Explicit Solution for a Class of Discrete‐Time Algebraic Riccati Equations

In the present paper we obtain an explicit closed‐form solution for the discrete‐time algebraic Riccati equation (DTARE) with vanishing state weight, whenever the unstable eigenvalues are distinct. We discuss links to current algorithmic solutions and observe that the AREs in such a class solve on one hand the infimal signal‐to‐noise ratio (SNR) problem, whilst on the other hand, in their dual form they solve a Kalman filter problem. We then extend the main result to an example case of the optimal linear quadratic regulator gain. We relax some of the assumptions behind the main result and conclude with possible future directions for the present work.     

区域极点约束下线性离散系统的Riccati鲁棒控制

讨论区域极点约束下,含结构参数扰动的不确定线性离散系统的鲁棒控制问题,即设计一鲁棒状态反馈控制器,使线性离散系统在可允许的参数扰动下,闭环矩阵极点始终位于一预先给定的圆形区域中,从而闭环系统具有期望的动态性能.上述控制目的可通过求解一含参数的代数离散Riccati方程达到.     

Robust Limits of Risk Sensitive Nonlinear Filters

Deterministic filter models are considered, and a criterion for a deterministic filter to be robust is introduced. Among the candidates for robust deterministic filters are so-called minimax estimators. In the second part of the paper, a risk sensitive stochastic approach to nonlinear filtering is considered, in which the traditional expected mean squared error criterion is replaced by an expected exponential-of-mean squared error. Minimax filters are obtained as totally risk averse limits of risk sensitive filters. Date received: July 22, 1999. Date revised: June 19, 2000.     

The Kleinman Iteration for Nonstabilizable Systems

 We consider the computation of Hermitian nonnegative definite solutions of algebraic Riccati equations. These solutions are the limit, P=lim_{i →∞} P _i, of a sequence of matrices obtained by solving a sequence of Lyapunov equations. The procedure parallels the well-known Kleinman technique but the stabilizability condition on the underlying linear time-invariant system is removed. The convergence of the constructed sequence {P _i }_i≥1 is guaranteed by the minimality of P _i in the set of Hermitian nonnegative definite solutions of the Lyapunov equation in the ith iteration step. Date received: October 21, 1999. Date revised: February 14, 2002. RID="*" ID="*"This work was supported by the Acciones Integradas programme of Deutscher Akademischer Austauschdienst (Germany) and Dirección General de Infraestructura y Relaciones Internacionales (Spain).     

On the solution of coupled Riccati equations used in mixed H2 and H∞ optimal control

This paper considers a set of three coupled Riccati equations. The solution of these equations constitutes necessary conditions for mixed H₂ and H_∞ control problems. The main contributions of the paper are related to the conditions under which the solution of these equations is unique as well as to characterize necessary and sufficient conditions for the existence of solutions.