Introduction

In a state–space model, the relationship between the input(s) and the output(s) of a system is described by means of an intermediate variable called the state. Recall that in a state-feedback-control system, the state information is required for computing the control input. In Chap. 8, we assume that the state of the system can be directly measured and used in the state-feedback-control law. This is certainly possible for a simple system (if enough sensors are available to measure all elements of the state vector), but usually not the case in practice. Fortunately, we can use a state estimator, otherwise known as an observer, to estimate the state from input and output measurements. The estimated state is then used in the state-feedback-control law as if it is the true state. Obviously it is desirable that the estimated state provided by an observer is as close as possible to the true state. Indeed, under ideal conditions, the estimated state can be the same as the true state (in theory at least).

The first part of this chapter addresses the problem of state estimation by use of an observer (Ref. [1–2]). The issue of integrating a state estimator into a state-feedback-control system and the stability analysis of the overall system will then be discussed. Under certain conditions, the determination of a state-feedback-control law and the design of an observer to be used can be treated as separate problems.