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

In this paper, we derive an expression for the loss of optimal performance (compared to the corresponding linear-quadratic optimal performance with the instantaneous full-state feedback) when the continuous-time finite-horizon linear-quadratic optimal controller uses the estimates of the state variables obtained via a reduced-order observer. It was shown that the loss of optimal performance value can be found by solving the differential Lyapunov equation whose dimensions are equal to dimensions of the reduced-order observer. A proton exchange membrane fuel cell example is included to demonstrate the loss of optimal performance as a function of the final time. It can be seen from the simulation results that the loss of optimal performance value can be very large. The loss of optimal performance value can be drastically reduced by using the proposed least-square formulas for the choice of the reduced-order observer initial conditions.