithm 2 to converge to the iterativesolution

P10

33299 minus03120 05202 01433

minus03120 96394 minus01292 01904

05202 minus01292 49731 00820

01433 01904 00820 09962

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(37)

with the residual AT(Pminus 1 + R)minus 1A + Q minus P 56438eminus

009 and Algorithm 2 has faster convergence speed thanAlgorithm 1 from Figure 1 Moreover from Table 1 we seethat Algorithm 2 is more efficient when tgt 1 Although weonly prove the convergence of Algorithm 2 when t isin 0 1] inthis paper Algorithm 2 works well in practical computationwhen tgt 1

Data Availability

All data generated or analyzed during this study are includedwithin this article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e work was supported in part by the National NaturalScience Foundation for Youths of China (11801164) Na-tional Natural Science Foundation of China (11971413) KeyProject of National Natural Science Foundation of China(91430213) General Project of Hunan Provincial NaturalScience Foundation of China (2015JJ2134) and the GeneralProject of Hunan Provincial Education Department ofChina (15C1320)

References

[1] H Kwakernaak and R Sivan Linear Optimal Control SystemsWiley-Interscience New York NY USA 1972

[2] L Wu and D W C Ho ldquoFuzzy filter design for Ito stochasticsystems with application to sensor fault detectionrdquo IEEETransactions on Fuzzy Systems vol 17 no 1 pp 233ndash2422009

[3] X Su P Shi L Wu and Y D Song ldquoA novel approach tofilter design for TCS fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20no 6 pp 1114ndash1129 2012

[4] W N Anderson T D Morley and G E Trapp ldquoLaddernetworks fixpoints and the geometric meanrdquo Circuits Sys-tems and Signal Processing vol 2 no 3 pp 259ndash268 1983

[5] M Z Q Chen L Zhang H Su and G Chen ldquoStabilizingsolution and parameter dependence of modified algebraicRiccati equation with application to discrete-time networksynchronizationrdquo IEEE Transactions on Automatic Controlvol 61 no 1 pp 228ndash233 2016

[6] H S Su H Wu and J Lam ldquoPositive edge-consensus fornodal networks via output feedbackrdquo IEEE Transactions onAutomatic Control vol 64 no 3 pp 1224ndash1249 2019

[7] M Lan and S Chand ldquoSolving linear quadratic discrete-timeoptimal controls usingneural networksrdquo in Proceedings of theIEEE Conference on Decision amp Control IEEE Honolulu HIUSA December 1990

[8] R A Kennedy ldquoLinear system theoryrdquo Automatica vol 30no 11 pp 1811ndash1813 1994

[9] C-H Lee ldquoMatrix bounds of the solutions of the continuousand discrete Riccati equationsmdasha unified approachrdquo Inter-national Journal of Control vol 76 no 6 pp 635ndash642 2003

[10] H H Choi ldquoUpper matrix bounds for the discrete algebraicRiccati matrix equationrdquo IEEE Transactions on AutomaticControl vol 46 no 3 pp 504ndash508 2001

[11] R Davies P Shi and R Wiltshire ldquoNew upper solutionbounds of the discrete algebraic Riccati matrix equationrdquoJournal of Computational and Applied Mathematics vol 213no 2 pp 307ndash315 2008

[12] J Liu L Wang and J Zhang ldquo+e solution bounds and fixedpoint iterative algorithm for the discrete coupled algebraicRiccati equation applied to automatic controlrdquo IMA Journalof Mathematical Control and Information pp 1ndash22 2016

[13] J Liu L Wang and J Zhang ldquoNew matrix bounds and it-erative algorithms for the discrete coupled algebraic Riccatiequationrdquo International Journal of Control vol 90 no 11pp 2326ndash2337 2017

[14] N Komaroff ldquoUpper bounds for the solution of the discreteRiccati equationrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1370ndash1373 1992

[15] J Zhang J Liu and Y Zha ldquo+e improved eigenvalue boundsfor the solution of the discrete algebraic Riccati equationrdquo

Table 1 Numerical results

t Iterations Residual t Iterations Residual06 35 90186e ndash 009 12 10 56438eminus 00908 24 54562eminus 009 13 12 92094eminus 00909 20 47874eminus 009 15 18 94721eminus 0091 17 20754eminus 009 18 40 86619eminus 00911 14 10160eminus 009 20 104 91141eminus 009

10minus10

10minus8

10minus6

10minus4

10minus2

100

102

Resid

ual

Algorithm 1Algorithm 2

2 4 6 8 10 12 14 16 180Iteration steps

Figure 1 +e relationship between iteration step and residual

Mathematical Problems in Engineering 5

IMA Journal of Mathematical Control and Informationpp 1ndash20 2016

[16] N Komaroff and B Shahian ldquoLower summation bounds forthe discrete Riccati and Lyapunov equationsrdquo IEEE Trans-actions on Automatic Control vol 37 no 7 pp 1078ndash10801992

[17] M T Tran and M E Sawan ldquoOn the discrete Riccati matrixequationrdquo SIAM Journal on Algebraic Discrete Methods vol 6no 1 pp 107-108 1985

[18] J Liu and J Zhang ldquo+e existence uniqueness and the fixediterative algorithm of the solution for the discrete coupledalgebraic Riccati equationrdquo International Journal of Controlvol 84 no 8 pp 1430ndash1441 2011

[19] R Huang J Z Liu and L Zhu ldquoAccurate solutions of di-agonally dominant tridiagonal linear systemsrdquo BIT NumericalMathematics vol 54 no 3 pp 711ndash727 2014

[20] Q H Liu X X Li and J Yan ldquoOn the large time behaviour ofsolutions for a class of time-dependent Hamilton-Jacobiequationsrdquo Science China Mathematics vol 59 no 5pp 875ndash8890 2016

[21] Z-H He ldquoSome new results on a system of Sylvester-typequaternion matrix equationsrdquo Linear and Multilinear Alge-bra pp 1ndash23 2019

[22] N Komaroff ldquoIterative matrix bounds and computationalsolutions to the discrete algebraic Riccati equationrdquo IEEETransactions on Automatic Control vol 39 no 8 pp 1676ndash1678 1994

[23] C-H Guo ldquoNewtonrsquos method for discrete algebraic Riccatiequations when the closed-loop matrix has eigenvalues on theunit circlerdquo SIAM Journal on Matrix Analysis and Applica-tions vol 20 no 2 pp 279ndash294 1998

[24] W-W Lin and S-F Xu ldquoConvergence analysis of structure-preserving doubling algorithms for riccati-type matrixequationsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 28 no 1 pp 26ndash39 2006

[25] E K-W Chu H-Y Fan W-W Lin and C-S WangldquoStructure-preserving algorithms for periodic discrete-timealgebraic Riccati equationsrdquo International Journal of Controlvol 77 no 8 pp 767ndash788 2004

[26] L-Z Lu W-W Lin and C E M Pearce ldquoAn efficient al-gorithm for the discrete-time algebraic Riccati equationrdquoIEEE Transactions on Automatic Control vol 44 no 6pp 1216ndash1220 1999

[27] T-M Hwang E K-W Chu and W-W Lin ldquoA generalizedstructure-preserving doubling algorithm for generalizeddiscrete-time algebraic Riccati equationsrdquo InternationalJournal of Control vol 78 no 14 pp 1063ndash1075 2005

[28] A Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24no 6 pp 913ndash921 1979

[29] H Dai and Z Z Bai ldquoOn eigenvalue bounds and iterationmethods for discrete algebraic Riccati equationsrdquo Journal ofComputational Mathematics vol 29 no 3 pp 341ndash366 2011

[30] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 2012

[31] X Zhan ldquoComputing the extremal positive definite solutionsof a matrix equationrdquo SIAM Journal on Scientific Computingvol 17 no 5 pp 1167ndash1174 1996

[32] A W Marshall and I Olkin Inequalities (eory of Majori-zation and its Applications Academic Press New York NYUSA 1979

[33] W Sun and Y Yuan Optimization (eory and MethodsSpringer Science and Business Media LLC New York NYUSA 2006

[34] M Monsalve and M Raydan ldquoA new inversion-free methodfor a rational matrix equationrdquo Linear Algebra and Its Ap-plications vol 433 no 1 pp 64ndash71 2010

[35] G Schulz ldquoIterative Berechung der reziproken MatrixrdquoZAMMmdashZeitschrift fur Angewandte Mathematik und Mech-anik vol 13 no 1 pp 57ndash59 1933

6 Mathematical Problems in Engineering

a different parameter twhen the process is stopped under therequired precision When t is near 1 we find that the steps ofiteration are less than [29] Especially when t 12 it onlyneeds 10 steps for Algorithm 2 to converge to the iterativesolution

P10

33299 minus03120 05202 01433

minus03120 96394 minus01292 01904

05202 minus01292 49731 00820

01433 01904 00820 09962

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(37)

with the residual AT(Pminus 1 + R)minus 1A + Q minus P 56438eminus

009 and Algorithm 2 has faster convergence speed thanAlgorithm 1 from Figure 1 Moreover from Table 1 we seethat Algorithm 2 is more efficient when tgt 1 Although weonly prove the convergence of Algorithm 2 when t isin 0 1] inthis paper Algorithm 2 works well in practical computationwhen tgt 1

Data Availability

All data generated or analyzed during this study are includedwithin this article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+e work was supported in part by the National NaturalScience Foundation for Youths of China (11801164) Na-tional Natural Science Foundation of China (11971413) KeyProject of National Natural Science Foundation of China(91430213) General Project of Hunan Provincial NaturalScience Foundation of China (2015JJ2134) and the GeneralProject of Hunan Provincial Education Department ofChina (15C1320)

References

[1] H Kwakernaak and R Sivan Linear Optimal Control SystemsWiley-Interscience New York NY USA 1972

[2] L Wu and D W C Ho ldquoFuzzy filter design for Ito stochasticsystems with application to sensor fault detectionrdquo IEEETransactions on Fuzzy Systems vol 17 no 1 pp 233ndash2422009

[3] X Su P Shi L Wu and Y D Song ldquoA novel approach tofilter design for TCS fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20no 6 pp 1114ndash1129 2012

[4] W N Anderson T D Morley and G E Trapp ldquoLaddernetworks fixpoints and the geometric meanrdquo Circuits Sys-tems and Signal Processing vol 2 no 3 pp 259ndash268 1983

[5] M Z Q Chen L Zhang H Su and G Chen ldquoStabilizingsolution and parameter dependence of modified algebraicRiccati equation with application to discrete-time networksynchronizationrdquo IEEE Transactions on Automatic Controlvol 61 no 1 pp 228ndash233 2016

[6] H S Su H Wu and J Lam ldquoPositive edge-consensus fornodal networks via output feedbackrdquo IEEE Transactions onAutomatic Control vol 64 no 3 pp 1224ndash1249 2019

[7] M Lan and S Chand ldquoSolving linear quadratic discrete-timeoptimal controls usingneural networksrdquo in Proceedings of theIEEE Conference on Decision amp Control IEEE Honolulu HIUSA December 1990

[8] R A Kennedy ldquoLinear system theoryrdquo Automatica vol 30no 11 pp 1811ndash1813 1994

[9] C-H Lee ldquoMatrix bounds of the solutions of the continuousand discrete Riccati equationsmdasha unified approachrdquo Inter-national Journal of Control vol 76 no 6 pp 635ndash642 2003

[10] H H Choi ldquoUpper matrix bounds for the discrete algebraicRiccati matrix equationrdquo IEEE Transactions on AutomaticControl vol 46 no 3 pp 504ndash508 2001

[11] R Davies P Shi and R Wiltshire ldquoNew upper solutionbounds of the discrete algebraic Riccati matrix equationrdquoJournal of Computational and Applied Mathematics vol 213no 2 pp 307ndash315 2008

[12] J Liu L Wang and J Zhang ldquo+e solution bounds and fixedpoint iterative algorithm for the discrete coupled algebraicRiccati equation applied to automatic controlrdquo IMA Journalof Mathematical Control and Information pp 1ndash22 2016

[13] J Liu L Wang and J Zhang ldquoNew matrix bounds and it-erative algorithms for the discrete coupled algebraic Riccatiequationrdquo International Journal of Control vol 90 no 11pp 2326ndash2337 2017

[14] N Komaroff ldquoUpper bounds for the solution of the discreteRiccati equationrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1370ndash1373 1992

[15] J Zhang J Liu and Y Zha ldquo+e improved eigenvalue boundsfor the solution of the discrete algebraic Riccati equationrdquo

Table 1 Numerical results

t Iterations Residual t Iterations Residual06 35 90186e ndash 009 12 10 56438eminus 00908 24 54562eminus 009 13 12 92094eminus 00909 20 47874eminus 009 15 18 94721eminus 0091 17 20754eminus 009 18 40 86619eminus 00911 14 10160eminus 009 20 104 91141eminus 009

10minus10

10minus8

10minus6

10minus4

10minus2

100

102

Resid

ual

Algorithm 1Algorithm 2

2 4 6 8 10 12 14 16 180Iteration steps

Figure 1 +e relationship between iteration step and residual

Mathematical Problems in Engineering 5

IMA Journal of Mathematical Control and Informationpp 1ndash20 2016

[16] N Komaroff and B Shahian ldquoLower summation bounds forthe discrete Riccati and Lyapunov equationsrdquo IEEE Trans-actions on Automatic Control vol 37 no 7 pp 1078ndash10801992

[17] M T Tran and M E Sawan ldquoOn the discrete Riccati matrixequationrdquo SIAM Journal on Algebraic Discrete Methods vol 6no 1 pp 107-108 1985

[18] J Liu and J Zhang ldquo+e existence uniqueness and the fixediterative algorithm of the solution for the discrete coupledalgebraic Riccati equationrdquo International Journal of Controlvol 84 no 8 pp 1430ndash1441 2011

[19] R Huang J Z Liu and L Zhu ldquoAccurate solutions of di-agonally dominant tridiagonal linear systemsrdquo BIT NumericalMathematics vol 54 no 3 pp 711ndash727 2014

[20] Q H Liu X X Li and J Yan ldquoOn the large time behaviour ofsolutions for a class of time-dependent Hamilton-Jacobiequationsrdquo Science China Mathematics vol 59 no 5pp 875ndash8890 2016

[21] Z-H He ldquoSome new results on a system of Sylvester-typequaternion matrix equationsrdquo Linear and Multilinear Alge-bra pp 1ndash23 2019

[22] N Komaroff ldquoIterative matrix bounds and computationalsolutions to the discrete algebraic Riccati equationrdquo IEEETransactions on Automatic Control vol 39 no 8 pp 1676ndash1678 1994

[23] C-H Guo ldquoNewtonrsquos method for discrete algebraic Riccatiequations when the closed-loop matrix has eigenvalues on theunit circlerdquo SIAM Journal on Matrix Analysis and Applica-tions vol 20 no 2 pp 279ndash294 1998

[24] W-W Lin and S-F Xu ldquoConvergence analysis of structure-preserving doubling algorithms for riccati-type matrixequationsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 28 no 1 pp 26ndash39 2006

[25] E K-W Chu H-Y Fan W-W Lin and C-S WangldquoStructure-preserving algorithms for periodic discrete-timealgebraic Riccati equationsrdquo International Journal of Controlvol 77 no 8 pp 767ndash788 2004

[26] L-Z Lu W-W Lin and C E M Pearce ldquoAn efficient al-gorithm for the discrete-time algebraic Riccati equationrdquoIEEE Transactions on Automatic Control vol 44 no 6pp 1216ndash1220 1999

[27] T-M Hwang E K-W Chu and W-W Lin ldquoA generalizedstructure-preserving doubling algorithm for generalizeddiscrete-time algebraic Riccati equationsrdquo InternationalJournal of Control vol 78 no 14 pp 1063ndash1075 2005

[28] A Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24no 6 pp 913ndash921 1979

[29] H Dai and Z Z Bai ldquoOn eigenvalue bounds and iterationmethods for discrete algebraic Riccati equationsrdquo Journal ofComputational Mathematics vol 29 no 3 pp 341ndash366 2011

[30] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 2012

[31] X Zhan ldquoComputing the extremal positive definite solutionsof a matrix equationrdquo SIAM Journal on Scientific Computingvol 17 no 5 pp 1167ndash1174 1996

[32] A W Marshall and I Olkin Inequalities (eory of Majori-zation and its Applications Academic Press New York NYUSA 1979

[33] W Sun and Y Yuan Optimization (eory and MethodsSpringer Science and Business Media LLC New York NYUSA 2006

[34] M Monsalve and M Raydan ldquoA new inversion-free methodfor a rational matrix equationrdquo Linear Algebra and Its Ap-plications vol 433 no 1 pp 64ndash71 2010

[35] G Schulz ldquoIterative Berechung der reziproken MatrixrdquoZAMMmdashZeitschrift fur Angewandte Mathematik und Mech-anik vol 13 no 1 pp 57ndash59 1933

6 Mathematical Problems in Engineering

IMA Journal of Mathematical Control and Informationpp 1ndash20 2016

[16] N Komaroff and B Shahian ldquoLower summation bounds forthe discrete Riccati and Lyapunov equationsrdquo IEEE Trans-actions on Automatic Control vol 37 no 7 pp 1078ndash10801992

[17] M T Tran and M E Sawan ldquoOn the discrete Riccati matrixequationrdquo SIAM Journal on Algebraic Discrete Methods vol 6no 1 pp 107-108 1985

[18] J Liu and J Zhang ldquo+e existence uniqueness and the fixediterative algorithm of the solution for the discrete coupledalgebraic Riccati equationrdquo International Journal of Controlvol 84 no 8 pp 1430ndash1441 2011

[19] R Huang J Z Liu and L Zhu ldquoAccurate solutions of di-agonally dominant tridiagonal linear systemsrdquo BIT NumericalMathematics vol 54 no 3 pp 711ndash727 2014

[20] Q H Liu X X Li and J Yan ldquoOn the large time behaviour ofsolutions for a class of time-dependent Hamilton-Jacobiequationsrdquo Science China Mathematics vol 59 no 5pp 875ndash8890 2016

[21] Z-H He ldquoSome new results on a system of Sylvester-typequaternion matrix equationsrdquo Linear and Multilinear Alge-bra pp 1ndash23 2019

[22] N Komaroff ldquoIterative matrix bounds and computationalsolutions to the discrete algebraic Riccati equationrdquo IEEETransactions on Automatic Control vol 39 no 8 pp 1676ndash1678 1994

[23] C-H Guo ldquoNewtonrsquos method for discrete algebraic Riccatiequations when the closed-loop matrix has eigenvalues on theunit circlerdquo SIAM Journal on Matrix Analysis and Applica-tions vol 20 no 2 pp 279ndash294 1998

[24] W-W Lin and S-F Xu ldquoConvergence analysis of structure-preserving doubling algorithms for riccati-type matrixequationsrdquo SIAM Journal on Matrix Analysis and Applica-tions vol 28 no 1 pp 26ndash39 2006

[25] E K-W Chu H-Y Fan W-W Lin and C-S WangldquoStructure-preserving algorithms for periodic discrete-timealgebraic Riccati equationsrdquo International Journal of Controlvol 77 no 8 pp 767ndash788 2004

[26] L-Z Lu W-W Lin and C E M Pearce ldquoAn efficient al-gorithm for the discrete-time algebraic Riccati equationrdquoIEEE Transactions on Automatic Control vol 44 no 6pp 1216ndash1220 1999

[27] T-M Hwang E K-W Chu and W-W Lin ldquoA generalizedstructure-preserving doubling algorithm for generalizeddiscrete-time algebraic Riccati equationsrdquo InternationalJournal of Control vol 78 no 14 pp 1063ndash1075 2005

[28] A Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24no 6 pp 913ndash921 1979

[29] H Dai and Z Z Bai ldquoOn eigenvalue bounds and iterationmethods for discrete algebraic Riccati equationsrdquo Journal ofComputational Mathematics vol 29 no 3 pp 341ndash366 2011

[30] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 2012

[31] X Zhan ldquoComputing the extremal positive definite solutionsof a matrix equationrdquo SIAM Journal on Scientific Computingvol 17 no 5 pp 1167ndash1174 1996

[32] A W Marshall and I Olkin Inequalities (eory of Majori-zation and its Applications Academic Press New York NYUSA 1979

[33] W Sun and Y Yuan Optimization (eory and MethodsSpringer Science and Business Media LLC New York NYUSA 2006

[34] M Monsalve and M Raydan ldquoA new inversion-free methodfor a rational matrix equationrdquo Linear Algebra and Its Ap-plications vol 433 no 1 pp 64ndash71 2010

[35] G Schulz ldquoIterative Berechung der reziproken MatrixrdquoZAMMmdashZeitschrift fur Angewandte Mathematik und Mech-anik vol 13 no 1 pp 57ndash59 1933

6 Mathematical Problems in Engineering