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Unit-Vector Control Design under Saturating Actuators

Andevaldo da Encarnação Vitório, Pedro Henrique Silva Coutinho, Iury Bessa, Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira

TL;DR

This work addresses robust unit-vector control (UVC) for polytopic uncertain systems with actuating saturation by deriving LMI-based conditions that guarantee finite-time stabilization of the origin. A transformed $z$-dynamics framework and a Lyapunov function $V(z)=z^{\top}Pz$ yield LMIs that produce a robust gain $K=ZX^{-1}$ and $L=YX^{-1}$, ensuring stability despite saturation and providing an estimate $\Omega=\{z: z^{\top}Pz\le1\}$ of the domain of attraction. Additionally, a convex optimization problem is proposed to enlarge the region of guaranteed reaching time $\rho$ by optimizing the decision variables under the LMIs, thereby increasing the set of initial conditions that converge within $\rho$ even with saturation. The approach is validated via two simulations—a planar kinematic manipulator and an underwater ROV—demonstrating finite-time convergence under saturation and illustrating how the region of attraction can be expanded without simply constraining the control gain. Together, these contributions offer a constructive, computable framework for UVC design under saturating actuators with uncertain, polytopic dynamics, with practical implications for precision robotics and autonomous underwater vehicles.

Abstract

This paper deals with unit vector control design for multivariable polytopic uncertain systems under saturating actuators. For that purpose, we propose LMI-based conditions to design the unit vector control gain such that the origin of the closed-loop system is finite-time stable. Moreover, an optimization problem is provided to obtain an enlarged estimate of the region of attraction of the equilibrium point for the closed-loop system, where the convergence of trajectories is ensured even in the presence of saturation functions. Numerical simulations illustrate the effectiveness of the proposed approach.

Unit-Vector Control Design under Saturating Actuators

TL;DR

This work addresses robust unit-vector control (UVC) for polytopic uncertain systems with actuating saturation by deriving LMI-based conditions that guarantee finite-time stabilization of the origin. A transformed -dynamics framework and a Lyapunov function yield LMIs that produce a robust gain and , ensuring stability despite saturation and providing an estimate of the domain of attraction. Additionally, a convex optimization problem is proposed to enlarge the region of guaranteed reaching time by optimizing the decision variables under the LMIs, thereby increasing the set of initial conditions that converge within even with saturation. The approach is validated via two simulations—a planar kinematic manipulator and an underwater ROV—demonstrating finite-time convergence under saturation and illustrating how the region of attraction can be expanded without simply constraining the control gain. Together, these contributions offer a constructive, computable framework for UVC design under saturating actuators with uncertain, polytopic dynamics, with practical implications for precision robotics and autonomous underwater vehicles.

Abstract

This paper deals with unit vector control design for multivariable polytopic uncertain systems under saturating actuators. For that purpose, we propose LMI-based conditions to design the unit vector control gain such that the origin of the closed-loop system is finite-time stable. Moreover, an optimization problem is provided to obtain an enlarged estimate of the region of attraction of the equilibrium point for the closed-loop system, where the convergence of trajectories is ensured even in the presence of saturation functions. Numerical simulations illustrate the effectiveness of the proposed approach.

Paper Structure

This paper contains 10 sections, 2 theorems, 35 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Preliminaries Results
  5. Control design condition
  6. Determining the set of guaranteed reaching time
  7. Numerical Results
  8. Example 1: Planar kinematic manipulator
  9. Example 2: Underwater ROV system
  10. Conclusion

Key Result

Lemma 1

Let the control input be defined as $u=Kz$ for all $z \in \mathbb{R}^{n}$ and a given $K \in \mathbb{R}^{m \times n}$, and the set $\mathcal{D}_{u}$ be defined as for any matrix $L \in \mathbb{R}^{m \times n}$. If $z \in \mathcal{D}_{u}$, then holds for any diagonal $U > 0$.

Figures (5)

  • Figure 1: Block diagram of the UVC system.
  • Figure 2: The set $\Omega$, defined in \ref{['eq:Dx']} for which the convergence occurs within the pre-specified reaching time $\rho$. The region $\Omega$ is contained in the region $\mathcal{D}_u$, given in \ref{['eq:Du']}.
  • Figure 3: Closed-loop trajectory for the initial condition $\sigma(0) = [0.0587 \; -0.7976]^\top$ in $\Omega$.
  • Figure 4: The set $\Omega$, defined in \ref{['eq:Dx']} for which the convergence occurs within the pre-specified reaching time $\rho$. The region $\Omega$ is contained in the region $\mathcal{D}_u$, given in \ref{['eq:Du']}.
  • Figure 5: Closed-loop trajectory for the initial condition $\sigma(0) = [0.0587 \; -0.7976]^\top$ in $\Omega$.

Theorems & Definitions (4)

  • Lemma 1: Adapted from Tarbouriech2011
  • proof
  • Theorem 1
  • proof