Sensitivity in Multivariable Control Systems

Definitions of system sensitivity for linear single variable systems have been extended, in the past, to linear multivariable systems in the form of a sensitivity matrix. The role of the sensitivity matrix in multivariable feedback control systems is studied further in this paper. The sensitivity matrix serves the dual function of governing the effects of plant parameter variation on the system transfer matrix and governing the effects of disturbances on the system output. The design implications of this are considered and it is shown that certain controllability/observability conditions are necessary if the system design is to be effective. By appropriate design of the loop gain matrix, L(s), a desired insensitivity to system error sources may be achieved. Unless the system has certain controllability/observability properties insensitivity cannot be achieved. It is shown that L(s) must have the property of functional reproducibility which is a relatively strong controllability/observability requirement.