2. Introduction to the Principles of Feedback
One particularly simple, yet insightful way of thinking about control problems is via inversion. To describe this idea we argue as follows:
In spite of the apparent naivety of this argument, its embellished ramifications play a profound role in control-system design. In particular, most of the real-world difficulties in control relate to the search for a strategy that captures the intent of the above inversion idea, while respecting a myriad of other considerations, such as insensitivity to model errors, disturbances, and measurement noise.
To be more specific, let us assume that the required behavior is specified by a scalar target signal or reference r(t), that has an additive disturbance d(t). Say we also have available a single manipulated variable, u(t). We denote by y(t) a function of time: .
In describing the prototype solution to the control problem below, we will make a rather general development that, in principle, can apply to general nonlinear dynamical systems. In particular, we will use a function, , to denote an operator mapping one function space to another. So as to allow this general interpretation, we introduce the following notation:
The symbol y (without brackets) will denote an element of a function space: . An operator, , will then represent a mapping from a function space, say , onto .
What we suggest is that the reader, on a first reading, simply interpret f as a static linear gain linking one real number, the input u to another real number, the output y. On a subsequent reading, the more general interpretation, using nonlinear dynamic operators, can be used.
Let us also assume (for the sake of argument) that the output is related to the input by a known functional relationship of the form
The control problem then requires us to find a way to generate y=r. In the spirit of inversion, a direct, although somewhat naive, approach to obtain a solution would thus be to set
This idea is illustrated in Figure 2.6.
This is a conceptual solution to the problem. However, a little thought indicates that the answer given in (2.5.3) presupposes certain stringent requirements for its success. For example, inspection of equations (2.5.1) and (2.5.3) suggests the following requirements:
Of course, these are very demanding requirements. Thus, a significant part of Automatic Control theory deals with the issue of how to change the control architecture so that inversion is achieved but in a more robust fashion and so that the stringent requirements set out above can be relaxed.
To illustrate the meaning of these requirements in practice, we briefly review a number of situations.
Example 2.1 (Heat exchanger) Consider the problem of a heat exchanger in which water is to be heated by steam having a fixed temperature. The plant output is the water temperature at the exchanger output and the manipulated variable is the air pressure (3 to 15 [psig]) driving a pneumatic valve that regulates the amount of steam feeding the exchanger.
In the solution of the associated control problem, the following issues should be considered:
Example 2.2 (Flotation in mineral processing) In copper processing, one crucial stage is the flotation process. In this process, the mineral pulp (water and ground mineral) is continuously fed to a set of agitated containers where chemicals are added to separate (by flotation) the particles with high copper concentration. From a control point of view, the goal is to determine the appropriate addition of chemicals and the level of agitation to achieve maximal separation.
Characteristics of this problem are as follows:
One could imagine various other practical cases where one or more of the requirements listed above cannot be satisfied. Thus, the only sensible way to proceed is to accept that there will inevitably be intrinsic limitations and to pursue the solution within those limitations. With this in mind, we will impose constraints that will allow us to solve the problem subject to the limitations that the physical set-up imposes. The most commonly used constraints are as follows:
In summary, we can conclude the following:
1 We introduce this term here loosely. A more rigorous treatment will be deferred to Chapter 19.