Abstract
After recalling the definitions of standard port-Hamiltonian systems and their algebraic constraints, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the Hamiltonian function by a general Lagrangian submanifold of the cotangent bundle of the state space manifold, motivated by developments in (Barbero-Linan et al., J. Geom. Mech. 11, 487–510, 2019) and extending the linear theory as developed in (van der Schaft and Maschke, Syst. Control Lett. 121, 31–37, 2018) and (Beattie et al., Math. Control Signals Syst. 30, 17, 2018). The resulting new type of algebraic constraints equations are called Lagrange algebraic constraints. It is shown how Dirac algebraic constraints can be converted into Lagrange algebraic constraints by the introduction of extra state variables, and, conversely, how Lagrange algebraic constraints can be converted into Dirac algebraic constraints by the use of Morse families.
Highlights
When modeling dynamical systems, the appearance of algebraic constraint equations is ubiquitous
The resulting new type of algebraic constraints equations are called Lagrange algebraic constraints. It is shown how Dirac algebraic constraints can be converted into Lagrange algebraic constraints by the introduction of extra state variables, and, how Lagrange algebraic constraints can be converted into Dirac algebraic constraints by the use of Morse families
We have shown how implicit energy storage relations locally can be represented by a Hamiltonian depending on part of the state variables and a complementary part of the co-state variables
Summary
The appearance of algebraic constraint equations is ubiquitous. In [3, 18], these were identified as arising from generalized port-based modeling with a state space that has higher dimension than the minimal number of energy variables, corresponding to implicit energy storage relations which can be formulated as Lagrangian subspaces. 0 ⎦ , J (x) = −J T (x), GT (x) 0 x ∈ X, from eS , eR, eP to fS , fR, fP , and a linear energy dissipation relation eR = −RfR for some matrix R = RT ≥ 0 This yields input-state-output port-Hamiltonian systems x = [J (x) − R(x)]∇H (x) + G(x)u, y = GT (x)∇H (x),. For a general Dirac structure algebraic constraints in the state variables x may appear; leading to port-Hamiltonian systems which are not of the form (5). The definition of the generalized portHamiltonian system (D, L, R) with dynamics (6) extends the definition in the linear case as recently given in [18] (partly motivated by [3])
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