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1. Optimal Steady-State Control for Linear Time-Invariant Systems

   https://doi.org/10.1109/CDC.2018.8619812  

   Lawrence, Liam S. ; Nelson, Zachary E. ; Mallada, Enrique ; Simpson-Porco, John W. ( December 2018 , 2018 IEEE Conference on Decision and Control (CDC))

   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 that determines the feasible set defined by the system equilibrium constraints. Our proposed design 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 equilibrium and outline two procedures for designing controllers that stabilize the closed-loop system. We explore key ideas through simple examples and simulations. 

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2. On Design of Robust Linear Quadratic Regulators

   https://doi.org/10.23919/ACC55779.2023.10156654  

   Komaee, Arash ( May 2023 , IEEE)

   Closed-loop stability of uncertain linear systems is studied under the state feedback realized by a linear quadratic regulator (LQR). Sufficient conditions are presented that ensure the closed-loop stability in the presence of uncertainty, initially for the case of a non-robust LQR designed for a nominal model not reflecting the system uncertainty. Since these conditions are usually violated for a large uncertainty, a procedure is offered to redesign such a non-robust LQR into a robust one that ensures closed-loop stability under a predefined level of uncertainty. The analysis of this paper largely relies on the concept of inverse optimal control to construct suitable performance measures for uncertain linear systems, which are non-quadratic in structure but yield optimal controls in the form of LQR. The relationship between robust LQR and zero-sum linear quadratic dynamic games is established. 

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3. Bridging Transient and Steady-State Performance in Voltage Control: A Reinforcement Learning Approach With Safe Gradient Flow

   https://doi.org/10.1109/LCSYS.2023.3289435  

   Feng, Jie ; Cui, Wenqi ; Cortés, Jorge ; Shi, Yuanyuan ( June 2023 , IEEE Control Systems Letters)

   Deep reinforcement learning approaches are becoming appealing for the design of nonlinear controllers for voltage control problems, but the lack of stability guarantees hinders their real-world deployment. This letter constructs a decentralized RL-based controller for inverter-based real-time voltage control in distribution systems. It features two components: a transient control policy and a steady-state performance optimizer. The transient policy is parameterized as a neural network, and the steady-state optimizer represents the gradient of the long-term operating cost function. The two parts are synthesized through a safe gradient flow framework, which prevents the violation of reactive power capacity constraints. We prove that if the output of the transient controller is bounded and monotonically decreasing with respect to its input, then the closed-loop system is asymptotically stable and converges to the optimal steady-state solution. We demonstrate the effectiveness of our method by conducting experiments with IEEE 13-bus and 123-bus distribution system test feeders. 

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4. 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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5. A Reinforcement Learning Look at Risk-Sensitive Linear Quadratic Gaussian Control

   Cui, Leilei ; Basar, Tamer ; Jiang, Zhong-Ping ( June 2023 , Proceedings of Machine Learning Research)

   Matni, N ; Morari, M ; Pappas, G J (Ed.)

   In this paper, we propose a robust reinforcement learning method for a class of linear discrete-time systems to handle model mismatches that may be induced by sim-to-real gap. Under the formulation of risk-sensitive linear quadratic Gaussian control, a dual-loop policy optimization algorithm is proposed to iteratively approximate the robust and optimal controller. The convergence and robustness of the dual-loop policy optimization algorithm are rigorously analyzed. It is shown that the dual-loop policy optimization algorithm uniformly converges to the optimal solution. In addition, by invoking the concept of small-disturbance input-to-state stability, it is guaranteed that the dual-loop policy optimization algorithm still converges to a neighborhood of the optimal solution when the algorithm is subject to a sufficiently small disturbance at each step. When the system matrices are unknown, a learning-based off-policy policy optimization algorithm is proposed for the same class of linear systems with additive Gaussian noise. The numerical simulation is implemented to demonstrate the efficacy of the proposed algorithm. 

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