Convergence analysis of the MCGNR algorithm for the least squares solution group of discrete-time periodic coupled matrix equations

Abstract

The conjugate gradient normal equations residual minimizing (CGNR) algorithm is a popular tool for solving large nonsymmetric linear systems. In this study, we propose the matrix form of the CGNR (MCGNR) algorithm to find the least squares solution group of the discrete-time periodic coupled matrix equations

${\begin{matrix} A_{1, t} X_{t} B_{1, t} + C_{1, t} X_{t + 1} D_{1, t} + E_{1, t} Y_{t} F_{1, t} = G_{1, t}, \\ A_{2, t} X_{t} B_{2, t} + C_{2, t} X_{t + 1} D_{2, t} + E_{2, t} Y_{t} F_{2, t} = G_{2, t}, \end{matrix} for t = 1, 2, \dots$

with periodic matrices. We prove that the MCGNR algorithm converges in a finite number of steps in the absence of round-off errors, that is, this algorithm has the finite termination property. Also we show that the norms of the residual matrices of the MCGNR algorithm decrease monotonically during its iteration, that is, this algorithm has the residual reducing property. To show the efficiency of the MCGNR algorithm, two numerical examples are presented.

Keywords

CGNR algorithm discrete-time periodic coupled matrix equations finite termination property residual reducing property iterative method

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References

Benner

Hossain

(2012) Structure preserving iterative solution of periodic projected Lyapunov equations. In: Proceedings of MATHMOD, Vienna, Austria, pp. 15–17.

Benner

Hossain

Stykel

(2014) Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems. Numerical Algorithms 67: 669–690.

Brezinski

(1997) Projection methods for linear systems. Journal of Computational and Applied Mathematics 77: 35–51.

Chu

Van Dooren

(2006) A novel numerical method for exact model matching problem with stability. Automatica 42: 1697–1704.

Chu

EKW

Fan

Lin

(1995) Reachability and observability of periodic descriptor systems. Preprint 2005-1-009 NCTS, National Tsing Hua University, Taiwan.

Chu

EKW

Fan

Lin

(2007) Projected generalized discrete-time periodic Lyapunov equations and balanced realization of periodic descriptor systems. SIAM Journal on Matrix Analysis and Applications 29: 982–1006.

Coll

Fullana

Sanchez

(2004) Reachability and observability indices of a discrete-time periodic descriptor system. Applied Mathematics and Computation 153: 485–496.

Dehghan

Hajarian

(2008) An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation. Applied Mathematics and Computation 202: 571–588.

Dehghan

Hajarian

(2010) The general coupled matrix equations over generalized bisymmetric matrices. Linear Algebra and its Applications 432: 1531–1552.

10.

Ding

Chen

(2005a) Iterative least squares solutions of coupled Sylvester matrix equations. Systems & Control Letters 54: 95–107.

11.

Ding

Chen

(2005b) On iterative solutions of general coupled matrix equations. SIAM Journal on Control and Optimization 44: 2269–2284.

12.

Ding

Chen

(2005c) Gradient based iterative algorithms for solving a class of matrix equations. IEEE Transactions on Automatic Control 50: 1216–1221.

13.

Ding

Huamin

(2014) Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems. IET Control Theory and Applications 8: 1588–1595.

14.

Granat

Jonsson

Kagstrom

(2006) Recursive blocked algorithms for solving periodic triangular Sylvester-type matrix equations. In: Proceedings of the 8th international conference on applied parallel computing: State of the art in scientific computing, pp. 531–539.

15.

Granat

Kagstrom

Kressner

(2007) Computing periodic deflating subspaces associated with a specified set of eigenvalues. BIT 47: 763–791.

16.

Hajarian

(2013a) Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. Journal of the Franklin Institute 350: 3328–3341.

17.

Hajarian

(2013b) Matrix form of the Bi-CGSTAB method for solving the coupled Sylvester matrix equations. IET Control Theory and Applications 7: 1828–1833.

18.

Hajarian

(2014a) Solving the general coupled and the periodic coupled matrix equations via the extended QMRCGSTAB algorithms. Computational & Applied Mathematics 33: 349–362.

19.

Hajarian

(2014b) A matrix LSQR algorithm for solving constrained linear operator equations. Bulletin of the Iranian Mathematical Society 40: 41–53.

20.

Hajarian

(2014c) Extending LSQR methods to solve the generalized Sylvester-transpose and periodic Sylvester matrix equations. Mathematical Methods in the Applied Sciences 37: 2017–2028.

21.

Hajarian

(2014d) Extending the GPBiCG algorithm for solving the generalized Sylvester-transpose matrix equation. International Journal of Control, Automation, and Systems 12: 1362–1365.

22.

Hajarian

(2014e) Matrix algorithms for solving the generalized coupled Sylvester and periodic coupled matrix equations. Transactions of the Institute of Measurement and Control 36: 963–970.

23.

Hajarian

(2014f) Matrix form of the CGS method for solving general coupled matrix equations. Applied Mathematics Letters 34: 3742.

24.

Huamin

Ding

(2014) A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations. Journal of the Franklin Institute 351: 340–357.

25.

Kenney

Hewer

(1990) The sensitivity of the algebraic and differential Riccati equations. SIAM Journal on Control and Optimization 28: 50–69.

26.

Lam

Yan

(1999) Pole assignment with eigenvalue and stability robustness. International Journal of Control 72: 1165–1174.

27.

LaSalle

Lefschetz

(1961) Stability by Liapunov’s Direct Method. New York, NY: Academic Press.

28.

(2001) CGNR is an error reducing algorithm. SIAM Journal on Scientific Computing 22: 2109–2112.

29.

Moore

(1981) Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control 26: 17–31.

30.

Stykel

(2006) On some norms for descriptor systems. IEEE Transactions on Automatic Control 51: 842–847.

31.

Varga

(2007) On computing minimal realizations of periodic descriptor systems. In: Proceedings of the IFAC workshop on periodic control systems, St. Petersburg, Russia, pp. 175–180.

32.

Wang

(2003) Functional Analysis and Optimization Theory. Beijing, China: Beijing University of Aeronautics and Astronautics Press.

33.

Zhou

Duan

(2006) Parametric approach for the normal Luenberger function observer design in second-order linear systems. In: Proceedings of the 45th IEEE conference on decision and control, pp. 1423–1428.

34.

Zhou

Duan

(2008) On the generalized Sylvester mapping and matrix equations. Systems & Control Letters 57: 200–208.

35.

Zhou

Duan

Lin

(2011) A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation. Automatica 47: 316–325.

36.

Zhou

Duan

(2009a) Gradient based iterative algorithm for solving coupled matrix equations. Systems & Control Letters 58: 327–333.

37.

Zhou

Duan

. (2009b) Weighted least squares solutions to general coupled Sylvester matrix equations. Journal of Computational and Applied Mathematics 224: 759–776.