Elsevier

Automatica

Volume 142, August 2022, 110284
Automatica

Stabilization of a class of mixed ODE–PDE port-Hamiltonian systems with strong dissipation feedback

https://doi.org/10.1016/j.automatica.2022.110284Get rights and content

Abstract

This paper deals with the asymptotic stabilization of a class of port-Hamiltonian (pH) 1-D Partial Differential Equations (PDE) with spatial varying parameters, interconnected with a class of linear Ordinary Differential Equations (ODE), with control input on the ODE. The class of considered ODE contains the effect of a proportional term, that can be considered as the proportional action of a controller or a spring in case of mechanical systems. In this particular case of study, it is not possible to directly add damping on the boundary of the PDE. To remedy this problem we propose a control law that makes use of a “strong feedback” term. We first prove that the closed-loop operator generates a contraction strongly continuous semigroup, then we address the asymptotic stability making use of a Lyapunov argument, taking advantage of the pH structure of the original system to be controlled. Furthermore, we apply the proposed control law for the stabilization of a vibrating string with a tip mass and we show the simulation results compared with the application of a simple PD controller.

Introduction

In this paper we are interested in the control design for a class of systems described by a set of hyperbolic Partial Differential Equations (PDE) coupled at the boundary with a set of Ordinary Differential Equations (ODE). We refer to this class of systems as mixed ODE–PDE (m-ODE–PDE). Control design and stability analysis for m-ODE–PDE has raised a significant attention from many researchers because of its wide range of applications. In particular, m-PDE–ODE equations enclose models of rotating and/or translating beams (Aoues et al., 2019, Banavar and Dey, 2010, Mattioni et al., 2020), controlled nanotweezer used for DNA manipulation (Ramirez, Le Gorrec, Macchelli, & Zwart, 2014) as well as electric transmission lines with load (Macchelli & Melchiorri, 2005).

In this work, we make use of the distributed parameter port-Hamiltonian (pH) approach introduced in van der Schaft and Maschke (2002) for the modelling and control of physical systems. This formalism has been adapted for the definition of pH Boundary Control Systems (Le Gorrec, Zwart, & Maschke, 2004), where a simple matrix condition suffices to characterize a well-posed (in the Hadamard sense) system (Le Gorrec, Zwart, & Maschke, 2005). A complete exposition with some further extensions of these first results can be found in Jacob and Zwart (2012) and Villegas (2007). Well-posedness and stabilization problems have been studied in case of static feedback (Villegas, Zwart, Le Gorrec, Maschke, & van der Schaft, 2005), dynamic linear feedback (Ramirez et al., 2014) and dynamic non-linear feedback (Ramirez, Zwart, & Le Gorrec, 2017). In case of dynamic linear feedback, the energy-shaping technique can be used to design the dynamic linear controller, assuring the asymptotic stability of the m-ODE–PDE closed-loop system (Macchelli, Gorrec, Ramirez, & Zwart, 2017).

The control design and the stabilization problem of m-ODE–PDE have been successfully tackled in different control scenarios using backstepping techniques. In particular, the stabilization problem for sandwiched parabolic m-PDE–ODE systems with control on the PDE boundaries and on the set of ODE has been solved on Deutscher (2015) and Wang and Krstic (2019), respectively. Moreover, backstepping control design has been applied to obtain the exponential stabilization of a class of heterodirectional hyperbolic m-ODE–PDE with actuation on the PDE boundaries (Meglio, Argomedo, Hu, & Krstic, 2018). Further, this result has been extended firstly to the same class of systems with space dependent parameters (Deutscher, Gehring, & Kern, 2019), and secondly for a class of heterodirectional m-ODE–PDE–ODE systems with actuation on one set of ODE (Deutscher, Gehring, & Kern, 2018). In this latter work, exponential stability is achieved through a control law that, to be implementable, needs the use of an observer. In this work, we propose an asymptotically stabilizing static control strategy for a different class of hyperbolic m-ODE–PDE systems, that in most applications does not need the implementation of a dynamical controller.

It has been proven that linear operator equations of the form ẋ=Ax+Bux(0)=x0y=Cxwith A generator of a bounded group (i.e. suptRT(t)<) on a infinite dimensional state space X, and input matrix BL(Rn,X), are not exponentially stabilizable with classical bounded linear feedback u=Fx with FL(X,Rn) (see in Lemma 8.4.1 of Curtain and Zwart (2020)). However, it has been shown that it is possible to use a “strong dissipation” feedback term u=Kpt(Cx) instead of the classical dissipation term. This type of feedback has already been applied and studied for specific sets of mixed ODE–PDE. In fact, the strong dissipation feedback has been used in Morgül, Rao, and Conrad (1994) to exponentially stabilize a wave equation with dynamic boundary conditions or in Conrad and Morgül (1998) for an Euler Bernoulli beam with a tip mass (see also de Queiroz, Dawson, Agarwal, and Zhang (1999) and de Queiroz, Dawson, and Zhang (1997) for other examples). Compared to the these previous works that use the strong dissipation feedback (Rao, 1995), we extend the class of linear systems that could be interconnected at the boundary, allowing the presence of a position control or, equivalently, the presence of a spring. The combined strong dissipation and position control has already been obtained using backstepping techniques in d’ Andréa-Novel and Coron (2000) for the specific case of a wave equation with dynamic boundary conditions. In d’ Andréa-Novel and Coron (2000), the authors carried out the analysis without position control term, concluding exponential stability of the closed-loop system. Besides, the strong dissipation with position control applied to a translating and rotating Timoshenko’s beam in contact scenario has already been studied in Endo, Sasaki, Matsuno, and Jia (2017), where exponential stability has been proved.

In this paper we generalize the concept of combined strong dissipation and position control for a class of m-pH systems that encloses a variety of practical applications. Moreover, we propose a Lyapunov argument to show the asymptotic stability of the closed-loop system. The use of a Lyapunov function instead of the classical frequency domain methods (Endo et al., 2017) opens the possibility of extending this method to the case of nonlinearities in the set of ODE. The stability proof makes use of the properties of infinite dimensional pH systems. Moreover, the effectiveness of the proposed control law is shown via the application on a clamped vibrating string with a tip mass on the free side, together with a simulation comparison with a simple PD control law. Since the strong dissipation feedback is obtained as the time derivative of a signal, we also propose some numerical simulations in case noise is added to the measurement.

The paper is organized as follows. In Section 2 we present the class of considered PDE–ODE together with the proposed control law and the consequent closed-loop system. Sections 3 Strongly continuous semigroup generation, 4 Asymptotic stabilization contain the C0-semigroup generation and the asymptotic stability theorems for the closed-loop operator respectively, and they correspond to the main contributions of the proposed work. The application and simulation results for a vibrating string with a tip mass are presented in Section 5. Finally, some concluding remarks and comments to future works are given in Section 6.

Section snippets

Preliminaries

In this paper we consider a plant composed of a set of PDE defined on a one-dimensional spatial domain interconnected with a set of ODE. In particular, we assume that the control actions are active on the ODE. This class of models is of practical interest because it includes moving vibrating beams or strings, where the control action acts on a boundary inertia. We begin by defining the set of first order 1D-spatial domain pH PDE zt(ξ,t)=P1ξHz(ξ,t)+(P0+G0)Hz(ξ,t),where ξ[a,b], P1Mn(R)1

Strongly continuous semigroup generation

In this section we investigate which type of solution function is generated by the closed-loop operator (19)–(20). The closed-loop operator is defined as the non-power preserving interconnection between an infinite and a finite dimensional linear pH system. Since the interconnection is not power preserving, it is not possible to show the contraction C0-semigroup generation in L2([a,b],Rn)×R2m equipped with the energy norm, as in classical interconnected ODE–PDE pH systems (Villegas, 2007).

Asymptotic stabilization

In this section we prove the asymptotic stability of the system described by Eqs. (19)–(20), that is equivalent to show the asymptotic stability of system (18). To do so, consider the state space X=L2([a,b],Rn)×R2m with inner product x1,x2X=z1,z2L2+v1Tv2 and associated norm (45). Before stating the main theorem on asymptotic stability, we need the following lemma that assures the compactness of the trajectories generated by the closed-loop operator.

Lemma 6

For x0X define the trajectory γ(x0)=t0T(

Example

To illustrate the applicability and the stability in closed-loop of the proposed control law (16), we propose the example of a clamped-free string with a mass connected at the free side, as shown in Fig. 2.

This system can be modelled by the following set of equations: twt(ξ,t)+q̇(t)=1ρ(ξ)ξT(ξ)wξ(ξ,t)m12wt2(0,t)=T(0)wξ(0,t)+f(t)w(ξ,0)=w0(ξ)w(L,t)=0for ξ[0,L].

We design a proportional control law such that f(t)=kw(0,t)+u(t), and define the state variables z1(ξ,t)=wξ, z2(ξ,t)=ρwt+q

Conclusions

In this paper we have developed a control strategy that asymptotically stabilizes a class of mixed PDE–ODE systems with actuation in the ODE part. The closed-loop operator has been obtained after an appropriate change of coordinates, and it has been shown to generate a contraction C0-semigroup in an appropriate space equipped with a weighted inner product, provided the correct selection of the control parameters. Afterwards, the weighted inner product has been shown to be equivalent to the

Acknowledgement

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 765579. This project has been supported by the EIPHI Graduate School, France (contract “ANR-17-EURE-0002”), by the ANR IMPACTS, France project (contract “ANR-21-CE48-0018”) and the MIAI@Grenoble Alpes, France (contract “ANR-19-P3IA-0003”).

Andrea Mattioni was born in Faenza, Italy, in 1993. He received his B.S. and M.S. degrees in Automation Engineering from the University of Bologna, Italy, in 2015 and 2018, respectively. From October to December 2019, he was a visiting student at the Department of Applied Mathematics, University of Twente, Enschede, The Netherlands. He received the Ph.D. degree from the University of Bourgogne Franche-Comté, Besançon, France in 2021. He currently holds a post-doctoral research position at

References (31)

  • ConradF. et al.

    On the stabilization of a flexible beam with a tip mass

    SIAM Journal on Control and Optimization

    (1998)
  • CurtainR.F. et al.

    An introduction to infinite-dimensional linear systems theory

    (1995)
  • CurtainR.F. et al.

    Introduction to infinite-dimensional linear systems theory, a state-space approach

    (2020)
  • de QueirozM.S. et al.

    Adaptive nonlinear boundary control of a flexible link robot arm

    IEEE Transactions on Robotics and Automation

    (1999)
  • de Queiroz, M. S., Dawson, D. M., & Zhang, F. (1997). Boundary control of a rotating flexible body-beam system. In...
  • Andrea Mattioni was born in Faenza, Italy, in 1993. He received his B.S. and M.S. degrees in Automation Engineering from the University of Bologna, Italy, in 2015 and 2018, respectively. From October to December 2019, he was a visiting student at the Department of Applied Mathematics, University of Twente, Enschede, The Netherlands. He received the Ph.D. degree from the University of Bourgogne Franche-Comté, Besançon, France in 2021. He currently holds a post-doctoral research position at Gipsa-lab, Grenoble, France. His research interests include control theory, port-Hamiltonian systems and control of partial differential equations.

    Yongxin Wu was born in Baoji, China in 1985. He received his engineer degree in Transportation Information and Control from the University of Chang’an, Xi’an, China in 2010 and his Master’s degree in Automatic Control from the University Claude Bernard of Lyon, Villeurbanne, France in 2012. He received his Ph.D. degree in Automatic Control in 2015 for his work on the model and controller reduction of port Hamiltonian systems at the Laboratory of Control and Chemical Engineering (LAGEP UMR CNRS 5007) of the University Claude Bernard of Lyon, Villeurbanne, France. From 2015 to 2016, He held a post-doctoral and teaching assistant position at LAGEP. Since 2016, he has been an Associate Professor of Automatic Control at the National Engineering Institute in Mechanics and Micro-technologies and affiliated to the AS2M department at FEMTO-ST institute (UMR CNRS 6174) in Besançon, France. His research interests include port-Hamiltonian systems, model and controller reduction, modelling and control of multi-physical systems.

    Yann Le Gorrec was graduated as engineer in « Control, Electronics, Computer Engineering » at the National Institute of Applied Sciences (INSA, Toulouse, France) in 1995. He received in 1998 his Ph. D. degree from the National Higher School of Aeronautics and Aerospace (Supaero, Toulouse, France). His field of interest was robust control and self-scheduled controller synthesis. From 1999 to 2008, he was Associate Professor in Automatic Control at the Laboratory of Control and Chemical Engineering of Lyon Claude Bernard University (LAGEP, Villeurbanne, France). He worked on port Hamiltonian systems and their use for the modelling and control of irreversible and distributed parameter systems with an application to physic-chemical processes. Since September 2008 he is full Professor at National Engineering Institute in Mechanics and Microtechnologies of Besançon, France. His current field of research is the control of nonlinear and distributed parameter systems with an application to smart material-based actuators and micro systems by using the port Hamiltonian framework.

    Hans Zwart was born in Hoogezand-Sappemeer, The Netherlands, in 1959. He received the Dr. degree in mathematics in 1984 and the Ph.D. degree in 1988, both from the University of Groningen, Groningen, The Netherlands. Both theses were written under the supervision of R. Curtain. Since 1988, he has been with the Department of Applied Mathematics, University of Twente, Enschede, The Netherlands. He is the (co-) author of four books, two with R. Curtain. The first one has become a standard reference on infinite-dimensional systems theory. He has published more than 100 research articles, treating various aspects of infinite-dimensional systems. His current research interests include stability, controllability, and control of infinite-dimensional systems, in particular for port-Hamiltonian systems. Since 2011, he also has a one-day per week position in the Dynamic and Control group, Department of Mechanical Engineering, Eindhoven University of Technology.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Rafael Vazquez under the direction of Editor Miroslav Krstic.

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