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Stability Analysis of Linear Partial Differential Equations With Generalized Energy Functions

IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2020年第5期65卷 1924-1939页

作者： Gahlawat, Aditya Valmorbida, Giorgio Univ Illinois Dept Mech Sci & Engn Urbana IL 61801 USA Univ Paris Saclay Univ Paris Sud CNRS Lab Signaux & SystCent Supelec Gif Sur Yvette 91192 France

We present a method for the stability analysis of a large class of linear partial differential equations (PDEs) in one spatial dimension. We rely on Lyapunov analysis to establish the exponential stability of the systems under consideration. The proposed test for the verification of the underlying Lyapunov inequalities relies on the existence of solutions of a system of coupled differential equations. We illustrate the application of this method using a PDE actuated by a backstepping computed feedback law. Furthermore, for the case of PDEs defined by polynomial data, we formulate a numerical methodology in the form of a convex optimization problem which can be solved algorithmically. We show the effectiveness of the proposed numerical methodology using examples of different types of PDEs.

关键词： Stability analysis Backstepping PD control Numerical stability Partial differential equations distributed parameter systems Lyapunov methods optimization stability analysis

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Prescribed-time estimation and output regulation of the linearized Schrodinger equation by backstepping

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EUROPEAN JOURNAL OF CONTROL 2020年 55卷 3-13页

作者： Steeves, Drew Krstic, Miroslav Vazquez, Rafael Univ Calif San Diego 9500 Gilman Dr La Jolla CA 92093 USA Univ Seville Avda Descubrimientos Seville 41092 Spain

We study state estimation of the linearized Schrodinger equation within a prescribed terminal time. We make use of a time-varying, complex-valued observer gain and boundary measurements to construct the observer, where the gain is designed such that the estimate error converges to zero within the terminal time. The observer gain proposed herein is developed via the backstepping method by selecting a target error equation that stabilizes to zero within the terminal time. Our time-varying observer gain diverges as time approaches the terminal time. Nevertheless, we can guarantee prescribed-time stabilization of the estimator error equation by characterizing the growth-in-time of the observer gain and comparing it to the stability of the target error equation. We develop the full-state feedback dual result, and we combine the boundary estimation and control results to develop prescribed-time output regulation. (C) 2020 European Control Association. Published by Elsevier Ltd. All rights reserved.

关键词： Prescribed-time estimation Output regulation Backstepping distributed parameter systems

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Singular Perturbation Analysis of a Coupled System Involving the Wave Equation

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2020年第11期65卷 4846-4853页

作者： Cerpa, Eduardo Prieur, Christophe Univ Tecn Federico Santa Maria Dept Matemat Av Espana 1680 Valparaiso Chile Univ Grenoble Alpes CNRS Grenoble INP GIPSA Lab F-38000 Grenoble France

This article considers a system coupling an ordinary differential equation with a wave equation through its boundary data. The existence of a small parameter in the wave equation (as a factor multiplying the time derivative) suggests the idea of applying a singular perturbation method to get the stability of the full system by analyzing the stability of some appropriate subsystems given by the method. However, for infinite-dimensional systems, it is known that in some cases, this method does not work. Indeed, one cannot be sure of the stability of the full system even if the given subsystems are stable. In this article, we prove that the singular perturbation method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which is the first of this kind for systems involving the wave equation. Simulations are performed to show the applicability of our results.

关键词： Stability analysis Propagation Mathematical model Lyapunov methods Boundary conditions Perturbation methods Couplings Automatic control distributed parameter systems partial differential equations

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Boundary control for a flexible string system with input saturation constraint

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ASIAN JOURNAL OF CONTROL 2020年第2期22卷 934-943页

作者： Zhao, Zhijia Ren, Zhigang Ma, Yonghao Guangzhou Univ Adv Technol Ctr Special Equipment Sch Mech & Elect Engn Guangzhou 510006 Peoples R China Guangzhou Univ Res Ctr Intelligent Equipment & Network Connect S Guangzhou 510006 Peoples R China Guangdong Univ Technol Sch Automat Guangzhou Peoples R China

In this study, we focus on the boundary control problem for a vibrating string system with saturated input under the condition of external disturbances. Based on the backstepping approach, a boundary vibration control scheme is proposed to globally stabilize the string around the equilibrium position. A smooth hyperbolic tangent function is exploited to restrict the control input, an auxiliary system and a Nussbaum function are adopted to cope with the nonlinear term derived from the input saturation, and a disturbance observer is employed to tackle the boundary disturbance. The developed controller can assure the convergence of the closed-loop system state to a small neighbourhood of zero. By the appropriate choice of control design parameters, numerical simulation is conducted to show the effectiveness of the derived control.

关键词： backstepping boundary control distributed parameter systems disturbance observer input saturation

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Backstepping design for a class of coupled parabolic PDEs with spatially varying coefficient

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ASIAN JOURNAL OF CONTROL 2020年第5期22卷 2017-2028页

作者： Ghaderi, Najmeh Keyanpour, Mohammad Univ Guilan Dept Appl Math Rasht Iran

In this paper, we concentrate on the state feedback stabilization of a class of coupled parabolic partial differential equations with space dependent diffusion coefficients. To stabilize the system, we design a state feedback controller using the backstepping technique. Namely, we convert the original system into a stable desired system called the target system to obtain a backstepping feedback controller. We prove that the feedback controller causes the main system to be exponential stable. Also, the numerical simulations show that validity and efficiency of theoretical results.

关键词： backstepping control design coupled parabolic partial differential equations distributed parameter systems Lyapunov function

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Disturbance observer-based adaptive boundary iterative learning control for a rigid-flexible manipulator with input backlash and endpoint constraint

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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING 2020年第9期34卷 1220-1241页

作者： Zhou, Xingyu Wang, Haoping Tian, Yang Zheng, Gang Nanjing Univ Sci & Technol Sch Automat Sino French Int Joint Lab Automat Control & Signa Nanjing Peoples R China INRIA Lille Nord Europe 40 Ave Halley Villeneuve Dascq France

In this article, an observer-based adaptive boundary iterative learning control law is developed for a class of two-link rigid-flexible manipulator with input backlash, the unknown external disturbance, and the endpoint constraint. To tackle the backlash nonlinearities and ensure the vibration suppression, the disturbance observers based upon the iterative learning conception are considered in the adaptive boundary control design. A barrier Lyapunov function is incorporated with boundary control law to restrict the endpoint state. Based on the defined barrier composite energy function, the tracking angle error convergence of the rigid part is guaranteed, and the vibrations of the flexible part are suppressed through the rigorous analysis. Finally, a numerical simulation is provided to illustrate the effectiveness of the proposed control.

关键词： adaptive control distributed parameter systems intelligent control learning systems

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Graph-based determination of structural controllability and observability for pressure and temperature dynamics during steam-assisted gravity drainage operation

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JOURNAL OF PROCESS CONTROL 2020年 86卷 65-80页

作者： Ganesh, Ajay Li, Zixuan Chalaturnyk, Richard J. Prasad, Vinay Univ Alberta Dept Chem & Mat Engn Edmonton AB T6G 1H9 Canada Univ Alberta Dept Civil & Environm Engn Edmonton AB T6G 1H9 Canada

Steam assisted gravity drainage (SAGD), which is used for the in-situ extraction and recovery of oil sands bitumen, is represented by a distributed parameter system (DPS). The problem of sensor placement and the control of steam chamber growth and oil production, respectively, require analysis of the observability and controllability of the system. In this type of system, parametric sensitivity is traditionally used in lieu of observability, and controllability has not been explored rigorously. In this work, we analyze the pressure and temperature fields of a SAGD model based on detailed reservoir simulations and present a data-driven technique to assess the structural controllability and observability of the system, with a view to determine optimal locations of actuators and sensors. An agglomerative hierarchical clustering technique is used to obtain a spanning tree of the clusters which is partitioned based on an objective function to arrive at a set of spatially contiguous clusters that display similar pressure/temperature dynamics. A Granger causality measure is used to create the linkage amongst the clusters to build a digraph model of the data. The driver nodes of the graph identify locations for actuation which provide full control over the graph, and the root strongly connected components indicate sensor locations which ensure structural observability over the entire graph. We demonstrate the method using data generated from SAGD simulations using the CMG-STARS simulator, identify the sensor and actuator locations required for complete structural observability and controllability of the system, and also provide a method of assessment of partial actuation and in-sensor ranges. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Structural controllability and observability distributed parameter systems Graph theory Driver nodes Root strongly connected components

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Infinite dimensional model of a double flexible-link manipulator: The Port-Hamiltonian approach

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APPLIED MATHEMATICAL MODELLING 2020年 83卷 59-75页

作者： Mattioni, Andrea Wu, Yongxin Le Gorrec, Yann Univ Bourgogne Franche Comte CNRS FEMTO ST 24 Rue Savary F-25000 Besancon France

This paper proposes a modular and control oriented model of a double flexible-link manipulator that stems from the modelling of a spatial flexible robot. The model consists of the power preserving interconnection between two infinite dimensional systems describing the beam's motion and deformation with a finite dimensional nonlinear system describing the dynamics of the actuated rotating joints. To derive the model, Timoshenko's assumptions are made for the flexible beams. Using Hamilton's principle, the dynamic equations of the system are derived and then written in the Port-Hamiltonian (PH) framework through a proper choice of the state variables. These so called energy variables allow to write the total energy as a quadratic form with respect to a state dependent energy matrix. The resulting model is shown to be a passive system, a convenient property for control design purposes. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Flexible arms Flexible robotics Port-Hamiltonian systems distributed parameter systems Boundary control systems Discretization Finite dimensional approximation

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Application of Koopman operator for model-based control of fracture propagation and proppant transport in hydraulic fracturing operation

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JOURNAL OF PROCESS CONTROL 2020年 91卷 25-36页

作者： Narasingam, Abhinav Sang-Il Kwon, Joseph Texas A&M Univ Artie McFerrin Dept Chem Engn College Stn TX 77845 USA Texas A&M Univ Texas A&M Energy Inst College Stn TX 77845 USA

This work explores the application of the recently developed Koopman operator approach for model identification and feedback control of a hydraulic fracturing process. Controlling fracture propagation and proppant transport with precision is a challenge due in large part to the difficulty of constructing approximate models that accurately capture the characteristic moving boundary and highly-coupled dynamics exhibited by the process. Koopman operator theory is particularly attractive here as it offers a way to explicitly construct linear representations for even highly nonlinear dynamics. The method is data-driven and relies on lifting the states to an infinite-dimensional space of functions called observables where the dynamics are governed by a linear Koopman operator. This work considers two problems: (a) fracture geometry control, and (b) proppant concentration control. In both cases, an approximate linear model of the corresponding dynamics is constructed and used to design a model predictive controller (MPC). The manuscript shows that in the case of highly nonlinear dynamics, as observed in the proppant concentration, use of canonical functions in the observable basis fails. In such cases, a priori system knowledge can be leveraged to choose the required basis. The numerical experiments demonstrate that the Koopman linear model shows excellent agreement with the real system and successfully achieves the desired target values maximizing the oil and gas productivity. Additionally, due to its linear structure, the Koopman models allow convex MPC formulations that avoid any issues associated with nonlinear optimization. Published by Elsevier Ltd.

关键词： Koopman operator Model predictive control distributed parameter systems Moving boundary problem Hydraulic fracturing Extended dynamic mode decomposition

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Input-to-State Stability of a Clamped-Free Damped String in the Presence of distributed and Boundary Disturbances

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2020年第3期65卷 1248-1255页

作者： Lhachemi, Hugo Saussi, David Zhu, Guchuan Shorten, Robert Univ Coll Dublin Sch Elect & Elect Engn Dublin 4 Ireland Polytech Montreal Dept Elect Engn Montreal PQ H3T 1J4 Canada Imperial Coll London Dyson Sch Design Engn London SW7 2AZ England

This note establishes the exponential input-to-state stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.

关键词： Perturbation methods Trajectory Hilbert space Stability criteria Tools distributed parameter systems Boundary disturbance distributed parameter systems input-to-state stability

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