arXiv subjects

Mohamed Maghenem

Publications and source records attributed to Mohamed Maghenem.

25 records · Page 2Linked to original sources

arXiv2023

On the Converse Safety Problem for Differential Inclusions: Solutions, Regularity, and Time-Varying Barrier Functions

This paper presents converse theorems for safety in terms of barrier functions for unconstrained continuous-time systems modeled as differential inclusions. Via a counterexample, we show the lack of existence of autonomous and continuous barrier functions certifying safety for a nonlinear system that is not only safe but also has a smooth right-hand side. Guided by converse Lyapunov theorems for (non-asymptotic) stability,time-varying barrier functions and appropriate infinitesimal conditions are shown to be both necessary as well as sufficient under mild regularity conditions on the right-hand side of the system. More precisely, we propose a general construction of a time-varying barrier function in terms of a marginal function involving the finite-horizon reachable set. Using techniques from set-valued and nonsmooth analysis, we show that such a function guarantees safety when the system is safe. Furthermore, we show that the proposed barrier function construction inherits the regularity properties of the proposed reachable set. In addition, when the system is safe and smooth, we build upon the constructed barrier function to show the existence of a smooth barrier function guaranteeing safety. Comparisons and relationships to results in the literature are also presented.

math.OC↗

arXiv2023

Hybrid Persistency of Excitation in Adaptive Estimation for Hybrid Systems

We propose a framework to analyze stability for a class of linear non-autonomous hybrid systems, where the continuous evolution of solutions is governed by an ordinary differential equation and the instantaneous changes are governed by a difference equation. Furthermore, the jumps are triggered by the influence of an external hybrid signal. The proposed framework builds upon a generalization of the well-known persistency of excitation (PE) and uniform observability (UO) notions to the realm of hybrid systems. That is, we establish conditions, under which, hybrid PE implies hybrid UO and, in turn, uniform exponential stability (UES) and input-to-state stability (ISS). Our proofs rely on an original statement for hybrid systems, expressed in terms of Lp bounds on the solutions. We demonstrate the utility of our results on generic adaptive estimation problems. The first one concerns the so-called gradient systems, reminiscent of the popular gradient-descent algorithm. The second one pertains to designing adaptive observers/identifiers for a class of hybrid systems that are nonlinear in the input and the output, and linear in the unknown parameters. In both cases, we illustrate through examples that the proposed hybrid framework succeeds when the classic purely continuous- or discrete-time counterparts fail.

math.OC↗

arXiv2023

Reconstructing Indistinguishable Solutions Via Set-Valued KKL Observer

KKL observer design consists in finding a smooth change of coordinates transforming the system dynamics into a linear filter of the output. The state of the original system is then reconstructed by implementing this filter from any initial condition and left-inverting the transformation, under a \textit{backward-distinguishability} property. In this paper, we consider the case where the latter assumption does not hold, namely when distinct solutions may generate the same output, and thus be indistinguishable. The KKL transformation is no longer injective and its ``left-inverse'' is thus allowed to be set-valued, yielding a set-valued KKL observer. Assuming the transformation is full-rank and its preimage has constant cardinality, we show the existence of a globally defined set-valued left-inverse that is Lipschitz in the Hausdorff sense. Leveraging on recent results linking this left-inverse with the \textit{backward-indistinguishable sets}, we show that the set-valued KKL observer converges in the Hausdorff sense to the backward-indistinguishable set of the system solution. When, additionally, a given output is generated by a specific number of solutions not converging to each other, we show that the designed observer asymptotically reconstructs each of those solutions. Finally, the different assumptions are discussed and illustrated via examples.

math.OC↗

arXiv2022

Certifying the LTL Formula p Until q in Hybrid Systems

In this paper, we propose sufficient conditions to guarantee that a linear temporal logic (LTL) formula of the form p Until q, denoted by $p \mathcal{U} q$, is satisfied for a hybrid system. Roughly speaking, the formula $p \mathcal{U} q$ is satisfied means that the solutions, initially satisfying proposition p, keep satisfying this proposition until proposition q is satisfied. To certify such a formula, connections to invariance notions such as conditional invariance (CI) and eventual conditional invariance (ECI), as well as finite-time attractivity (FTA) are established. As a result, sufficient conditions involving the data of the hybrid system and an appropriate choice of Lyapunov-like functions, such as barrier functions, are derived. The considered hybrid system is given in terms of differential and difference inclusions, which capture the continuous and the discrete dynamics present in the same system, respectively. Examples illustrate the results throughout the paper.

eess.SY↗

arXiv2022

Singular-Perturbations-Based Analysis of Dynamic Consensus in Directed Networks of Heterogeneous Nonlinear Systems

We analyze networked heterogeneous nonlinear systems, with diffusive coupling and interconnected over a generic static directed graph. Due to the network's hetereogeneity, complete synchronization is impossible, in general, but an emergent dynamics arises. This may be characterized by two dynamical systems evolving in two time-scales. The first, "slow", corresponds to the dynamics of the network on the synchronization manifold. The second, "fast", corresponds to that of the synchronization errors. We present a framework to analyse the emergent dynamics based on the behavior of the slow dynamics. Firstly, we give conditions under which if the slow dynamics admits a globally asymptotically stable equilibrium, so does the networked systems. Secondly, we give conditions under which, if the slow dynamics admits an asymptotically stable orbit and a single unstable equilibrium point, there exists a unique periodic orbit that is almost-globally asymptotically stable. The emergent behavior is thus clear, the systems asymptotically synchronize in frequency and, in the limit, as the coupling strength grows, the emergent dynamics approaches that of the slow system. Our analysis is established using singular-perturbations theory. In that regard, we contribute with original statements on stability of disconnected invariant sets and limit cycles.

math.OC↗

arXiv2022

Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions

This paper studies set invariance and contractivity in hybrid systems modeled by hybrid inclusions using barrier functions. After introducing the notion of a multiple barrier functions, we investigate the tightest possible sufficient conditions to guarantee different forward invariance and contractivity notions of a closed set for hybrid systems with nonuniqueness of solutions and solutions terminating prematurely. More precisely, we consider forward (pre-)invariance of sets, which guarantees solutions to stay in a set, and (pre-)contractivity, which further requires solutions that reach the boundary of the set to evolve (continuously or discretely) towards its interior. Our conditions for forward invariance and contractivity involve infinitesimal conditions in terms of multiple barrier functions. Examples illustrate the results. Keywords: Forward invariance, contractivity, barrier functions, hybrid dynamical systems.

math.OC↗

arXiv2022

Necessary and Sufficient Conditions for the Nonincrease of Scalar Functions Along Solutions to Constrained Differential Inclusions

In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable.

math.OC↗