PID Controllers, 2nd Edition

Chapter 7.5 - Control Paradigms: Nonlinear Elements

Nonlinear elements have been discussed before. In Section 3.5 we used a limiter to avoid integral windup, in Section 3.4 we discussed the addition of nonlinearities to obtain "error squared on proportional" and similar control functions. In Chapter 6 it was shown that performance could be improved by gain scheduling. In this section we describe more nonlinear elements and also present some control paradigms that guide the use of these elements.

Limiters

Since all physical values are limited, it is useful to have limiting devices in control systems too. A simple amplitude limiter is shown in Figure 7.11. The limiter can mathematically be described as the

07_5-1.jpg

Figure 7.11 Block diagram of a simple amplitude limiter.

07_5-2.jpg

It is also useful to limit the rate of change of signals. A system for doing this is shown in Figure 7.12. This circuit is called a rate limiter or a ramp unit. The output will attempt to follow the input signals. Since there is integral action in the system, the inputs and the outputs will be identical in steady state. Since the output is generated by an integrator with limited input signal, the rate of change of the output will be limited to the bounds given by the limiter. Rate limiters are used, for example, in model-following control of the type shown in Section 7.4. A more sophisticated limiter is shown in Figure 7.13. This limiter is called a jump and rate limiter. The output will follow the input for small changes in the input signal. At large changes the output will follow the input with a limited rate. The system in Figure 7.13 can be described by the following equations

07_5-3.jpg

If u - x = a it follows from the equations describing the system that y = u, and if u = x + a it follows that dx/dt = a. Thus, the output

07_5-4.jpg

Figure 7.12 Block diagram of a rate limiter or a ramp unit.

07_5-5.jpg

Figure 7.13 Jump and rate limiter.

signal will approach the input signal at the rate a.

Limiters are used in many different ways. They can be used to limit the command signals so that we are not generating setpoints that are demanding faster changes than a system can cope with. In Section 3.5 it was shown how amplitude limiters may be used to avoid integral windup in PID controllers.

 

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