Restricted structure control loop performance assessment for state-space systems

A novel H₂ optimal control performance assessment and benchmarking problem is considered for discrete-time state-space multivariable systems, where the structure of the controller is assumed to be fixed a priori. The controller structure may be specified to be of reduced order, lead/lag, or PID forms. The theoretical problem considered is to represent the state-space model in a discrete polynomial matrix form and then to obtain the causal, stabilising controller of a prespecified form, that minimises an H₂ criterion. This then provides the performance measure against which other controllers can be judged. The underlying practical problem of importance is to obtain a simple method of performance assessment and benchmarking low order controllers. The main theoretical step is to derive a simpler cost-minimization problem whose solution can provide both the full order and restricted structure optimal benchmark cost values. This problem involves the introduction of spectral factor and diophantine equations and is solved via a Wiener type of cost-function expansion and simplification. The numerical solution of this problem is straightforward and involves approximating the simplified integral criterion by a fixed number of frequency points. The main benchmarking theorem applies to multivariable systems that may be unstable, non-minimum phase and non-square.