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11. Dealing with Constraints

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An ubiquitous problem in control is that all real actuators have limited authority. This implies that they are constrained in amplitude and/or rate of change. If one ignores this possibility then serious degradation in performance can result in the event that the input reaches a constraint limit. This is clearly a very important problem. There are two ways of dealing with it:

  1. reduce the performance demands so that a linear controller never violates the limits, or
  2. modify the design to account for the limits

Option (i) above is the most common strategy. However, this implies that either the actuator was oversized in the first place or one is unnecessarily compromising the performance. Anyway, we will see below that option (ii) is quite easy to achieve.

This chapter gives a first treatment of option (ii) based on modifying a given linear design. This will usually work satisfactorily for modest violations of the constraint (up to say 100%). If more serious violations of the constraints occur then we would argue that the actuator has been undersized for the given application.

We will also show how the same ideas can be used to avoid simple kinds of state constraint.

Here we assume that the control laws are bi-proper and minimum phase. This will generally be true in SISO systems. Bi-properness can be achieved by adding extra zeros if necessary. Techniques that do not depend on those assumptions will be described in section 18.? of Chapter 18.

Also, in a later chapter (Chapter 25) we will describe another technique for dealing with control and state constraints based on constraint optimal control theory. These latter techniques are generally called "model predictive control", and one of the major success stories in modern control.

Summary
  • Constraints are ubiquitous in real control systems
     
  • There are two possible strategies for dealing with them
    • limit the performance so that the constraints are never violated
    • carry out a design with the constraints in mind
       
  • Here we have given a brief introduction to the latter idea
     
  • A very useful insight is provided by the arrangement shown in Figure 19.19, which can be set in a general inversion set-up:

\begin{figure} \hangcaption{Implicit inversion $X^{-1}$\space } \end{figure}

    • X is a bi-proper, stable, minimum-phase transfer function
    • $x_\infty $ is the high-frequency gain of $x_\infty \neq 0$
    • given that X is bi-proper, then $% x_\infty ^{-1}$ and hence $% x_\infty ^{-1}$ is finite
    • the overall system is equivalent to X-1
    • the fascinating and useful point about the implementation in Figure 11.14 is that the circuit effectively inverts X without that X was inverted explicitly
       
  • Here we have used this idea to describe ant-windup mechanisms for achieving input saturation and slew rate limits
     
  • We have also shown how state constraints might be achieved
     
  • In later chapters we will use the same idea for many other purposes