Port-Hamiltonian Differential-Algebraic Systems
The basic starting point of port-Hamiltonian systems theory is network modeling; considering the overall physical system as the interconnection of simple subsystems, mutually influencing each other via energy flow. As a result of the interconnections alge
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Abstract The basic starting point of port-Hamiltonian systems theory is network modeling; considering the overall physical system as the interconnection of simple subsystems, mutually influencing each other via energy flow. As a result of the interconnections algebraic constraints between the state variables commonly arise. This leads to the description of the system by differential-algebraic equations (DAEs), i.e., a combination of ordinary differential equations with algebraic constraints. The basic point of view put forward in this survey paper is that the differential-algebraic equations that arise are not just arbitrary, but are endowed with a special mathematical structure; in particular with an underlying geometric structure known as a Dirac structure. It will be discussed how this knowledge can be exploited for analysis and control. Keywords Port-Hamiltonian systems · Passivity · Algebraic constraints · Kinematic constraints · Casimirs · Switching systems · Dirac structure · Interconnection Mathematics Subject Classification (2010) 34A09 · 37J05 · 70G45 · 93B10 · 93B27 · 93C10
1 Introduction to Port-Hamiltonian Differential-Algebraic Systems The framework of port-Hamiltonian systems is intended to provide a systematic approach to the modeling, analysis, simulation and control of, possibly large-scale, multi-physics systems; see [9, 15, 19, 20, 24, 25, 29, 31–34, 38, 39] for some key references. Although the framework includes distributed-parameter systems as well, we will focus in this paper on lumped-parameter, i.e., finite-dimensional, systems. A.J. van der Schaft (B) Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands e-mail: [email protected] A. Ilchmann, T. Reis (eds.), Surveys in Differential-Algebraic Equations I, Differential-Algebraic Equations Forum, DOI 10.1007/978-3-642-34928-7_5, © Springer-Verlag Berlin Heidelberg 2013
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The basic starting point of port-Hamiltonian systems theory is (power-based) network modeling; considering the overall system as the interconnection of simple subsystems, mutually influencing each other via energy flow [27]. As a result of the interconnections algebraic constraints between the state variables commonly arise. This leads to the description of the system by differential-algebraic equations (DAEs), i.e., a combination of ordinary differential equations with algebraic constraints. However, the basic point of view put forward in this paper is that the differential-algebraic equations that arise are not just arbitrary differentialalgebraic equations, but are endowed with a special mathematical structure, which may be fruitfully used for analysis, simulation and control. As a motivating and guiding example for the theory surveyed in this paper we will start with the following example.
1.1 A Motivating Example Consider an LC-circuit consisting of two capacitors and one inductor, all in parallel. Naturally this system can be seen as the interconn
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