On computing the stabilizing solution of the discrete-time Riccati equation
The discrete-time Riccati equation (DTARE) is \[ A^ TXA-X-(L+A^ TXB)(R+B^ TXB)^{-1}(B^ TXA+L^ T)+Q=0, \] where \(A\in\mathbb{R}^{n\times n}\), \(B\in\mathbb{R}^{n\times m}\), \(R=R^ T\in\mathbb{R}^{m\times m}\), \(L\in\mathbb{R}^{n\times m}\), and \(Q=Q^ T\in\mathbb{R}^{n\times n}\). The matrix \(X=X^ T\in\mathbb{R}^{n\times n}\) is said to be a stabilizing solution of the DTARE if \(R+B^ TXB\) is invertible. A necessary and sufficient condition for the existence of a stabilizing solution is given which does not include invertibility or positivity assumptions. The result involves a special associated matrix pencil. An algorithm and a numerical example are given.
scientific article; zbMATH DE number 69531
scientific article