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Multi-layer barrier function-based adaptive super-twisting controller

Antoine Thibault Vié, Leonid Fridman, Roberto Galeazzi, Dimitrios Papageorgiou

TL;DR

This work addresses adaptive control for uncertain first-order systems with rate-bounded disturbances of unknown magnitude, where discrete-time operation can jeopardize trajectory bounds. It develops an adaptive Super-Twisting Sliding Mode Controller augmented with positive semidefinite barrier functions and a nested multi-layer barrier architecture, enabling two modes of gain modulation: inside-barrier adaptation and outside-barrier adaptation. Key contributions include (i) barrier-function-based adaptive gains within a predefined barrier, (ii) a complementary outside-barrier mode for large initial conditions and perturbations, and (iii) a multi-layer design that uses inner and outer barrier sets to handle varying perturbation bounds and sampling effects with Lyapunov guarantees and finite-time convergence to barrier sets. The approach reduces conservatism, improves boundedness and robustness for discrete-time implementations, and demonstrates effectiveness through simulations with step and sinusoidal disturbances.

Abstract

This article presents an adaptive Super-Twisting Sliding Mode Control framework for uncertain first-order systems, with rate-bounded perturbations, where the bound is constant but unknown. Positive definite barrier functions, when used in self-tuning super-twisting controllers may introduce some conservatism in relation to initial estimations of the perturbation rate bound. Moreover, discrete time implementation of the algorithm does not necessarily guarantee the boundedness of the closed-loop trajectories when sudden changes in the perturbation occur in between two time samples. The salient features of the proposed methodology pertain to extending the use of positive semidefinite barrier functions to Super-Twisting controller adaptation and the employment of a "nested barriers" scheme that ensures boundedness of the solutions even for "unfavourable" perturbations-to-sampling time ratios. The stability of the closed-loop system is assessed via Lyapunov analysis and simulations demonstrate the efficacy of the proposed framework.

Multi-layer barrier function-based adaptive super-twisting controller

TL;DR

This work addresses adaptive control for uncertain first-order systems with rate-bounded disturbances of unknown magnitude, where discrete-time operation can jeopardize trajectory bounds. It develops an adaptive Super-Twisting Sliding Mode Controller augmented with positive semidefinite barrier functions and a nested multi-layer barrier architecture, enabling two modes of gain modulation: inside-barrier adaptation and outside-barrier adaptation. Key contributions include (i) barrier-function-based adaptive gains within a predefined barrier, (ii) a complementary outside-barrier mode for large initial conditions and perturbations, and (iii) a multi-layer design that uses inner and outer barrier sets to handle varying perturbation bounds and sampling effects with Lyapunov guarantees and finite-time convergence to barrier sets. The approach reduces conservatism, improves boundedness and robustness for discrete-time implementations, and demonstrates effectiveness through simulations with step and sinusoidal disturbances.

Abstract

This article presents an adaptive Super-Twisting Sliding Mode Control framework for uncertain first-order systems, with rate-bounded perturbations, where the bound is constant but unknown. Positive definite barrier functions, when used in self-tuning super-twisting controllers may introduce some conservatism in relation to initial estimations of the perturbation rate bound. Moreover, discrete time implementation of the algorithm does not necessarily guarantee the boundedness of the closed-loop trajectories when sudden changes in the perturbation occur in between two time samples. The salient features of the proposed methodology pertain to extending the use of positive semidefinite barrier functions to Super-Twisting controller adaptation and the employment of a "nested barriers" scheme that ensures boundedness of the solutions even for "unfavourable" perturbations-to-sampling time ratios. The stability of the closed-loop system is assessed via Lyapunov analysis and simulations demonstrate the efficacy of the proposed framework.

Paper Structure

This paper contains 12 sections, 2 theorems, 47 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. Barrier function-based adaptive STSMC
  4. Adaptation inside the barrier
  5. Adaptation outside the barrier
  6. Multi-layered barrier function design
  7. Basic concept and mechanism
  8. Multi-layered barrier function algorithm
  9. Simulation
  10. Sequence of steps
  11. Varying frequency sinusoid
  12. CONCLUSIONS AND FUTURE WORKS

Key Result

Proposition 1

The control law eq:nonHomogeneousSTSMC with $k_1 = k_{1,\alpha}(s,\epsilon)$, $k_2 = k_{2,\alpha}(s,\epsilon)$ ensures that the trajectories of the closed loop system eq:STA, once started in the set $\vert s \vert < \epsilon$, they will remain there for all future times.

Figures (7)

  • Figure 1: $k_{1,\alpha}(s,\epsilon)$ for $\epsilon \in \left[1e^{-4},1e^{-1}\right], \lvert s \rvert < 0.99\epsilon$, $\alpha = \frac{1}{2}$
  • Figure 2: Conceptual illustration of three-layer barrier functions architecture (left) with step disturbances compared to single-layer barrier function (right). In the latter case the system solely relies on dynamic adaptation outside $\vert s \vert \leq \epsilon_1$.
  • Figure 3: Perturbation as a sequence of step changes and its impulse-like derivative $\delta(t)$.
  • Figure 4: Sliding variable $s$ for step sequence perturbation with single barrier function (top) and double-layered architecture (bottom). The dashed black lines depict the inner layer $\epsilon_1$ and the dashed green lines depict the outer layer $\epsilon_2$.
  • Figure 5: Control signal for sequence of step changes in single-barrier (top) and double-barrier (bottom) architecture.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof