INTRODUCTION TO THE PRINCIPLES OF FEEDBACK
2.5 Prototype Solution to the Control Problem via Inversion
One particularly simple, yet insightful way of thinking about control problems is via inversion. To describe this idea we argue as follows:
• say that we know what effect an action at the input of a system produces at the output, and
• say that we have a desired behavior for the system output, then one simply needs to invert the relationship between input and output to determine what input action is necessary to achieve the desired output behavior.
In spite of the apparent naivity 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, whilst respecting a myriad of other considerations such as insensitivity to model errors, disturbances, measurement noise, etc.
To be more specific, let us assume that the required behavior is specified by a scalar target signal, or reference r(t), for a particular process variable, y(t) which has an additive disturbance d(t). Say we also have available a single manipulated variable,u(t). We denote byy a function of time i.e. y={y(t) :t∈R}.
In describing the prototype solution to the control problem below, we will make a rather general development which, in principle, can apply to general nonlinear dynamical systems. In particular, we will use a function,f◦ , 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, i.e. y ={y(t) : R→R}. An operatorf◦ will then represent a mapping from a function space, sayχ, ontoχ.
What we suggest is that the reader, on a first reading, simply interpretf as a static linear gain linking one real number, the inputu, to another real number, the outputy. 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:
y=fu +d (2.5.1) where f is a transformation or mapping (possibly dynamic) which describes the input-output relations in the plant.1 We call a relationship of the type given in (2.5.1), amodel.
The control problem then requires us to find a way to generateuin such a way thaty=r. In the spirit of inversion, a direct, although somewhat naive, approach to obtain a solution would thus be to set
y=r=fu +d (2.5.2)
from which we could derive acontrol law, by solving foru. This leads to
u=f−1r−d (2.5.3)
This idea is illustrated in Figure 2.6
r +
−
f−1◦ u
f◦ +
+
y d
Conceptual controller Plant
Figure 2.6. Conceptual controller
This is a conceptual solution to the problem. However a little thought indi- cates 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) suggest the following requirements:
R1 The transformationf clearly needs to describe the plant exactly.
R2 The transformationf should be well formulated in the sense that a bounded output is produced when uis bounded–we then say that the transformation isstable.
1We introduce this term here loosely. A more rigorous treatment will be deferred to Chapter 19.
Section 2.5. Prototype Solution to the Control Problem via Inversion 31 R3 The inverse f−1 should also be well formulated in the sense used inR2.
R4 The disturbance needs to be measurable, so thatuis computable.
R5 The resulting actionushould be realizable and not violate any constraint.
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 to15 [psig]) driving a pneumatic valve which regulates the amount of steam feeding the exchanger.
In the solution of the associated control problem,the following issues should be considered:
• Pure time delays might be a significant factor,since this plant involves mass and energy transportation. However a little thought indicates that a pure time delay does not have a realizable inverse (otherwise we could predict the future) and hence R3 will not be met.
• It can easily happen that,for a given reference input,the control law (2.5.3) leads to a manipulated variable outside the allowable input range (3to15[psig]
in this example). This will lead to saturationin the plant input. Condition R5 will then not be met.
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Example 2.2 (Flotation in mineral processing). In copper processing one cru- cial 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 con- trol 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:
• The process is complex (physically distributed,time varying,highly nonlinear, multivariable and so on) and hence it is difficult to obtain an accurate model for it. Thus R1 is hard to satisfy.
• One of the most significant disturbances in this process is the size of the min- eral particles in the pulp. This disturbance is actually the output of a previous
stage (grinding). To apply a control law derived from (2.5.3)one would need to measure the size of all these particles,or at least to obtain some average measure of this. Thus,condition R4 is hard to satisfy.
• Pure time delays are also present in this process and thus condition R3cannot be satisfied.
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One could imagine various other practical cases where one or more of the require- ments 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 which will allow us to solve the problem subject to the limitations which the physical set-up imposes. The most commonly used constraints are:
L1 To restrict attention to those problems where the prescribed behavior (ref- erence signals) belong to restricted classes and where the desired behavior is achieved only asymptotically.
L2 To seekapproximate inverses.
In summary, we may conclude
In principle, all controllers implicitly generate an inverse of the process, in so far that this is feasible. Controllers differ with respect to the mechanism used to generate the required approximate inverse.