Discrete-Time Linear-Quadratic Optimal Controller Driven by a Reduced-Order Observer Steady State Performance Loss

This paper formulates and solves a problem encountered in engineering practice when a discrete-time linear-quadratic optimal feedback controller uses state estimates obtained via a discrete-time reduced-order observer. Due to the use of state estimates instead of the actual state variables, the optimal quadratic performance is degraded in a pretty complex manner. The paper shows how to find the exact expression for the optimal performance degradation for the steady state case in terms of a solution of a reduced-order discrete-time algebraic Lyapunov equation. The quantities that impact the performance criterion loss are identified. Simulation results show that the optimal performance degradation can be considerably reduced by using the proposed least square formula to set up the reduced-order observer initial condition.