On the equivalence of two nonlinear control approaches: Immersion and invariance and IDA-PBC Paul Kotyczka a,n , Ioannis Sarras b a Lehrstuhl für Regelungstechnik, Technische Universität München, Boltzmannstr. 15, 85748 Garching, Germany b IFP New Energy, 1 and 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France article info Article history: Received 1 February 2013 Accepted 26 September 2013 Recommended by A. Astolfi Available online 10 October 2013 Keywords: Underactuated mechanical systems Port-Hamiltonian systems Passivity based control Immersion and invariance abstract In this paper we compare the two well-known nonlinear control design techniques Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) and Immersion and Invariance (I&I) at the example of the so-called Acrobot underactuated mechanical system. The immersion and matching equations in both approaches have a similar structure which is exploited to derive equivalent control laws, each of them providing a different perspective on the stabilization problem. In particular, the coordinate change which renders the potential energy matching PDE in IDA-PBC an ordinary differential equation is used to define the immersion map in I&I. It is shown that the energy shaping part of the IDA-PBC controller makes the closed-loop system an interconnection of two lower-dimensional port-Hamiltonian (pH) systems in the on- and off-manifold coordinates that appear in the I&I framework. The effect of damping injection output feedback can be identified with dissipation in the off-manifold part of the inter- connected system. Dissipation is propagated to the on-manifold part which results in asymptotic stability of the system's equilibrium. The particular choice of the I&I design parameters in the present example, including the unconventional definition of coordinates on the invariant manifold, provides an interesting re-interpretation of the IDA-PBC control law from the I&I perspective. Finally, a discussion on the equivalence of the two approaches is presented by examining the cases of linear mechanical systems with one unactuated pivot as well as of general linear mechanical systems. & 2013 European Control Association. Published by Elsevier Ltd. All rights reserved. 1. Introduction The I&I methodology was originally introduced for the stabili- zation of general nonlinear systems in [3]. This work was further developed in a series of publications that have been summarized in [2]. In the I&I approach the desired behavior of the system to be controlled is captured by the choice of a target dynamical system of lower dimension than the original system. The control objective is to find a controller which guarantees that the closed-loop system asymptotically behaves like the target system achieving asymptotic model matching. This should be contrasted with the more restrictive exact matching techniques such as the IDA-PBC methodology. The success of I&I is witnessed by the wide range of applications such as electrical, mechanical and power systems. Recently, there have been several developments, for example in the approach for stabilization [15–17] and speed observation [4] of mechanical systems as well as for the adaptive control of nonlinearly parameterized systems [11]. The IDA-PBC controller design technique on the other hand, cf. [12] for an overview, aims at transforming a nonlinear dyna- mical system by state feedback into a (full order) port-Hamiltonian (pH) system. The target pH structure, cf. the book [20] on the pH state representation, allows for an easy passivity/Lyapunov based stability proof of the closed-loop equilibrium. By the shape of the energy function which is constructed in the course of the design process analytically, an estimate of the domain of attraction is given. A wide range of application examples are solved using IDA-PBC. Recently, some effort has been made in finding sets of reasonable design parameters to enhance transparency with respect to dynamics assignment [8]. Especially for mechanical systems it is convenient in the I&I approach to represent the dynamics on the target manifold as a pH system (which, in contrast to IDA-PBC is lower-dimensional) in order to keep its physical structure. Both design procedures of IDA-PBC and I&I require solutions of partial differential equations, which have certain structural similarities. Note however that I&I can be seen as a relaxation of the IDA-PBC approach, in the sense that less structure is prespecified in advance for the closed-loop system. The goal of the present note is to identify some of these similar structures, in the design procedures, such as coordinate changes and part of the immersion mapping, and based thereon to elaborate on Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ejcon European Journal of Control 0947-3580/$ - see front matter & 2013 European Control Association. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejcon.2013.09.008 n Corresponding author. Tel.: þ49 8928915663. E-mail addresses: kotyczka@tum.de (P. Kotyczka). Ioannis.Sarras@ifpen.fr (I. Sarras). European Journal of Control 19 (2013) 445–453