On some iterations for optimal control of jump linear equations. (English) Zbl 1162.65020
The author describes some iterative methods for solving coupled algebraic Riccati equations of the optimal control problem for jump linear systems. Some of these methods are known from the literature. Two new methods are proposed: an accelerated Lyapunov method and an accelerated Riccati method. The first method is investigated theoretically. Some numerical experiments are presented to show the effectiveness of the different methods.
Reviewer: Michael Jung (Dresden)
MSC:
| 65F30 | Other matrix algorithms (MSC2010) |
| 15A24 | Matrix equations and identities |
Keywords:
coupled algebraic Riccati equation; Lyapunov iterations, Newton’s method; positive semidefinite matrix; convergence acceleration; optimal control problem; jump linear systems; Riccati method; numerical experimentsReferences:
| [1] | Rami, M. A.; Zhou, X. Y.; Moore, J. B., Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon, Systems Control Lett., 41, 123-133 (2000) · Zbl 0985.93060 |
| [2] | Rami, M. A.; Moore, J. B.; Zhou, X. Y., Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim., 40, 1296-1311 (2001) · Zbl 1009.93082 |
| [3] | Freiling, G.; Hochhaus, A., On a class of rational matrix differential equations arising in stochastic control, Linear Algebra Appl., 379, 43-68 (2004) · Zbl 1070.34054 |
| [4] | Costa, E.; do Val, J., An algorithm for solving a perturbed algebraic Riccati equation, Eur. J. Control, 10, 576-580 (2004) · Zbl 1293.93776 |
| [5] | Gajic, Z.; Borno, I., Lyapunov iterations for optimal control of jump linear systems at steady state, IEEE Trans. Automat. Control, 40, 1971-1975 (1995) · Zbl 0837.93073 |
| [6] | Gajic, Z.; Losada, R., Monotonicity of algebraic Lyapunov iterations for optimal control of jump parameter linear systems, Systems Control Lett., 41, 175-181 (2000) · Zbl 0985.93017 |
| [7] | Damm, T.; Hinrichsen, D., Newton’s method for a rational matrix equation occurring in stochastic control, Linear Algebra Appl., 332-334, 81-109 (2001) · Zbl 0982.65050 |
| [8] | Gajic, Z.; Losada, R., Solution of the state-dependent noise optimal control problem in terms of Lyapunov iterations, Automatica, 35, 951-954 (1999) · Zbl 0934.93070 |
| [9] | Guo, C., Iterative solution of a matrix Riccati equation arising in stochastic control, Oper. Theory Adv. Appl., 130, 209-221 (2001) · Zbl 1020.65024 |
| [10] | Gajic, Z.; Quershi, The Lyapunov Matrix Equation in Systems Stability and Control (1995), Academic Press: Academic Press San Diego · Zbl 1153.93300 |
| [11] | Lancaster, P.; Rodman, L., Algebraic Riccati Equations (1995), Clarendon Press: Clarendon Press Oxford · Zbl 0836.15005 |
| [12] | Ivanov, I., Iterations for solving a rational Riccati equation arising in stochastic control, Comput. Math. Appl., 53, 977-988 (2007) · Zbl 1127.65025 |
| [13] | do Val, J. B.; Geromel, J. C.; Costa, O. L.V., Solutions for the linear quadratic control problem of Markov jump linear systems, J. Optim. Theory Appl., 103, 2, 283-311 (1999) · Zbl 0948.49018 |
| [14] | Wonham, W., Random difference equations in control theory, (Bharucha-Ried, A., Probabilistic Methods in Applied Mathematics (1971), Academic: Academic New York), 131-212 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
