| Summary: | This paper presents both sufficient and necessary conditions for polyhedral sets and symmetric polyhedral sets to be robust positively invariant sets within perturbed linear discrete-time systems. These conditions are derived through the application of optimization and dual optimization theory. By leveraging the definition of a robust positively invariant set and employing the Pontryagin difference, we have obtained robust positively invariant conditions in optimized forms. Through the use of dual optimization theory, various equivalent forms are introduced, offering additional tools for verifying that polyhedral sets are indeed robust positively invariant sets for perturbed linear discrete-time dynamic systems. The efficacy of these conclusions is further evidenced by numerical examples.
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