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Finite-Dimensional Boundary Control of a Wave Equation With Viscous Friction and Boundary Measurements

IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2024年第5期69卷 3182-3189页

作者： Selivanov, Anton Fridman, Emilia Univ Sheffield Dept Automat Control & Syst Engn Sheffield S10 2TN England Tel Aviv Univ Sch Elect Engn IL-69978 Tel Aviv Israel

Recently, a constructive approach to the design of finite-dimensional observer-based controller has been proposed for parabolic partial differential equations (PDEs). This article extends it to hyperbolic PDEs. Namely, we design a finite-dimensional, output-feedback, boundary controller for a wave equation with in-domain viscous friction. The control-free system is unstable for any friction coefficient due to an external force. Our approach is based on modal decomposition: an observer-based controller is designed for a finite-dimensional projection of the wave equation on N eigenfunctions (modes) of the Sturm-Liouville operator. The danger of this approach is the "spillover" effect: such a controller may have a deteriorating effect on the stability of the unconsidered modes and cause instability of the full system. Our main contribution is an appropriate Lyapunov-based analysis leading to linear matrix inequalities (LMIs) that allow one to find a controller gain and number of modes, N, guaranteeing that the "spillover" effect does not occur. An important merit of the derived LMIs is that their complexity does not change when N grows. Moreover, we show that appropriate N always exists and, if the LMIs are feasible for some N, they remain so for N + 1.

关键词： distributed parameter systems LMIs modal decomposition wave equation

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Backstepping neural operators for 2 x 2 hyperbolic PDEs☆

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AUTOMATICA 2025年 178卷

作者： Wang, Shanshan Diagne, Mamadou Krstic, Miroslav Univ Shanghai Sci & Technol Dept Control Sci & Engn Shanghai 200093 Peoples R China Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled hyperbolic PDEs, coupled Goursat-form PDEs govern two or more gain kernels - a structure unaddressed thus far with DeepONet. In this contribution, we open the subject of approximating systems of gain kernel PDEs by considering a counter-convecting 2 x 2 hyperbolic system whose backstepping boundary controller and observer gains are the solutions to 2 x 2 kernel PDE systems in Goursat form. We establish the continuity of the mapping from (a total of five) functional coefficients of the plant to the kernel PDEs solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and ensure that the DeepONet-based approximated gains guarantee stabilization when replacing the exact backstepping gain kernel functions. Taking into account anti-collocated boundary actuation and sensing, our L2-globally-exponentially stabilizing (GES) control law requires the deep learning of both the controller and the observer gains. Moreover, the encoding of the feedback law into DeepONet ensures semi-global practical exponential stability (SG-PES), as established in our result. The neural operators (NOs) speed up the computation of both controller and observer gains by multiple orders of magnitude. Its theoretically proved stabilizing capability is demonstrated through simulations. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： PDE backstepping Deep learning Neural networks distributed parameter systems

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A control-oriented model of a distributed parameter heating process 4th

A control-oriented model of a distributed parameter heating ...

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4th IFAC Workshop on Control of systems Governed by Partial Differential Equations (CPDE)

作者： Weiss, Ruven Diehl, Moritz Rieger, Harald Reuter, Johannes Hsch Konstanz Inst Syst Dynam Constance Germany Univ Freiburg Dept Microsyst Engn Freiburg Germany HOMAG AG Schopfloch Germany

In many industrial applications a workpiece is continuously fed through a heating zone in order to reach a desired temperature to obtain specific material properties. Many examples of such distributed parameter systems exist in heavy industry and also in furniture production such processes can be found. In this paper, a real-time capable model for a heating process with application to industrial furniture production is modeled. As the model is intended to be used in a Model Predictive Control (MPC) application, the main focus is to achieve minimum computational runtime while maintaining a sufficient amount of accuracy. Thus, the governing Partial Differential Equation (PDE) is discretized using finite differences on a grid, specifically tailored to this application. The grid is optimized to yield acceptable accuracy with a minimum number of grid nodes such that a relatively low order model is obtained. Subsequently, an explicit Runge-Kutta ODE (Ordinary Differential Equation) solver of fourth order is compared to the Crank-Nicolson integration scheme presented in Weiss et al. (2022) in terms of runtime and accuracy. Finally, the unknown thermal parameters of the process are estimated using real-world measurement data that was obtained from an experimental setup. The final model yields acceptable accuracy while at the same time shows promising computation time, which enables its use in an MPC controller. Copyright (C) 2022 The Authors.

关键词： distributed parameter systems heat conduction transportation real-time furniture production

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Asymptotic regulation for distributed parameter systems via zero dynamics inverse design

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 2013年第3期23卷 305-333页

作者： Byrnes, C. I. Gilliam, D. S. Hu, C. Shubov, V. I. Washington Univ St Louis MO 63130 USA Texas Tech Univ Lubbock TX 79409 USA Univ Louisville Dept Math Louisville KY 40292 USA Univ Massachusetts Lowell Dept Math Lowell MA 01854 USA

In this paper, we introduce a new approach, zero dynamics inverse (ZDI) design, for designing a feedback compensation scheme achieving asymptotic regulation for a linear or nonlinear distributed parameter system in the case when the value w(t) at time t of the signal w to be tracked or rejected is a measured variable. Following the nonequilibrium formulation of output regulation, we formulate the problem of asymptotic regulation by requiring zero steady-state error together with ultimate boundedness of the state of the system and the controller(s), with a bound determined by bounds on the norms of the initial data and w. Because a controller solving this problem depends only on a bound on the norm of w not on the particular choice of w, this formulation is in sharp contrast to both exact tracking, asymptotic tracking or dynamic inversion of a completely known trajectory and to output regulation with a known exosystem. The ZDI design consists of the interconnection, via a memoryless filter, of a stabilizing feedback compensator and a cascade controller, designed in a simple, universal way from the zero dynamics of the closed-loop feedback system. This design philosophy is illustrated with a problem of asymptotic regulation for a boundary controlled viscous Burgers' equation, for which we prove that the ZDI is input-to-state stable. In infinite dimensions, however, input-to-state stable compactness arguments are supplanted by smoothing arguments to accommodate crucial technical details, including the global existence, uniqueness, and regularity of solutions to the interconnected systems. Copyright (C) 2011 John Wiley & Sons, Ltd.

关键词： asymptotic tracking cascade and feedback interconnections distributed parameter systems input-to-state stability output regulation

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Necessary Stability Conditions for Reaction-Diffusion-ODE systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2024年第11期69卷 7940-7947页

作者： Bajodek, Mathieu Lhachemi, Hugo Valmorbida, Giorgio Univ Paris Saclay CNRS Cent Supelec INRIALab Signaux & Syst F-91190 Gif Sur Yvette France

This article reports necessary stability conditions for a parabolic partial differential equation (PDE) interconnected through the boundaries to an ordinary differential equation (ODE). We intend to propose numerical certificates for the instability of such interconnections. From one side, using spectral methods, we derive a necessary analytical condition based on root locus analysis, which can be tested in the parameters space. On the other side, using Lyapunov direct and converse approaches, two necessary conditions of stability are established in terms of matrix inequalities. The novelties lie both in the type of system studied and in the converse stability methods that are used. The numerical results demonstrate the performance of the different criteria set up in this article.

关键词： Numerical stability Stability criteria Asymptotic stability Vectors Transfer functions Mathematical models Boundary conditions distributed parameter systems reaction-diffusion semidefinite programming stability of linear systems

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Singular Perturbation Analysis for a Coupled KdV-ODE System

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2024年第8期69卷 5326-5337页

作者： Marx, Swann Cerpa, Eduardo Ecole Cent Nantes Nantes Lab Digital Sci LS2N F-44000 Nantes France CNRS UMR 6004 F-44000 Nantes France Pontificia Univ Catolica Chile Fac Matemat Inst Ingn Matemat & Computac Santiago 8320165 Chile

Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different time-scales appear. The singular perturbation method allows to decouple a full system into what are called the reduced-order system and the boundary layer system to get simpler stability conditions for the original system. In the infinite-dimensional setting, we do not have a general result making sure this strategy works. This article is devoted to this analysis for some systems coupling the Korteweg-de Vries equation and an ordinary differential equation with different time scales. More precisely, we obtain stability results and Tikhonov-type theorems.

关键词： Automatic control distributed parameter systems partial differential equations (PDEs)

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Rapid Stabilization of Timoshenko Beam by PDE Backstepping

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2024年第2期69卷 1141-1148页

作者： Chen, Guangwei Vazquez, Rafael Krstic, Miroslav Beijing Univ Technol Fac Informat Technol Beijing 100124 Peoples R China Univ Seville Dept Ingn Aerosp & Mecan Fluidos Camino Descubrimiento S-N Seville 41092 Spain Univ Calif San Diego Dept Mech & Aerosp Engn San Diego CA 92093 USA

In this article, we present rapid boundary stabilization of a Timoshenko beam with antidamping and antistiffness at the uncontrolled boundary, by using infinite-dimensional backstepping. We introduce a Riemann transformation to map the Timoshenko beam states into a set of coordinates that verify a 1-D hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the $L<^>{2}$ sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters, and has not been achieved in previous results. Finally, a numerical simulation shows the effectiveness of the proposed controller. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, by allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

关键词： Boundary control distributed parameter systems hyperbolic systems PDE backstepping Timoshenko beam

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Delay-independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay

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JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 2013年第2期350卷 397-418页

作者： Xing, Hailong Li, Donghai Gao, Cunchen Kao, Yonggui Qingdao Agr Univ Sci & Informat Coll Qingdao 266109 Peoples R China Tsinghua Univ Dept Thermal Engn State Key Lab Power Syst Beijing 100084 Peoples R China Ocean Univ China Sch Math Sci Qingdao 266100 Peoples R China Harbin Inst Technol Sch Sci Weihai 264200 Peoples R China

The sliding mode control (SMC) problem for a class of quasi-linear parabolic partial differential equation (PDE) systems with time-varying delay is considered. Firstly, the stability problem for the reduced order sliding dynamical equations is investigated and a sufficient condition for the stability of sliding motion is given. Then the SMC law, which forces the system state from any initial state to reach the sliding manifold within finite time, is designed. At last a simulation example is presented to illustrate effectiveness of the proposed method. Crown Copyright (C) 2012 Published by Elsevier Ltd. on behalf of The Franklin Institute All rights reserved.

关键词： SLIDING mode control QUASILINEARIZATION distributed parameter systems TIME-varying systems TIME delay systems PARTIAL differential equations STABILITY theory SIMULATION methods & models

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Moving horizon estimation for pipeline leak detection, localization, and constrained size estimation

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COMPUTERS & CHEMICAL ENGINEERING 2024年 188卷

作者： Xie, Junyao Huang, Biao Dubljevic, Stevan Univ Alberta Dept Chem & Mat Engn Edmonton AB T6G 2V4 Canada

Advanced pipeline leak detection and localization techniques are needed to reduce greenhouse gas emissions from hydrocarbon transportation pipelines. Developing effective leak detection and localization methods is challenging due to the spatiotemporal dynamics of process variables, the presence of process/measurement disturbances and constraints, and the limited measurement data. To address this issue, this manuscript proposes a novel moving horizon estimation design for pipeline leak detection, constrained estimation of leak size and location by using an infinite-dimensional pipeline hydraulic model. Based on the mass and momentum balance laws and the Cayley-Tustin time-discretization method, an infinite-dimensional discrete-time pipeline hydraulic model is proposed considering (unknown but bounded) disturbance and leak. By introducing a coordinate transformation, we decouple the leak size and location estimation problems. The implementable discrete-time moving horizon estimator and observer are designed for constrained leak size and location estimation. The effectiveness of the proposed designs is validated via simulation examples.

关键词： distributed parameter systems Hyperbolic partial differential equations Moving horizon estimation Pipeline leak detection and localization Constrained estimation

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Efficiency improvement in multi-sensor wireless network based estimation algorithms for distributed parameter systems with application at the heat transfer

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EURASIP JOURNAL ON ADVANCES IN SIGNAL PROCESSING 2013年第1期2013卷 1-17页

作者： Volosencu, Constantin Curiac, Daniel-Ioan Politehn Univ Timisoara Dept Automat & Appl Informat Fac Automat & Comp Timisoara 300223 Romania

This paper gives a technical solution to improve the efficiency in multi-sensor wireless network based estimation for distributed parameter systems. A complex structure based on some estimation algorithms, with regression and autoregression, implemented using linear estimators, neural estimators and ANFIS estimators, is developed for this purpose. The three kinds of estimators are working with precision on different parts of the phenomenon characteristic. A comparative study of three methods - linear and nonlinear based on neural networks and adaptive neuro-fuzzy inference system - to implement these algorithms is made. The intelligent wireless sensor networks are taken in consideration as an efficient tool for measurement, data acquisition and communication. They are seen as a "distributed sensor", placed in the desired positions in the measuring field. The algorithms are based on regression using values from adjacent and also on auto-regression using past values from the same sensor. A modelling and simulation for a case study is presented. The quality of estimation is validated using a quadratic criterion. A practical implementation is made using virtual instrumentation. Applications of this complex estimation system are in fault detection and diagnosis of distributed parameter systems and discovery of malicious nodes in wireless sensor networks.

关键词： Intelligent sensor networks distributed parameter systems Estimation techniques System monitoring Virtual instrumentation

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