The united stable solution set of interval continuous-time algebraic Riccati equation and verified numerical computation of its outer estimation

Authors: TAYYEBE HAQIRI, MAHMOUD MOHSENI MOGHADAM, AZIM RIVAZ

Abstract: This paper introduces the interval continuous-time algebraic Riccati equation $\mathbf{A}^* X + X\mathbf{A} + \mathbf{Q} -X \mathbf{G} X=0$, where $\mathbf{A}, \mathbf{G}$, and $\mathbf{Q}$ are known $n \times n$ complex interval matrices, $\mathbf{G}$ and $\mathbf{Q}$ are Hermitian, and $X$ is an unknown matrix of the same size, and develops two approaches for enclosing the united stable solution set of this interval equation. We first discuss the united stable solution set and then derive a nonlinear programming method in order to find an enclosure for the united stable solution set. We also advance an efficient technique for enclosing the united stable solution set based on a variant of the Krawczyk method together with some modifications. These modifications enable us to reduce the computational complexity significantly. Various numerical experiments established upon a number of standard benchmark examples are also given to show the efficiency of this modified Krawczyk technique.

Keywords: Interval continuous-time algebraic Riccati equation, united stable solution set, Krawczyk's method, verified computation, interval analysis, preconditioning

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