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Set invariance in control. (English) Zbl 0935.93005
This is an overview paper on the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems, which contains the following sections. (1) Introduction. (2) Basic definitions of (robustly) positively invariant sets, (robustly) controlled invariant sets for systems and some other related concepts. (3) Basic results including some classical positive invariance conditions and the relation between positively invariant sets and Lyapunov functions. In particular, it is shown how a convex and compact invariant set containing the origin in its interior can “shape” a Lyapunov function. (4) Special families of positively invariant sets, which presents the main properties of the two most currently used families of candidate invariant sets: ellipsoids and polytopes. Invariant sets of some other types are also roughly reviewed. (5) Construction of invariant sets and control synthesis, in which some basic construction techniques are reviewed and the structure of the controllers that can be associated to controlled-invariant sets are investigated. (6) Applications of set invariance, which was classified into six subsections: Invariant set as a theoretical tool; invariance and control problems with time-domain constraints; invariance and robustness; disturbance rejection; performance analysis via invariant sets and the receding horizon control. (7) Concluding remarks with some current research directions. There are 176 publications as references, most of them are very recently published.
Reviewer: W.Bian (Glasgow)

##### MSC:
 93-02 Research monographs (systems and control) 93D30 Scalar and vector Lyapunov functions 93B50 Synthesis problems
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