Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Control of Nonlinear Systems Under Multiple Time-Varying Output Constraints: A Single Funnel Approach

Farhad Mehdifar, Lars Lindemann, Charalampos P. Bechlioulis, Dimos V. Dimarogonas

TL;DR

The paper addresses uncertain nonlinear systems subject to multiple time-varying output constraints by introducing a single scalar $\alpha(t,x)$, representing the signed distance to the constraint boundary, and smoothing it with a log-sum-exp to form a tractable inner-approximation. A single $\alpha$-funnel constraint is designed, along with a robust, low-complexity prescribed performance control (PPC) law that drives $\alpha$ to stay within the funnel without requiring state estimation. Theoretical guarantees include coercivity-based boundedness of the constraint sets and conditions ensuring a unique global maximizer of $\alpha$, together with a proof of forward completeness of the closed-loop signals. A simulation demonstrates finite-time convergence to the time-varying constraint set and invariance thereafter, highlighting the method’s ability to handle coupled constraints in an optimization-free, model-free framework. This work extends prescribed performance and funnel control concepts to general, potentially coupled, time-varying constraints and offers a practical tool for safety-critical nonlinear control tasks.

Abstract

This paper proposes a novel control framework for handling (potentially coupled) multiple time-varying output constraints for uncertain nonlinear systems. First, it is shown that the satisfaction of multiple output constraints boils down to ensuring the positiveness of a scalar variable (the signed distance from the time-varying output-constrained set's boundary). Next, a single funnel constraint is designed properly, whose satisfaction ensures convergence to and invariance of the time-varying output-constrained set. Then a robust and low-complexity funnel-based feedback controller is designed employing the prescribed performance control method. Finally, a simulation example clarifies and verifies the proposed approach.

Control of Nonlinear Systems Under Multiple Time-Varying Output Constraints: A Single Funnel Approach

TL;DR

The paper addresses uncertain nonlinear systems subject to multiple time-varying output constraints by introducing a single scalar , representing the signed distance to the constraint boundary, and smoothing it with a log-sum-exp to form a tractable inner-approximation. A single -funnel constraint is designed, along with a robust, low-complexity prescribed performance control (PPC) law that drives to stay within the funnel without requiring state estimation. Theoretical guarantees include coercivity-based boundedness of the constraint sets and conditions ensuring a unique global maximizer of , together with a proof of forward completeness of the closed-loop signals. A simulation demonstrates finite-time convergence to the time-varying constraint set and invariance thereafter, highlighting the method’s ability to handle coupled constraints in an optimization-free, model-free framework. This work extends prescribed performance and funnel control concepts to general, potentially coupled, time-varying constraints and offers a practical tool for safety-critical nonlinear control tasks.

Abstract

This paper proposes a novel control framework for handling (potentially coupled) multiple time-varying output constraints for uncertain nonlinear systems. First, it is shown that the satisfaction of multiple output constraints boils down to ensuring the positiveness of a scalar variable (the signed distance from the time-varying output-constrained set's boundary). Next, a single funnel constraint is designed properly, whose satisfaction ensures convergence to and invariance of the time-varying output-constrained set. Then a robust and low-complexity funnel-based feedback controller is designed employing the prescribed performance control method. Finally, a simulation example clarifies and verifies the proposed approach.
Paper Structure (12 sections, 4 theorems, 36 equations, 2 figures)

This paper contains 12 sections, 4 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. A Scalar Variable for Constraints Satisfaction
  5. Designing A Single Funnel Constraint
  6. Controller Design and Stability Analysis
  7. Simulation Results
  8. Conclusions
  9. Proof of Lemma \ref{['lem:Omegab_bounded']}
  10. Proof of Lemma \ref{['lem:alphb_coercive_h']}
  11. Proof of Lemma \ref{['lem:global_max_suffi']}
  12. Proof of Theorem \ref{['th:main']}

Key Result

Lemma 1

Under Assumption assum:coercive_alphabar, $\bar{\Omega}(t)$ (resp. $\Omega(t)$) is a bounded set for all $t \geq 0$.

Figures (2)

  • Figure 1: Snapshots of $\bar{\Omega}(t)$ and its corresponding inner-approximation under \ref{['smooth_alph']} for two different examples.
  • Figure 2: Evolution of $\alpha(t,x(t;x_0))$ (top-left) and $x(t)$ under \ref{['eq:PPC_control_explicit']}.

Theorems & Definitions (18)

  • Remark 1
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 8 more