Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimization-based control by interconnection of nonlinear port-Hamiltonian systems

Hannes Gernandt, Till Preuster, Manuel Schaller

TL;DR

The paper addresses stabilizing nonlinear port-Hamiltonian ($pH$) systems under input constraints by embedding a finite-horizon optimal-control problem into a continuous-time primal-dual gradient flow that is itself cast as a port-Hamiltonian system. By interconnecting this optimizer dynamics with the plant in a structure-preserving way, the authors derive a suboptimal MPC-type controller that preserves passivity and energy balance. They develop the theory for both unconstrained and inequality-constrained OCPs, proving existence and uniqueness of solutions, weak convergence to optima via LaSalle-type arguments, and, under observability, asymptotic stabilization of the closed-loop. A numerical example demonstrates the practical viability and highlights the energy-dissipation behavior of the suboptimal MPC interconnection, offering a pathway toward suboptimal MPC for PDEs within a modular, energy-based framework.

Abstract

In this paper, we formulate an optimization-based control-by-interconnection approach to the stabilization problem of nonlinear port-Hamiltonian systems. Motivated by model predictive control, the feedback is defined as an initial part of a suboptimal solution of a finite horizon optimal control problem. To this end, we write the optimization method given by a primal-dual gradient dynamics arising from a possibly control-constrained optimal control problem as a port-Hamiltonian system. Then, using the port-Hamiltonian structure of the plant, we show that the MPC-type feedback law is indeed a structure-preserving interconnection of two port-Hamiltonian systems. We prove that, under an observability assumption, the interconnected system asymptotically stabilizes the plant dynamics. We illustrate the theoretical results by means of a numerical example.

Optimization-based control by interconnection of nonlinear port-Hamiltonian systems

TL;DR

The paper addresses stabilizing nonlinear port-Hamiltonian () systems under input constraints by embedding a finite-horizon optimal-control problem into a continuous-time primal-dual gradient flow that is itself cast as a port-Hamiltonian system. By interconnecting this optimizer dynamics with the plant in a structure-preserving way, the authors derive a suboptimal MPC-type controller that preserves passivity and energy balance. They develop the theory for both unconstrained and inequality-constrained OCPs, proving existence and uniqueness of solutions, weak convergence to optima via LaSalle-type arguments, and, under observability, asymptotic stabilization of the closed-loop. A numerical example demonstrates the practical viability and highlights the energy-dissipation behavior of the suboptimal MPC interconnection, offering a pathway toward suboptimal MPC for PDEs within a modular, energy-based framework.

Abstract

In this paper, we formulate an optimization-based control-by-interconnection approach to the stabilization problem of nonlinear port-Hamiltonian systems. Motivated by model predictive control, the feedback is defined as an initial part of a suboptimal solution of a finite horizon optimal control problem. To this end, we write the optimization method given by a primal-dual gradient dynamics arising from a possibly control-constrained optimal control problem as a port-Hamiltonian system. Then, using the port-Hamiltonian structure of the plant, we show that the MPC-type feedback law is indeed a structure-preserving interconnection of two port-Hamiltonian systems. We prove that, under an observability assumption, the interconnected system asymptotically stabilizes the plant dynamics. We illustrate the theoretical results by means of a numerical example.
Paper Structure (16 sections, 18 theorems, 200 equations, 5 figures)

This paper contains 16 sections, 18 theorems, 200 equations, 5 figures.

Table of Contents

  1. Introduction
  2. M-monotone pH systems
  3. Shifted passivity and stability
  4. Structure-preserving interconnection
  5. Monotonicity in optimal control
  6. Stability of open-loop primal-dual gradient dynamics
  7. Stability of unconstrained optimizer dynamics
  8. Stability of constrained optimizer dynamics
  9. Closed-loop: Suboptimal MPC-type controller
  10. The case without control constraints.
  11. Inequality constrained problems
  12. Numerical examples
  13. Conclusion
  14. Appendix
  15. Monotone perturbations of skew-adjoint operators
  16. ...and 1 more sections

Key Result

Proposition 2.2

Let $(M,B)$ be a maximally monotone pH system and $(\overline{x},\overline{u}) \in D(M) \times U$ a steady state pair of $(M,B)$. Then, for all $u\in L^1([0,t_f],U)$ and for almost every $t\in[0,t_f]$ we have the passivity inequality where $\overline{y} = B^*\overline{x}$ is the steady-state output.

Figures (5)

  • Figure 1: Illustration of Model Predictive Control
  • Figure 2: Proposed MPC-type control-by-interconnection scheme. The subscript $\mathrm{p}$ refers to the plant to be controlled where $\mathrm{opt}$ denotes the dynamics of state, control and adjoint in the optimization scheme.
  • Figure 3: Norm of the state component corresponding to the plant with initial value $x_0=(-0.5,-3)^\top$,
  • Figure 4: Visualization of plant inputs on symmetric logarithmic scale: $u_{\mathrm{unconstr}}=\tfrac{1}{\alpha} B^\top \lambda_0$ as in \ref{['eq:intercon']} and saturated input $u_{\mathrm{constr}}$ as in \ref{['eq:coupling']} with $\overline{u}=-\underline{u}=1$, $\alpha = 1.5$
  • Figure 5: Norm of the coupled optimizer-plant dynamics over time.

Theorems & Definitions (35)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof
  • Corollary 4.1
  • ...and 25 more