The Optimal Performance Loss of the Finite-Horizon Discrete-Time Linear-Quadratic Controller Driven by a Reduced-Order Observer

In this paper we formulate and solve a problem encountered in engineering practice when a discrete-time linear-quadratic optimal feedback controller uses the 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. In the paper, we derive the exact formula for the optimal performance degradation for the finite time horizon optimization problem in terms of solution of a reduced-order difference Lyapunov equation. An aircraft example demonstrates that the optimal performance loss can be significant when a reduced-order observer is used. The optimal performance degradation can be considerably reduced by using the least square method to set-up the reduced-order observer initial condition.