Minimal and non-minimal optimal fixed-order compensators for time-varying discrete-time systems

Using the minimality property of finite-horizon time-varying compensators, established in this paper, and the Moore–Penrose pseudo-inverse instead of the standard inverse, strengthened discrete-time optimal projection equations (SDOPE) and associated boundary conditions are derived for finite-horizon fixed-order LQG compensation. They constitute a two-point boundary value problem explicit in the LQG problem parameters which is equivalent to first-order necessary optimality conditions and which is suitable for numerical solution. The minimality property implies that minimal compensators have time-varying dimensions and that the finite-horizon optimal full-order compensator is not minimal. The use of the Moore–Penrose pseudo-inverse is further exploited to reveal that the optimal projection approach can be generalised, but only to partially include non-minimal compensators. Furthermore, the structure of the space of optimal compensators with arbitrary dimensions is revealed to a large extent. Max–min compensator dimensions are introduced and their significance in solving numerically the two-point boundary value problem is explained. The numerical solution is presented in a recently published companion paper, which relies on the results of this paper.