Abstract
In this chapter as well as in the next chapters one shows how the mathematical results derived in the previous chapters are involved in the design of stabilizing controllers with some imposed performances for a wide class of linear stochastic systems. The design problem of some stabilizing controls minimizing quadratic performance criteria is studied. More precisely, this chapter deals with the so-called linear quadratic optimization problem (LQOP). LQOP has received much attention in control literature due to its wide area of applications.
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Bibliography
M. Ait-Rami, L. El-Ghaoui, Robust stabilization of jump linear systems using linear matrix inequalities, in Proceedings of IFAC Symposium on Robust Control Design Rio de Janeiro, RJ (1994), pp. 148–151
M. Ait-Rami, X.Y. Zhou, Linear matrix inequalities, Riccati equations and indefinite stochastic linear quadratic controls. IEEE Trans. Automat. Contr. 45(6), 1131–1143 (2000)
J.M. Bismut, Linear quadratic optimal control with random coefficients. SIAM J. Contr. Optim. 14, 419–444 (1976)
S. Chen, X. Li, Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Contr. Optim. 36(5), 1685–1702 (1998)
G. Da Prato, A. Ichikawa, Stability and quadratic control for linear stochastic equation with unbounded coefficients. Boll. Un Mat. 6, 989–1001 (1985)
G. Da Prato, A. Ichikawa, Quadratic control for linear time-varying stochastic systems. SIAM J. Contr. Optim. 28, 359–381 (1990)
V. Dragan, T. Morozan, Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping. J. Differ. Equat. 194(1), 1–38 (2003)
V. Dragan, T. Morozan, The linear quadratic optimization problem and tracking problem for a class of linear stochastic systems with multiplicative white noise and Markovian jumping. IEEE-T-AC 49, 665–676 (2004)
A. Halanay, Differential Equations, Stability, Oscillations, Time Lag (Academic, New York, 1966)
A. Haurie, A. Leizarowitz, Overtracking optimal regulation and tracking of piecewise diffusion linear systems. SIAM J. Contr. Optim. 30(4), 816–832 (1992)
U.G. Hausmann, Optimal stationary control with state and control dependent noise. SIAM J. Contr. Optim. 9, 184–198 (1971)
A. Ichikawa, Optimal control of linear stochastic evolution equation with state and control dependent noise, in Proceedings of IMA Conference on Recent Theoretical Developments in Control, Leicester (Academic, New York, 1978), pp. 383–401
A. Ichikawa, Filtering and control of stochastic differential equations with unbounded coefficients. Stoch. Anal. Appl. 4, 187–212 (1986)
Y. Ji, H.J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Trans. Automat. Contr. 35(7), 777–788 (1990)
Y. Ji, H.J. Chizeck, Jump linear quadratic Gaussian control: steady-state solution and testable conditions. Contr. Theor. Adv. Technol. 7(2), 247–270 (1991)
H. Kushner, Stochastic Stability and Control (Academic, New York, 1967)
T. Morozan, Optimal stationary control for dynamic systems with Markov perturbations. Stoch. Anal. Appl. 1(3), 299–325 (1983)
T. Morozan, Stochastic uniform observability and Riccati equations of stochastic control. Rev. Roum. Math. Pures Appl. 38(9), 771–781 (1993)
T. Morozan, Stability and control for linear systems with jump Markov perturbations. Stoch. Anal. Appl. 13(1), 91–110 (1995)
T. Morozan, Linear quadratic and tracking problems for time-invariant stochastic differential systems perturbed by Markov chain. Rev. Roum. Math. Pures Appl. 25(6), 783–804 (2001)
A.J. Pritchard, J. Zabczyk, Stability and stabilizability of infinite dimensional systems. SIAM Rev. 23, 25–52 (1981)
T. Sasagawa, LP-stabilization problem for linear-stochastic control systems with multiplicative noise. J. Optim. Theor. Appl. 61(3), 451–471 (1989)
T. Sasagawa, Stabilization of linear stochastic systems with delay by using a certain class of controls, in Transactions of the 8th Prague Conference on Information Theory, Statistical Decision Functions 28 August-1 September 1978, pp. 295–300
G. Tessitore, Some remarks on the Riccati equation arising in an optimal control problem with state and control dependent noise. SIAM J. Contr. Optim. 30(3), 714–744 (1992)
J.L. Willems, Mean square stability criteria for linear white noise stochastic systems. Probl. Contr. Inf. Theor. 2(3–4), 199–217 (1973)
J.L. Willems, J.C. Willems, Feedback stabilization for stochastic systems with state and control dependent noise. Automatica 12, 277–283 (1976)
J.L. Willems, J.C. Willems, Robust stabilization of uncertain systems. SIAM J. Contr. 21, 395–409 (1983)
W.M. Wonham, Random differential equations in control theory, in Probabilistic Methods in Applied Mathematics, vol. 2 (Academic, New York, 1970), pp. 131–212
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Dragan, V., Morozan, T., Stoica, AM. (2013). Linear Quadratic Optimization Problems for Linear Stochastic Systems. In: Mathematical Methods in Robust Control of Linear Stochastic Systems., vol 52. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8663-3_6
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DOI: https://doi.org/10.1007/978-1-4614-8663-3_6
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