Interconnection of Lagrange–Dirac systems through nonstandard interaction structures

Abstract

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The Lagrange–Dirac interconnection theory has been developed for primitive subsystems coupled by a standard interaction Dirac structure, i.e. a structure of the form D int = Σ int ⊕ Σ ∘ int , where Σ int ⊂ T ( T ∗ Q ) is a regular distribution, Σ ∘ int ⊂ T ∗ ( T ∗ Q ) is its annihilator and Q is the configuration manifold of the theory. In this work, we extend this theory to allow for parameter-dependent subsystems coupled by nonstandard interaction Dirac structures. This is done, first, by using the Dirac tensor product and, then, by using interaction forces. Both approaches are shown to be equivalent, and also equivalent to a variational principle. After that, we demonstrate the relevance of this generalization by investigating three applications. First, an electromechanical system is modeled; namely, a piston driven by an ideal DC motor through a scotch-yoke mechanism. Second, we relate the interconnection theory to the Euler–Poincaré–Suslov reduction. More specifically, we show that the reduced system may be regarded as an interconnected Lagrange–Dirac system with parameters. The nonholonomic Euler top is presented as a particular instance of this situation. Lastly, control interconnected systems are defined and a control for a planar rigid body with wheels is designed.