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Optimal control of port-Hamiltonian systems: energy, entropy, and exergy

Friedrich Philipp, Manuel Schaller, Karl Worthmann, Timm Faulwasser, Bernhard Maschke

Abstract

We consider irreversible and coupled reversible-irreversible nonlinear port-Hamiltonian systems and the respective sets of thermodynamic equilibria. In particular, we are concerned with optimal state transitions and output stabilization on finite-time horizons. We analyze a class of optimal control problems, where the performance functional can be interpreted as a linear combination of energy supply, entropy generation, or exergy supply. Our results establish the integral turnpike property towards the set of thermodynamic equilibria providing a rigorous connection of optimal system trajectories to optimal steady states. Throughout the paper, we illustrate our findings by means of two examples: a network of heat exchangers and a gas-piston system.

Optimal control of port-Hamiltonian systems: energy, entropy, and exergy

Abstract

We consider irreversible and coupled reversible-irreversible nonlinear port-Hamiltonian systems and the respective sets of thermodynamic equilibria. In particular, we are concerned with optimal state transitions and output stabilization on finite-time horizons. We analyze a class of optimal control problems, where the performance functional can be interpreted as a linear combination of energy supply, entropy generation, or exergy supply. Our results establish the integral turnpike property towards the set of thermodynamic equilibria providing a rigorous connection of optimal system trajectories to optimal steady states. Throughout the paper, we illustrate our findings by means of two examples: a network of heat exchangers and a gas-piston system.
Paper Structure (14 sections, 11 theorems, 100 equations, 5 figures, 1 table)

This paper contains 14 sections, 11 theorems, 100 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Coupled reversible-irreversible pH-Systems
  3. Definition of reversible-irreversible pH-Systems
  4. Tutorial Examples
  5. Optimal state transitions
  6. Optimal steady states
  7. Turnpikes towards thermodynamic equilibria
  8. Optimal output stabilization
  9. Numerical results
  10. Stabilization of a heat exchanger
  11. Set-point transition for a gas-piston system
  12. Network of heat exchangers
  13. Conclusions and future work
  14. Distances

Key Result

Lemma 2.4

The set of thermodynamic equilibria $\mathcal{T}$ is closed in $\mathbb X$. If $S(x) = e^\top x$ with some $e\in\mathbb R^n$ and for all $x\in\mathbb X$, we have then $\mathcal{T}$ is a $C^1$-submanifold of $\mathbb X$. It is empty if and only if $H_x(\mathbb X)\cap\bigcap_{k=1}^N(J_ke)^\perp = \varnothing$. Otherwise, its dimension is given by

Figures (5)

  • Figure 1: Simple model of a heat exchanger
  • Figure 2: Depiction of the optimal intensive variable, entropy bracket and control over time for output stabilization.
  • Figure 3: Depiction of the optimal variables for the gas-piston system.
  • Figure 4: Network of five compartments exchanging heat.
  • Figure 5: Entropies in the individual compartments for the network of heat exchangers and generated entropy.

Theorems & Definitions (33)

  • Definition 2.1: Reversible-irreversible pH-Systems
  • Remark 2.2: Linear and affine input structures
  • Remark 2.3: Local existence of solutions
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Example 2.6: Heat exchanger
  • Example 2.7: Gas-piston system, cf. vdS23Ramirez_EJC13
  • Proposition 3.1
  • ...and 23 more