TY - JOUR

T1 - H∞ control for discrete-time nonlinear stochastic systems

AU - Berman, Nadav

AU - Shaked, Uri

N1 - Funding Information:
Manuscript received April 15, 2004; revised April 21, 2005 and December 31. 2005. Recommended by Associate Editor A. Garulli. This work was supported by C&M Maus Chair at Tel Aviv University, Israel. N. Berman is with the Department of Mechanical Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel (e-mail: [email protected]). U. Shaked is with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.876808

PY - 2006/6

Y1 - 2006/6

N2 - In this note, we develop an H∞-type theory for a large class of discrete-time nonlinear stochastic systems. In particular, we establish a bounded real lemma (BRL) for this case. We introduce the notion of stochastic dissipative systems, analogously to the familiar notion of dissipation associated with deterministic systems, and utilize it in the derivation of the BRL. In particular, this BRL establishes a necessary and sufficient condition, in terms of a certain Hamilton Jacobi inequality (HJI), for a discrete-time nonlinear stochastic system to have l2 - gain ≤ γ. The time-invariant case is also considered as a special case. In this case, the BRL guarantees necessary and sufficient conditions for the system to have l2 - gain ≤ γ, by means of a solution to a certain algebraic HJI. An application of this theory to a special class of systems which is a characteristic of numerous physical systems, yields a more tractable HJI which serves as a sufficient condition for the underlying system to possess l2 - gain ≤ γ. Stability in both the mean square sense and in probability, is also discussed. Systems that possess a special structure (norm-bounded) of uncertainties in their model are considered. Application of the BRL to this class of systems yields a linear state-feedback stabilizing controller which achieves l2 - gain ≤ γ, by means of certain linear matrix inequalities (LMIs).

AB - In this note, we develop an H∞-type theory for a large class of discrete-time nonlinear stochastic systems. In particular, we establish a bounded real lemma (BRL) for this case. We introduce the notion of stochastic dissipative systems, analogously to the familiar notion of dissipation associated with deterministic systems, and utilize it in the derivation of the BRL. In particular, this BRL establishes a necessary and sufficient condition, in terms of a certain Hamilton Jacobi inequality (HJI), for a discrete-time nonlinear stochastic system to have l2 - gain ≤ γ. The time-invariant case is also considered as a special case. In this case, the BRL guarantees necessary and sufficient conditions for the system to have l2 - gain ≤ γ, by means of a solution to a certain algebraic HJI. An application of this theory to a special class of systems which is a characteristic of numerous physical systems, yields a more tractable HJI which serves as a sufficient condition for the underlying system to possess l2 - gain ≤ γ. Stability in both the mean square sense and in probability, is also discussed. Systems that possess a special structure (norm-bounded) of uncertainties in their model are considered. Application of the BRL to this class of systems yields a linear state-feedback stabilizing controller which achieves l2 - gain ≤ γ, by means of certain linear matrix inequalities (LMIs).

KW - Discrete-time H control

KW - Dissipative systems

KW - Nonlinear stochastic systems

KW - Stochastic BRL

UR - http://www.scopus.com/inward/record.url?scp=33745714910&partnerID=8YFLogxK

U2 - 10.1109/TAC.2006.876808

DO - 10.1109/TAC.2006.876808

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AN - SCOPUS:33745714910

SN - 0018-9286

VL - 51

SP - 1041

EP - 1046

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 6

ER -