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"Control Theory"
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A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria
We analyze a class of nonlinear partial differential equations (PDEs) defined on
eBook
Tunneling estimates and approximate controllability for hypoelliptic equations
This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator
The first result is the tunneling estimate
The main
result is a stability estimate for solutions to the hypoelliptic wave equation
We then prove the approximate controllability of the
hypoelliptic heat equation
We also explain how the analyticity
assumption can be relaxed, and a boundary
Most results turn out to be optimal on a family of Grushin-type operators.
The main proof relies on the
general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).
eBook
A Novel Active Fault-Tolerant Tracking Control for Robot Manipulators with Finite-Time Stability
Many terminal sliding mode controllers (TSMCs) have been suggested to obtain exact tracking control of robotic manipulators in finite time. The ordinary method is based on TSMCs that secure trajectory tracking under the assumptions such as the known robot dynamic model and the determined upper boundary of uncertain components. Despite tracking errors that tend to zero in finite time, the weakness of TSMCs is chattering, slow convergence speed, and the need for the exact robot dynamic model. Few studies are handling the weakness of TSMCs by using the combination between TSMCs and finite-time observers. In this paper, we present a novel finite-time fault tolerance control (FTC) method for robotic manipulators. A finite-time fault detection observer (FTFDO) is proposed to estimate all uncertainties, external disturbances, and faults accurately and on time. From the estimated information of FTFDO, a novel finite-time FTC method is developed based on a new finite-time terminal sliding surface and a new finite-time reaching control law. Thanks to this approach, the proposed FTC method provides a fast convergence speed for both observation error and control error in finite time. The operation of the robot system is guaranteed with expected performance even in case of faults, including high tracking accuracy, small chattering behavior in control input signals, and fast transient response with the variation of disturbances, uncertainties, or faults. The stability and finite-time convergence of the proposed control system are verified that they are strictly guaranteed by Lyapunov theory and finite-time control theory. The simulation performance for a FARA robotic manipulator proves the proposed control theory’s correctness and effectiveness.
Journal Article
Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems
We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems. [PUBLICATION ABSTRACT]
Journal Article
Reinforcement Learning for Decentralized Robust Optimal Voltage Control of Uncertain Islanded DC Microgrid Under ZIP Load
This paper delves into the application of robust optimal control theory for voltage regulation in DC microgrids with uncertain ZIP loads. The primary challenge in DC microgrids with local ZIP loads is addressed through a two‐phase approach encompassing classical robust control and data‐driven control methodologies. Initially, the robust control problem for voltage regulation is tackled using an undiscounted optimal approach. Subsequently, the classical structure of the proposed robust optimal control scheme is converted into a data‐driven control strategy employing a reinforcement learning (RL) algorithm. Given the system's unmatched uncertainties, a virtual control input is necessary during the robust control problem‐solving process, preventing the extension to a model‐free control strategy. By converting the unmatched uncertainties into matched ones in the first phase, a data‐driven robust control strategy is achieved using the RL‐based algorithm in the second phase. The simulation results which are obtained using MATLAB/SimPowerSystems toolbox showcase the effectiveness of the data‐driven approach in achieving stability and adaptability in uncertain DC microgrid environments. This paper explores robust optimal control for voltage regulation in DC microgrids with uncertain ZIP loads, addressing challenges through a two‐phase approach. Initially, robust control is applied using an undiscounted optimal control method, followed by a conversion to a data‐driven strategy with reinforcement learning. The transition from unmatched to matched uncertainties enables a data‐driven robust control strategy, showcasing stability and adaptability in uncertain DC microgrid environments through simulation results.
Journal Article