TY - JOUR
T1 - Structural properties of minimax controllers for a class of differential games arising in nonlinear H∞ control
AU - Didinsky, Garry
AU - Başar, Tamer
AU - Bernhard, Pierre
N1 - Funding Information:
During the last few years, several authors have obtained results on various nonlinear extensions of the linear H ~ theory \[2, 6, 7\]. These results employed established game-theoretic methods used for linear systems, where the H °~ optimal control problem is treated as a minimax game with a soft-constrained kernel, and the controller and disturbance acting as minimizing and maximizing players, respectively. It was shown in \[3\] (and later in \[2\])t hat if there exists a full-state information saddle-point controller, and if a set of truncated optimization problems admit unique solutions, then the measurement feedback game admits a saddle-point controller, which satisfies (and can be computed through) a certainty-equivalence principle. In \[6, 7\], a somewhat different direction was followed. The authors considered games where the system dynamics are affine in control and disturbance, and the costs are quadratic, which implies existence of a saddle point whenever the upper value is bounded. Furthermore, they take the controller to be a full-state information controller, with the state variable replaced by a state estimator, A by-product of their Correspondence to: Prof. Tamer Ba~ar, Coordinated Science Laboratory, University of lllinois, 1308 West Main Street, Urbana, IL 61801, USA; Fax: 217-244-1653; E-mail: [email protected]. * This work was supported in part by the National Science Foundation under Grant ECS91-13153, and in part by the Joint Services Electronics Program under Contract N00014-84-C-0149.
PY - 1993/12
Y1 - 1993/12
N2 - This paper introduces, in precise mathematical terms, two properties (named, certainty equivalence and generalized certainty equivalence) that nonlinear minimax controller problems might possess. The certainty equivalence is a generalization of the one introduced earlier by Başar and Bernhard (1991) and Bernhard (1990), which applies to problems where the 'worst-case disturbance' may not be unique (but the worst-case state trajectory is). The generalized certainty equivalance, on the other hand, extends this to accomodate nonunique worst-case state trajectories, and leads to the constrollers that guarantee a bounded upper value for the underlying game. The paper also shows that for a large class of games (and under certain conditions), certainty-equivalent (as well as generalized certainty-equivalent) controllers admit (infinite-dimensional) estimator (Kalman-filter) structures, where the estimator gain depends on the state of the estimator.
AB - This paper introduces, in precise mathematical terms, two properties (named, certainty equivalence and generalized certainty equivalence) that nonlinear minimax controller problems might possess. The certainty equivalence is a generalization of the one introduced earlier by Başar and Bernhard (1991) and Bernhard (1990), which applies to problems where the 'worst-case disturbance' may not be unique (but the worst-case state trajectory is). The generalized certainty equivalance, on the other hand, extends this to accomodate nonunique worst-case state trajectories, and leads to the constrollers that guarantee a bounded upper value for the underlying game. The paper also shows that for a large class of games (and under certain conditions), certainty-equivalent (as well as generalized certainty-equivalent) controllers admit (infinite-dimensional) estimator (Kalman-filter) structures, where the estimator gain depends on the state of the estimator.
KW - H control
KW - certainty equivalence
KW - differential games
KW - minimax controllers
KW - nonlinear systems
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U2 - 10.1016/0167-6911(93)90048-B
DO - 10.1016/0167-6911(93)90048-B
M3 - Article
AN - SCOPUS:0027882609
SN - 0167-6911
VL - 21
SP - 433
EP - 441
JO - Systems and Control Letters
JF - Systems and Control Letters
IS - 6
ER -