arXiv subjects

Dimitri Peaucelle

Publications and source records attributed to Dimitri Peaucelle.

12 recordsLinked to original sources

arXiv2026

Linear Stochastic Systems with i.i.d. uncertainties: Exact Covariance Characterization, Stability Analysis and State-feedback Design

This paper studies linear discrete-time systems affected by independent and identically distributed (i.i.d.) multiplicative uncertainties and additive noise. It establishes the main links between covariance recursions, the spectral properties of associated Kronecker-based matrices, and mean-square stability, and exploits these links to derive tractable conditions for controller synthesis. We first derive a deterministic covariance recursion within the tube-based Stochastic Model Predictive Control (SMPC) framework using a Kronecker product based matrix augmentation. For linear stochastic systems with multiplicative uncertainty and without additive noise, we show that the full-space matrix representation arising from the covariance recursion has the same spectral radius as its symmetric-space counterpart. Combined with the existing symmetric-space characterization, this establishes that Schur stability of the full-space augmented matrix is equivalent to mean-square stability. For state-feedback design, we propose new sufficient Linear Matrix Inequality (LMI) conditions that are numerically more tractable owing to their reduced size compared with the conventional necessary and sufficient conditions. Numerical tests illustrate the usefulness of the covariance characterization for recursively estimating the covariance without relying on sampling-based methods. We also assess the computational burden of the proposed LMI conditions and their conservatism relative to the necessary and sufficient ones.

eess.SY↗

arXiv2025

Covariance Stabilization for a class of Stochastic Discrete-time Linear Systems using the S-Variable Approach

This paper deals with the problem of covariance stabilization for a class of linear stochastic discrete-time systems in the Stochastic Model Predictive Control (SMPC) framework. The considered systems are affected by independent and identically distributed (i.i.d.) additive and parametric stochastic uncertainties (potentially unbounded), in addition to polytopic deterministic uncertainties bounding the mean of the state and input parameters. The design conditions presented in this paper are formulated as Linear Matrix Inequalities (LMIs), using the S-variable approach in order to reduce the potential conservatism. These conditions are derived using a deterministic exact characterization of the covariance dynamics, the latter involves bilinear terms in the control gain. A technique to linearize such dynamics is presented, it results in a descriptor representation allowing to derive sufficient conditions for the design of a covariance-stabilizing controller. The derived condition is first compared with a known necessary and sufficient stability condition for systems without deterministic uncertainties and additive stochastic noise. Although more conservative, the proposed condition is more numerically tractable, with an LMI size scaling as O(n^2) instead of O(n^3). Then, the same condition is used to design controllers that are robust to both deterministic and stochastic uncertainties. Several numerical examples are presented for comparison and illustration.

eess.SY↗

arXiv2025

Detecting Destabilizing Nonlinearities in Absolute Stability Analysis of Discrete-Time Feedback Systems

This paper is concerned with the absolute stability analysis of discrete-time feedback systems with slope-restricted nonlinearities. By employing static O'Shea-Zames-Falb multipliers in the framework of integral quadratic constraints, we can obtain a certificate for the absolute stability in the form of a linear matrix inequality (LMI). However, since this LMI certificate is only a sufficient condition, we cannot draw any definite conclusion if the LMI turns out to be infeasible. To address this issue, we focus on the dual LMI that is feasible if and only if the original (primal) LMI is infeasible. As the main result, if the dual solution satisfies a certain rank condition, we prove that we can detect a destabilizing nonlinearity within the assumed class of slope-restricted nonlinearities as well as a non-zero equilibrium point of the resulting feedback system, thereby we can conclude that the system of interest is never absolutely stable. The effectiveness of the technical results is demonstrated through numerical examples.

math.OC↗

arXiv2024

On Dual of LMIs for Absolute Stability Analysis of Nonlinear Feedback Systems with Static O'Shea-Zames-Falb Multipliers

This study investigates the absolute stability criteria based on the framework of integral quadratic constraint (IQC) for feedback systems with slope-restricted nonlinearities. In existing works, well-known absolute stability certificates expressed in the IQC-based linear matrix inequalities (LMIs) were derived, in which the input-to-output characteristics of the slope-restricted nonlinearities were captured through static O'Shea-Zames-Falb multipliers. However, since these certificates are only sufficient conditions, they provide no clue about the absolute stability in the case where the LMIs are infeasible. In this paper, by taking advantage of the duality theory of LMIs, we derive a condition for systems to be not absolutely stable when the above-mentioned LMIs are infeasible. In particular, we can identify a destabilizing nonlinearity within the assumed class of slope-restricted nonlinearities as well as a non-zero equilibrium point of the resulting closed-loop system, by which the system is proved to be not absolutely stable. We demonstrate the soundness of our results by numerical examples.

math.OC↗

arXiv2024

LMI hierarchies for stability analysis of ReLU feedback systems

We consider the stability analysis of feedback systems with rectified linear unit (ReLU) activations, and model this problem with polynomial optimization. Stability can be certified by means of copositive multipliers in the framework of integral quadratic constraints. Based on a duality argument, we show how to certify instability by considering a complete hierarchy of linear matrix inequalities. This hierarchy is obtained by leveraging the specific equality constraints arising from the ReLU encoding. We illustrate the effectiveness of the proposed approach through several numerical examples.

math.OC↗

arXiv2024

Induced Norm Analysis of Linear Systems for Nonnegative Input Signals

This paper is concerned with the analysis of the $L_p\ (p\in[1,\infty), p=\infty)$ induced norms of continuous-time linear systems where input signals are restricted to be nonnegative. This norm is referred to as the $L_{p+}$ induced norm in this paper. It has been shown recently that the $L_{2+}$ induced norm is effective for the stability analysis of nonlinear feedback systems where the nonlinearity returns only nonnegative signals. However, the exact computation of the $L_{2+}$ induced norm is essentially difficult. To get around this difficulty, in the first part of this paper, we provide a copositive-programming-based method for the upper bound computation by capturing the nonnegativity of the input signals by copositive multipliers. Then, in the second part of the paper, we derive uniform lower bounds of the $L_{p+}\ (p\in[1,\infty), p=\infty)$ induced norms with respect to the standard $L_{p}$ induced norms that are valid for all linear systems including infinite-dimensional ones. For each linear system, we finally derive a computation method of the lower bounds of the $L_{2+}$ induced norm that are larger than (or equal to) the uniform one. The effectiveness of the upper/lower bound computation methods are fully illustrated by numerical examples.

math.OC↗

arXiv2024

$L_{2+}$ Induced Norm Analysis of Continuous-Time LTI Systems Using Positive Filters and Copositive Programming

This paper is concerned with the analysis of the $L_{2}$ induced norm of continuous-time LTI systems where the input signals are restricted to be nonnegative. This induced norm is referred to as the $L_{2+}$ induced norm in this paper. It has been shown very recently that the $L_{2+}$ induced norm is particularly useful for the stability analysis of nonlinear feedback systems constructed from linear systems and static nonlinearities where the nonlinear elements only provide nonnegative signals. For the upper bound computation of the $L_{2+}$ induced norm, an approach with copositive programming has also been proposed. It is nonetheless true that this approach becomes effective only for multi-input systems, and for single-input systems this approach does not bring any improvement over the trivial upper bound, the standard $L_2$ norm. To overcome this difficulty, we newly introduce positive filters to increase the number of positive signals. This enables us to enlarge the size of the copositive multipliers so that we can obtain better (smaller) upper bounds with copositive programming.

math.OC↗

arXiv2023

Local Lipschitz Constant Computation of ReLU-FNNs: Upper Bound Computation with Exactness Verification

This paper is concerned with the computation of the local Lipschitz constant of feedforward neural networks (FNNs) with activation functions being rectified linear units (ReLUs). The local Lipschitz constant of an FNN for a target input is a reasonable measure for its quantitative evaluation of the reliability. By following a standard procedure using multipliers that capture the behavior of ReLUs,we first reduce the upper bound computation problem of the local Lipschitz constant into a semidefinite programming problem (SDP). Here we newly introduce copositive multipliers to capture the ReLU behavior accurately. Then, by considering the dual of the SDP for the upper bound computation, we second derive a viable test to conclude the exactness of the computed upper bound. However, these SDPs are intractable for practical FNNs with hundreds of ReLUs. To address this issue, we further propose a method to construct a reduced order model whose input-output property is identical to the original FNN over a neighborhood of the target input. We finally illustrate the effectiveness of the model reduction and exactness verification methods with numerical examples of practical FNNs.

math.OC↗

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.

math.OC↗

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.

math.OC↗

arXiv2019

Distribution Modeling and Stabilization Control for Discrete-Time Linear Random Dynamical Systems Using Ensemble Kalman Filter

This paper studies an output feedback stabilization control framework for discrete-time linear systems with stochastic dynamics determined by an independent and identically distributed (i.i.d.) process. The controller is constructed with an ensemble Kalman filter (EnKF) and a feedback gain designed with our earlier result about state feedback control. The EnKF is also used for modeling the distribution behind the system, which is required in the feedback gain synthesis. The effectiveness of our control framework is demonstrated with numerical experiments. This study will become the first step toward the realization of learning type control using our stochastic systems control theory.

eess.SY↗

arXiv2016

Randomized and robust methods for uncertain systems using R-RoMulOC, with applications to DEMETER satellite benchmark

R-RoMulOC is a freely distributed toolbox which aims at making easily available to the users different optimization-based methods for dealing with uncertain systems. It implements both deterministic LMI-based results, that provide guaranteed performances for all values of the uncertainties, and probabilistic randomization-based approaches, that guarantee performances for all values of the uncertainties except for a subset with arbitrary small probability measure. The paper is devoted to the description of these two approaches for analysis and control design when applied to a satellite benchmark proposed by CNES, the French Space Agency. The paper also describes the modeling of the DEMETER satellite and its integration into the R-RoMulOC toolbox as a challenging test example. Design of state-feedback controllers and closed-loop performance analysis are carried out with the randomized and robust methods available in the R-RoMulOC toolbox.

math.OC↗