Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems

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.

Differential-algebraic systems are generically controllable and stabilizable

“…Port-Hamiltonian models arise in various formulations, such as geometric (by means of, e.g., Dirac structures [6,7,25] and Lagrangian subspaces [29]) or explicit state space models, both in finite- [38] and infinite-dimensional [20,34] spaces. This is, so far, the state of the art in relative genericity of controllability/stabilizability of descriptor port-Hamilonian systems.…”

Linear Boundary Port Hamiltonian Systems defined on Lagrangian submanifolds

“…5.3] [43]. The same basic idea was recently employed for finite-dimensional linear and nonlinear port-Hamiltonian systems in [30,31].…”

“…In this paper, we shall extend the definition of boundary port-Hamiltonian systems in order to include these two cases. This will be done by starting from the recently introduced definition of port-Hamiltonian systems defined on Lagrangian subspaces [30,32] or Lagrangian submanifolds [31] in the finite-dimensional case. In this definition, the energy is no longer defined by a function on the state space, but instead by reciprocal constitutive relations between the state and co-state variables.…”

Differential-algebraic systems with dissipative Hamiltonian structure

“…Constraints typically arise in electrical or transport networks, where Kirchhoff's laws constrain the models at network nodes, as balance equations in chemical engineering, or as holonomic or non-holonomic constraints in mechanical multibody systems. Furthermore, they arise in the interconnection of pH systems when the interface conditions are explicitly formulated and enforced via Lagrange multipliers, see [4,29,36,37,38,39], or the recent survey [30].…”