zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Infinite horizon stochastic $H_2/H_\infty $control for discrete-time systems with state and disturbance dependent noise. (English) Zbl 1153.93030
Summary: This paper studies the infinite horizon mixed $H_{2}/H_\infty $ control for discrete-time stochastic systems with state and disturbance dependent noise. We shall first establish a version of Stochastic Bounded Real Lemma (SBRL) which by itself has theoretical importance. Based on the SBRL, it is shown that under the condition of exact observability, the existence of a static state feedback $H_{2}/H_\infty $ controller is equivalent to the solvability of four coupled matrix-valued equations. A suboptimal $H_{2}/H_\infty $ controller design is given based on a convex optimization approach and an iterative algorithm is proposed to solve the four coupled matrix-valued equations.

93E03General theory of stochastic systems
93C55Discrete-time control systems
90C25Convex programming
Full Text: DOI
[1] Anderson, B. D. O.; Moore, J. B.: Optimal control-linear quadratic methods, (1989)
[2] Rami, M. Ait; Chen, X.; Moore, J. B.; Zhou, X. Y.: Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls, IEEE transactions on automatic control 46, 428-440 (2001) · Zbl 0992.93097 · doi:10.1109/9.911419
[3] Basar, T.; Bernhar, P.: H\infty-optimal control and related minimax design problems: A dynamic game approach, (1995)
[4] Bernstein, D. S.; Haddad, W. M.: LQG control with an H$\infty $performance bound: A Riccati equation approach, IEEE transactions on automatic control 34, 293-305 (1989) · Zbl 0674.93069 · doi:10.1109/9.16419
[5] Chen, B. S.; Zhang, W.: Stochastic H2/H$\infty $control with state-dependent noise, IEEE transactions on automatic control 49, 45-57 (2004)
[6] Costa, O. L. V.; Marques, R. P.: Mixed H2/H$\infty $control of discrete-time Markovian jump linear systems, IEEE transactions on automatic control 43, 95-100 (1998) · Zbl 0907.93062 · doi:10.1109/9.654895
[7] Chen, X.; Zhou, K.: Multiobjective H2/H$\infty $control design, SIAM journal on control and optimization 40, 628-660 (2001) · Zbl 0997.93033 · doi:10.1137/S0363012998346372
[8] De Souza, C. E.; Xie, L.: On the discrete-time bounded real lemma with application in the characterization of static state feedback H$\infty $controllers, Systems and control letters 18, 61-71 (1992) · Zbl 0743.93038 · doi:10.1016/0167-6911(92)90108-5
[9] De Oliviera, M. C., Geromel, J. C., & Bernussou, J. (1999). An LMI optimization approach to multiobjective controller design for discrete-time systems. In Proc. 38th IEEE conf. decision and control (pp. 3611-3616)
[10] Du, C.; Xie, L.; Teoh, J. N.; Guo, G.: An improved mixed H2/H$\infty $control design for hard disk drives, IEEE transactions on control systems technology 13, 832-839 (2005)
[11] El Bouhtouri, A.; Hinrichsen, D.; Pritchard, A. J.: H\infty-type control for discrete-time stochastic systems, International journal of robust nonlinear control 9, 923-948 (1999) · Zbl 0934.93022 · doi:10.1002/(SICI)1099-1239(199911)9:13<923::AID-RNC444>3.0.CO;2-2
[12] Gershon, E.; Shaked, U.; Yaesh, U.: Control and estimation of state-multiplicative linear systems, (2005) · Zbl 1116.93003
[13] Huang, Y., Zhang, W., &amp; Zhang, H. (2006). Infinite horizon LQ optimal control for discrete-time stochastic systems. In Proc. of the 6th world congress on control and automation (pp. 252-256)
[14] Khargonekar, P. P.; Rotea, M. A.: Mixed H2/H$\infty $control: A convex optimization approach, IEEE transactions on automatic control 36, 824-837 (1991) · Zbl 0748.93031 · doi:10.1109/9.85062
[15] Limebeer, D. J. N.; Anderson, B. D. O.; Hendel, B.: A Nash game approach to mixed H2/H$\infty $control, IEEE transactions on automatic control 39, 69-82 (1994) · Zbl 0796.93027 · doi:10.1109/9.273340
[16] Muradore, R.; Picci, G.: Mixed H2/H$\infty $control: the discrete-time case, Systems and control letters 54, 1-13 (2005) · Zbl 1129.93435
[17] Qian, L., &amp; Gajic, Z. (2002). Variance minimization stochastic power control in CDMA systems. In IEEE international conference on communications, 3 (pp. 1763-1767)
[18] Sweriduk, G. D.; Calise, A. J.: Differential game approach to the mixed H2-H$\infty $problem, Journal of guidance control, and dynamics 20, 1229-1234 (1997) · Zbl 0904.93007 · doi:10.2514/2.4181
[19] Sznaier, M., &amp; Rotsein, H. (1994). An exact solution to general 4-blocks discrete-time mixed H2/H\infty problems via convex optimization. In Proc. American control (pp. 2251-2256)
[20] Zhang, W.; Chen, B. S.: On stabilizability and exact observability of stochastic systems with their applications, Automatica 40, 87-94 (2004) · Zbl 1043.93009 · doi:10.1016/j.automatica.2003.07.002
[21] Zhang, W.; Huang, Y.; Zhang, H.: Stochastic H2/H$\infty $control for discrete-time systems with state and disturbance dependent noise, Automatica 43, 513-521 (2007) · Zbl 1137.93057 · doi:10.1016/j.automatica.2006.09.015
[22] Zhang, W.; Zhang, H.; Chen, B. S.: Stochastic H2/H$\infty $control with (x,u,v)-dependent noise: finite horizon case, Automatica 42, 1891-1898 (2006) · Zbl 1114.93093 · doi:10.1016/j.automatica.2006.05.025