arXiv subjects

Raphaël M. Jungers

Publications and source records attributed to Raphaël M. Jungers.

At least 73 records · Page 4Linked to original sources

arXiv2020

Stability of Planar Switched Systems under Delayed Event Detection

In this paper, we analyse the impact of delayed event detection on the stability of a 2-mode planar hybrid automata. We consider hybrid automata with a unique equilibrium point for all the modes, and we find the maximum delay that preserves stability of that equilibrium point. We also show for the class of hybrid automata treated that the instability of the equilibrium point for the equivalent hybrid automaton with delay in the transitions is equivalent to the existence of a closed orbit in the hybrid state space, a result that is inspired by the Joint Spectral Radius theorem. This leads to an algorithm for computing the maximum stable delay exactly. Other potential applications of our technique include co-simulation, networked control systems and delayed controlled switching with a state feedback control.

math.DS↗

arXiv2020

Piecewise semi-ellipsoidal control invariant sets

Computing control invariant sets is paramount in many applications. The families of sets commonly used for computations are ellipsoids and polyhedra. However, searching for a control invariant set over the family of ellipsoids is conservative for systems more complex than unconstrained linear time invariant systems. Moreover, even if the control invariant set may be approximated arbitrarily closely by polyhedra, the complexity of the polyhedra may grow rapidly in certain directions. An attractive generalization of these two families are piecewise semi-ellipsoids. We provide in this paper a convex programming approach for computing control invariant sets of this family.

math.OC↗

arXiv2020

Optimal measurement budget allocation for particle filtering

Particle filtering is a powerful tool for target tracking. When the budget for observations is restricted, it is necessary to reduce the measurements to a limited amount of samples carefully selected. A discrete stochastic nonlinear dynamical system is studied over a finite time horizon. The problem of selecting the optimal measurement times for particle filtering is formalized as a combinatorial optimization problem. We propose an approximated solution based on the nesting of a genetic algorithm, a Monte Carlo algorithm and a particle filter. Firstly, an example demonstrates that the genetic algorithm outperforms a random trial optimization. Then, the interest of non-regular measurements versus measurements performed at regular time intervals is illustrated and the efficiency of our proposed solution is quantified: better filtering performances are obtained in 87.5% of the cases and on average, the relative improvement is 27.7%.

eess.SY↗

arXiv2020

On the Quality of First-Order Approximation of Functions with Hölder Continuous Gradient

We show that Hölder continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the first-order Taylor approximation. We also relate this global upper bound to the Hölder constant of the gradient. This relation is expressed as an interval, depending on the Hölder constant, in which the error of the first-order Taylor approximation is guaranteed to be. We show that, for the Lipschitz continuous case, the interval cannot be reduced. An application to the norms of quadratic forms is proposed, which allows us to derive a novel characterization of Euclidean norms.

math.OC↗

arXiv2019

An entropy-based bound for the computational complexity of a switched system

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We analyse the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the p-radius of the system and the entropy of the language of allowed switching sequences. We end this paper with a method to reduce the computation of the JSR of low rank matrices to the computation of the constrained JSR of matrices of small dimension.

math.OC↗

arXiv2019

Optimal Intermittent Measurements for Tumor Tracking in X-ray Guided Radiotherapy

In radiation therapy, tumor tracking is a challenging task that allows a better dose delivery. One practice is to acquire X-ray images in real-time during treatment, that are used to estimate the tumor location. These informations are used to predict the close future tumor trajectory. Kalman prediction is a classical approach for this task. The main drawback of X-ray acquisition is that it irradiates the patient, including its healthy tissues. In the classical Kalman framework, X-ray measurements are taken regularly, i.e. at a constant rate. In this paper, we propose a new approach which relaxes this constraint in order to take measurements when they are the most useful. Our aim is for a given budget of measurements to optimize the tracking process. This idea naturally brings to an optimal intermittent Kalman predictor for which measurement times are selected to minimize the mean squared prediction error over the complete fraction. This optimization problem can be solved directly when the respiratory model has been identified and the optimal sampling times can be computed at once. These optimal measurement times are obtained by solving a combinatorial optimization problem using a genetic algorithm. We created a test benchmark on trajectories validated on one patient. This new prediction method is compared to the regular Kalman predictor and a relative improvement of 9:8% is observed on the root mean square position estimation error.

stat.AP↗

arXiv2019

The Synchronizing Probability Function for Primitive Sets of Matrices

Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding Černý conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitivity process in the setting of a two-player probabilistic game and we make use of convex optimization techniques to describe its behavior. We develop a tool for approximating and upper bounding the exponent of any primitive set and supported by numerical results we state a conjecture that, if true, would imply a quadratic upper bound on the reset threshold of a new class of automata.

cs.FL↗

arXiv2018

On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs

We tackle the problem of the randomized generation of slowly synchronizing deterministic automata (DFAs) by generating random primitive sets of matrices. We show that when the randomized procedure is too simple the exponent of the generated sets is O(n log n) with high probability, thus the procedure fails to return DFAs with large reset threshold. We extend this result to random nondeterministic automata (NDFAs) by showing, in particular, that a uniformly sampled NDFA has both a 2-directing word and a 3-directing word of length O(n log n) with high probability. We then present a more involved randomized algorithm that manages to generate DFAs with large reset threshold and we finally leverage this finding for exhibiting new families of DFAs with reset threshold of order $ Ω(n^2/4) $.

cs.FL↗

arXiv2018

Minimally Constrained Stable Switched Systems and Application to Co-simulation

We propose an algorithm to restrict the switching signals of a constrained switched system in order to guarantee its stability, while at the same time attempting to keep the largest possible set of allowed switching signals. Our work is motivated by applications to (co-)simulation, where numerical stability is a hard constraint, but should be attained by restricting as little as possible the allowed behaviours of the simulators. We apply our results to certify the stability of an adaptive co-simulation orchestration algorithm, which selects the optimal switching signal at run-time, as a function of (varying) performance and accuracy requirements.

math.OC↗

arXiv2018

Path-complete $p$-dominant switching linear systems

The notion of path-complete $p$-dominance for switching linear systems (in short, path-dominance) is introduced as a way to generalize the notion of dominant/slow modes for LTI systems. Path-dominance is characterized by the contraction property of a set of quadratic cones in the state space. We show that path-dominant systems have a low-dimensional dominant behavior, and hence allow for a simplified analysis of their dynamics. An algorithm for deciding the path-dominance of a given system is presented.

math.OC↗

arXiv2018

On randomized generation of slowly synchronizing automata

Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank $ n\!-\!1 $. We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on \textit{primitive} sets of matrices. We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of $ Ω(n^2/4) $. We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a $ 0 $ entry changed into a $ 1 $ is primitive and has exponent of $ O(n\log n) $ with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to $ 1 $ with probability $ p $ and equal to $ 0 $ with probability $ 1-p $, when $ np-\log n\rightarrow\infty $ as $ n\rightarrow\infty $.

cs.FL↗

arXiv2018

On Completely Reachable Automata and Subset Reachability

This article focuses on subset reachability in synchronizing automata. First, we provide families of synchronizing automata with subsets which cannot be reached with short words. These families do not fulfil Don's Conjecture about subset reachability. Moreover, they show that some subsets need exponentially long words to be reached, and that the restriction of the conjecture to included subsets also does not hold. Second, we analyze completely reachable automata and provide a counterexample to the conjecture of Bondar and Volkov about the so-called $Γ_1$-graph. We finally prove an alternative version of this conjecture.

cs.FL↗

arXiv2018

Computing controlled invariant sets for hybrid systems with applications to model-predictive control

In this paper, we develop a method for computing controlled invariant sets using Semidefinite Programming. We apply our method to the controller design problem for switching affine systems with polytopic safe sets. The task is reduced to a semidefinite programming problem by enforcing an invariance relation in the dual space of the geometric problem. The paper ends with an application to safety critical model predictive control.

math.OC↗

arXiv2017

Sets of Stochastic Matrices with Converging Products: Bounds and Complexity

An SIA matrix is a stochastic matrix whose sequence of powers converges to a rank-one matrix. This convergence is desirable in various applications making use of stochastic matrices, such as consensus, distributed optimization and Markov chains. We study the shortest SIA products of sets of matrices. We observe that the shortest SIA product of a set of matrices is usually very short and we provide a first upper bound on the length of the shortest SIA product (if one exists) of any set of stochastic matrices. We also provide an algorithm that decides the existence of an SIA product. When particularized to automata, the problem becomes that of finding periodic synchronizing words, and we develop the consequences of our results in relation with the celebrated Cerny conjecture in automata theory. We also investigate links with the related notions of positive-column, Sarymsakov, and scrambling matrices.

cs.DM↗

arXiv2017

On Path-Complete Lyapunov Functions: Geometry and Comparison

We study optimization-based criteria for the stability of switching systems, known as Path-Complete Lyapunov Functions, and ask the question "can we decide algorithmically when a criterion is less conservative than another". Our contribution is twofold. First, we show that a Path-Complete Lyapunov Function, which is a multiple Lyapunov function by nature, can always be expressed as a common Lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it. Geometrically, our results provide for each Path-Complete criterion an implied invariant set. Second, we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given Path-Complete Lyapunov functions.

math.DS↗

arXiv2017

On the interplay between Babai and Cerny's conjectures

Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with $n$ states in this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$ and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters of the pair digraphs of permutation automata and construct $n$-state permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.

cs.FL↗

arXiv2016

Path-Complete Graphs and Common Lyapunov Functions

A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called its pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time switching systems under arbitrary switching. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. The converse, however, is not true even for discrete-time linear systems: we present such a system where a max-of-2 quadratics Lyapunov function exists while no corresponding Path-Complete Lyapunov function with 2 quadratic pieces exists. In light of this, we investigate when it is possible to decide if a Path-Complete Lyapunov function is less conservative than another. By analyzing the combinatorial and algebraic structure of the graph and the pieces respectively, we provide simple tools to decide when the existence of such a Lyapunov function implies that of another.

math.DS↗

arXiv2016

Tight Bounds for Consensus Systems Convergence

We analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set, by looking only at finite or periodic products. It is known that when the matrices of the set have a common nonincreasing polyhedral norm, all infinite products converge to zero if and only if all infinite periodic products with period smaller than a certain value converge to zero, and bounds exist on that value. We provide a stronger bound holding for both polyhedral norms and polyhedral seminorms. In the latter case, the matrix products do not necessarily converge to 0, but all trajectories of the associated system converge to a common invariant space. We prove our bound to be tight, in the sense that for any polyhedral seminorm, there is a set of matrices such that not all infinite products converge, but every periodic product with period smaller than our bound does converge. Our technique is based on an analysis of the combinatorial structure of the face lattice of the unit ball of the nonincreasing seminorm. The bound we obtain is equal to half the size of the largest antichain in this lattice. Explicitly evaluating this quantity may be challenging in some cases. We therefore link our problem with the Sperner property: the property that, for some graded posets, -- in this case the face lattice of the unit ball -- the size of the largest antichain is equal to the size of the largest rank level. We show that some sets of matrices with invariant polyhedral seminorms lead to posets that do not have that Sperner property. However, this property holds for the polyhedron obtained when treating sets of stochastic matrices, and our bound can then be easily evaluated in that case. In particular, we show that for the dimension of the space $n \geq 8$, our bound is smaller than the previously known bound by a multiplicative factor of $\frac{3}{2 \sqrt{πn}}$.

cs.DM↗