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Provably Safe Control for Constrained Nonlinear Systems with Bounded Input

Saurabh Kumar, Shashi Ranjan Kumar, Abhinav Sinha

TL;DR

This paper addresses safe control of a class of nonlinear systems subject to actuator saturation and output constraints. It introduces a smooth asymmetric input saturation model that keeps the actual actuator input $u$ within a safe set while preserving stability, and couples it with a backstepping-based output-tracking controller to achieve exact tracking under bounded inputs. For constrained outputs, it leverages barrier Lyapunov functions to enforce time-varying output constraints, yielding semi-global stability results and guaranteeing boundedness of all closed-loop signals. Numerical simulations on a second-order nonlinear system validate that the proposed approach achieves robust tracking while respecting both actuator and output constraints, demonstrating practical safety advantages in constrained settings.

Abstract

In real-world control applications, actuator constraints and output constraints (specifically in tracking problems) are inherent and critical to ensuring safe and reliable operation. However, generally, control strategies often neglect these physical limitations, leading to potential instability, degraded performance, or even system failure when deployed on real-world systems. This paper addresses the control design problem for a class of nonlinear systems under both actuator saturation and output constraints. First, a smooth asymmetric saturation model (a more generic representative of practical scenarios) is proposed to model actuator saturation, which ensures that the control inputs always remain confined within a predefined set to ensure safety. Based on the proposed model, we develop a nonlinear control framework that guarantees output tracking while ensuring that system output remains confined to the predefined set. Later, we integrate this design with the constrained output tracking control problem, wherein we show that the system output tracks its desired trajectory by simultaneously satisfying input and output constraints. The global stabilization of the tracking error is achieved in the presence of input constraints, while semi-global stabilization is achieved in the presence of both input and output constraints. Additionally, we rigorously establish the boundedness of all closed-loop signals under the proposed design. Simulation results demonstrate the effectiveness of the proposed methods in handling asymmetric constraints while achieving desirable tracking performance.

Provably Safe Control for Constrained Nonlinear Systems with Bounded Input

TL;DR

This paper addresses safe control of a class of nonlinear systems subject to actuator saturation and output constraints. It introduces a smooth asymmetric input saturation model that keeps the actual actuator input within a safe set while preserving stability, and couples it with a backstepping-based output-tracking controller to achieve exact tracking under bounded inputs. For constrained outputs, it leverages barrier Lyapunov functions to enforce time-varying output constraints, yielding semi-global stability results and guaranteeing boundedness of all closed-loop signals. Numerical simulations on a second-order nonlinear system validate that the proposed approach achieves robust tracking while respecting both actuator and output constraints, demonstrating practical safety advantages in constrained settings.

Abstract

In real-world control applications, actuator constraints and output constraints (specifically in tracking problems) are inherent and critical to ensuring safe and reliable operation. However, generally, control strategies often neglect these physical limitations, leading to potential instability, degraded performance, or even system failure when deployed on real-world systems. This paper addresses the control design problem for a class of nonlinear systems under both actuator saturation and output constraints. First, a smooth asymmetric saturation model (a more generic representative of practical scenarios) is proposed to model actuator saturation, which ensures that the control inputs always remain confined within a predefined set to ensure safety. Based on the proposed model, we develop a nonlinear control framework that guarantees output tracking while ensuring that system output remains confined to the predefined set. Later, we integrate this design with the constrained output tracking control problem, wherein we show that the system output tracks its desired trajectory by simultaneously satisfying input and output constraints. The global stabilization of the tracking error is achieved in the presence of input constraints, while semi-global stabilization is achieved in the presence of both input and output constraints. Additionally, we rigorously establish the boundedness of all closed-loop signals under the proposed design. Simulation results demonstrate the effectiveness of the proposed methods in handling asymmetric constraints while achieving desirable tracking performance.

Paper Structure

This paper contains 8 sections, 7 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Asymmetric Input Saturation Model
  5. Output Tracking with Bounded Control
  6. Constrained Output Tracking with Bounded Control
  7. Numerical Simulations
  8. Conclusions

Key Result

Lemma 1

(doi:10.1016/j.automatica.2008.11.017) Let $\mathbb{T}:=\{ \chi \in \mathbb{R}:\lvert \chi \rvert < 1 \} \subset \mathbb{R}$ and $\mathbb{Q}:=\mathbb{R}^{\ell} \times \mathbb{T} \subset \mathbb{R}^{\ell+1}$ be open sets. Consider the system $\dot{\lambda}=f(t,\lambda)$, where $\lambda=[\omega, \chi]

Figures (4)

  • Figure 1: A depiction of bounded input scenario.
  • Figure 2: Output tracking with bounded input.
  • Figure 3: Constrained (time-varying) output tracking with bounded input.
  • Figure 4: Constrained (constant) output tracking with bounded input.

Theorems & Definitions (17)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 7 more