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Stabilization of monotone control systems with input constraints

Till Preuster, Hannes Gernandt, Manuel Schaller

Abstract

We present a stabilizing output-feedback controller for nonlinear finite and infinite-dimensional control systems governed by monotone operators that respects given input constraints. In particular, we show that if a system is stabilizable with unconstrained controls, and if the control corresponding to the desired equilibrium is in the interior of the control constraint set, then a saturated version of the controller also achieves stabilization under any given control constraints. The class of monotone systems under consideration encompasses port-Hamiltonian systems, and we demonstrate our findings using a heat equation, a wave equation, and a finite-dimensional nonlinear Hamiltonian system.

Stabilization of monotone control systems with input constraints

Abstract

We present a stabilizing output-feedback controller for nonlinear finite and infinite-dimensional control systems governed by monotone operators that respects given input constraints. In particular, we show that if a system is stabilizable with unconstrained controls, and if the control corresponding to the desired equilibrium is in the interior of the control constraint set, then a saturated version of the controller also achieves stabilization under any given control constraints. The class of monotone systems under consideration encompasses port-Hamiltonian systems, and we demonstrate our findings using a heat equation, a wave equation, and a finite-dimensional nonlinear Hamiltonian system.
Paper Structure (9 sections, 13 theorems, 102 equations, 7 figures)

This paper contains 9 sections, 13 theorems, 102 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Constrained stabilization of monotone control systems
  4. Main result: Control-constrained stabilization
  5. Detectability and the VOVS property
  6. Numerical examples
  7. A finite-dimensional example
  8. Two-dimensional heat equation
  9. Two-dimensional wave equation

Key Result

Lemma 3.1

Let $F \subset U$ be closed and convex. Then, it holds that $P_F$ is firmly nonexpansive, i.e., for all $x,z \in U$.

Figures (7)

  • Figure 1: Top: Closed-loop state trajectories $x_1(t)$, $x_2(t)$ of \ref{['eq:ex_closed_loop:state']} with initial value $x_0=(2.0,-3.0)$. Bottom: Saturated control input $u(t)=P_{[a,b]}(-(x_1(t)-x_1^\star))$.
  • Figure 2: The controlled equilibrium $(x^\star,u^\star)$ with $\Omega_c=(0.2, 0.8)^2$.
  • Figure 3: The plot $(b)$ shows the controller \ref{['eq:heat_control']} in action. The corresponding state at time $t=5.0$ is shown in $(a)$.
  • Figure 4: Top: Convergence of the closed-loop state and control. Bottom: Feasibility of the control.
  • Figure 5: Dogbone domain $\Omega$ with collars ($\Omega_1,\Omega_4$) in both lobes and two strips ($\Omega_2,\Omega_3$) along the neck edges. The control region is $\Omega_c=\cup_{i=1,\ldots,4} \Omega_i$.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Definition 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Lemma 3.5
  • Theorem 3.6
  • ...and 17 more