| On the one hand, as for its simple structure and characteristics of easy controlled, the PID controller always occupies a dominant position in industrial process. On the other hand, strictly speaking, almost all of the control systems are nonlinear and have various kinds of forms, just as the plants with common typical nonlinearities and interval plants whose parameters are variable. However, stability is the basic requirement for system design, to some extent, it is the key and difficult points to obtain the PID parameter region of nonlinear systems, and it will make important theory value in engineer applications.Compared with the linear systems, there are some essential differences for nonlinear systems. Thus, it is no longer applicable to use methods which are useful in linear systems. And we have to analyze the original nonlinear system directly with nonlinear methods.The paper is divided into two parts:for the PID controlling system with saturation, according to Popov criterion, for the several given plants of which its linear parts have been concreted to several common component, we will give the PID parameters range which may make the system stable in theory and then use Simulink examples to verify it. While for the stability analysis of interval plants part, two methods for judging stability of nonlinear systems with PID controller are given:the method of scanning segment plant, the method of virtual polynomials. The former is a necessary and sufficient condition, but it needs to traverse all over the convex parameters λ on the interval [0,1] and has some complexity. The latter is a sufficient condition and it shows that as long as the PID controller can make the32virtual polynomials stable, then the entire family will be stable. It does not need to sweep λ, but has certain conservation. At the same time, we attempt to draw the stability domain using MATLAB GUI with the former one, and it is convenient to adjust parameters online and analyze other system performances. |
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