The optimal projection approach to designing optimal finite-dimensional controllers for distributed-parameter systems

One of the major difficulties in designing implementable active controllers for distributed parameter systems such as flexible space structures is that such systems are inherently infinite dimensional while controller dimension is severely constrained by on-line computing capability. Suboptimal approaches to this problem usually either seek a nonrealizable distributed parameter control law or a low-order dynamic controller based upon a discretized model. This paper presents a more direct approach based upon explicit optimality conditions for finite-dimensional steady-state fixed-order dynamic compensation of infinite-dimensional systems. In contrast to the pair of operator Riccati equations for the "full-order" LQG case, the optimal fixed-order dynamic compensator is characterized by four operator equations (two modified Riccati equations and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains. The coupling represents a graphic portrayal of the demise of the classical separation principle for the finite-dimensional reduced-order controller case. The results obtained apply to a semigroup formulation in Hilbert space and thus are applicable to control problems involving a broad range of specific partial and hereditary differential equations.