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On $\ell_2$-performance of weakly-hard real-time control systems

Marc Seidel, Simon Lang, Frank Allgöwer

TL;DR

This work addresses $\ell_2$-performance guarantees for linear plants with unreliable feedback governed by weakly-hard real-time (WHRT) constraints. It models the problem as a graph-constrained switched system (CSS) and derives graph-aware, LMI-based conditions for $\ell_2$-gain $\gamma$, leveraging a lifting method to enable tractable controller synthesis. The main contributions are (i) a sufficient $\ell_2$-performance condition for CSSs under WHRT switching, (ii) a lifted-system formulation that preserves $\ell_2$-gain while reducing graph size and enabling LMIs-based controller synthesis with $K=RG^{-1}$, and (iii) a numerical example showing significant performance improvements and computational savings, along with discussions on extensions to switched controllers and broader dissipativity frameworks. The results provide practical tools for ensuring performance in networked and real-time control where packet losses and deadlines follow WHRT patterns, with potential applicability to general graph-constrained switched systems and future work toward nonlinear and output-feedback settings.

Abstract

This paper considers control systems with failures in the feedback channel, that occasionally lead to loss of the control input signal. A useful approach for modeling such failures is to consider window-based constraints on possible loss sequences, for example that at least r control attempts in every window of s are successful. A powerful framework to model such constraints are weakly-hard real-time constraints. Various approaches for stability analysis and the synthesis of stabilizing controllers for such systems have been presented in the past. However, existing results are mostly limited to asymptotic stability and rarely consider performance measures such as the resulting $\ell_2$-gain. To address this problem, we adapt a switched system description where the switching sequence is constrained by a graph that captures the loss information. We present an approach for $\ell_2$-performance analysis involving linear matrix inequalities (LMI). Further, leveraging a system lifting method, we propose an LMI-based approach for synthesizing state-feedback controllers with guaranteed $\ell_2$-performance. The results are illustrated by a numerical example.

On $\ell_2$-performance of weakly-hard real-time control systems

TL;DR

This work addresses -performance guarantees for linear plants with unreliable feedback governed by weakly-hard real-time (WHRT) constraints. It models the problem as a graph-constrained switched system (CSS) and derives graph-aware, LMI-based conditions for -gain , leveraging a lifting method to enable tractable controller synthesis. The main contributions are (i) a sufficient -performance condition for CSSs under WHRT switching, (ii) a lifted-system formulation that preserves -gain while reducing graph size and enabling LMIs-based controller synthesis with , and (iii) a numerical example showing significant performance improvements and computational savings, along with discussions on extensions to switched controllers and broader dissipativity frameworks. The results provide practical tools for ensuring performance in networked and real-time control where packet losses and deadlines follow WHRT patterns, with potential applicability to general graph-constrained switched systems and future work toward nonlinear and output-feedback settings.

Abstract

This paper considers control systems with failures in the feedback channel, that occasionally lead to loss of the control input signal. A useful approach for modeling such failures is to consider window-based constraints on possible loss sequences, for example that at least r control attempts in every window of s are successful. A powerful framework to model such constraints are weakly-hard real-time constraints. Various approaches for stability analysis and the synthesis of stabilizing controllers for such systems have been presented in the past. However, existing results are mostly limited to asymptotic stability and rarely consider performance measures such as the resulting -gain. To address this problem, we adapt a switched system description where the switching sequence is constrained by a graph that captures the loss information. We present an approach for -performance analysis involving linear matrix inequalities (LMI). Further, leveraging a system lifting method, we propose an LMI-based approach for synthesizing state-feedback controllers with guaranteed -performance. The results are illustrated by a numerical example.
Paper Structure (16 sections, 5 theorems, 26 equations, 3 figures)

This paper contains 16 sections, 5 theorems, 26 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Notation
  3. SETUP
  4. WHRT constraints
  5. Switched system
  6. $\ell_2$-performance
  7. $\ell_2$-PERFORMANCE ANALYSIS FOR CSSs
  8. Graph representation
  9. $\ell_2$-performance analysis result
  10. LIFTED SYSTEM
  11. Main idea and lifting
  12. Application to WHRT control systems
  13. $\ell_2$-performance for the lifted system
  14. CONTROLLER SYNTHESIS
  15. NUMERICAL EXAMPLE
  16. ...and 1 more sections

Key Result

Theorem 1

The CSS eq:switchedSystem, whose switching is captured by a graph $\mathcal{G}$ according to Definition def:switchingGraph, has $\ell_2$-performance with gain $\gamma$ if there exist symmetric matrices $S_i \in \mathbb{R}^{n \times n}$ and matrices $G_i \in \mathbb{R}^{n \times n}$, $i = 1,...,n_\ma A Lyapunov function is then given by

Figures (3)

  • Figure 1: The WHRT control system with losses in the controller-actuator connection.
  • Figure 2: An exemplary loss sequence $\mu_k = 1 \, 0 \, 0 \, 1 \, 1 \, 1 \, 0 \hdots$ satisfying the constraint $\lambda = \genfrac{(}{)}{0pt}{}{2}{4}$: In every window of length 4, there are at least 2 successful control attempts.
  • Figure 3: Left: non-lifted WHRT graph with labels representing the loss sequence $\mu$; right: its lifted counterpart with labels representing $\alpha$.

Theorems & Definitions (18)

  • Definition 1: Bernat2001a
  • Definition 2: Bernat2001a
  • Definition 3: Bernat2001a
  • Definition 4: Bernat2001a
  • Definition 5: Bernat2001a
  • Definition 6: WHRT control system
  • Definition 7: $\ell_2$-performance
  • Definition 8: WHRT graph
  • Theorem 1: $\ell_2$-performance analysis
  • proof
  • ...and 8 more