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## 2. Introduction to the Principles of Feedback

### 2.7 From Open- to Closed-Loop Architectures

A particular scheme has been suggested in Figure 2.7 for realizing an approximate inverse of a plant model. Although the controller in this scheme is implemented as a feedback system, the control is actually applied to the plant in open loop. In particular, we see that the control signal u(t) is independent of what is actually happening in the plant. This is a serious drawback, because the methodology will not lead to a satisfactory solution to the control problem unless

• the model on which the design of the controller has been based is a very good representation of the plant,
• the model and its inverse are stable, and
• disturbances and initial conditions are negligible.

We are thus motivated to find an alternative solution to the problem, one that retains the key features but does not suffer from the above drawback. This is indeed possible by changing the scheme slightly so that feedback is placed around the plant itself rather than around the model.

To develop this idea, we begin with the basic feedback structure as illustrated in Figure 2.9. We proceed as follows. If we assume, for the moment, that the model in Figure 2.9 is perfect, then we can rearrange the diagram to yield the alternative scheme shown in Figure 2.10 This scheme, which has been derived from an open-loop architecture, is the basis of feedback control. The key feature of this scheme is that the controller output depends not only on the a-priori data provided by the model but also on what is actually happening at the plant output at every instant. It has other interesting features, which are discussed in detail below. However, at this point, it will be worthwhile to carry out an initial discussion of the similarities and differences between open- and closed-loop architectures of the types shown in Figure 2.9 and Figure 2.10.

• The first thing to note is that, provided that the model represents the plant exactly, and that all signals are bounded (i.e., the loop is stable), then the schemes are equivalent regarding the relation between r(t) and y(t). The key differences are due to disturbances and different initial conditions.

• In the open-loop control scheme the controller incorporates feedback internally: a signal at point A is fed back. In closed-loop control, the signal at A' is fed back. The fundamental difference is that, in the first case, everything happens inside the controller, either in a computer or in some external hardware connection. In the second case, the signal fed back is a process variable: measuring devices are used to determine what is actually happening. The heuristic advantages of the latter alternative are undoubtedly clear to the reader. We will develop the formal background to these advantages as we proceed.