Constructive Interconnection and Damping Assignment for Port-controlled Hamiltonian K. Nunna, M. Sassano and A. Astolfi Abstract— The Interconnection and Damping Assignment passivity-based control method for port-controlled Hamiltonian systems is discussed. We propose a novel construction which exploits the notion of algebraic solution of the so-called matching equation. The latter notion is instrumental in constructing an energy function defined on an extended state-space without involving the solution of any partial differential equation. This results, differently from the classical solution, in a dynamic state feedback that stabilizes a desired equilibrium point. Finally we show that, in the linear time-invariant case and under standard assumptions, the proposed methodology provides the standard passivity-based controller. I. I NTRODUCTION Energy-based frameworks for modeling mechanical, elec- trical and electromechanical systems have been extensively developed over the past few decades. In these approaches a complex nonlinear system is considered to be built up of simpler lumped parameter physical subsystems. Each individual subsystem consists of energy storage elements, resistive elements and ports. The ports of a system model the interaction with the environment while the resistive elements capture the dissipation of the system. These subsystems are assumed to be interconnected to one another through their ports in a power preserving manner [2]. A comprehensive discussion of this topic may be found, for instance, in [5], [7], [10], [11]. Port-controlled Hamiltonian (PCH) models characterize a class of finite dissipation systems stabilizable with passivity based control (PBC). A formulation of PBC known as Inter- connection and Damping Assignment (IDA) was introduced in [4], [5]. In this method, energy is shaped by modifying the interconnection structure and/or adding damping to the system. A useful advantage of the method is that there is a This work is partially supported by the EPSRC Programme Grant Control For Energy and Sustainability EP/G066477 and by the MIUR under PRIN Project Advanced Methods for Feedback Control of Uncertain Nonlinear Systems. K. Nunna is with the Department of Electrical and Electronic En- gineering, Imperial College London, London SW7 2AZ, UK (Email: kameswarie.nunna09@imperial.ac.uk). M. Sassano is with the Dipartimento di Ingegneria Civile e Ingegneria Informatica, Universit` a di Roma “Tor Vergata”, Via del Politecnico, 1 00133 Rome, Italy (Email: mario.sassano@uniroma2.it). A. Astolfi is with the Department of Electrical and Electronic Engi- neering, Imperial College London, London SW7 2AZ, UK and with the Dipartimento di Ingegneria Civile e Ingegneria Informatica, Universit` a di Roma “Tor Vergata”, Via del Politecnico, 1 00133 Rome, Italy (Email: a.astolfi@ic.ac.uk). physical interpretation of the control action as insertion of virtual springs, dampers and constraints [6]. The procedure for IDA as described in [4] and [3] involves assigning a closed-loop energy function from which a feedback law can be developed by solving a set of partial differential equations (PDEs), in the nonlinear case, and a set of linear matrix equations, in the linear case. The main contribution of the paper consists in a method- ology that permits the application of the Interconnection and Damping Assignment technique for port-controlled Hamil- tonian systems without involving the solution of any partial differential equation. The result is achieved making use of a dynamic extension and introducing the notion of alge- braic solution of the so-called matching equation. A similar approach is explored in [8] and [9] in a different context, namely optimal and robust control of nonlinear input-affine systems. This notion is instrumental for the construction of an auxiliary energy function defined on the extended state-space. The proposed approach provides a dynamic state feedback that stabilizes the desired equilibrium while imposing a (possibly perturbed) Hamiltonian structure to the closed-loop system. The rest of the paper is organized as follows. Section II introduces the problem under examination. The topic of Section III is the definition of the notion of algebraic solution of the matching equation together with some basic notation. The main result, namely the proposed dynamic state feedback, is discussed in Section IV for the case of nonlinear port-controlled Hamiltonian systems and then specialized to the case of linear systems in Section V. The paper is concluded by Section VII with some final comments and suggestions for future extensions. II. I NTERCONNECTION AND DAMPING ASSIGNMENT We consider nonlinear systems described by equations of the form ˙ x = J (x) ∂H ∂x ⊤ + g(x)u, y = g(x) ⊤ ∂H ∂x ⊤ , (1) where x(t) ∈ R n denotes the state of the system, u(t) ∈ R m is the input and y(t) ∈ R m is the output, J : R n → R n×n , J (x)= −J (x) ⊤ for all x, is the interconnection matrix and H : R n → R is a continuously differentiable function. The 2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013 978-1-4799-0178-4/$31.00 ©2013 AACC 1810