A Relationship Between Sensitivity and Stability of Multivariable Feedback Systems

U NDER the assumptions that a plant is completely deterministic and perfectly modeled, either open-loop compensation or closed-loop compensation can be used to achieve a desired overall design. However, the assumptions of determinism and perfect modeling are never satisfied in practice. When these assumptions do not hold, feedback control systems can possess properties such as stability, sensitivity reduction, disturbance attenuation and rejection, and linearization of nonlinear elements which cannot be achieved by open-loop compensation. For single-input single-output systems, the relationship of these properties to the return difference of the closed-loop system has been understood for some time [l], [2]. The intuition and insight which has resulted from understanding this relationship forms the basis for most classical design procedures. The extension of the intuition and insight, and hence the design procedures, to multivariable feedback systems has not been straightforward. Although it is clear that the return difference matrix and its inverse play an important role in multivariable design and analysis, it has not been clear which quantities associated with these matrices wi l provide the appropriate generalizations. Several generalizations have been proposed. One of the earliest was the extension of the concept of sensitivity reduction to multivariable systems by Cruz and Perluns [3]. The main result was a development of a linear relationship between closed-loop error and open-loop error via a return difference matrix and a sufficient condition for the closed-loop control system design to perform bet-