Superiority of transfer function over state-variable methods in linear time-invariant feedback system design

Abstract

The objectives and achievements of state-variable methods in linear time-invariant feedback system synthesis are examined. It is argued that the philosophy and objectives associated with eigenvalue realization by state feedback, with or without observers, are highly naive and incomplete in the practical context of control systems. Furthermore, even the objectives undertaken have not really been attained by the state-variable techniques which have been developed. The extremely important factors of sensor noise and loop bandwidths are obscured by the state-variable formulation and have been ignored in the state-variable literature. The basic fundamental problem of sensitivity in the face of significant plant parameter uncertainty has hardly received any attention. Instead, the literature has concentrated primarily on differential sensitivity functions and even those results are so highly obscured in the state-variable formation as to lead to incorrect conclusions. In contrast, the important practical considerations and constraints have been clearly revealed and considered in the transfer function formulation. Differential sensitivity results are simple and transparent. For single input-output systems, there exists an exact design technique for achieving quantitative sensitivity specifications in the face of significant parameter uncertainty, which is optimum in an important practical sense. This problem is much more difficult and has not been completely solved for multivariable systems, but it has at least been realistically attacked by some transfer function methods. Finally, the concepts of controllability and observability so much emphasized in the state-variable literature are examined. It is argued that their importance in this problem class has been greatly exaggerated. On the one hand, transfer function methods can be used to check for their existence. On the other hand, nothing is lost when they are ignored, if the synthesis problem is treated as one with parameter uncertainty by transfer function methods.