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1. A simple hierarchy for computing controlled invariant sets

   https://doi.org/10.1145/3365365.3382205  

   Anevlavis, Tzanis; Tabuada, Paulo (April 2020, HSCC '20: Proceedings of the 23rd International Conference on Hybrid Systems: Computation and Control)

   null (Ed.)

   In this paper we revisit the problem of computing controlled invariant sets for controllable discrete-time linear systems and present a novel hierarchy for their computation. The key insight is to lift the problem to a higher dimensional space where the maximal controlled invariant set can be computed exactly and in closed-form for the lifted system. By projecting this set into the original space we obtain a controlled invariant set that is a subset of the maximal controlled invariant set for the original system. Building upon this insight we describe in this paper a hierarchy of spaces where the original problem can be lifted into so as to obtain a sequence of increasing controlled invariant sets. The algorithm that results from the proposed hierarchy does not rely on iterative computations. We illustrate the performance of the proposed method on a variety of scenarios exemplifying its appeal. 

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2. Opportunistic Safety Outside the Maximal Controlled Invariant Set

   https://doi.org/10.1109/LCSYS.2023.3347334  

   Liu, Zexiang; Chen, Hao; Gao, Yulong; Ozay, Necmiye (December 2023, IEEE Control Systems Letters)

   Existing safety control methods for non-stochastic systems become undefined when the system operates outside the maximal robust controlled invariant set (RCIS), making those methods vulnerable to unexpected initial states or unmodeled disturbances. In this work, we propose a novel safety control framework that can work both inside and outside the maximal RCIS, by identifying a worst-case disturbance that can be handled at each state and constructing the control inputs robust to that worst-case disturbance model. We show that such disturbance models and control inputs can be jointly computed by considering an invariance problem for an auxiliary system. Finally, we demonstrate the efficacy of our method both in simulation and in a drone experiment. 

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3. Controlled Invariant Sets: Implicit Closed-Form Representations and Applications

   https://doi.org/10.1109/TAC.2023.3336819  

   Anevlavis, Tzanis; Liu, Zexiang; Ozay, Necmiye; Tabuada, Paulo (January 2023, IEEE Transactions on Automatic Control)

   We revisit the problem of computing (robust) controlled invariant sets for discrete-time linear systems. Departing from previous approaches, we consider implicit, rather than explicit, representations for controlled invariant sets. Moreover, by considering such representations in the space of states and finite input sequences we obtain closed-form expressions for controlled invariant sets. An immediate advantage is the ability to handle high-dimensional systems since the closed-form expression is computed in a single step rather than iteratively. To validate the proposed method, we present thorough case studies illustrating that in safety-critical scenarios the implicit representation suffices in place of the explicit invariant set. The proposed method is complete in the absence of disturbances, and we provide a weak completeness result when disturbances are present. 

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4. Computing controlled invariant sets in two moves

   https://doi.org/10.1109/CDC40024.2019.9029610  

   Anevlavis, Tzanis; Tabuada, Paulo (December 2019, 2019 IEEE 58th Conference on Decision and Control (CDC))

   null (Ed.)

   In this paper we revisit the problem of computing controlled invariant sets for controllable discrete-time linear systems. We propose a novel algorithm that does not rely on iterative computations. Instead, controlled invariant sets are computed in two moves: 1) we lift the problem to a higher dimensional space where a controlled invariant set is computed in closed-form; 2) we project the resulting set back to the original domain to obtain the desired controlled invariant set. One of the advantages of the proposed method is the ability to handle larger systems. 

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5. Automaton-based Implicit Controlled Invariant Set Computation for Discrete-Time Linear Systems

   https://doi.org/10.1109/CDC45484.2021.9683574  

   Liu, Zexiang; Anevlavis, Tzanis; Ozay, Necmiye; Tabuada, Paulo (December 2021, IEEE Conference on Decision and Control)

   In this paper, we derive closed-form expressions for implicit controlled invariant sets for discrete-time controllable linear systems with measurable disturbances. In particular, a disturbance-reactive (or disturbance feedback) controller in the form of a parameterized finite automaton is considered. We show that, for a class of automata, the robust positively invariant sets of the corresponding closed-loop systems can be expressed by a set of linear inequality constraints in the joint space of system states and controller parameters. This leads to an implicit representation of the invariant set in a lifted space. We further show how the same parameterization can be used to compute invariant sets when the disturbance is not available for measurement. 

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