Can we stay where we are when we don’t know ‘everything’ exactly?
In this chapter, the Viability Control Problem is considered for hybrid systems under time–independent state constraints under the three forms of uncertainty transition dynamics, structural uncertainty, and parametric uncertainty. Ensuring that viability remains satisfied under uncertainty will be referred to as the robust viability problem. The nominal dynamics are taken as the collection of constituent control systems having single–valued nominal dynamics with no uncertainty. Two approaches are used for considering uncertainty relative to viability. In the first approach the effect of uncertainty on the nominal design is examined. In the second approach an uncertainty operator is used to determine the effect of uncertainty on the nominal design. To ensure that viability remains satisfied under uncertainty, two possibilities are considered. The uncertainty can be either taken into account at each iteration of the Controllability Operator Fixed Point Approximation Algorithm or compensated by an appropriate nominal design of the control automaton. In the treatment of the former case, we require that the admissible control law class is the same for the nominal and the uncertain case. In the latter case, we allow for either a larger subset of control law classes in the uncertain case or a different set.
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