Fixed-structure optimality conditions for nonminimal dynamic compensators

Researchers have shown that when it is assumed that a fixed-order optimal compensator is minimal, the necessary conditions can be characterized in terms of coupled Riccati and Lyapunov equations, usually termed optimal projection equations. The authors relax the minimality assumption on the compensator and derive necessary conditions for fixed-structure H₂-optimal control that reduce to the standard optimal projections when the compensator is assumed to be minimal. The results are then specialized to the full-order case. It is shown that under standard stabilizability and detectability assumptions, an extremal compensator always exists that is characterized by a pair of decoupled equations. It is also shown that this same compensator is characterized by a set of coupled equations that is essentially a set of optimal projection equations. The results are expected to lead to the first general fixed-structure proof of the global optimality of the standard linear-quadratic-Gaussian (LQG) compensator