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1. State Constrained Controller Design for Uncertain Linear Systems using Polynomial Chaos

   Nandi, Souransu; Singh, Tarunraj (June 2016, 2016 American Control Conference (ACC))

   The focus of this paper is on the design of state constrained controllers which are robust to time invariant uncertain variables. Polynomial Chaos spectral expansion is used to parameterize the uncertain variables, which permits evaluation of the evolution of the uncertain states. The coefficients of the truncated polynomial chaos expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A mapping into Bernstein polynomial space permits determination of bounds on the evolving states. Linear programming is used on the deterministic set of equation with constraints as the predetermined bounds to determine controllers which are robust to the epistemic uncertainties. Numerical examples are used to illustrate the benefit of the proposed technique for the design of rest-to-rest controllers subject to deformation constraints; which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass system. 

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2. Chance Constraint based Design of Controllers for Linear Uncertain Systems

   Nandi, Souransu; Singh, Tarunraj (May 2017, 2017 American Control Conference)

   This paper considers the problem of state-tostate transition with state and control constraints, for a linear system with model parameter uncertainties. Polynomial chaos is used to transform the stochastic model to a deterministic surrogate model. This surrogate model is used to pose a chance constrained optimal control problem where the state constraints and the residual energy cost are represented in terms of the mean and variance of the stochastic states. The resulting convex optimization is illustrated on the problem of rest-to-rest maneuver of the benchmark floating oscillator. 

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

   Liu, Zexiang; Anevlavis, Tzanis; Ozay, Necmiye; Tabuada, Paulo (January 2021, IEEE 60th 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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5. The Role of Strategic Load Participants in Two-Stage Settlement Electricity Markets

   https://doi.org/10.1109/CDC40024.2019.9029514  

   You, Pengcheng; Gayme, Dennice F.; Mallada, Enrique (December 2019, Conference on Decision and Control)

   We consider the problem of designing a feedback controller that guides the input and output of a linear time-invariant system to a minimizer of a convex optimization problem. The system is subject to an unknown disturbance, piecewise constant in time, which shifts the feasible set defined by the system equilibrium constraints. Our proposed design combines proportional-integral control with gradient feedback, and enforces the Karush-Kuhn-Tucker optimality conditions in steady-state without incorporating dual variables into the controller. We prove that the input and output variables achieve optimality in steady-state, and provide a stability criterion based on absolute stability theory. The effectiveness of our approach is illustrated on a simple example system. 

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