arXiv subjects

Jean Lévine

Publications and source records attributed to Jean Lévine.

13 recordsLinked to original sources

arXiv2026

Polytopic Inner Approximation of Admissible Sets for Linear Systems

This paper presents a method for computing inner polytopic approximations of admissible sets for continuous-time linear control systems subject to multiple affine state constraints, with a particular concern on computational tractability for large dimensional problems. In place of globally computing the admissible set and the part of its boundary called the barrier, we compute the so-called individual admissible sets and the corresponding barriers for each single constraint. We then use the exact sampling of linear systems and generate polytopes in half-space representation that provide an approximation of the individual admissible sets, to finally intersect them. We provide a complexity analysis of the whole procedure to evaluate its efficiency. The approach is illustrated by two examples -a triple integrator and a mass-spring-damper chain considered in 4, 6, 8, and 10 dimensions- with corresponding runtimes evaluated for both.

math.OC↗

arXiv2026

On the Boundary of the Robust Admissible Set in State and Input Constrained Nonlinear Systems

In this paper, we consider nonlinear control systems subject to bounded disturbances and to both state and input constraints. We introduce the definition of robust admissible set - the set of all initial states from which the state and input constraints can be satisfied for all times against all admissible disturbances. We focus on its boundary that can be decomposed into the usable part on the state constraint boundary and the barrier, interior to the state constraints. We show that, at the intersection of these two components, the boundary of the robust admissible set must be tangent to the state constraint set and separate the interior of the robust admissible set and its complement, a property that we call the ultimate locally separating hyperplane condition. Moreover, we prove that the barrier must satisfy a saddle-point principle on a Hamiltonian, based on Pontryagin's maximum principle, whose final condition is precisely the ultimate locally separating condition, thus providing a set of differential equations made of the system and its adjoint for a direct construction of the barrier. Lastly, we illustrate our results by calculating the robust admissible set for an adaptive cruise control example.

math.OC↗

arXiv2025

On Driftless Systems with m controls and 2m or 2m-1 states that are Flat by Pure Prolongation

It is widely recognized that no tractable necessary and sufficient conditions exist for determining whether a system is, in general, differentially flat. However, specific cases do provide such conditions. For instance, driftless systems with two inputs have known necessary and sufficient conditions. For driftless systems with three or more inputs, the available conditions are only sufficient. This paper presents new findings on determining whether a system with m inputs and $2m$ or $2m-1$ states is flat by pure prolongation, a specific subclass of differential flatness. While this condition is more restrictive than general differential flatness, the algorithm for computing flat outputs remains remarkably simple, and the verification requirements are relatively lenient. Moreover, the conditions proposed in this work broaden the class of systems recognized as differentially flat, as our sufficient condition differs from existing criteria.

math.OC↗

arXiv2023

Differential Flatness by Pure Prolongation: Necessary and Sufficient Conditions

In this article, we introduce the notion of differential flatness by pure prolongation: loosely speaking, a system admits this property if, and only if, there exists a pure prolongation of finite order such that the prolonged system is feedback linearizable. We obtain Lie-algebraic necessary and sufficient conditions for a general nonlinear multi-input system to satisfy this property. These conditions are comprised of the involutivity and relative invariance of a pair of filtrations of distributions of vector fields. An algorithm computing the minimal prolongation lengths of the input channels that achieve the system linearization, yielding the associated flat outputs, is deduced. Examples that show the efficiency and computational tractability of the approach are then presented.

math.OC↗

arXiv2021

Barriers and Potentially Safe Sets in Hybrid Systems: Pendulum with Non-Rigid Cable

This paper deals with an application of the notion of barrier in mixed constrained nonlinear systems to an example of a pendulum mounted on a cart with non-rigid cable, whose dynamics may switch to free-fall when the tension of the cable vanishes. We present a direct construction of the boundary of the potentially safe set in which there always exists a control such that the cable never goes slack. A discussion on the dependence of this set with respect to the pendulum and cart masses is then sketched.

math.OC↗

arXiv2021

Epidemic Management with Admissible and Robust Invariant Sets

We present a detailed set-based analysis of the well-known SIR and SEIR epidemic models subjected to hard caps on the proportion of infective individuals, and bounds on the allowable intervention strategies, such as social distancing, quarantining and vaccination. We describe the admissible and maximal robust positively invariant (MRPI) sets of these two models via the theory of barriers. We show how the sets may be used in the management of epidemics, for both perfect and imperfect/uncertain models, detailing how intervention strategies may be specified such that the hard infection cap is never breached, regardless of the basic reproduction number. The results are clarified with detailed examples.

math.OC↗

arXiv2020

On Singularities of Flat Affine Systems With $n$ States and $n-1$ Controls

We study the set of intrinsic singularities of flat affine systems with $n-1$ controls and $n$ states using the notion of Lie-Bäcklund atlas, previously introduced by the authors. For this purpose, we prove two easily computable sufficient conditions to construct flat outputs as a set of independent first integrals of distributions of vector fields, the first one in a generic case, namely in a neighborhood of a point where the $n-1$ control vector fields are independent, and the second one at a degenerate point where $p-1$ control vector fields are dependent of the $n-p$ others, with $p>1$. We show that the set of intrinsic singularities includes the set of points where the system does not satisfy the strong accessibility rank condition and is included in the set where the distribution of vector fields, introduced in the generic case, is singular. We conclude this analysis by three examples of apparent singularites of flat systems in generic and non generic degenerate cases.

math.OC↗

arXiv2020

Maintaining Hard Infection Caps in Epidemics via the Theory of Barriers

Research in epidemiology often focusses on designing interventions that result in the number of infected individuals asymptotically approaching zero, without considering that this number may peak at high values during transients. Recent research has shown that a set-based approach could be used to address the problem, and we build on this idea by applying the theory of barriers to construct admissible and invariant sets for an epidemic model. We describe how these sets may be used to choose intervention strategies that maintain infection caps during epidemics. We also derive algebraic conditions of the model parameters that classify a system as being either comfortable, comfortable-viable, viable, or desperate.

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arXiv2015

Flatness For Linear Fractional Systems With Application to a Thermal System

This paper is devoted to the study of the flatness property of linear time-invariant fractional systems. In the framework of polynomial matrices of the fractional derivative operator, we give a characterization of fractionally flat outputs and a simple algorithm to compute them. We also obtain a characterization of the so-called fractionnally $0$-flat outputs. We then present an application to a two dimensional heated metallic sheet, whose dynamics may be approximated by a fractional model of order 1/2. The trajectory planning of the temperature at a given point of the metallic sheet is obtained thanks to the fractional flatness property, without integrating the system equations. The pertinence of this approach is discussed on simulations.

physics.comp-ph↗

arXiv2015

Barriers in Nonlinear Control Systems with Mixed Constraints

In this paper, we propose an extension to mixed multidimensional constraints of the problem of state and input constrained control introduced in DeDona and Lévine, SIAM J. Control and Optimiz., Vol. 51, 2013, where the admissible set, namely the subset of the state space where the state and input constraints can be satisfied for all times, was studied, with focus on its boundary. The latter may be divided in two parts, one of them being called barrier, a semipermeable surface. We extend this notion of barrier to the mixed case and prove that it can be constructed via a minimum-like principle involving the Karush-Kühn-Tucker multipliers associated to the constraints and a generalised gradient condition at its endpoints.

math.OC↗

arXiv2013

On Barriers in State and Input Constrained Nonlinear Systems

In this paper, the problem of state and input constrained control is addressed, with multidimensional constraints. We obtain a local description of the boundary of the admissible subset of the state space where the state and input constraints can be satisfied \emph{for all times}. This boundary is made of two disjoint parts: the subset of the state constraint boundary on which there are trajectories pointing towards the interior of the admissible set or tangentially to it; and a barrier, namely a semipermeable surface which is constructed via a minimum-like principle.

math.OC↗

arXiv2011

On Necessary and Sufficient Conditions for Differential Flatness

This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-Bäcklund equivalence, introduced in Fliess et al. (1999), to this implicit context and focus attention on Lie-Bäcklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate $\ddt$, whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.

math.OC↗

arXiv2010

On the Computation of $π$-Flat Outputs for Linear Time-Delay Systems

This paper deals with linear time-varying, delay systems. Extensions of the concept of differential flatness \cite{Fliess_95} to this context have been first proposed in \cite{Mounier_95,Fliess_96} (see also \cite{Rudolph_03,Chyzak_05}), by the introduction of $π$-flat output. Roughly speaking, it means that every system variable may be expressed as a function of a particular output $y$, a finite number of its time derivatives, time delays, and predictions, the latter resulting from the prediction operator $π^{-1}$. We propose a simple and constructive algorithm for the computation of $π$-flat outputs based on concepts of polynomial algebra, in particular Smith-Jacobson decomposition of polynomial matrices. Some examples are provided to illustrate the proposed methodology.

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