Theory and computation of disturbance invariant sets for discrete-time linear systems

This paper considers the characterization and computation of invariant sets for discrete-time, time-invariant, linear systems with disturbance inputs whose values are confined to a specified compact set but are otherwise unkn own. The emphasis is on determining maximal disturbance-invariant sets X th at belong to a specified subset Gamma of the state space. Such d-invariant sets have important applications in control problems where there are pointw ise-in-time state constraints of the form x(t)is an element of T. One purpo se of the paper is to unite and extend in a rigorous way disparate results from the prior literature, In addition there are entirely new results, Spec ific contributions include: exploitation of the Pontryagin set difference t o clarify conceptual matters and simplify mathematical developments, specia l properties of maximal invariant sets and conditions for their finite dete rmination, algorithms for generating concrete representations of maximal in variant sets, practical computational questions, extension of the main resu lts to general Lyapunov stable systems, applications of the computational t echniques to the bounding of state and output response. Results on Lyapunov stable systems ape applied to the implementation of a logic-based, nonline ar multimode regulator. For plants with disturbance inputs and state-contro l constraints it enlarges the constraint-admissible domain of attraction. N umerical examples illustrate the various theoretical and computational resu lts.