Synthesis of static output feedback SPR systems via LQR weighting matrix design

In this paper we propose an approach using linear quadratic regulator (LQR) weighting matrices to synthesize strictly positive real (SPR) systems by static output feedback. The systems being considered are linear time-invariant (LTI). We first recall full state feedback LQR design. The two weighting matrices for state and control input respectively in the performance index are then used as two free parameters to design the SPR controller. By connecting strictly positive realness with full state feedback LQR through the algebraic Riccati equation associated with the latter and imposing well-posed condition in terms of positive definiteness on the weighting matrices, we show that the proposed formula for weighting matrices in this paper can render the resulting closed loop system SPR. The stabilizing static output feedback gain which is designed to make the closed loop system SPR becomes readily available once the two LQR weighting matrices are determined. Moreover, from the derived explicit form of control gain, we can achieve SPR synthesis even when system matrices are partially known. We provide in the end a numerical example to validate the approach.