In this paper we propose a structure-preserving doubling algorithm (SDA) for computing the minimal nonnegative solutions to the nonsymmetric algebraic Riccati equation (NARE) based on the techniques developed in the symmetric cases. This method allows the simultaneous approximation of the minimal nonnegative solutions of the NARE and its dual equation, only requires the solutions of two linear systems, and does not need to choose any initial matrix, thus it overcomes all the defaults of the Newton iteration method and the fixed-point iteration methods. Under suitable conditions, we establish the convergence theory by using only the knowledge from elementary matrix theory. The theory shows that the SDA iteration matrix sequences are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation, respectively. Numerical experiments show that the SDA algorithm is feasible and effective, and can outperform the Newton iteration method and the fixed-point iteration methods.
This note deals with the problems of robust stability and stabilization for uncertain discrete-time singular systems. The parameter uncertainties are assumed to be time-invariant and norm-bounded appearing in both the state and input matrices. A new necessary and sufficient condition for a discrete-time singular system to be regular, causal and stable is proposed in terms of a strict linear matrix inequality (LMI). Based on this, the concepts of generalized quadratic stability and generalized quadratic stabilization for uncertain discrete-time singular systems are introduced. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are obtained in terms of a strict LMI and a set of matrix inequalities, respectively. With these conditions, the problems of robust stability and robust stabilization are solved. An explicit expression of a desired state feedback controller is also given, which involves no matrix decomposition. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.
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