Modeling and Control of Complex Physical Systems

AbstractMany engineering activities, in particular mechatronic design, require that a multi-domain or ‘multi-physics’ system and its control system be designed as an integrated system. This chapter discusses the background and concepts of a portbased approach to integrated modeling and simulation of physical systems and their controllers, with parameters that are directly related to the real-world system, thus improving insight and direct feedback on modeling decisions. It serves as the conceptual motivation from a physical point of view that is elaborated mathematically and applied to particular cases in the remaining chapters.

In this chapter, we will show how the representation of a lumped-parameter physical system as a bond graph naturally leads to a dynamical system endowed with a geometric structure, called a port-Hamiltonian system. The dynamics are determined by the storage elements in the bond graph (cf. Sect. 1.6.3), as well as the resistive elements (cf. Sect. 1.6.4), while the geometric structure arises from the generalized junction structure of the bond graph. The formalization of this geometric structure as a Dirac structure is introduced as the key mathematical concept to unify the description of complex interactions in physical systems. It will also allow to extend the definition of a finite-dimensional port-Hamiltonian systems as given in this chapter to the infinite-dimensional case in Chapter 4, thus dealing with distributed-parameter physical systems. We will show how this port-Hamiltonian formulation offers powerful methods for the analysis of complex multi-physics systems, also paving the way for the results on control of port-Hamiltonian systems in Chapter 5 and in Chapter 6. Furthermore, we describe how the port-Hamiltonian structure relates to the classical Hamiltonian structure of physical systems as being prominent in e.g. classical mechanics, as well as to the Brayton-Moser description of RLC-circuits.

In this Chapter we present some detailed examples of modelling in several domains using port and port-Hamiltonian concepts, as have been presented in the previous chapters. We start with the electromechanical domain in Sect. 3.1, while in Sect. 3.2 it is shown how port-Hamiltonian systems can be fruitfully used for the structured modelling of robotics mechanisms. In Sect. 3.3, it is show how to model simple elastic systems either in the Lagrangian and Hamiltonian framework, while, in Sect. 3.4, an expressions of the models representing momentum, heat and mass transfer as well as chemical reactions within homogeneous fluids in the port-based formalism is proposed. To this end, the entropy balance and the associated source terms are systematically written in accordance with the principle of irreversible thermodynamics. Some insights are also given concerning the constitutive equations and models allowing to calculate transport and thermodynamic properties. As it will be shown, for each physical domain, these port-based models can be translated into bond-graph models, in the case of distributed as well as lumped parameters models.

This chapter presents the formulation of distributed parameter systems in terms of port-Hamiltonian system. In the first part it is shown, for different examples of physical systems defined on one-dimensional spatial domains, how the Dirac structure and the port-Hamiltonian formulation arise from the description of distributed parameter systems as systems of conservation laws. In the second part we consider systems of two conservation laws, describing two physical domains in reversible interaction, and it is shown that they may be formulated as port-Hamiltonian systems defined on a canonical Dirac structure called canonical Stokes-Dirac structure. In the third part, this canonical Stokes-Dirac structure is generalized for the examples of the Timoshenko beam, a nonlinear flexible link, and the ideal compressible fluid in order to encompass geometrically complex configurations and the convection of momentum.

We discuss in this chapter a number of approaches to exploit the model structure of port-Hamiltonian systems for control purposes. Actually, the formulation of physical control systems as port-Hamiltonian systems may lead in some cases to a re-thinking of standard control paradigms. Indeed, it opens up the way to formulate control problems in a way that is different and perhaps broader than usual. For example, formulating physical systems as port-Hamiltonian systems naturally leads to the consideration of ‘impedance’ control problems, where the behavior of the system at the interaction port is sought to be shaped by the addition of a controller system, and it suggests energy-transfer strategies, where the energy is sought to be transferred from one part the system to another. Furthermore, it naturally leads to the investigation of a particular type of dynamic controllers, namely those that can be also represented as port-Hamiltonian systems and that are attached to the given plant system in the same way as a physical system is interconnected to another physical system. As an application of this strategy of ‘control by interconnection’ within the port-Hamiltonian setting we consider the problem of (asymptotic) stabilization of a desired equilibrium by shaping the Hamiltonian into a Lyapunov function for this equilibrium. From a mathematical point of view we will show that the mathematical formalism of port-Hamiltonian systems provides various useful techniques, ranging from Casimir functions, Lyapunov function generation, shaping of the Dirac structure by composition, and the possibility to combine finitedimensional and infinite-dimensional systems.

Infinite dimensional port Hamiltonian systems have been introduced in Chapter 4 as a novel framework for modeling and control distributed parameter systems. In this chapter, some results regarding control applications are presented. In some sense, it is more correct to speak about preliminary results in control of distributed port Hamiltonian systems, since a general theory, as the one discussed in Chapter 5 for the finite dimensional port Hamiltonian systems, has not been completely developed, yet. We start with a short overview on the stability problem for distributed parameter systems in Sect. 6.2, together with some simple but useful stability theorems. Then, in Sect. 6.3, the control by damping injection is generalized to the infinite dimensional case and an application to the boundary and distributed control of the Timoshenko beam is presented. In Sect. 6.4, a simple generalization of the control by interconnection and energy shaping to the infinite dimensional framework is discussed. In particular, the control scheme is developed in order to cope with a simple mixed finite and infinite dimensional port Hamiltonian system. Then, an application to the dynamical control of a Timoshenko beam is discussed in Sect. 6.5.