arXiv subjects

Raphaël M. Jungers

Publications and source records attributed to Raphaël M. Jungers.

At least 55 records · Page 3Linked to original sources

arXiv2022

Equivalent Polyadic Decompositions of Matrix Multiplication Tensors

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de~Groote, who showed that for the multiplication of $2\times2$ matrices with $7$ active multiplications, all algorithms are essentially equivalent to Strassen's algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices, (e.g., $2\times3$ by $3\times2$ or $3\times3$ by $3\times3$ matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.

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arXiv2022

Stabilization of rank-deficient continuous-time switched affine systems

This paper treats the global stabilization problem of continuous-time switched affine systems that have rank-deficient convex combinations of their dynamic matrices. For these systems, the already known set of attainable equilibrium points has higher dimensionality than in the full-rank case due to the existence of what we define as singular equilibrium points. Our main goal is to design a state-dependent switching function to ensure global asymptotic stability of a chosen point inside this set with conditions expressed in terms of linear matrix inequalities. For this class of systems, global exponential stability is generally impossible to be guaranteed. Hence, the proposed switching function is shown to ensure global asymptotic and local exponential stability of the desired equilibrium point. The position control and the velocity control with integral action of a dc motor driven by an h-bridge fed via a boost converter are used for validation. This practical application example is composed of eight subsystems, and all possible convex combinations of the dynamic matrices are singular.

math.OC↗

arXiv2022

Optimal Intermittent Particle Filter

The problem of the optimal allocation (in the expected mean square error sense) of a measurement budget for particle filtering is addressed. We propose three different optimal intermittent filters, whose optimality criteria depend on the information available at the time of decision making. For the first, the stochastic program filter, the measurement times are given by a policy that determines whether a measurement should be taken based on the measurements already acquired. The second, called the offline filter, determines all measurement times at once by solving a combinatorial optimization program before any measurement acquisition. For the third one, which we call online filter, each time a new measurement is received, the next measurement time is recomputed to take all the information that is then available into account. We prove that in terms of expected mean square error, the stochastic program filter outperforms the online filter, which itself outperforms the offline filter. However, these filters are generally intractable. For this reason, the filter estimate is approximated by a particle filter. Moreover, the mean square error is approximated using a Monte-Carlo approach, and different optimization algorithms are compared to approximately solve the combinatorial programs (a random trial algorithm, greedy forward and backward algorithms, a simulated annealing algorithm, and a genetic algorithm). Finally, the performance of the proposed methods is illustrated on two examples: a tumor motion model and a common benchmark for particle filtering.

eess.SY↗

arXiv2022

Stability of Switched Affine Systems: Arbitrary and Dwell-Time Switching

The dynamical behavior of switched affine systems is known to be more intricate than that of the well-studied switched linear systems, essentially due to the existence of distinct equilibrium points for each subsystem. First, under arbitrary switching rules, the stability analysis must be generally carried out with respect to a compact set with non-empty interior rather than to a singleton. We provide a novel proof technique for existence and outer approximation of attractive invariant sets of a switched affine system, under the hypothesis of global uniform stability of its linearization. On the other hand, considering dwell-time switching signals, forward invariant sets need not exist for this class of switched systems, even for stable ones. Hence, more general notions of stability/boundedness are introduced and studied, highlighting the relations of these concepts to the uniform stability of the linear part of the system under the same class of dwell-time switching signals. These results reveal the main differences and specificities of switched affine systems with respect to linear ones, providing a first step for the analysis of switched systems composed by subsystems not sharing the same equilibrium. Numerical methods based on linear matrix inequalities and sum-of-squares programming are presented and illustrate the developed theory.

eess.SY↗

arXiv2021

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function.

math.OC↗

arXiv2021

Comparison of Path-Complete Lyapunov Functions via Template-Dependent Lifts

This paper investigates, in the context of discrete-time switching systems, the problem of comparison for path-complete stability certificates. We introduce and study abstract operations on path-complete graphs, called lifts, which allow us to recover previous results in a general framework. Moreover, this approach highlights the existing relations between the analytical properties of the chosen set of candidate Lyapunov functions (the template) and the admissibility of certain lifts. This provides a new methodology for the characterization of the order relation of path-complete Lyapunov functions criteria, when a particular template is chosen. We apply our results to specific templates, notably the sets of primal and dual copositive norms, providing new stability certificates for positive switching systems. These tools are finally illustrated with the aim of numerical examples.

math.OC↗

arXiv2021

Complexity of the LTI system trajectory boundedness problem

We study the algorithmic complexity of the problem of deciding whether a Linear Time Invariant dynamical system with rational coefficients has bounded trajectories. Despite its ubiquitous and elementary nature in Systems and Control, it turns out that this question is quite intricate, and, to the best of our knowledge, unsolved in the literature. We show that classical tools, such as Gaussian Elimination, the Routh--Hurwitz Criterion, and the Euclidean Algorithm for GCD of polynomials indeed allow for an algorithm that is polynomial in the bit size of the instance. However, all these tools have to be implemented with care, and in a non-standard way, which relies on an advanced analysis.

eess.SY↗

arXiv2021

Data-driven stability analysis of switched affine systems

We consider discrete-time switching systems composed of a finite family of affine sub-dynamics. First, we recall existing results and present further analysis on the stability problem, the existence and characterization of compact attractors, and the relations these problems have with the joint spectral radius of the set of matrices composing the linear part of the subsystems. Second, we tackle the problem of providing probabilistic certificates of stability along with the existence of forward invariant sets, assuming no knowledge on the system data but only observing a finite number of sampled trajectories. Some numerical examples illustrate the advantages and limits of the proposed conditions.

eess.SY↗

arXiv2021

Optimal Control for Linear Networked Control Systems with Information Transmission Constraints

This paper addresses the problem of robust control of a linear discrete-time system subject to bounded disturbances and to measurement and control budget constraints. Using Q-parameterization and a polytope containment method, we prove that the co-design of an affine feedback controller, a measurement schedule and a control schedule can be exactly formulated as a mixed integer linear program with 2 binary variables per time step. As a consequence, this problem can be solved efficiently, even when an exhaustive search for measurement and control times would have been impossible in a reasonable amount of time.

math.OC↗

arXiv2021

Data-driven computation of invariant sets of discrete time-invariant black-box systems

We consider the problem of computing the maximal invariant set of discrete-time black-box nonlinear systems without analytic dynamical models. Under the assumption that the system is asymptotically stable, the maximal invariant set coincides with the domain of attraction. A data-driven framework relying on the observation of trajectories is proposed to compute almost-invariant sets, which are invariant almost everywhere except a small subset. Based on these observations, scenario optimization problems are formulated and solved. We show that probabilistic invariance guarantees on the almost-invariant sets can be established. To get explicit expressions of such sets, a set identification procedure is designed with a verification step that provides inner and outer approximations in a probabilistic sense. The proposed data-driven framework is illustrated by several numerical examples.

eess.SY↗

arXiv2021

Data-driven stability analysis of switched linear systems with Sum of Squares guarantees

We present a new data-driven method to provide probabilistic stability guarantees for black-box switched linear systems. By sampling a finite number of observations of trajectories, we construct approximate Lyapunov functions and deduce the stability of the underlying system with a user-defined confidence. The number of observations required to attain this confidence level on the guarantee is explicitly characterized. Our contribution is twofold: first, we propose a novel approach for common quadratic Lyapunov functions, relying on sensitivity analysis of a quasi-convex optimization program. By doing so, we improve a recently proposed bound. Then, we show that our new approach allows for extension of the method to Sum of Squares Lyapunov functions, providing further improvement for the technique. We demonstrate these improvements on a numerical example.

math.OC↗

arXiv2021

Immersion-based model predictive control of constrained nonlinear systems: Polyflow approximation

In the framework of Model Predictive Control (MPC), the control input is typically computed by solving optimization problems repeatedly online. For general nonlinear systems, the online optimization problems are non-convex and computationally expensive or even intractable. In this paper, we propose to circumvent this issue by computing a high-dimensional linear embedding of discrete-time nonlinear systems. The computation relies on an algebraic condition related to the immersibility property of nonlinear systems and can be implemented offline. With the high-dimensional linear model, we then define and solve a convex online MPC problem. We also provide an interpretation of our approach under the Koopman operator framework.

math.OC↗

arXiv2021

Geometric control of algebraic systems

In this paper, we present a geometric approach for computing the controlled invariant set of a continuous-time control system. While the problem is well studied for in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function. We compare it with the common algebraic approach for polynomial sublevel sets and show that it is significantly more conservative than our method.

math.OC↗

arXiv2021

Chance-constrained quasi-convex optimization with application to data-driven switched systems control

We study quasi-convex optimization problems, where only a subset of the constraints can be sampled, and yet one would like a probabilistic guarantee on the obtained solution with respect to the initial (unknown) optimization problem. Even though our results are partly applicable to general quasi-convex problems, in this work we introduce and study a particular subclass, which we call "quasi-linear problems". We provide optimality conditions for these problems. Thriving on this, we extend the approach of chance-constrained convex optimization to quasi-linear optimization problems. Finally, we show that this approach is useful for the stability analysis of black-box switched linear systems, from a finite set of sampled trajectories. It allows us to compute probabilistic upper bounds on the JSR of a large class of switched linear systems.

math.OC↗

arXiv2020

A data-driven method for computing polyhedral invariant sets of black-box switched linear systems

In this paper, we consider the problem of invariant set computation for black-box switched linear systems using merely a finite set of observations of system trajectories. In particular, this paper focuses on polyhedral invariant sets. We propose a data-driven method based on the one step forward reachable set. For formal verification of the proposed method, we introduce the concepts of $λ$-contractive sets and almost-invariant sets for switched linear systems. The convexity-preserving property of switched linear systems allows us to conduct contraction analysis on the computed set and derive a probabilistic contraction property. In the spirit of non-convex scenario optimization, we also establish a chance-constrained guarantee on set invariance. The performance of our method is then illustrated by numerical examples.

eess.SY↗

arXiv2020

Computation of the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints

We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other smooth constraints with Lipschitz gradient. With these quadratic relaxations, a sufficient condition for set invariance is derived and it can be formulated as a set of linear matrix inequalities. Based on the sufficient condition, a new algorithm is presented with finite-time convergence to the actual maximal invariant set under mild assumptions. This algorithm can be also extended to switched linear systems and some special nonlinear systems. The performance of this algorithm is demonstrated on several numerical examples.

eess.SY↗

arXiv2020

Abstraction-based branch and bound approach to Q-learning for hybrid optimal control

In this paper, we design a theoretical framework allowing to apply model predictive control on hybrid systems. For this, we develop a theory of approximate dynamic programming by leveraging the concept of alternating simulation. We show how to combine these notions in a branch and bound algorithm that can further refine the Q-functions using Lagrangian duality. We illustrate the approach on a numerical example.

math.OC↗

arXiv2020

Finite Data-Rate Feedback Stabilization of Continuous-Time Switched Linear Systems with Unknown Switching Signal

In this paper, we study the problem of stabilizing switched linear systems when only limited information about the state and the mode of the system is available, which occurs in many applications involving networked switched systems (such as cyber-physical systems, IoT, etc.). First, we show that switched linear systems with arbitrary switching, i.e., with no constraint on the switching signal, are in general not stabilizable with a finite data rate. Then, drawing on this result, we restrict our attention to systems satisfying a fairly mild slow-switching assumption, in the sense that the switching signal has an average dwell time bounded away from zero. We show that under this assumption, switched linear systems that are stabilizable in the classical sense remain stabilizable with a finite data rate. A practical coder-controller that stabilizes the system is presented and its applicability is demonstrated on numerical examples.

math.OC↗