The continuous closed form controllability gramian and its inverse

The continuous controllability Gramian is the solution of an input Lyapunov equation in the controller (companion) form or equivalently the infinite integral of an outer product of a vector containing the impulse response and its derivatives corresponding to a unity numerator transfer function. In this paper we make use of both these viewpoints in order to derive the simple zero plaid structure of this Gramian and present the interesting links that the entries of the Gramian have to the entries of the Routh table. Moreover, an expression for the inverse of the Gramian is derived as a simple function of the coefficients of the characteristic polynomial from the fact that it is the solution of a Riccati equation. We show how the controllability Gramian forms the core part of closed form expressions of Gramians of more general MIMO systems as well as solutions of general Sylvester equations. The controllability Gramian also appears in certain zero optimization problems, either in a PID like controller setting or in a model reduction setting. The inverse of the controllability Gramian is a key element in such zero optimization. While much of the results presented can be found in closely related forms in published papers, we believe that they deserve more attention as an effective tool in numerical computations of small to mid-size systems.