CONTROL BY INTERCONNECTION FOR DISTRIBUTED PORT HAMILTONIAN SYSTEMS Alessandro Macchelli * Arjan van der Schaft ** Claudio Melchiorri * * CASY–DEIS, University of Bologna, viale Risorgimento 2, 40136 Bologna, Italy ** Dept. of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands Abstract: In this paper, new results on the control of distributed parameter systems in port Hamiltonian form are presented. The control by interconnection and energy shaping is applied to the stabilization of a distributed parameter system by means of a finite dimensional controller. The regulator acts on the system through the boundary or the distributed port. The key point is the generalization of Casimir function for systems resulting from the power conserving interconnection of an infinite and a finite dimensional part. Copyright c  2005 IFAC Keywords: Port-Hamiltonian systems, Infinite-dimensional systems, Energy-shaping 1. INTRODUCTION The port Hamiltonian description of finite dimen- sional systems (Maschke and van der Schaft, 1992) has been generalized to the distributed parameter case, (Macchelli et al., 2004a; van der Schaft and Maschke, 2002) by extending the notion of Dirac structure (Dalsmo and van der Schaft, 1999) on an infinite dimensional power space. In this paper, the control by interconnection is generalized to the regulation of an infinite dimen- sional system by means of a finite dimensional boundary or distributed controller. The main re- sult concerns the conditions for a real-valued func- tion defined over the state space of the closed loop system to be a structural invariant (Casimir function). Once these conditions are deduced, by choosing a proper family of Casimir functions, the control by interconnection and energy shaping methodology can be applied as in the finite di- mensional case. In this way, the open-loop energy function can be shaped by introducing a new min- imum at the desired equilibrium configuration. This paper is organized as follows. In Sect. 2, the control by interconnection and energy shaping for finite dimensional systems is briefly introduced. Then, the boundary control by interconnection for infinite dimensional systems is discussed in Sect. 3, while the distributed control is presented in Sect. 4. In both cases, conditions for the ex- istence of Casimir functions in the closed loop system are deduced and the application in the energy shaping procedure is described. Finally, conclusions are presented in Sect. 5. 2. CONTROL BY INTERCONNECTION IN FINITE DIMENSIONS Denote by X an n-dimensional space of state (energy) variables and by H : X → R the en- ergy (Hamiltonian) function, bounded from be-