Observer Design For a Class of Nonlinear Port Hamiltonian Systems

Port-Hamiltonian systems provide a powerful energy-based modeling framework for multiphysics systems, particularly in the context of nonlinear Electromechanical devices such as soft electrostatic actuators. While passivity-based control techniques are well established for this class of systems, the nonlinear observer design remains significantly less developed, especially in the presence of state-dependent input matrices and nonlinear Hamiltonians. This thesis addresses the observer design problem for a class of nonlinear port-Hamiltonian systems arising from Mechanical and Electromechanical models inspired by soft actuation devices. A first contribution consists in a rigorous nonlinear observability analysis of both Mechanical and Electromechanical structures. A contraction-based framework is then employed to derive structure-preserving observers and to certify exponential convergence of the estimation error within bounded operating domains. The limitations of contraction-based observer synthesis are identified, particularly in the presence of nonlinear input matrices and more general energy functions. To overcome these restrictions, a novel formulation of the estimation error dynamics is developed using an integral Jacobian representation combined with polytopic embedding techniques. This reformulation enables the derivation of convex Linear Matrix Inequality (LMI) conditions for robust observer synthesis based on common quadratic Lyapunov functions. The proposed methodology generalizes existing contraction-based structure-preserving observers and extends their applicability to a broader class of nonlinear port-Hamiltonian systems. Finally, the theoretical results are validated on a nonlinear electromechanical model representative of soft actuator dynamics (HASEL), illustrating both theoretical guarantees and practical performance.