arXiv subjects

Michael Margaliot

Publications and source records attributed to Michael Margaliot.

At least 37 records · Page 2Linked to original sources

arXiv2024

Analysis of the Identifying Regulation with Adversarial Surrogates Algorithm

Given a time-series of noisy measured outputs of a dynamical system z[k], k=1...N, the Identifying Regulation with Adversarial Surrogates (IRAS) algorithm aims to find a non-trivial first integral of the system, namely, a scalar function g() such that g(z[i]) = g(z[j]), for all i,j. IRAS has been suggested recently and was used successfully in several learning tasks in models from biology and physics. Here, we give the first rigorous analysis of this algorithm in a specific setting. We assume that the observations admit a linear first integral and that they are contaminated by Gaussian noise. We show that in this case the IRAS iterations are closely related to the self-consistent-field (SCF) iterations for solving a generalized Rayleigh quotient minimization problem. Using this approach, we derive several sufficient conditions guaranteeing local convergence of IRAS to the correct first integral.

eess.SY↗

arXiv2024

Multiplicative and additive compounds via Kronecker products and Kronecker sums

Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use. Here, we derive new formulas for the multiplicative and additive compounds of a matrix using Kronecker products and sums. This provides a new approach to matrix compounds based on the well-known and powerful theory of Kronecker products and sums. We demonstrate several applications of these new formulas, including deriving a new expression for the additive compound of the product of two matrices.

math.CA↗

arXiv2023

The $k$-Compound of a Difference-Algebraic System

The multiplicative and additive compounds of a matrix have important applications in geometry, linear algebra, and the analysis of dynamical systems. In particular, the $k$-compounds allow to build a $k$-compound dynamical system that tracks the evolution of $k$-dimensional parallelotopes along the original dynamics. This has recently found many applications in the analysis of non-linear systems described by ODEs and difference equations. Here, we introduce the $k$-compound system corresponding to a differential-algebraic system, and describe several applications to the analysis of discrete-time dynamical systems described by difference-algebraic equations.

eess.SY↗

arXiv2023

On special quadratic Lyapunov functions for linear dynamical systems with an invariant cone

We consider a continuous-time linear time-invariant dynamical system that admits an invariant cone. For the case of a self-dual and homogeneous cone we show that if the system is asymptotically stable then it admits a quadratic Lyapunov function with a special structure. The complexity of this Lyapuonv function scales linearly with the dimension of the dynamical system. In the particular case when the cone is the nonnegative orthant this reduces to the well-known and important result that a positive system admits a diagonal Lyapunov function. We demonstrate our theoretical results by deriving a new special quadratic Lyapunov function for systems that admit the ice-cream cone as an invariant set.

math.DS↗

arXiv2023

On the gain of entrainment in a class of weakly contractive bilinear control systems with applications to the master equation and the ribosome flow model

We consider a class of bilinear weakly contractive systems that entrain to periodic excitations. Entrainment is important in many natural and artificial processes. For example, in order to function properly synchronous generators must entrain to the frequency of the electrical grid, and biological organisms must entrain to the 24h solar day. A dynamical system has a positive gain of entrainment (GOE) if entrainment also yields a larger output, on average. This property is important in many applications from the periodic operation of bioreactors to the periodic production of proteins during the cell cycle division process. We derive a closed-form formula for the GOE to first-order in the control perturbation. This is used to show that in the class of systems that we consider the GOE is always a higher-order phenomenon. We demonstrate the theoretical results using two applications: the master equation and a model from systems biology called the ribosome flow model, both with time-varying and periodic transition rates.

math.DS↗

arXiv2023

Contraction and $k$-contraction in Lurie systems with applications to networked systems

A Lurie system is the interconnection of a linear time-invariant system and a nonlinear feedback function. We derive a new sufficient condition for $k$-contraction of a Lurie system. For $k=1$, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. However, Lurie systems often have more than a single equilibrium and are thus not contractive with respect to any norm. For $k=2$, our condition guarantees a well-ordered asymptotic behaviour of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We demonstrate our results by deriving a sufficient condition for $k$-contraction of a general networked system, and then applying it to guarantee $k$-contraction in a Hopfield neural network, a nonlinear opinion dynamics model, and a 2-bus power system.

eess.SY↗

arXiv2023

A sufficient condition for $k$-contraction in Lurie systems

We consider a Lurie system obtained via a connection of a linear time-invariant system and a nonlinear feedback function. Such systems often have more than a single equilibrium and are thus not contractive with respect to any norm. We derive a new sufficient condition for $k$-contraction of a Lurie system. For $k=1$, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. For $k=2$, our condition guarantees well-ordered asymptotic behaviour of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We apply our results to derive a sufficient condition for $k$-contractivity of a networked system.

eess.SY↗

arXiv2023

On singularly perturbed systems that are monotone with respect to a matrix cone of rank $k$

We derive a sufficient condition guaranteeing that a singularly perturbed linear time-varying system is strongly monotone with respect to a matrix cone $C$ of rank $k$. This implies that the singularly perturbed system inherits the asymptotic properties of systems that are strongly monotone with respect to $C$, which include convergence to the set of equilibria when $k=1$, and a Poincaré-Bendixson property when $k=2$. We extend this result to singularly perturbed nonlinear systems with a compact and convex state-space. We demonstrate our theoretical results using a simple numerical example.

math.OC↗

arXiv2023

On the gain of entrainment in the $n$-dimensional ribosome flow model

The ribosome flow model (RFM) is a phenomenological model for the flow of particles along a 1D chain of $n$ sites. It has been extensively used to study ribosome flow along the mRNA molecule during translation. When the transition rates along the chain are time-varying and jointly $T$-periodic the RFM entrains, i.e., every trajectory of the RFM converges to a unique $T$-periodic solution that depends on the transition rates, but not on the initial condition. In general, entrainment to periodic excitations like the 24h solar day or the 50Hz frequency of the electric grid is important in numerous natural and artificial systems. An interesting question, called the gain of entrainment (GOE) in the RFM, is whether proper coordination of the periodic translation rates along the mRNA can lead to a larger average protein production rate. Analyzing the GOE in the RFM is non-trivial and partial results exist only for the RFM with dimensions $n=1,2$. We use a new approach to derive several results on the GOE in the general $n$-dimensional RFM. Perhaps surprisingly, we rigorously characterize several cases where there is no GOE, so to maximize the average production rate in these cases, the best choice is to use constant transition rates all along the chain.

math.OC↗

arXiv2022

Verifying $k$-Contraction without Computing $k$-Compounds

Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for $k$-contraction is that a logarithmic norm (also called matrix measure) of the $k$-additive compound of the Jacobian is uniformly negative. However, this may be difficult to check in practice because the $k$-additive compound of an $n\times n$ matrix has dimensions $\binom{n}{k}\times \binom{n}{k}$. For an $n\times n$ matrix $A$, we prove a duality relation between the $k$ and $(n-k)$ compounds of $A$. We use this duality relation to derive a sufficient condition for $k$-contraction that does not require the computation of any $k$-compounds. We demonstrate our results by deriving a sufficient condition for $k$-contraction of an $n$-dimensional Hopfield network that does not require to compute any compounds. In particular, for $k=2$ this sufficient condition implies that the network is $2$-contracting and this implies a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.

math.DS↗

arXiv2022

On matrices whose exponential is a P-matrix

A matrix is called a P-matrix if all its principal minors are positive. P-matrices have found important applications in functional analysis, mathematical programming, and dynamical systems theory. We introduce a new class of real matrices denoted~$\EP$. A matrix is in~$\EP$ if and only if its matrix exponential is a P-matrix for all positive times. In other words, $A\in \EP$ if and only if the transition matrix of the linear system~$\dot x=Ax$ is a P-matrix for any positive time~$t$. We analyze the properties of this new class of matrices and describe an application of our theoretical results to opinion dynamics.

math.CA↗

arXiv2022

Compound matrices in systems and control theory: a tutorial

The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a growing interest in applications of these compounds, and their generalizations, in systems and control theory. The goal of this tutorial paper is to provide a gentle and self-contained introduction to these topics with an emphasis on the geometric interpretation of the compounds, and to describe some of their recent applications including several non-trivial generalizations of positive systems, cooperative systems, contracting systems, and more.

math.OC↗

arXiv2022

Minimum effort decentralized control design for contracting network systems

We consider the problem of making a networked system contracting by designing minimal effort local controllers. Our method combines a hierarchical contraction characterization and a matrix-balancing approach to stabilizing a Metzler matrix via minimal diagonal perturbations. We demonstrate our approach by designing local controllers that render contractive a network of FitzHugh-Nagumo neurons with a general topology of interactions.

math.OC↗

arXiv2021

Serial interconnections of 1-contracting and 2-contracting systems

The flow of contracting systems contracts 1-dimensional polygons (i.e. lines) at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an overall contracting system. A recent generalization of contracting systems is called $k$-contracting systems, where $k\in\{1,\dots,n\}$. The flow of such systems contracts $k$-dimensional polygons at an exponential rate, and in particular they reduce to contracting systems when $k=1$. Here, we analyze serial interconnections of $1$-contracting and $2$-contracting systems. We provide conditions guaranteeing that such interconnections have a well-ordered asymptotic behaviour, and demonstrate the theoretical results using several examples.

math.DS↗

arXiv2021

A sufficient condition for $k$-contraction of the series connection of two systems

The flow of contracting systems contracts 1-dimensional parallelotopes, i.e., line segments, at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an overall contracting system. A generalization of contracting systems is $k$-contracting systems, where $k\in\{1,\dots,n\}$. The flow of such systems contracts the volume of $k$-dimensional parallelotopes at an exponential rate, and in particular they reduce to contracting systems when $k=1$. It was shown by Muldowney and Li that time-invariant $2$-contracting systems have a well-ordered asymptotic behaviour: all bounded trajectories converge to the set of equilibria. Here, we derive a sufficient condition guaranteeing that the system obtained from the series interconnection of two sub-systems is $k$-contracting. This is based on a new formula for the $k$th multiplicative and additive compounds of a block-diagonal matrix, which may be of independent interest. As an application, we find conditions guaranteeing that $2$-contracting systems with an exponentially decaying input retain the well-ordered behaviour of time-invariant 2-contracting systems.

math.DS↗

arXiv2021

Compound matrices in systems and control theory

The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a growing interest in applications of these compounds, and their generalizations, in systems and control theory. This tutorial paper provides a gentle introduction to these topics with an emphasis on the geometric interpretation of the compounds, and surveys some of their recent applications.

math.OC↗

arXiv2021

Variability in mRNA Translation: A Random Matrix Theory Approach

The rate of mRNA translation depends on the initiation, elongation, and termination rates of ribosomes along the mRNA. These rates depend on many "local" factors like the abundance of free ribosomes and tRNA molecules in the vicinity of the mRNA molecule. All these factors are stochastic and their experimental measurements are also noisy. An important question is how protein production in the cell is affected by this considerable variability. We develop a new theoretical framework for addressing this question by modeling the rates as identically and independently distributed random variables and using tools from random matrix theory to analyze the steady-state production rate. The analysis reveals a principle of universality: the average protein production rate depends only on the of the set of possible values that the random variable may attain. This explains how total protein production can be stabilized despite the overwhelming stochasticticity underlying cellular processes.

q-bio.SC↗

arXiv2021

On the Diagonal Stability of $k$-Positive Linear Systems

We consider $k$-positive linear systems, that is, systems that map the set of vectors with up to $k-1$ sign variations to itself. For $k=1$, this reduces to positive linear systems. It is well-known that stable positive linear time invariant (LTI) systems admit a diagonal Lyapunov function. This property has many important implications. A natural question is whether stable $k$-positive systemsalso admit a diagonal Lyapunov function. This paper shows that, in general, the answer is no. However, for both continuous-time and discrete-time $n$-dimensional systems that are $(n-1)$-positive we provide a sufficient condition for diagonal stability.

math.DS↗