Systems & Control Letters 60 (2011) 579–589 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation Alessandro Macchelli ∗ University of Bologna, Department of Electronics, Computer Science and Systems (DEIS), viale del Risorgimento 2, 40136 Bologna, Italy article info Article history: Received 8 June 2010 Received in revised form 21 April 2011 Accepted 22 April 2011 Available online 18 May 2011 Keywords: Passivity-based control Energy shaping control Port-Hamiltonian systems Casimir functions abstract The main contribution of this paper is a procedure for the control by energy shaping via Casimir generation of infinite dimensional port-Hamiltonian systems based on a particular finite element approximation. The proposed approach is justified by the fact that the adopted spatial discretization technique is able to preserve Casimir functions in the closed-loop system when going from the distributed to the (approximated) lumped parameter system. Besides the intrinsic difficulties related to the large number of state variables, the finite element model is generally given in terms of a Dirac structure and is completely a-causal, which implies that the plant dynamics is not given in standard input-state-output form, but as a set of DAEs. Consequently, the classical energy Casimir method has to be extended in order to deal with dynamical systems with constraints, usually appearing in the form of Lagrangian multipliers. The general methodology is illustrated with the help of an example in which the distributed parameter system is a lossless transmission line. © 2011 Elsevier B.V. All rights reserved. 1. Introduction This paper illustrates a novel procedure for the control by energy shaping of distributed port-Hamiltonian systems [1,2]. In recent works [3–8], this task has been accomplished by looking at or generating a set of Casimir functions in the closed-loop system that robustly (i.e. independently from the Hamiltonian function) relates the state of the infinite dimensional port-Hamiltonian system with the state of the controller. The controller has been usually modeled as a finite dimensional port-Hamiltonian system which has to be interconnected in a power conserving way to the boundary of the distributed parameter system. The shape of the energy function of the closed-loop system can be changed by properly choosing the Hamiltonian function of the controller in order to introduce a (possibly global) minimum in a desired configuration. This procedure is basically the generalization to the distributed parameter case of the control by interconnection via Casimir generation developed for finite dimensional port- Hamiltonian systems, [9–11]. In case of plants modeled as distributed parameter systems, it is relatively easy to shape the energy function. The main difficulties arise in proving that the new minimum of the closed- loop Hamiltonian function corresponds also to an asymptotically stable equilibrium point. Only stability can be verified by means of ∗ Tel.: +39 051 20 93031; fax: +39 051 20 93073. E-mail address: alessandro.macchelli@unibo.it. relatively simple techniques, as reported in [12]. This is because, even if the extension to the distributed parameter system of La Salle’s Invariance Principle exists, its application is not immediate due to several technical problems, mostly related to the analysis of the solution of linear or nonlinear PDEs, [13]. Instead of working on the full-order (i.e. infinite dimensional) dynamics of the plant, the idea is to rely on a finite dimensional approximation provided by the spatial discretization procedure presented in [14,15]. In particular, given a distributed parame- ter system in port-Hamiltonian form, from the analysis of its ge- ometric structure (i.e. of its Stokes–Dirac structure, [1]), a finite dimensional approximation still in port-Hamiltonian form can be obtained. This discretization technique is able to preserve the physical properties of the original infinite dimensional system, since the dynamics is given in terms of a Dirac structure [16,17], that describes both the internal exchange of energy and the power flow between the system and the environment through the bound- ary, and of discrete constitutive equations. Moreover, the ap- proximating system is completely a-causal, which means that (boundary) inputs and outputs are determined by the causality of the interconnected subsystems (including the boundary condi- tions), and able to deal with time varying boundary conditions gen- erated, for example, by a state-feedback controller. The main contribution of this paper is to show how this particular spatial discretization technique is able to preserve Casimir functions in the closed-loop system when the infinite dimensional dynamics is ‘‘replaced’’ by its finite dimensional approximation, and also how the control by energy shaping via Casimir generation, developed for finite dimensional systems 0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.04.016