IFAC PapersOnLine 52-15 (2019) 495–500 ScienceDirect ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.11.724 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 1. INTRODUCTION Ionic polymer metal composites (IPMCs) are electro- active systems that can be used either as an actuator or a sensor. Among the diversity of electro-active materials such as piezoelectric materials, magnetostrictive materials etc., IPMCs are more and more used in different applica- tion fields, e.g biomedical applications, bio-manipulation and micro- or macro-electromechanical systems (Shahin- poor, 2016) due to their low-cost voltage, large deforma- tion, wide working frequency ranges and their capability of working in aqueous environments. IPMCs consist of a double electrode layer filled with a polyelectrolyte gel. Cations and solvent molecules migrate toward the cath- ode when a difference in the electric potential is imposed across the two terminals of its double electrode layer. As a consequence, the cathode side swells while the anode side shrinks, entailing a bending effect to the anode side (Park et al., 2010). Based on its physical structure and working principle, various models for IPMCs have been proposed in the literature, going from the black box model (Xiao and Bhattacharya, 2001) to models using more physical insight (Shahinpoor, 2016; Branco et al., 2012). A powerful tool for the modeling and control of complex multi-physical nonlinear systems, called port-Hamiltonian approach, has been introduced and developed in the last decade (Maschke and van der Schaft, 1992). The first port-Hamiltonian modeling of IPMC actua- tors has been proposed in (Nishida et al., 2011). This model consists in three sub-components which are multi- scale, and are all described by distributed parameter sys-  This work is supported by the INFIDHEM project and the Bourgogne-Franche-comt´e Region ANER project under the reference code ANR-16-CE92-0028 and 2018Y-06145, respectively. tems interconnected each other using boundary multi- scale (BMS) coupling elements. By considering the out- domain variables as uniform (Nishida et al., 2011), the BMS works as a differential gyrator, which lets the out- domain variables be multiplied by a characteristic func- tion, meanwhile, makes the in-domain variables be inte- grated spatially. However, there exists a conflict of causal- ity due to the coupling of the mechanical properties of the gel and the mechanical structure (passive moment coupling of equation (54) in (Nishida et al., 2011)). To deal with this conflict, we consider a multiscale model including Lagrange multiplier to account for these me- chanical constraints, and numerically simulate the model more precisely, which includes all coupling relations. The resulting system of differential algebraic equation (DAE) is reduced to an ordinary differential equation (ODE) using coordinates projection. The present paper is organized as follows. In Section 2 is given the constrained port Hamiltonian model of the multiscale IPMC. In Section 3, a finite difference method on staggered grids is applied to discretize the system and the final model is reduced by using coordinates projection. Numerical simulation and conclusions are given in Section 4 and 5, respectively. 2. MODELING OF IPMC The IPMC under investigation (cf. Fig. 1) is of length L, width b and thickness h. It consists of three sub-systems at different scales as shown in Fig. 1. First, an electrical model, which is at a scale of nanometer, is used to represent the fractal-like structure of the double electrical layers. The dynamics of the polyelectolyte gel, at a scale of 100 μm, is described by an electro-stress diffusion coupling model. At last, the global mechanical Copyright © 2019 IFAC 1242 Keywords: Constrained port Hamiltonian system, infinite dimensional system, multi-scale modeling, model reduction, IPMC actuator Abstract: In this paper, a constrained distributed parameter port-Hamiltonian model of the ionic polymer metal composite actuator is proposed. This model describes the multiscale structure of the system. Submodels are coupled by boundary multi-scale elements. In order to preserve the causality of the system, Lagrangian multipliers are introduced to deal with the coupling between the electro-stress diffusion in the polymer and the flexible beam structure of the actuator. Finally, a structure-preserving discretization scheme and some appropriate projections are used to derive an explicit model suitable for simulation. The accuracy of the model is verified using experimental data. * FEMTO-ST CNRS UMR 6174, Universit´e Bourgogne Franche-Comt´e, 26 chemin de l’´epitaphe, F-25030 Besan¸con, France. (e-mail: ning.liu@femto-st.fr; yongxin.wu@femto-st.fr; yann.le.gorrec@ens2m.fr). Ning Liu, * Yongxin Wu, * Yann Le Gorrec * Constrained port Hamiltonian formulation of multiscale distributed parameter IPMC systems 