# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stochastic linear quadratic regulation for discrete-time linear systems with input delay. (English) Zbl 1175.93246
Summary: This paper considers the stochastic linear Quadratic Regulation (LQR) problem for systems with input delay and stochastic parameter uncertainties in the state and input matrices. The problem is known to be difficult due to the presence of interactions among the delayed input channels and the stochastic parameter uncertainties in the channels. The key to our approach is to convert the LQR control problem into an optimization one in a Hilbert space for an associated backward stochastic model and then obtain the optimal solution to the stochastic LQR problem by exploiting the dynamic programming approach. Our solution is given in terms of two generalized Riccati difference equations of the same dimension as that of the plant.
##### MSC:
 93E20 Optimal stochastic control (systems) 49N10 Linear-quadratic optimal control problems 93C55 Discrete-time control systems
Full Text:
##### References:
 [1] Rami, M. Ait; Chen, X.; Zhou, X. Y.: Discrete-time indefinite LQ control with state and control dependent noises, Journal of global optimization 43, 245-265 (2002) · Zbl 1035.49024 · doi:10.1023/A:1016578629272 [2] Basin, M. V., & Rodriguez Gonzalez, J. (2003). Optimal control for linear systems with time delay in control input based on the duality principle. In Proc. American control Conf. (pp. 2144-2148) [3] Carravetta, F.; Mavelli, G.: Suboptimal stochastic linear feedback control of linear systems with state-and control-dependent noise: the incomplete information case, Automatica 43, 751-757 (2007) · Zbl 1117.93337 · doi:10.1016/j.automatica.2006.09.010 [4] Costa, O. L. V.; Kubrusly, C. S.: State-feedback H$\infty$control for discrete-time infinite-dimensional stochastic bilinear systems, Journal of mathematical systems. Estimation and control 6, 1-32 (1996) · Zbl 0844.93036 [5] Ghaoui, L. Ei: State-feedback control of systems with multiplicative noise via linear matrix inequalities, Systems & control letters 24, 223-228 (1995) · Zbl 0877.93076 · doi:10.1016/0167-6911(94)00045-W [6] Gershon, E.; Shaked, U.; Yaesh, I.: H$\infty$control and filtering of discrete-time stochastic systems with multiplicative noise, Automatica 37, 409-417 (2001) · Zbl 0989.93030 · doi:10.1016/S0005-1098(00)00164-3 [7] Hassibi, B.; Sayed, A. H.; Kailath, T.: Indefinite quadratic estimation and control: A unified approach to H2 and H$\infty$theories, SIAM studies in applied mathematics series (1998) · Zbl 0997.93506 [8] Huang, Y., Zhang, W., & Zhang, H. (2006). Infinite horizon LQ optimal control for discrete-time stochastic systems. In Proc. of the sixth world congress on intelligent control and automation. Vol. 10 (pp. 252-256) [9] Kalman, R. E.: Contributions to the theory of optimal control, Boletin de la sociedad matematica mexicana 5, 102-119 (1960) · Zbl 0112.06303 [10] Kojima, A.; Ishijima, S.: H$\infty$performance of preview control systems, Automatica 39, 693-701 (2003) · Zbl 1029.93021 · doi:10.1016/S0005-1098(02)00286-8 [11] Meditch, J. S.: Stochastic optimal linear estimation and control, (1969) · Zbl 0225.93045 [12] Meditch, J. S.: On optimal control of linear systems in the presence of multiplicative noise, IEEE transactions on aerospace and electronic systems 12, 80-85 (1976) [13] Meinsma, G.; Mirkin, L.: H$\infty$control of systems with multiple i/o delays via decomposition to adobe problems, IEEE transactions on automatic control 50, 199-211 (2005) [14] Mohler, R.; Kolodziej, W.: An overview of stochastic bilinear control processes, IEEE transactions on systems, man and cybernetics 10, 913-919 (1980) · Zbl 0475.93054 · doi:10.1109/TSMC.1980.4308421 [15] Wang, F.; Balakrishnan, V.: Robust Kalman filters for linear time-varying systems with stochastic parameter uncertainties, IEEE transactions on signal processing 50, 803-813 (2002) [16] Wonham, W. M.: On a matrix Riccati equation of stochastic control, SIAM journal on control and optimization 6, 681-697 (1968) · Zbl 0182.20803 [17] Zhang, H.; Duan, G.; Xie, L.: Linear quadratic regulation for linear time-varying systems with multiple input delays, Automatica 42, 1465-1476 (2006) · Zbl 1128.49304 · doi:10.1016/j.automatica.2006.04.007 [18] Zhang, H.; Xie, L.; Duan, G.: H$\infty$control of discrete-time systems with multiple input delays, IEEE transactions on automatic control 52, 271-283 (2007)