Discrete-Time Optimal Quadratic Performance Loss of a Linear Reduced Order Observer Based Controller Designed via a Sylvester Equation

This paper considers the problem for an optimal feedback controller based on a reduced-order discrete-time observer designed via a Sylvester algebraic equation. Since linear observers only estimate state variables, using them for feedback control instead of the actual system state variables, the optimal performance gets degraded. In this paper, an analytical expression was obtained for the discrete-time linear quadratic (LQ) optimal performance loss for discrete-time steady state optimization. In general, the optimal performance loss can be significant, and in some cases the optimal performance loss is enormous. The paper derives a least square formula for the discrete-time observer initial conditions that can significantly reduce the optimal performance loss.