arXiv subjects

Sophie Tarbouriech

Publications and source records attributed to Sophie Tarbouriech.

28 records · Page 2Linked to original sources

arXiv2022

Anti-windup-like Compensator Synthesis for Discrete-Time Quantized Control Systems

This paper addresses the problem of designing an anti-windup like compensator for discrete-time linear control systems with quantized input. The proposed compensator provides a correction signal proportional to the quantization error that fed to the controller. The compensator is designed to ensure that solutions to the closed-loop systems converge in finite time into a compact set containing the origin that can be tuned by the designer. A numerically tractable algorithm with feasibility guarantees is provided for the design of the compensator. The proposed results are illustrated on an academic example and an open-loop unstable aircraft system.

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arXiv2022

Stability Analysis of Recurrent Neural Networks by IQC with Copositive Mutipliers

This paper is concerned with the stability analysis of the recurrent neural networks (RNNs) by means of the integral quadratic constraint (IQC) framework. The rectified linear unit (ReLU) is typically employed as the activation function of the RNN, and the ReLU has specific nonnegativity properties regarding its input and output signals. Therefore, it is effective if we can derive IQC-based stability conditions with multipliers taking care of such nonnegativity properties. However, such nonnegativity (linear) properties are hardly captured by the existing multipliers defined on the positive semidefinite cone. To get around this difficulty, we loosen the standard positive semidefinite cone to the copositive cone, and employ copositive multipliers to capture the nonnegativity properties. We show that, within the framework of the IQC, we can employ copositive multipliers (or their inner approximation) together with existing multipliers such as Zames-Falb multipliers and polytopic bounding multipliers, and this directly enables us to ensure that the introduction of the copositive multipliers leads to better (no more conservative) results. We finally illustrate the effectiveness of the IQC-based stability conditions with the copositive multipliers by numerical examples.

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arXiv2021

On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming

Stability and stabilization for linear state feedback control systems in the presence of sensor quantization are studied. As the closed-loop system is described by a discontinuous right-hand side differential equation, Krasovskii solutions (to the closed-loop system) are considered. Sufficient conditions in the form of matrix inequalities are proposed to characterize uniform global asymptotic stability of a compact set containing the origin. Such conditions are shown to be always feasible whenever the quantization-free closed-loop system is asymptotically stable. Building on the obtained conditions, computationally affordable algorithms for the solution to the considered problems are illustrated. The effectiveness of the proposed methodology is shown in three examples.

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arXiv2021

Event-triggering mechanism to damp the linear wave equation

This paper aims at proposing a sufficient matrix inequality condition to carry out the global exponential stability of the wave equation under an event-triggering mechanism that updates a damping source term. The damping is distributed in the whole space but sampled in time. The wellposedness of the closed-loop event-triggered control system is shown. Furthermore, the avoidance of Zeno behavior is ensured provided that the initial data are more regular. The interest of the results is drawn through some numerical simulations.

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arXiv2021

Lyapunov Stability Analysis of a Mass-Spring system subject to Friction

This paper deals with the stability analysis of a mass-spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick-slip phenomenon, the mass may then periodically sticks to the ground. The objective consists of developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The proposed approach merges two intermediate results: The first one relies on the characterization of an attractor around the origin, to which converges the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed from the second result. An illustrative example draws the interest of the approach.

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arXiv2020

$l_2$ Induced Norm Analysis of Discrete-Time LTI Systems for Nonnegative Input Signals and Its Application to Stability Analysis of Recurrent Neural Networks

In this paper, we focus on the "positive" $l_2$ induced norm of discrete-time linear time-invariant systems where the input signals are restricted to be nonnegative. To cope with the nonnegativity of the input signals, we employ copositive programming as the mathematical tool for the analysis. Then, by applying an inner approximation to the copositive cone, we derive numerically tractable semidefinite programming problems for the upper and lower bound computation of the "positive" $l_2$ induced norm. This norm is typically useful for the stability analysis of feedback systems constructed from an LTI system and nonlinearities where the nonlinear elements provide only nonnegative signals. As a concrete example, we illustrate the usefulness of the "positive" $l_2$ induced norm for the stability analysis of recurrent neural networks with activation functions being rectified linear units.

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arXiv2020

$\mathcal{L}_2$ State Estimation with Guaranteed Convergence Speed in the Presence of Sporadic Measurements

This paper deals with the problem of estimating the state of a linear time-invariant system in the presence of sporadically available measurements and external perturbations. An observer with a continuous intersample injection term is proposed. Such an intersample injection is provided by a linear dynamical system, whose state is reset to the measured output estimation error at each sampling time. The resulting system is augmented with a timer triggering the arrival of a new measurement and analyzed in a hybrid system framework. The design of the observer is performed to achieve global exponential stability with a given decay rate to a set wherein the estimation error is equal to zero. Robustness with respect to external perturbations and $\mathcal{L}_2$-external stability from the plant perturbation to a given performance output are considered. Moreover, computationally efficient algorithms based on the solution to linear matrix inequalities are proposed to design the observer. Finally, the effectiveness of the proposed methodology is shown in three examples.

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arXiv2019

Event-triggered damping stabilization of a linear wave equation

The paper addresses the design of an event-triggering mechanism for a partial differential wave equation posed in a bounded domain. The wave equation is supposed to be controlled through a first order time derivative term distributed in the whole domain. Sufficient conditions based on the use of suitable Lyapunov functional are proposed to guarantee that an event-triggered distributed control still ensures the exponential stability of the closed-loop system. Moreover, the designed event-triggering mechanism allows to avoid the Zeno behavior. The 'existence and regularity' prerequisite properties of solutions for the closed loop system are also proven.

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arXiv2018

Some results on exponential synchronization of nonlinear systems (long version)

Based on recent works on transverse exponential stability, we establish some necessary and sufficient conditions for the existence of a (locally) exponential synchronizing control law. We show that the existence of a structured synchronizer is equivalent to the existence of a stabilizer for the individual linearized systems (on the synchronization manifold) by a linear state feedback. This, in turn, is also equivalent to the existence of a symmetric covariant tensor field, which satisfies a Control Matrix Function inequality. Based on this result, we provide the construction of such synchronizer via backstepping approaches. In some particular cases, we show how global exponential synchronization may be obtained.

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arXiv2017

Disturbance-to-State Stabilization and Quantized Control for Linear Hyperbolic Systems

We consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. Because of the disturbances in the measurement, the problem of designing dynamic controllers is considered so that the closed-loop system is robust with respect to measurement errors. Assuming that the disturbance is a locally essentially bounded measurable function of time, we derive a disturbance-to-state estimate which provides an upper bound on the maximum norm of the state (with respect to the spatial variable) at each time in terms of $\mathcal{L}^\infty$-norm of the disturbance up to that time. The analysis is based on constructing a Lyapunov function for the closed-loop system, which leads to controller synthesis and the conditions on system dynamics required for stability. As an application of this stability notion, the problem of quantized control for hyperbolic PDEs is considered where the measurements sent to the controller are communicated using a quantizer of finite length. The presence of quantizer yields practical stability only, and the ultimate bounds on the norm of the state trajectory are also derived.

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