Stable hermitian solutions of discrete algebraic Riccati equations
The authors consider the discrete algebraic Riccati equation \[ X=A^* XA+Q-A^* XB(R+B^* XB)^{-1}B^* XA, \] where \(R\) and \(Q\) are (real or complex) Hermitian matrices of sizes \(m\times m\) and \(n\times n\) respectively, \(R\) is positive definite, \(A\) and \(B\) are (real or complex) matrices of sizes \(n\times n\) and \(n\times m\) respectively, and * denotes conjugate transpose. It is assumed that \(R+B^* XB\) is invertible. The unknown matrix \(X\) is Hermitian and \(n\times n\). Algebraic Riccati equations of this form arise in many applications including filter design, discrete time control theory, and systems theory. A Hermitian solution of the equation above is called stable if it persists under small perturbations of \(A\), \(B\), \(Q\) and \(R\). The authors provide a complete characterization of stable solutions under the conditions that the matrix \((A,B)\) is controllable, \(A\) is non-singular, and that an additional technical condition holds. The authors also discuss the stronger stability condition known as Lipschitz stability. The main tools in the authors' investigations are geometric descriptions of the solutions of discrete algebraic Riccati equations, as well as appropriately modified results on the stability of invariant Lagrangian subspaces.
scientific article; zbMATH DE number 58438
scientific article