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On the convergence of the accelerated Riccati iteration method. (English) Zbl 1453.65085

Summary: In this paper, we establish results fully addressing two open problems proposed recently by I. G. Ivanov [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 11, 4012–4024 (2008; Zbl 1162.65020)], with respect to the convergence of the accelerated Riccati iteration method for solving the continuous coupled algebraic Riccati equation, or CCARE for short. These results confirm several desirable features of that method, including the monotonicity and boundedness of the sequences it produces, its capability of determining whether the CCARE has a solution, the extremal solutions it computes under certain circumstances, and its faster convergence than the regular Riccati iteration method.

MSC:

65F45 Numerical methods for matrix equations
15A24 Matrix equations and identities
15B48 Positive matrices and their generalizations; cones of matrices

Citations:

Zbl 1162.65020

Software:

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References:

[1] Abou-Kandil, H.; Freiling, G.; Jank, G., Solution and asymptotic behavior of coupled Riccati equations in jump linear systems, IEEE Transactions on Automatic Control, 39, 1631-1636 (1994) · Zbl 0925.93387
[2] Arnold III, W.; Laub, A., Generalized eigenproblem algorithms and software for algebraic Riccati equations, Proceedings of the IEEE, 72, 1746-1754 (1984)
[3] Bini, D.; Lannazzo, B.; Meini, B., Numerical solution of algebraic Riccati equations (2012), Philadelphia, Pennsylvania: SIAM, Philadelphia, Pennsylvania
[4] Costa, E.; do Val, J., An algorithm for solving a perturbed algebraic Riccati equation, European Journal of Control, 10, 576-580 (2004) · Zbl 1293.93776
[5] Costa, O.; Fragoso, M.; Todorov, M., Continuous-time Markov jump linear systems (2013), Berlin, Heidelberg: Springer-Verlag, Berlin, Heidelberg · Zbl 1277.60003
[6] Czornik, A.; Swierniak, A., Upper bounds on the solution of coupled algebraic Riccati equation, Journal of Inequalities and Applications, 6, 373-385 (2001) · Zbl 1006.93035
[7] Damm, T.; Hinrichsen, D., Newton’s method for a rational matrix equation occurring in stochastic control, Linear Algebra and Its Applications, 332/334, 81-109 (2001) · Zbl 0982.65050
[8] Davies, R.; Shi, P.; Wiltshire, R., Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations, Automatica, 44, 1088-1096 (2008) · Zbl 1283.93134
[9] do Val, J.; Costa, E., Stabilizability and positiveness of solutions of the jump linear quadratic problem and the coupled algebraic Riccati equation, IEEE Transactions on Automatic Control, 50, 691-695 (2005) · Zbl 1365.93422
[10] do Val, J.; Geromel, J.; Costa, O., Solutions for the linear-quadratic control problem of Markov jump linear systems, Journal of Optimization Theory and Applications, 103, 283-311 (1999) · Zbl 0948.49018
[11] Gajic, Z.; Borno, I., Lyapunov iterations for optimal control of jump linear systems at steady state, IEEE Transactions on Automatic Control, 40, 1971-1975 (1995) · Zbl 0837.93073
[12] Guo, C., Iterative methods for a linearly perturbed algebraic matrix Riccati equation arising in stochastic control, Numerical Functional Analysis and Optimization, 34, 516-529 (2013) · Zbl 1275.65023
[13] Ivanov, I., Iterations for solving a rational Riccati equation arising in stochastic control, Computers and Mathematics with Applications, 53, 977-988 (2007) · Zbl 1127.65025
[14] Ivanov, I., On some iterations for optimal control of jump linear equations, Nonlinear Analysis: Theory, Methods & Applications, 69, 4012-4024 (2008) · Zbl 1162.65020
[15] Ivanov, I.; Hasanov, V.; Minchev, B., On matrix equations \(####\), Linear Algebra and Its Applications, 326, 27-44 (2001) · Zbl 0979.15007
[16] Kleinman, D., On an iterative technique for Riccati equation computations, IEEE Transactions on Automatic Control, AC-13, 114-115 (1968)
[17] Kučera, V., A review of the matrix Riccati equation, Kybernetika, 9, 42-61 (1973) · Zbl 0279.49015
[18] Mariton, M., Jump linear systems in automatic control (1990), New York: Marcel Dekker, New York
[19] Rami, M.; Ghaoui, L., LMI optimization for nonstandard Riccati equations arising in stochastic control, IEEE Transactions on Automatic Control, 41, 1666-1671 (1996) · Zbl 0863.93087
[20] Ran, A.; Vreugdenhil, R., Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems, Linear Algebra and Its Applications, 99, 63-83 (1988) · Zbl 0637.15008
[21] Sandell, N., On Newton’s method for Riccati equation solution, IEEE Transactions on Automatic Control, AC-19, 254-255 (1974) · Zbl 0278.65096
[22] Willems, J., Least squares stationary optimal control and the algebraic Riccati equation, IEEE Transactions on Automatic Control, 16, 621-634 (1971)
[23] Williams II, R.; Lawrence, D., Linear state-space control systems (2007), Hoboken, New Jersey: John Wiley & Sons, Inc, Hoboken, New Jersey
[24] Xu, J., Unified, improved matrix upper bound on the solution of the continuous coupled algebraic Riccati equation, Journal of the Franklin Institute, 350, 1634-1648 (2013) · Zbl 1293.93375
[25] Xu, J.; Rajasingam, P., New unified matrix upper bound on the solution of the continuous coupled algebraic Riccati equation, Journal of the Franklin Institute, 353, 1233-1247 (2016) · Zbl 1336.93176
[26] Xu, J.; Xiao, M., On the iterative refinement of matrix upper bounds for the solution of continuous coupled algebraic Riccati equations, Automatica, 49, 2168-2175 (2013) · Zbl 1364.93251
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