### Abstract

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 in [2] and [3], which applies to problems where the `worst-case disturbance' may not be unique (but the worst-case state trajectory is). The generalized certainty equivalence, on the other hand, extends this to accommodate nonunique worst-case state trajectories, and leads to the construction of controllers 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. These results are then applied to the nonlinear minimax filtering problem, which is treated here as a special case of the general control problem.

Original language | English (US) |
---|---|

Pages (from-to) | 184-189 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 1 |

State | Published - Dec 1 1993 |

Event | Proceedings of the 32nd Conference on Decision and Control - San Antonio, TX, USA Duration: Dec 15 1993 → Dec 15 1993 |

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### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

^{∞}-control and filtering.

*Proceedings of the IEEE Conference on Decision and Control*,

*1*, 184-189.

**Structural properties of minimax policies for a class of differential games arising in nonlinear H ^{∞}-control and filtering.** / Didinsky, Garry; Basar, Tamer; Bernhard, Pierre.

Research output: Contribution to journal › Conference article

^{∞}-control and filtering',

*Proceedings of the IEEE Conference on Decision and Control*, vol. 1, pp. 184-189.

^{∞}-control and filtering. Proceedings of the IEEE Conference on Decision and Control. 1993 Dec 1;1:184-189.

}

TY - JOUR

T1 - Structural properties of minimax policies for a class of differential games arising in nonlinear H∞-control and filtering

AU - Didinsky, Garry

AU - Basar, Tamer

AU - Bernhard, Pierre

PY - 1993/12/1

Y1 - 1993/12/1

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 in [2] and [3], which applies to problems where the `worst-case disturbance' may not be unique (but the worst-case state trajectory is). The generalized certainty equivalence, on the other hand, extends this to accommodate nonunique worst-case state trajectories, and leads to the construction of controllers 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. These results are then applied to the nonlinear minimax filtering problem, which is treated here as a special case of the general control problem.

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 in [2] and [3], which applies to problems where the `worst-case disturbance' may not be unique (but the worst-case state trajectory is). The generalized certainty equivalence, on the other hand, extends this to accommodate nonunique worst-case state trajectories, and leads to the construction of controllers 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. These results are then applied to the nonlinear minimax filtering problem, which is treated here as a special case of the general control problem.

UR - http://www.scopus.com/inward/record.url?scp=0027750982&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027750982&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:0027750982

VL - 1

SP - 184

EP - 189

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

ER -