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Koopman-based feedback design with stability guarantees

Robin Strässer, Manuel Schaller, Karl Worthmann, Julian Berberich, Frank Allgöwer

Abstract

We present a method to design a state-feedback controller ensuring exponential stability for nonlinear systems using only measurement data. Our approach relies on Koopman-operator theory and uses robust control to explicitly account for approximation errors due to finitely many data samples. To simplify practical usage across various applications, we provide a tutorial-style exposition of the feedback design and its stability guarantees for single-input systems. Moreover, we extend this controller design to multi-input systems and more flexible nonlinear state-feedback controllers using gain-scheduling techniques to increase the guaranteed region of attraction. As the proposed controller design is framed as a semidefinite program, it allows for an efficient solution. Further, we enhance the geometry of the region of attraction through a heuristic algorithm that establishes a connection between the employed Koopman lifting and the dynamics of the system. Finally, we validate the proposed feedback design procedure by means of numerical examples.

Koopman-based feedback design with stability guarantees

Abstract

We present a method to design a state-feedback controller ensuring exponential stability for nonlinear systems using only measurement data. Our approach relies on Koopman-operator theory and uses robust control to explicitly account for approximation errors due to finitely many data samples. To simplify practical usage across various applications, we provide a tutorial-style exposition of the feedback design and its stability guarantees for single-input systems. Moreover, we extend this controller design to multi-input systems and more flexible nonlinear state-feedback controllers using gain-scheduling techniques to increase the guaranteed region of attraction. As the proposed controller design is framed as a semidefinite program, it allows for an efficient solution. Further, we enhance the geometry of the region of attraction through a heuristic algorithm that establishes a connection between the employed Koopman lifting and the dynamics of the system. Finally, we validate the proposed feedback design procedure by means of numerical examples.
Paper Structure (16 sections, 4 theorems, 98 equations, 6 figures, 2 algorithms)

This paper contains 16 sections, 4 theorems, 98 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem setting
  4. Data-driven system representation using the Koopman operator
  5. Koopman operator and its approximation
  6. EDMD error bounds
  7. Data-driven control of nonlinear systems
  8. Reformulation as robust linear control problem
  9. Solution via linear matrix inequalities
  10. General state-feedback laws reducing conservatism
  11. On the geometry of the region of attraction
  12. Numerical examples
  13. Nonlinear system with invariant Koopman lifting
  14. Nonlinear system without invariant Koopman lifting
  15. Inverted pendulum
  16. ...and 1 more sections

Key Result

Proposition 3

Suppose that Assumption ass:invariance-of-dictionary holds and the data samples are i.i.d. Further, let an error bound $c_r>0$ and a probabilistic tolerance $\delta \in (0,1)$ be given. Then, there is an amount of data $d_0\in\mathbb{N}$ such for all $d\geq d_0$ and all $u\in\mathbb{U}$ we have the with probability $1-\delta$.

Figures (6)

  • Figure 1: Illustration of the regions $\mathbf{\Delta}_x$ (), $\mathbf{\Delta}_{\Phi,1}$ (), and $\mathbf{\Delta}_{\Phi,2}$ ().
  • Figure 2: RoA $\mathcal{X}_\mathrm{RoA}$ corresponding to Section \ref{['exmp:cooked-up-eigenfunctions']} and the controller $\mu$.
  • Figure 3: Region containing all $x$ with $\hat{\Phi}(x)\in\mathbf{\Delta}_\Phi$ () corresponding to Section \ref{['exmp:cooked-up-overapprox']} and the RoA $\mathcal{X}_\mathrm{RoA}$ () for the controller $\mu_1$.
  • Figure 4: Region containing all $x$ with $\hat{\Phi}(x)\in\mathbf{\Delta}_\Phi$ () corresponding to Section \ref{['exmp:cooked-up-overapprox']} and the RoA $\mathcal{X}_\mathrm{RoA}$ () for the controller $\mu_2$.
  • Figure 5: Region containing all $x$ with $\hat{\Phi}(x)\in\mathbf{\Delta}_\Phi$ () corresponding to Section \ref{['exmp:inverse-pendulum-sin']} and the RoA $\mathcal{X}_\mathrm{RoA}$ for the controllers $\mu_1$ () and $\mu_2$ ().
  • ...and 1 more figures

Theorems & Definitions (10)

  • Remark 1
  • Proposition 3: schaller:worthmann:philipp:peitz:nuske:2023
  • Remark 4
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Remark 9