8.9.  EXTENSION OF COMBINED COMPENSATOR DESIGN INCLUDING A CONTROLLER AND AN ESTIMATOR FOR SYSTEMS CONTAINING A REFERENCE INPUT

How can we extend the concepts developed in the previous section for regulator design, shown in Figure 8.19 (where the reference input r(t) = 0), to the more general problem where the reference input exists? Several methods exist which can be used to design such a system [8,9]. This section will consider the configuration illustrated in Figure 8.22. The design goal of the approach to be presented is to have the state-estimation error (t) be independent of the reference input r(t) (e.g., should be uncontrollable from r(t)). This is a very important consideration, as we do not want the state-estimation error to be dependent on the type and level of the input.

Let us reconsider the controller equation (8.143),

Figure 8.22  Addition of the reference input to the system shown in Figure 8.19 containing a combined controller and estimator.

and the estimator equation (8.152)

The reference input r(t) will be introduced to these equations by adding a term Ar(t) to the controller equation (8.184), and a term Nr(t) to the estimator equation (8.185) (where N is a vector). Therefore, Eqs. (8.184) and (8.185) become

What kind of system do Eqs. (8.186) and (8.187) infer? Does it result in the configuration of Figure 8.22? To answer this question, let us first substitute

into Eq. (8.187) and, thereby, eliminate the output c(t) from Eq. (8.187):

To find the estimation error (which we want to be independent of r(t)), let us difference [from Eq. (8.189)] and the state equation

We will first substitute Eq. (8.186) into Eq. (8.190):

Substracting Eq. (8.189) from Eq. (8.191), we obtain the following for the derivative of the estimation error:

(t) = (t)− (t) = Px(t) + b(−K(t) + Ar(t))
               − (P − Kb − ML)(t) − MLx(t) − Nr(t).

Simplifying, we obtain the following equation:

In order to eliminate r(t) from Eq. (8.192), it is necessary that

Therefore, the design criterion of the control-system engineer is to invoke Eq. (8.193). Substituting Eqs. (8.186) and (8.193) into Eq. (8.187), we obtain the following:

Simplifying, we find that

or

Notice that Eq. (8.196) is the same estimator equation defined in Eq. (8.120). It is important to emphasize that this occurs only when bA = N as defined in Eq. (8.193). Therefore we conclude that the introduction of the reference input by adding the term Ar(t) in the controller equation (8.186) and a term Nr(t) to Eq. (8.187) results in the configuration shown in Figure 8.22.

Complete design examples for the design of the controller, estimator, and compensator, with their associated root-locus and Bode-diagram analyses of the resulting design are found in Chapter 7 of the accompanying volume. In Section 7.5, the state-variable design for the controller and full-order estimator for a space vehicle is presented. In Problem 7.6, the state-variable design for the controller and full-order estimator of a chemical process control system is analyzed.