Introduction

Abstract

A common engineering practice is to assume that a system can be described by a set of linear differential equations for some operating range of interest as follows

$$\dot x = Ax + Bu$$

((1.1))

wherex(t)denotes the states of the system,A,Bdenote time invariant matrices, and u(t) denotes the control input. Based on the assumption that (1.1) accurately describes the system behavior, the control practitioner can then exploit various properties from linear control theory. As stated in [10], in the absence of the input signal (i.e., the unforced system), these properties include (1) a unique equilibrium point if the A matrix is nonsingular, (2) the equilibrium point is stable if the eigenvalues of A have negative real roots, and (3) the linear differential equations can be solved analytically, allowing the transient response to be explicitly determined When a control inputu(t)is present, linear time invariant systems exhibit properties including (1) superposition, (2) asymptotic stability of the unforced system ensures bounded-input bounded-output stability, and (3) a sinusoidal input leads to a sinusoidal output of the same frequency.