Control of port-Hamiltonian systems with minimal energy supply

Abstract

We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded in dissipative directions. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in a non-dissipative subspace, which is induced by the kernel of the dissipation operator. We prove that all solutions of the optimal control problem exhibit the turnpike phenomenon with respect to this subspace, i.e., for varying initial conditions and varying horizon length the optimal state trajectories evolve close to the conservative subspace for the majority of the time. We conclude this paper by illustrating our finding by a numerical example.

Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems

“…Intuitively, the (not-necessarily strict) dissipativity of such OCPs is clear, as pH-systems are passive with respect to the supplied energy. However, in [20] we have shown that in case of linear finitedimensional pH systems, the minimization of energy supply induces a singular OCP which is strictly dissipative with respect to a specific subspace. More precisely, the analysis in [20] leverages the port-Hamiltonian structure to verify strict dissipativity with respect to the kernel of the dissipation matrix of the pH system.…”

“…In [20] we considered the port-Hamiltonian OCP (1.1) in the finite-dimensional setting, i.e., dim H < ∞, and proved that optimal solutions exhibit a turnpike towards the conservative subspace ker R, i.e., they stay close to this subspace for the majority of the time. Since the situation in the infinite-dimensional case is obviously more involved, let us first evaluate some numerical experiments.…”

“…As the above problem is singular, i.e., there is no quadratic term of the control in the objective functional, established methods to derive turnpike properties, i.e., strict dissipativity with respect to a steady state or Riccati theory cannot be applied. Hence the present paper transfers the subspace approach [20] to the infinite-dimensional setting of (1.1).…”

“…Motivated by the results and observations in [20], we expected a turnpike behavior of optimal solutions of (1.1) towards the subspace ker R 1 2 , which obviously consists of the constant functions. In other words, we expected R 1 2 x(t) to perform a turnpike towards zero.…”

Manifold Turnpikes, Trims and Symmetries

“…Moreover, in [24,25] we analysed a class of timevarying turnpike properties induced by symmetry in a specific class of OCPs formulated on Euler-Lagrange equations. Recently in [26] it was shown that minimization of supplied energy in the context of port-Hamiltonian systems gives rise to an entire linear subspace of turnpikes.…”