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Existence of solutions to port-Hamiltonian systems: initial value problems and optimal control

Willem Esterhuizen, Bernhard Maschke, Till Preuster, Manuel Schaller, Karl Worthmann

Abstract

We investigate the existence of solutions of reversible and irreversible port-Hamiltonian systems. To this end, we utilize the associated exergy, a function that is composed of the system's Hamiltonian and entropy, to prove global existence in time for bounded control functions. The results are then leveraged to prove existence of solutions of energy- and entropy-optimal control problems. Last, we explore model predictive control tailored to irreversible port-Hamiltonian systems by means of a numerical case study with a heat exchanger network.

Existence of solutions to port-Hamiltonian systems: initial value problems and optimal control

Abstract

We investigate the existence of solutions of reversible and irreversible port-Hamiltonian systems. To this end, we utilize the associated exergy, a function that is composed of the system's Hamiltonian and entropy, to prove global existence in time for bounded control functions. The results are then leveraged to prove existence of solutions of energy- and entropy-optimal control problems. Last, we explore model predictive control tailored to irreversible port-Hamiltonian systems by means of a numerical case study with a heat exchanger network.

Paper Structure

This paper contains 16 sections, 6 theorems, 103 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Coupled Reversible-Irreversible Port-Hamiltonian Systems
  3. Heat Exchanger Network
  4. Gas-Piston System
  5. RIPHS: Existence of Solutions to Initial-Value Problems
  6. A global existence result for RIPHS
  7. A corollary for dissipative port-Hamiltonian systems
  8. RIPHS: Existence of Solutions to Optimal Control Problems
  9. Applications
  10. Networks of Heat Exchangers
  11. Gas-Piston System
  12. Numerical examples
  13. Optimal control of heat exchanger network
  14. Model predictive control of the heat exchanger network
  15. Conclusion and Outlook
  16. ...and 1 more sections

Key Result

Proposition 3.2

There exists a unique solution to $\mathrm{(IVP)}$ on the maximal interval of existence $I_{x_0,u} = [0,t_{x_0,u}) \subseteq [0,\infty)$, with $t_{x_0,u} > 0$.

Figures (6)

  • Figure 1: A simple heat exchanger.
  • Figure 2: A simple heat exchanger network with thermal conductivity between compartment $i$ and $j$ indicated by $\Lambda_{ij} = \Lambda_{ji}$ and corresponding incidence matrix.
  • Figure 3: A gas-piston system with control as entropy flow into the cylinder.
  • Figure 4: The state trajectories, $x^\star$, the solutions to $(\mathrm{OCP}_T^{\mathrm{heat}})$, with $x_0 = \mathbf{0}$, $\bar{u} = 10$, $\Lambda_{12} = \Lambda_{23}=\Lambda_{34}= \Lambda_{35}= 1$, $\alpha_1 = \alpha_2 = \alpha_3 = 1$ and different horizon lengths, $T$. The turnpike, $x^{\mathrm{tp}} \approx (3.75, 5.57, 6.18, 5.37, 6.88)^\top$, is indicated by the horizontal dashed lines.
  • Figure 5: The state trajectories, $x^\star$, the solution to $(\mathrm{OCP}_T^{\mathrm{heat}})$, showing the turnpike with large $\alpha_2$. Turnpike indicated with dashed horizontal lines.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3: Global existence of solutions
  • proof
  • Remark 3.4
  • Corollary 3.5
  • proof
  • Proposition 3.6
  • ...and 7 more