Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust Control of Uncertain Switched Affine Systems via Scenario Optimization

Negar Monir, Mahdieh S. Sadabadi, Sadegh Soudjani

TL;DR

The paper tackles robust switching control for uncertain switched affine systems with parametric uncertainty. It proposes a data-driven scenario optimization framework that designs a switching law based on a quadratic Lyapunov function $V(\xi)$ to create a minimal invariant set of attraction and reduce chattering, without relying on invariant-set relaxations. The approach reformulates the robust constraints as scenario programs SP_N^i and SP_M^ii with probabilistic guarantees via a wait-and-judge theorem, and validates the method on MOIMDPs and power-electronic converters. The results demonstrate robust stabilization within a small invariant set, improved accuracy, and tighter sets than prior methods under uncertainty, enabling scalable application to multi-mode systems.

Abstract

Switched affine systems are often used to model and control complex dynamical systems that operate in multiple modes. However, uncertainties in the system matrices can challenge their stability and performance. This paper introduces a new approach for designing switching control laws for uncertain switched affine systems using data-driven scenario optimization. Instead of relaxing invariant sets, our method creates smaller invariant sets with quadratic Lyapunov functions through scenario-based optimization, effectively reducing chattering effects and regulation error. The framework ensures robustness against parameter uncertainties while improving accuracy. We validate our method with applications in multi-objective interval Markov decision processes and power electronic converters, demonstrating its effectiveness.

Robust Control of Uncertain Switched Affine Systems via Scenario Optimization

TL;DR

The paper tackles robust switching control for uncertain switched affine systems with parametric uncertainty. It proposes a data-driven scenario optimization framework that designs a switching law based on a quadratic Lyapunov function to create a minimal invariant set of attraction and reduce chattering, without relying on invariant-set relaxations. The approach reformulates the robust constraints as scenario programs SP_N^i and SP_M^ii with probabilistic guarantees via a wait-and-judge theorem, and validates the method on MOIMDPs and power-electronic converters. The results demonstrate robust stabilization within a small invariant set, improved accuracy, and tighter sets than prior methods under uncertainty, enabling scalable application to multi-mode systems.

Abstract

Switched affine systems are often used to model and control complex dynamical systems that operate in multiple modes. However, uncertainties in the system matrices can challenge their stability and performance. This paper introduces a new approach for designing switching control laws for uncertain switched affine systems using data-driven scenario optimization. Instead of relaxing invariant sets, our method creates smaller invariant sets with quadratic Lyapunov functions through scenario-based optimization, effectively reducing chattering effects and regulation error. The framework ensures robustness against parameter uncertainties while improving accuracy. We validate our method with applications in multi-objective interval Markov decision processes and power electronic converters, demonstrating its effectiveness.
Paper Structure (10 sections, 3 theorems, 27 equations, 5 figures, 1 algorithm)

This paper contains 10 sections, 3 theorems, 27 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Setup
  3. Data-driven Design of Robust Switching Control Law
  4. Stability Analysis
  5. Scenario Optimization to Design the Invariant Set of Attraction
  6. Case studies
  7. MOIMDPs
  8. Power Electronic Systems
  9. Comparison with an alternative approach
  10. Conclusion

Key Result

theorem 2

Consider the error dynamics of the switched affine system in eq. error dy. From the optimal solution $F > 0$, $W > 0$, and $h$ of the robust convex programming with $Q_\lambda = \Sigma_\pi \lambda_\pi(F - A_\pi^\top F A_\pi)$, $\varrho_\lambda = \Sigma_\pi \lambda_\pi(A_\pi^\top h + A_\pi^\top F L_\pi -h)$ and $c_\lambda = \Sigma_\pi \lambda_\pi(2h^\top L_\pi + L_\pi^\top F L_\pi)$, determine th

Figures (5)

  • Figure 1: IMDP adopted from hahn2019intervalmonir2024lyapunov
  • Figure 2: The value function evolution is plotted for IMDP, considering $\underbar{A}_\pi$ and $\underbar{L}_\pi$ in (a), and $\mkern 1.5mu\overline{\mkern-1.5mu\text{A}\mkern-1.5mu}\mkern 1.5mu_\pi$ and $\mkern 1.5mu\overline{\mkern-1.5mu\text{L}\mkern-1.5mu}\mkern 1.5mu_\pi$ in (b). The switching law is indicated in pink in both (a) and (b). The invariant set of attraction with error trajectories for both lower- and upper-bound matrices is shown in (c). The Lyapunov function in (d) shows that the trajectories approach the invariant set of attraction and remain within the set.
  • Figure 3: The evolution of the value function for MOIMDP considering $\underbar{A}_\pi$ and $\underbar{L}_\pi$ in (a), and $\mkern 1.5mu\overline{\mkern-1.5mu\text{A}\mkern-1.5mu}\mkern 1.5mu_\pi$ and $\mkern 1.5mu\overline{\mkern-1.5mu\text{L}\mkern-1.5mu}\mkern 1.5mu_\pi$ in (b). The switching law is indicated in pink in both (a) and (b). The Lyapunov function in (c) shows that the trajectories approach the invariant set of attraction and remain within the set.
  • Figure 4: The evolution of $x_k$ for lower- and upper-bound matrices is illustrated in (a) and (b). The desired operating point $x_e$ is indicated by light blue in both (a) and (b). The switching law is shown in pink in both (a) and (b), in steady state, it is switching between $\pi = 2$ and $\pi =3$. The invariant set of attraction with error trajectories for both lower- and upper-bound matrices is expressed in (d). The Lyapunov function in (c) shows that the trajectories approach the invariant set of attraction and remain within the set.
  • Figure 5: The evolution of $x_k$ for $\delta = 0.2$ is illustrated in (a). The desired operating point $x_e$ is indicated by red, and the switching law is shown in pink in (a). The invariant set of attraction with error trajectories for $\delta = 0.2$ is expressed in (b). The Lyapunov function in (c) shows that the trajectories approach the invariant set of attraction and remain within the set.

Theorems & Definitions (10)

  • remark 1
  • definition 1
  • theorem 2
  • proof
  • remark 2
  • theorem 3
  • proof
  • remark 3
  • theorem 4
  • proof