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second edition; International audience; Dissipative Systems Analysis and Control (second edition) presents a fully revised and expanded treatment of dissipative systems theory, constituting a self-contained, advanced introduction for graduate students, researchers and practising engineers. It examines linear and nonlinear systems with examples of both in each chapter; some infinite-dimensional examples are also included. Throughout, emphasis is placed on the use of the dissipative properties of a system for the design of stable feedback control laws. The theory is substantiated by experimental results and by reference to its application in illustrative physical cases (Lagrangian and Hamiltonian systems and passivity-based and adaptive controllers are covered thoroughly). The second edition is substantially reorganized both to accommodate new material and to enhance its pedagogical properties. Some of the changes introduced are: * Complete proofs of the main theorems and lemmas. * The Kalman-Yakubovich-Popov Lemma for non-minimal realizations, singular systems, and discrete-time systems (linear and nonlinear). * Passivity of nonsmooth systems (differential inclusions, variational inequalities, Lagrangian systems with complementarity conditions). * Sections on optimal control and H-infinity theory. * An enlarged bibliography with more than 550 references, and an augmented index with more than 500 entries. * An improved appendix with introductions to viscosity solutions, Riccati equations and some useful matrix algebra.

This paper discusses the mixed finite-element based Hamiltonian discretization of a distributed RC circuit aimed at modeling the impedance of rough electrodes of ionic polymer actuators. The accuracy of the discretized model is discussed by comparing the frequency responses between the discretized and the exact solution in the case of the constant impedance distribution. The convergence analysis and the numerical example are shown in order to evaluate the approximation.

International audience

An energy balance equation with respect to a control contact system provides port outputs which are conjugated to inputs. These conjugate variables are used to define the composition of port contact systems in the contact framework. We then propose a power-conserving interconnection structure, which generalizes the Dirac interconnection in the Hamiltonian formalism. Furthermore, the composed system is again a port contact system, as illustrated on the example of a gas-piston system undergoing some irreversible transformation.

Port-based network modeling of a large class of complex physical systems leads to dynamical systems known as port-Hamiltonian systems. The key ingredient of any port-Hamiltonian system is a power-conserving interconnection structure (mathematically formalized by the geometric notion of a Dirac structure) linking the pairs of conjugate port variables of the various ports corresponding to energy storage (defined by a Hamiltonian function depending on energy variables), resistive effects, external interaction, etc. The interconnection of port-Hamiltonian systems defines a new port-Hamiltonian system with Dirac structure determined by the Dirac structures of the constituent parts. For thermodynamic systems this framework needs modification by extending the space of energy variables, as used for port-Hamiltonian systems, into a space of energy and co-energy variables together with an additional coordinate needed for the formulation of the energy. Geometrically this extended space is formalized as a contact manifold. The thermodynamic properties of the system are given by a Legendre submanifold of the contact manifold. Furthermore a contact Hamiltonian is defined, related to the internal power-conserving interconnection structure, whose resulting dynamics leaves invariant the Legendre submanifold. Finally, interaction contact Hamiltonians are defined together with port-conjugated pairs of input and output variables modeling the interaction of the system with its environment. Interconnection of such thermodynamic systems is shown to lead to a thermodynamic system with the same structure.

Associated with a skew-symmetric linear operator on the spatial domain $[a,b]$ we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated to this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko Beam.