The Class of Controllability Gramians Assignable by State Feedback

Abstract

The controllability Gramian in linear time-invariant systems is closely related to control performances such as disturbance decoupling and perfect regulation. Skelton and his colleague, and Ohara and Kitamori have formulated the controllability Gramian assignment problem and have individually derived necessary and sufficient conditions for a given positive definite matrix to be the controllability Gramian of the state feedback system.In this paper, all positive difinite matrices which can be assigned as the controllability Gramian of the closed-loop system by state feedback are parametrized. First, such a parametrization for the c-controllability Gramian which is from control input to state variable, is provided; especially for the systems of controllable standard form, the class of assignable Gramians is indicated in the explicit matrix representation. It is also shown that its magnitude in the sense of matrix norm can be unconditionally realized. Second, the assignable d-controllability Gramian being from disturbance is characterized as the sum of a c-controllability Gramian and the negative semidifinite matrix satisfying a Lyapunov equation. Because of the negative semi-difinite matrix, the d-controllability Gramian cannot generally be made arbitrarily small.Finally, a numerical example is illustrated to demonstrate the availability of the controllability Gramian assignment technique in control system design taking account of the disturbance decoupling property.