A Separation Principle for Affine Systems

Abstract

In control problems for dynamical systems, the full system state vector is often unknown, and only some functions of state variables, called system outputs, can be measured. One way to estimate the state vector on the basis of the output is to construct an observer, that is, a special dynamical system whose state approaches (either asymptotically or exponentially) the state of the original system in the course of time. Suppose that we have found a solution of the stabilization problem for the dynamical system in the form of a state feedback and the state estimation via an observer is available. Consider the control obtained from the feedback control by replacing the system state with the estimate provided by the observer. We arrive at the problem as to whether the resulting state estimate feedback control solves the stabilization problem. For linear stationary systems, the answer is “yes”; this is the well-known separation principle [1, p. 157]. More precisely, once we construct an exponential observer and a linear state feedback globally stabilizing a given equilibrium of a linear stationary system, the stability of the equilibrium is preserved for the corresponding state estimate feedback. For general nonlinear systems, the answer is negative, and there are examples of nonlinear systems to which the separation principle cannot be applied [2]. In the following, we consider the global stabilization problem for a given equilibrium of an affine stationary system with output