arXiv subjects

Rodolphe Sepulchre

Publications and source records attributed to Rodolphe Sepulchre.

At least 55 records · Page 3Linked to original sources

arXiv2022

Open-loop contraction design

Given a non-contracting trajectory of a nonlinear system, we consider the question of designing an input perturbation that makes the perturbed trajectory contracting. This paper stresses the analogy of this question with the classical question of feedback stabilization. In particular, it is shown that the existence of an output variable that ensures contraction of the inverse system facilitates the design of a contracting input perturbation. We illustrate the relevance of this question in parameter estimation.

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arXiv2022

Splitting algorithms and circuit analysis

The splitting algorithms of monotone operator theory find zeros of sums of relations. This corresponds to solving series or parallel one-port electrical circuits, or the negative feedback interconnection of two subsystems. One-port circuits with series and parallel interconnections, or block diagrams with multiple forward and return paths, give rise to current-voltage relations consisting of nested sums and inverses. In this extended abstract, we present new splitting algorithms specially suited to these structures, for interconnections of monotone and anti-monotone relations.

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arXiv2022

On the incremental form of dissipativity

Following the seminal work of Zames, the input-output theory of the 70s acknowledged that incremental properties (e.g. incremental gain) are the relevant quantities to study in nonlinear feedback system analysis. Yet, non-incremental analysis has dominated the use of dissipativity theory in nonlinear control from the 80s. Results connecting dissipativity theory and incremental analysis are scattered and progress has been limited. This abstract investigates whether this theoretical gap is of fundamental nature and considers new avenues to circumvent it.

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arXiv2022

Loop Shaping with Scaled Relative Graphs

The Scaled Relative Graph (SRG) is a generalization of the Nyquist diagram that may be plotted for nonlinear operators, and allows nonlinear robustness margins to be defined graphically. This abstract explores techniques for shaping the SRG of an operator in order to maximize these robustness margins.

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arXiv2022

Affine-Invariant Midrange Statistics

We formulate and discuss the affine-invariant matrix midrange problem on the cone of $n\times n$ positive definite Hermitian matrices $\mathbb{P}(n)$, which is based on the Thompson metric. A particular computationally efficient midpoint of this metric is investigated as a highly scalable candidate for an average of two positive definite matrices within this context, before studying the $N$-point problem in the vector and matrix settings.

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arXiv2022

Adaptive Conductance Control

Neuromodulation is central to the adaptation and robustness of animal nervous systems. This paper explores the classical paradigm of indirect adaptive control to design neuromodulatory controllers in conductance-based neuronal models. The adaptive control of maximal conductance parameters is shown to provide a methodology aligned with the central concepts of neuromodulation in physiology and of impedance control in robotics.

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arXiv2022

Spiking Control Systems

Spikes and rhythms organize control and communication in the animal world, in contrast to the bits and clocks of digital technology. As continuous-time signals that can be counted, spikes have a mixed nature. This paper reviews ongoing efforts to develop a control theory of spiking systems. The central thesis is that the mixed nature of spiking results from a mixed feedback principle, and that a control theory of mixed feedback can be grounded in the operator theoretic concept of maximal monotonicity. As a nonlinear generalization of passivity, maximal monotonicity acknowledges at once the physics of electrical circuits, the algorithmic tractability of convex optimization, and the feedback control theory of incremental passivity. We discuss the relevance of a theory of spiking control systems in the emerging age of event-based technology.

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arXiv2021

Reliability of Event Timing in Silicon Neurons

Analog, low-voltage electronics show great promise in producing silicon neurons (SiNs) with unprecedented levels of energy efficiency. Yet, their inherently high susceptibility to process, voltage and temperature (PVT) variations, and noise has long been recognised as a major bottleneck in developing effective neuromorphic solutions. Inspired by spike transmission studies in biophysical, neocortical neurons, we demonstrate that the inherent noise and variability can coexist with reliable spike transmission in analog SiNs, similarly to biological neurons. We illustrate this property on a recent neuromorphic model of a bursting neuron by showcasing three different relevant types of reliable event transmission: single spike transmission, burst transmission, and the on-off control of a half-centre oscillator (HCO) network.

q-bio.NC↗

arXiv2021

Training Lipschitz continuous operators using reproducing kernels

This paper proposes that Lipschitz continuity is a natural outcome of regularized least squares in kernel-based learning. Lipschitz continuity is an important proxy for robustness of input-output operators. It is also instrumental for guaranteeing closed-loop stability of kernel-based controlllers through small incremental gain arguments. We introduce a new class of nonexpansive kernels that are shown to induce Hilbert spaces consisting of only Lipschitz continuous operators. The Lipschitz constant of estimated operators within such Hilbert spaces can be tuned by suitable selection of a regularization parameter. As is typical for kernel-based models, input-output operators are estimated from data by solving tractable systems of linear equations. The approach thus constitutes a promising alternative to Lipschitz-bounded neural networks, that have recently been investigated but are computationally expensive to train.

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arXiv2021

Kernel-based models for system analysis

This paper introduces a computational framework to identify nonlinear input-output operators that fit a set of system trajectories while satisfying incremental integral quadratic constraints. The data fitting algorithm is thus regularized by suitable input-output properties required for system analysis and control design. This biased identification problem is shown to admit the tractable solution of a regularized least squares problem when formulated in a suitable reproducing kernel Hilbert space. The kernel-based framework is a departure from the prevailing state-space framework. It is motivated by fundamental limitations of nonlinear state-space models at combining the fitting requirements of data-based modeling with the input-output requirements of system analysis and physical modeling.

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arXiv2021

Scaled relative graphs for system analysis

Scaled relative graphs were recently introduced to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this paper, we connect scaled relative graphs to the classical theory of input/output systems. It is shown that the Nyquist diagram of an LTI system on $L_2$ is the convex hull of its scaled relative graph under a particular change of coordinates. The SRG may be used to visualize approximations of static nonlinearities such as the describing function and quadratic constraints, allowing system properties to be verified or disproved. Interconnections of systems correspond to graphical manipulations of their SRGs. This is used to provide a simple, graphical proof of the classical incremental passivity theorem.

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arXiv2021

Oscillations in Mixed-Feedback Systems

A new method is presented for the analysis of limit cycle oscillations in mixed-feedback systems. The calculation of the limit cycle is reformulated as the zero finding of a mixed-monotone relation, that is, of the difference of two maximally monotone relations. The problem can then be solved efficiently by borrowing existing algorithms that minimize the difference of two convex functions. The potential of the method is illustrated on the classical Van der Pol oscillator.

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arXiv2021

Target formation on the circle by monotone system design

Positivity and Perron-Frobenius theory provide an elegant framework for the convergence analysis of linear consensus algorithms. Here we consider a generalization of these ideas to the analysis of nonlinear consensus algorithms on the circle and establish tools for the design of consensus protocols that monotonically converge to target formations on the circle.

math.OC↗

arXiv2021

Neuromorphic Control

Neuromorphic engineering is a rapidly developing field that aims to take inspiration from the biological organization of neural systems to develop novel technology for computing, sensing, and actuating. The unique properties of such systems call for new signal processing and control paradigms. The article introduces the mixed feedback organization of excitable neuronal systems, consisting of interlocked positive and negative feedback loops acting in distinct timescales. The principles of biological neuromodulation suggest a methodology for designing and controlling mixed-feedback systems neuromorphically. The proposed design consists of a parallel interconnection of elementary circuit elements that mirrors the organization of biological neurons and utilizes the hardware components of neuromorphic electronic circuits. The interconnection structure endows the neuromorphic systems with a simple control methodology that reframes the neuronal control as an input-output shaping problem. The potential of neuronal control is illustrated on elementary network examples that suggest the scalability of the mixed-feedback principles.

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arXiv2021

Inductive Geometric Matrix Midranges

Covariance data as represented by symmetric positive definite (SPD) matrices are ubiquitous throughout technical study as efficient descriptors of interdependent systems. Euclidean analysis of SPD matrices, while computationally fast, can lead to skewed and even unphysical interpretations of data. Riemannian methods preserve the geometric structure of SPD data at the cost of expensive eigenvalue computations. In this paper, we propose a geometric method for unsupervised clustering of SPD data based on the Thompson metric. This technique relies upon a novel "inductive midrange" centroid computation for SPD data, whose properties are examined and numerically confirmed. We demonstrate the incorporation of the Thompson metric and inductive midrange into X-means and K-means++ clustering algorithms.

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arXiv2021

Monotone RLC Circuits

The circuit-theoretic origins of maximal monotonicity are revisited using modern optimization algorithms for maximal monotone operators. We present an algorithm for computing the periodic behavior of an interconnection of maximal monotone systems using a fixed point iteration. The fixed point iteration may be split according to the interconnection structure of the system. In this preliminary work, the approach is demonstrated on port interconnections of maximal monotone resistors and LTI capacitors and inductors.

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arXiv2021

Variation diminishing linear time-invariant systems

This paper studies the variation diminishing property of $k$-positive linear time-invariant (LTI) systems, which map inputs with $k-1$ sign changes to outputs with at most the same variation. We characterize this property for the Toeplitz and Hankel operators of finite-dimensional systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of $k$ first order positive systems. This is shown by expressing the $k$-positivity of a LTI system as the external positivity (that is, $1$-positivity) of $k$ compound LTI systems. Our characterization generalizes well known properties of externally positive systems ($k=1$) and totally positive systems ($k=\infty$; also known as relaxation systems).

math.OC↗

arXiv2021

Balanced truncation of $k$-positive systems

This paper considers balanced truncation of discrete-time Hankel $k$-positive systems, characterized by Hankel matrices whose minors up to order $k$ are nonnegative. Our main result shows that if the truncated system has order $k$ or less, then it is Hankel totally positive ($\infty$-positive), meaning that it is a sum of first order lags. This result can be understood as a bridge between two known results: the property that the first-order truncation of a positive system is positive ($k=1$), and the property that balanced truncation preserves state-space symmetry. It provides a broad class of systems where balanced truncation is guaranteed to result in a minimal internally positive system.

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