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Robust Aperiodic Sampled-Data Washout Control for Uncertain Affine Systems

Folco Giorgetti, Francesco Crocetti, Mario Luca Fravolini, Francesco Ferrante

TL;DR

Problem: stabilize the unknown unforced equilibrium of an uncertain affine plant under aperiodic sampling. Approach: model the closed-loop as a hybrid system with a timer-triggered sampler and memory states, deriving necessary and sufficient conditions for global exponential stability of the unknown equilibrium set $\mathcal{X}_d$ and providing a constructive SOS-based design via clock-dependent LMIs. Contributions: hybrid modeling of aperiodic washout control, exact stability conditions in terms of clock-dependent LMIs, SOS-based algorithm to compute controller gains, and numerical demonstration showing convergence of the plant state to $-A^{-1}d$ and the control to zero. Significance: enables robust stabilization under parametric uncertainty and irregular sampling with a practical design workflow for washout-type controllers in networked settings.

Abstract

In this paper, we address the problem of designing an aperiodic sampled-data controller stabilizing the zero-input equilibrium of an uncertain affine plant. The closed-loop system is modeled as a hybrid dynamical system incorporating a timer triggering the occurrence of the sampling events and two memory states storing the value of the controller state and controller output at each sampling time. Necessary and sufficient conditions on the controller parameters are given to establish the sought property. A constructive controller design algorithm based on sum-of-squares programming is given. A numerical example illustrates the effectiveness of the approach.

Robust Aperiodic Sampled-Data Washout Control for Uncertain Affine Systems

TL;DR

Problem: stabilize the unknown unforced equilibrium of an uncertain affine plant under aperiodic sampling. Approach: model the closed-loop as a hybrid system with a timer-triggered sampler and memory states, deriving necessary and sufficient conditions for global exponential stability of the unknown equilibrium set and providing a constructive SOS-based design via clock-dependent LMIs. Contributions: hybrid modeling of aperiodic washout control, exact stability conditions in terms of clock-dependent LMIs, SOS-based algorithm to compute controller gains, and numerical demonstration showing convergence of the plant state to and the control to zero. Significance: enables robust stabilization under parametric uncertainty and irregular sampling with a practical design workflow for washout-type controllers in networked settings.

Abstract

In this paper, we address the problem of designing an aperiodic sampled-data controller stabilizing the zero-input equilibrium of an uncertain affine plant. The closed-loop system is modeled as a hybrid dynamical system incorporating a timer triggering the occurrence of the sampling events and two memory states storing the value of the controller state and controller output at each sampling time. Necessary and sufficient conditions on the controller parameters are given to establish the sought property. A constructive controller design algorithm based on sum-of-squares programming is given. A numerical example illustrates the effectiveness of the approach.

Paper Structure

This paper contains 12 sections, 5 theorems, 66 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Contributions
  4. Notation
  5. Preliminaries on hybrid systems
  6. Problem Statement
  7. Problem setting
  8. Hybrid modeling
  9. Main Results
  10. Controller Design
  11. Numerical example
  12. Conclusion

Key Result

Lemma 1

Let $d\in\mathbb{{R}}^{n_p}$. Suppose that $(L,G,K, R)$ solves Problem prob:state_feedback. If the pair $(L, K)$ is observable, then there exists $\bar{\xi}\in\mathbb{{R}}^{n_c}$ such that for all $\phi\in\mathcal{S}_{\mathcal{H}}$ where, for all $x\in\mathcal{C}$ $\triangle$

Figures (2)

  • Figure 1: Evolution of the plant state.
  • Figure 2: Evolution of the control input $u$.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Proposition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Proposition 2
  • Remark 4
  • ...and 2 more