In this chapter, a relationship is established between comparison systems and control invariance or contractivity. It is shown that the control invariance and contractivity of sets \(\mathcal {P}(v,d)\) represented by sublevel sets or sets \(\mathcal {Q}(W,g,d)\) represented in dual form is equivalent to the existence of an admissible control law and a v-comparison system or a dual W-comparison system admitting the polyhedral set \(\mathcal {P}(I,d)\) or, respectively, the set \( \mathcal {P}(g,d)\) as a positively invariant or contractive set. These general results are then used to develop methods for determining control laws making sets of particular form positively invariant or contractive. Finally, methods of enlargement or size reduction of control invariant or contractive sets are presented.

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Control Invariance and Contractivity

  • George Bitsoris

摘要

In this chapter, a relationship is established between comparison systems and control invariance or contractivity. It is shown that the control invariance and contractivity of sets \(\mathcal {P}(v,d)\) represented by sublevel sets or sets \(\mathcal {Q}(W,g,d)\) represented in dual form is equivalent to the existence of an admissible control law and a v-comparison system or a dual W-comparison system admitting the polyhedral set \(\mathcal {P}(I,d)\) or, respectively, the set \( \mathcal {P}(g,d)\) as a positively invariant or contractive set. These general results are then used to develop methods for determining control laws making sets of particular form positively invariant or contractive. Finally, methods of enlargement or size reduction of control invariant or contractive sets are presented.