On Structural Invariants in the Energy-Based Control of Infinite-Dimensional Port-Hamiltonian Systems with In-Domain Actuation (original) (raw)
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Systems & Control Letters, 2020
This contribution deals with energy-based in-domain control of systems governed by partial differential equations with spatial domain up to dimension two. We exploit a port-Hamiltonian system description based on an underlying jet-bundle formalism, where we restrict ourselves to systems with 2nd-order Hamiltonian. A certain power-conserving interconnection enables the application of a dynamic control law based on structural invariants. Furthermore, we use various examples such as beams and plates with in-domain actuation to demonstrate the capability of our approach.
Energy-Based In-Domain Control and Observer Design for Infinite-Dimensional Port-Hamiltonian Systems
IFAC-PapersOnLine, 2021
In this paper, we consider infinite-dimensional port-Hamiltonian systems with indomain actuation by means of an approach based on Stokes-Dirac structures as well as in a framework that exploits an underlying jet-bundle structure. In both frameworks, a dynamic controller based on the energy-Casimir method is derived in order to stabilise certain equilibrias. Moreover, we propose distributed-parameter observers deduced by exploiting damping injection for the observer error. Finally, we compare the approaches by means of an in-domain actuated vibrating string and show the equivalence of the control schemes derived in both frameworks.
2020
In this paper, we consider infinite-dimensional port-Hamiltonian systems with indomain actuation by means of an approach based on Stokes-Dirac structures as well as in a framework that exploits an underlying jet-bundle structure. In both frameworks, a dynamic controller based on the energy-Casimir method is derived in order to stabilise certain equilibrias. Moreover, we propose distributed-parameter observers deduced by exploiting damping injection for the observer error. Finally, we compare the approaches by means of an in-domain actuated vibrating string and show the equivalence of the control schemes derived in both frameworks.
Energy-Based Control of Nonlinear Infinite-Dimensional Port-Hamiltonian Systems with Dissipation
2018 IEEE Conference on Decision and Control (CDC), 2018
In this paper, we consider nonlinear PDEs in a port-Hamiltonian setting based on an underlying jet-bundle structure. We restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian including certain dissipation models that can be incorporated in the port-Hamiltonian framework by means of appropriate differential operators. For this system class, energy-based control by means of Casimir functionals as well as energy balancing is analysed and demonstrated using a nonlinear Euler-Bernoulli beam.
arXiv (Cornell University), 2021
In this paper, we present a control-design method based on the energy-Casimir method for infinite-dimensional, boundary-actuated port-Hamiltonian systems with two-dimensional spatial domain and second-order Hamiltonian. The resulting control law depends on distributed system states that cannot be measured, and therefore, we additionally design an infinitedimensional observer by exploiting the port-Hamiltonian system representation. A Kirchhoff-Love plate serves as an example in order to demonstrate the proposed approaches.
Port Hamiltonian formulation of infinite dimensional systems II. Boundary control by interconnection
Decision and Control (CDC 2004). Proceedings of the 43rd IEEE Conference on, 2004
In this paper, some new results concerning the boundary control of distributed parameter systems in port Hamiltonian form are presented. The classical finite dimensional port Hamiltonian formulation of a dynamical system has been generalized to the distributed parameter and multivariable case by extending the notion of finite dimensional Dirac structure in order to deal with an infinite dimensional space of power variables. Consequently, it seems natural that also finite dimensional control methodologies developed for finite dimensional port Hamiltonian systems can be extended in order to cope with infinite dimensional systems. In this paper, the control by interconnection and energy shaping methodology is applied to the stabilization problem of a distributed parameter system by means of a finite dimensional controller. The key point is the generalization of the definition of Casimir function to the hybrid case, i.e. when the dynamical system to be considered results from the power conserving interconnection of an infinite and a finite dimensional part. A simple application concerning the stabilization of the one-dimensional heat equation is presented.
2021 Proceedings of the Conference on Control and its Applications
In this paper we consider in-domain control of distributed parameter port-Hamiltonian systems defined on a one dimensional spatial domain. Through an early lumping approach we extend the control by interconnection and energy shaping approach to the use of distributed control over the spatial domain. With the established finite dimensional controller, the closed-loop performances can be modified over a given range of frequencies while guaranteeing the closedloop stability of the infinite dimensional system. Two cases are investigated, the ideal case where the controller acts on the complete spatial domain (infinite dimensional distributed control), and the more realistic one where the control is piecewise homogeneous (finite rank distributed control). The proposed control strategies are illustrated through simulations on the stabilization of a vibrating Timoshenko beam.
Energy-shaping of port-controlled Hamiltonian systems by interconnection
Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), 1999
Passivity-based control (PBC) has shown to be very powerful to design robust controllers for physical systems described by Euler-Lagrange (EL) equations of motion. The application of PBC in regulation problems of mechanical systems yields controllers that have a clear physical interpretation in terms of interconnection of the system with its environment. In particular, the total energy of the closed-loop is the difference between the energy of the system and the energy supplied by the controller. Furthermore, since the EL structure is preserved in closed-loop, PBC is robust vis ci vis unmodeled dissipative effects. Unfortunately, these nice properties are sometimes lost when PBC is used in other applications, for instance, in electrical and electromechanical systems. In this paper we further contribute to develop a new PBC theory encompassing a broader class of systems, and preserving the aforementioned energy-balancing stabilization mechanism and the structure invariance, continuing upon our work in [14], [9] and [17]. Towards this end we consider port-controlled Hamiltonian systems with dissipation (PCHD), which result from the network modeling of energy-conserving lumped-parameter physical systems with independent storage elements, and strictly contain the class of EL models. 'Recall that if 51 + R I = 52 + R 2 , & with Ji skew-symmetric and Ri symmetric, i = 1,2, then JI = 52, RI = R2.
2000
It is shown how network modeling of lumped-parameter physical systems naturally leads to a geometrically defined class of systems, called port-controlled Hamiltonian systems with dissipation. The structural properties of these systems are discussed, in particular the existence of Casimir functions and their implications for stability. It is shown how a power-conserving interconnection of port-controlled Hamiltonian systems defines another port-controlled Hamiltonian system, and how this may be used for design and for control by shaping the internal energy. § § q k¨T are generalized configuration coordinates for the system with k degrees of freedom, the Lagrangian L equals the difference K © P between kinetic energy K and potential energy P, and ¦ § § p k¨T the k second-order equations (1) transform into 2k first-order equations ¤ q " H
Energy-Based In-Domain Control of a Piezo-Actuated Euler-Bernoulli Beam
IFAC-PapersOnLine, 2019
The main contribution of this paper is the extension of the well-known boundary-control strategy based on structural invariants to the control of infinite-dimensional systems with in-domain actuation. The systems under consideration, governed by partial differential equations, are described in a port-Hamiltonian setting making heavy use of the underlying jet-bundle structure, where we restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian. To show the applicability of the proposed approach, we develop a dynamic controller for an Euler-Bernoulli beam actuated with a pair of piezoelectric patches and conclude the article with simulation results.