The Controllability Gramian, the Hadamard Product, and the Optimal Actuator/Leader and Sensor Selection Problem

Abstract

We show that for LTI systems, when the system matrix $A$ is diagonalizable, the controllability Gramian can be expressed as a Hadamard product of two positive semi-definite matrices. Using the Hadamard decomposition, we show how to tackle the optimal actuator/leader selection problem for single input systems using the determinant of the controllability Gramian and the trace of the inverse of the controllability Gramian as a controllability metric. For multi input systems, we use the trace of the controllability Gramian as a controllability metric. We show that one can reduce the amount of computations by bypassing the explicit computation of the Gramian in finite horizon problems for continuous/discrete time systems when the determinant of the Gramian is used as a controllability metric. This is done by using the algebraic properties of the Hadamard product. We show that the other two metrics do not share this property. The corresponding optimal sensor selection problems can be handled in a similar manner.