Guaranteed Time Control using Linear Matrix Inequalities

TL;DR

The paper tackles guaranteeing finite-time convergence to a target region containing the origin for constrained and uncertain systems. It introduces a guaranteed time control framework built on a harmonic transformation of a Lyapunov function and a novel piecewise quadratic representation over a simplicial state-space partition, solved via policy iteration with LMIs under structural relaxation. The approach yields an upper-bound on the reach time, along with an estimate of the origin’s domain of attraction, and extends to piecewise polytopic and Takagi-Sugeno nonlinear models. Through three diverse examples (linear, unstable polytopic, and nonlinear chaotic systems), the method demonstrates improved time-to-origin performance and larger DoA estimates compared with existing linear, finite-time, and time-optimal approaches, while maintaining state/input constraints. The authors also discuss limitations (notably dimensionality) and propose future directions such as domain decomposition and adaptive grids to scale the framework to higher-dimensional problems.

Abstract

This paper presents a synthesis approach aiming to guarantee a minimum upper-bound for the time taken to reach a target set of non-zero measure that encompasses the origin, while taking into account uncertainties and input and state constraints. This approach is based on a harmonic transformation of the Lyapunov function and a novel piecewise quadratic representation of this transformed Lyapunov function over a simplicial partition of the state space. The problem is solved in a policy iteration fashion, whereas the evaluation and improvement steps are formulated as linear matrix inequalities employing the structural relaxation approach. Though initially formulated for uncertain polytopic systems, extensions to piecewise and nonlinear systems are discussed. Three examples illustrate the effectiveness of the proposed approach in different scenarios.

Figures (12)

• Figure 1: Illustration of a Simplicial partition for $\mathcal{X}$ in the 2-dimensional case. As illustrated in the figure, we always consider that the origin, $(0,0)$ in this case, is a part of the grid of points used to partition the space. As illustrated by the blue colored region in the figure, we consider that every simplex which has the origin as a vertex is a part of the set $\mathcal{X}_g$ towards which we guarantee the upper-bound on convergence time.
• Figure 2: Double Tank System.
• Figure 3: A comparison of the closed loop behavior for the double tank system, from the empty tanks initial condition, obtained from the Guaranteed Time Control (solid lines), using Theorems \ref{['thm:policy_evaluation']} and \ref{['thm:policy_improvement']}, a linear control law that imposes the maximum decay rate without saturating the control input (dashed lines), and a saturated linear control law Tarbouriech2011 that imposes the maximum decay rate while guaranteeing the largest estimated domain of attraction (dotted lines).
• Figure 4: A comparison of the estimation of the closed loop's Domain of Attraction for the double tank system obtained from the Guaranteed Time Control approach (blue solid line), using Theorems \ref{['thm:policy_evaluation']} and \ref{['thm:policy_improvement']} and a saturated linear control law Tarbouriech2011 that imposes the maximum decay rate while guaranteeing the largest estimated domain of attraction (black dashed line).
• Figure 5: A comparison of the closed loop behavior for the double tank system, using the Guaranteed Time Control approach (solid lines, blue control signal) against a finite-time control law based on an Implicit Lyapunov function approach Polyakov2015 (dashed lines, black control signal), and a finite-time control law based on Prescribed Performance Control Guo2022 (dash-dotted lines, red control signal).

Theorems & Definitions (9)

• Theorem 1
• proof
• Remark 1
• Theorem 2: Policy Evaluation
• proof
• Theorem 3: Policy Improvement
• proof
• Remark 2
• Remark 3