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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. 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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3. Large-Scale Invariant Sets for Safe Coordination of Thermostatic Loads

   https://doi.org/10.23919/ACC50511.2021.9483385  

   Jang, Sunho ; Ozay, Necmiye ; Mathieu, Johanna L. ( January 2021 , American Control Conference (ACC) 2021)

   Systems often face constraints at multiple levels. For example, in coordinating a collection of thermostatically controlled loads to provide grid services, the controller must ensure temperature constraints for each load (local constraints) and distribution network constraints (global constraints) are satisfied. In this paper, we leverage invariant sets to ensure safe coordination of systems with both local and global constraints. Specifically, we develop a method for constructing a controlled invariant set for a collection of subsystems, modeled as transition systems, to ensure they indefinitely satisfy the constraints, based on cycles in individual transition systems. Then, we develop a control algorithm that keeps the state inside the maximal controlled invariant set.We apply these algorithms to a demand response problem, specifically, the tracking of a power trajectory (e.g., a frequency regulation signal) by a population of homogeneous air conditioners. The algorithm simultaneously maintains local temperature requirements and aggregate power consumption limits, ensuring the control is nondisruptive to consumers and benign to the distribution network. 

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4. Model Predictive Real-Time Monitoring of Linear Systems

   https://doi.org/10.1109/RTSS.2017.00035  

   Chen, Xin ; Sankaranarayanan, Sriram ( December 2017 , 2017 IEEE Real-Time Systems Symposium (RTSS))

   The predictive monitoring problem asks whether a deployed system is likely to fail over the next T seconds under some environmental conditions. This problem is of the utmost importance for cyber-physical systems, and has inspired real-time architectures capable of adapting to such failures upon forewarning. In this paper, we present a linear model-predictive scheme for the real-time monitoring of linear systems governed by time-triggered controllers and time-varying disturbances. The scheme uses a combination of offline (advance) and online computations to decide if a given plant model has entered a state from which no matter what control is applied, the disturbance has a strategy to drive the system to an unsafe region. Our approach is independent of the control strategy used: this allows us to deal with plants that are controlled using model-predictive control techniques or even opaque machine-learning based control algorithms that are hard to reason with using existing reachable set estimation algorithms. Our online computation reuses the symbolic reachable sets computed offline. The real-time monitor instantiates the reachable set with a concrete state estimate, and repeatedly performs emptiness checks with respect to a safety property. We classify the various alarms raised by our approach in terms of what they imply about the system as a whole. We implement our real-time monitoring approach over numerous linear system benchmarks and show that the computation can be performed rapidly in practice. Furthermore, we also examine the alarms reported by our approach and show how some of the alarms can be used to improve the controller. 

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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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