On the solution of matrix inequalities in the Kalman-Yakubovich theorem via hidden parameters of positive rational functions (Q1816924)

In the stability theory of control systems the frequency theorem of Kalman and Yakubovich states a necessary and sufficient condition for the existence of a nonnegative matrix \(H\) solving the matrix inequality \[ HA+A^*H\leq 0,\qquad C=B^*H \tag{1} \] which guarantees the absolute stability of the linear dynamical system defined by \((A,B,C)\) \[ {dx\over dt}=Ax(t)+Bu(t), \quad y(t)=Cx(t); \qquad t\geq 0. \] In this paper it is shown how a solution \(H\) of (1) can be found constructively by a new algebraic method, which is called the method of hidden parameters. In that way, a new constructive proof of the Kalman-Yakubovich frequency theorem is obtained.