Linear System Theory Research Articles

Interpretable Design of Reservoir Computing Networks Using Realization Theory.

The reservoir computing networks (RCNs) have been successfully employed as a tool in learning and complex decision-making tasks. Despite their efficiency and low training cost, practical applications of RCNs rely heavily on empirical design. In this article, we develop an algorithm to design RCNs using the realization theory of linear dynamical systems. In particular, we introduce the notion of α -stable realization and provide an efficient approach to prune the size of a linear RCN without deteriorating the training accuracy. Furthermore, we derive a necessary and sufficient condition on the irreducibility of the number of hidden nodes in linear RCNs based on the concepts of controllability and observability from systems theory. Leveraging the linear RCN design, we provide a tractable procedure to realize RCNs with nonlinear activation functions. We present numerical experiments on forecasting time-delay systems and chaotic systems to validate the proposed RCN design methods and demonstrate their efficacy.

On the Admissibility and Stability of Multiagent Nonlinear Interconnected Positive Systems With Heterogeneous Delays

Many multi-agent interconnected systems include typical nonlinearities which are highly sensitive to inevitable communication delays. This makes their analysis challenging and the generalization of results from linear interconnected systems theory to those nonlinear interconnected systems very limited. This paper deals with the analysis of Multi-Agent Nonlinear Interconnected Positive Systems (MANIPS). The main contributions of this work are two fold. Based on Perron-Frobenius theorem, we first study the “admissibility” property for MANIPS, and show that it is a fundamental requirement for this category of systems. Then, using admissibility/positivity properties and sequences of functions theory, we propose a suitable Lyapunov function candidate to conduct the analysis of the dynamical behavior of such systems. We show that the stability of MANIPS is reduced to the positiveness property (i.e. negative or positive definiteness) of a new specific matrix-valued function ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {Z}$</tex-math></inline-formula> ) that we derive in this paper. Furthermore, the obtained results generalize the existing theory. The quality of the results achieved are demonstrated through the applications of the developed theory on cells with multi-stage maturation process dynamical models.

Формування математичної моделі турбовального двигуна

The subject of the study is the process of forming a mathematical model (MM) of a turboshaft gas turbine engine and a twin-engine helicopter power plant, which provides the determination of parameters of the working process in steady and transient operating modes for use in the estimation of dynamic characteristics, in the analysis and synthesis of engine and helicopter automatic control systems. The goal is to substantiate the structure and methodology of MM formation intended for use in real and accelerated time scale systems. Tasks: implementation of the previously proposed MM structure taking into account the turboshaft engine performances, development of a methodology for determining the MM coefficients based on known information about the static and dynamic properties of the engine, and formation of the MM structure of a two-engine helicopter power plant. For this, the methods of the theory of airjet engines and the theory of linear dynamic systems are used. The following results were obtained: the structure of a multimode high-speed MM of a turboshaft engine and a two-engine power plant was formed and tested. The scientific and practical novelty of the obtained results is as follows: the structure of the multimode linearized MM of the turboshaft engine is formed, which consists of static and dynamic submodels implemented in corrected parameters; the modeling technique was worked out on a simplified model, compiled considering expert information about the static and dynamic properties of the engine in the considered operation area. Formulas were obtained that relate the coefficients of the linear dynamic model to the values of the time constants of the rotors and the sensitivities obtained from the static characteristics; transient characteristics of the engine based on changes in fuel consumption and load power are determined, which correspond to physical knowledge about the engine; the modeling methodology and MM structure of a two-engine power plant were formed, which is distinguished by the combination of individual static and linear dynamic models of two engines with a single nonlinear dynamic model of the helicopter rotor; a simplified MM load necessary for testing the MM of the engine installation is proposed, which provides the calculation of the power consumed by the rotor, depending on the angular position of the blades.

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

In this paper, we provide a system-theoretic treatment of certain continuous-time homogeneous polynomial dynamical systems (HPDS) via tensor algebra. In particular, if a system of homogeneous polynomial differential equations can be represented by an orthogonally decomposable (odeco) tensor, we can construct its explicit solution by exploiting tensor Z-eigenvalues and Z-eigenvectors. We refer to such HPDS as odeco HPDS. By utilizing the form of the explicit solution, we are able to discuss the stability properties of an odeco HPDS. We illustrate that the Z-eigenvalues of the corresponding dynamic tensor can be used to establish necessary and sufficient stability conditions, similar to these from linear systems theory. In addition, we are able to obtain the complete solution to an odeco HPDS with constant control. Finally, we establish results which enable one to determine if a general HPDS can be transformed to or approximated by an odeco HPDS, where the previous results can be applied. We demonstrate our framework with simulated and real-world examples.

Quaternion algebra approach to nonlinear Schrödinger equations with nonvanishing boundary conditions

In this article we apply quaternionic linear algebra and quaternionic linear system theory to develop the inverse scattering transform theory for the nonlinear Schrödinger equation with nonvanishing boundary conditions. We also determine its soliton solutions by using triplets of quaternionic matrices.

Stochastic linear quadratic optimal control problems with expectation-type linear equality constraints on the terminal states

A stochastic linear quadratic (LQ) optimal control problem with an expectation-type linear equality constraint on the terminal state is considered. Under the solvability condition on a stochastic Riccati equation and a surjectivity condition on the linear constraint mapping, the constrained stochastic LQ problem is solved completely by the Lagrangian duality theory. Some equivalent characterizations of the surjectivity condition are discussed by the controllability theory of linear control systems. Especially, the equivalence between the surjectivity condition and a Kalman-type rank condition is proved under proper conditions.

Binary Restructuring Particle Swarm Optimization and Its Application.

Restructuring Particle Swarm Optimization (RPSO) algorithm has been developed as an intelligent approach based on the linear system theory of particle swarm optimization (PSO). It streamlines the flow of the PSO algorithm, specifically targeting continuous optimization problems. In order to adapt RPSO for solving discrete optimization problems, this paper proposes the binary Restructuring Particle Swarm Optimization (BRPSO) algorithm. Unlike other binary metaheuristic algorithms, BRPSO does not utilize the transfer function. The particle updating process in BRPSO relies solely on comparison results between values derived from the position updating formula and a random number. Additionally, a novel perturbation term is incorporated into the position updating formula of BRPSO. Notably, BRPSO requires fewer parameters and exhibits high exploration capability during the early stages. To evaluate the efficacy of BRPSO, comprehensive experiments are conducted by comparing it against four peer algorithms in the context of feature selection problems. The experimental results highlight the competitive nature of BRPSO in terms of both classification accuracy and the number of selected features.

Point-Counterpoint: Transfer function analysis of dynamic cerebral autoregulation: To band or not to band?

Transfer function analysis (TFA) of dynamic cerebral autoregulation (dCA) is based on linear system theory to examine the relationship between changes in blood pressure and cerebral blood flow. With TFA, dCA is characterized as a frequency-dependent phenomenon quantified by gain, phase, and coherence in the distinctive frequency bands. These frequency bands likely reflect the underlying regulatory mechanisms of the cerebral vasculature. In addition, obtaining TFA metrics over a specific frequency band facilitates reliable spectral estimation and statistical data analysis to reduce random noise. This commentary discusses the benefits and cautions of banding TFA parameters in dCA studies.

Data‐driven control of dynamic event‐triggered systems with delays

AbstractThis article studies data‐driven control of unknown sampled‐data linear systems with communication delays under an event‐triggering transmission mechanism. While event‐triggered control has received much attention in the existing literature, its design and implementation typically require detailed model knowledge. Due to the difficulties in finding accurate models and the abundance of data in many practical applications, we propose a novel data‐driven event‐triggered control scheme for unknown, delayed systems with theoretical guarantees. To this end, as a first step, a model‐based stability condition for the resulting event‐triggered time‐delay system is established using a novel looped‐functional approach. Combining this model‐based condition with the data‐driven representation of linear systems, a data‐based stability condition is derived. Further, methods for co‐designing the controller gain and the event‐triggering matrix are subsequently provided without using any prior knowledge of the system matrices. Finally, numerical examples are presented to corroborate the merits of the proposed data‐driven event‐triggered control schemes relative to existing results.

On the invertibility indices and the Morse normal form for linear multivariable systems

In this paper, by extending the controllability and Brunovsky indices for a matrix pair to a matrix quadruple, three kinds of indices, say, the invertibility indices, the left invertibility indices, and the right invertibility indices, for linear time-invariant systems are introduced and studied. Based on properties of these indices, a neat expression for the Morse lists of a linear system is given without transforming it into its Morse normal form, which is a deep and fundamental result in linear systems theory, and takes an important role in many analysis and design problems for linear systems. Furthermore, the Morse normal form with only algebraic equivalence transformation is also investigated and is simplified.

A METHOD FOR SYNTHESIZING HARDWARE AND SOFTWARE TOOLS TO ENSURE THE STABILITY OF A CORPORATE COMPUTER NETWORK

The paper represents a method for ensuring the resilience of a corporate computer network under the influence of various types of threats. This article will provide an overview of the aspects of resilience and existing approaches to ensuring resilient routing. This article is the result of many studies and experiments, and evaluating the final result, it can be noted that this method can successfully reflect the possible importance of a node when it comes to epidemic dynamics for various network models to ensure network resilience. A possible way to solve the problem was to use the theory of linear stationary systems and the phenomenon of propagation in networks as the basis of the method. Complex interdependencies between their elements characterize various systems. The method of synthesizing hardware and software means of ensuring the stability of a corporate computer network consists of such steps as representing networks as a linear stationary system, modelling the stability of a computer network in the context of epidemics by using virtual network expansion, studying the stability of a computer network in the context of uncertain data transmission and virtual network expansion, processing input data received from the modelled computer network, etc. To solve the problem, the method involves the theory of linear stationary systems and the use of the NiR metric, which can successfully reflect the possible importance of a node when it comes to the dynamics of an epidemic for various network models to ensure network resilience. The method is tested by simulations, the results of which show a high correlation with the actual propagation dynamics modeled by SI and SIR processes. NiR also shows a small variance, which means it is reliable for different computer network topologies. The method also involves finding the most critical nodes in a computer network, for which a cascading failure model was used, which models overloaded nodes as non-functional.

A low‐dimensional realization algorithm for periodic input/output behavioral systems

The state‐space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time‐invariant case was solved, particular attention was paid to the case of linear periodic systems. Recently, such systems have regained importance, for instance, in the context of coding theory where periodic convolutional encoders play an important role. The majority of the contributions within this area only concern the realization of transfer functions (or impulse responses), thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. In previous research conducted by the authors, some results on the realization of periodic behavioral systems were obtained, which, despite their importance, are conservative with respect to the dimension of the obtained realizations. In this paper, we revisit the behavioral periodic realization problem and give further insight into this question. This allows to derive new results and set up an algorithm to compute low‐dimensional state‐space realizations.

Wavenumber-Division Multiplexing in Line-of-Sight Holographic MIMO Communications

Starting from first principles of wave propagation, we consider a multiple-input multiple-output (MIMO) representation of a communication system between two spatially-continuous volumes. This is the concept of holographic MIMO communications. The analysis takes into account the electromagnetic interference, generated by external sources, and the constraint on the physical radiated power. The electromagnetic MIMO model is particularized for a pair of parallel line segments in line-of-sight conditions. Inspired by orthogonal-frequency division-multiplexing, we assume that the spatially-continuous transmit currents and received fields are represented using the Fourier basis functions. In doing so, a wavenumber-division multiplexing (WDM) scheme is obtained, which is not optimal but can be efficiently implemented. The interplay among the different system parameters (e.g., transmission range, wavelength, and sizes of source and receiver) in terms of number of communication modes and level of interference among them is studied with conventional tools of linear systems theory. Due to the non-finite support (in the spatial domain) of the electromagnetic channel, WDM cannot provide non-interfering communication modes. The interference decreases as the receiver size grows, and goes to zero only asymptotically. Different digital processing architectures, operating in the wavenumber domain, are thus used to deal with the interference. The simplest implementation provides the same spectral efficiency of a singular-value decomposition architecture with water-filling when the receiver size is comparable to the transmission range. The developed framework is also used to represent a communication scheme that performs only an integration over short spatial segments. This is equivalent to a classical MIMO system with uniform linear arrays made of electrically small dipoles. Numerical comparisons show that better performance than WDM can be achieved only when a higher number of radio-frequency chains is used.

A vehicle cloud control system considering communication quantization and stochastic delay

AbstractA novel kind of networked feedback controller is designed for the vehicle cloud control system in the presence of both sensor‐controller/controller‐actuator delay and communication quantization. Firstly, the vehicle cloud control system with delayed and quantized communication is modeled using the lateral/longitudinal vehicle dynamic and Markovian jump linear system (MJLS) theory. Then, some efficient stabilization conditions in matrix inequalities forms are derived for the considered cloud‐controlled vehicles. Furthermore, a cone complementary linearization method is utilized to solve the nonlinear matrix inequalities involved in the stabilization conditions. Simulation tests on connected vehicle lateral and longitudinal control are conducted to verify the effectiveness of the analytical results. Compared with the commonly used constant‐parameter controllers, the proposed method is practical to design vehicle cloud controllers under stochastic delay and communication quantization scenarios, with known delay distribution. Lastly, the stability of the vehicle cloud control system is guaranteed under the infection of unreliable communication factors with the proposed control method.

Transient Growth of Accelerated Optimization Algorithms

Optimization algorithms are increasingly being used in applications with limited time budgets. In many real-time and embedded scenarios, only a few iterations can be performed and traditional convergence metrics cannot be used to evaluate performance in these nonasymptotic regimes. In this article, we examine the transient behavior of accelerated first-order optimization algorithms. For convex quadratic problems, we employ tools from linear systems theory to show that transient growth arises from the presence of nonnormal dynamics. We identify the existence of modes that yield an algebraic growth in early iterations and quantify the transient excursion from the optimal solution caused by these modes. For strongly convex smooth optimization problems, we utilize the theory of integral quadratic constraints to establish an upper bound on the magnitude of the transient response of Nesterov’s accelerated algorithm. We show that both the Euclidean distance between the optimization variable and the global minimizer and the rise time to the transient peak are proportional to the square root of the condition number of the problem. Finally, for problems with large condition numbers, we demonstrate tightness of the bounds that we derive up to constant factors.

Tom Chaffey [PhDs in Control

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Tom:</i> My dissertation lies at the intersection of nonlinear systems theory, electrical network theory, and monotone operator theory. All three of these areas have common roots in the study of passive networks in the 1920s and 1930s. However, they have grown apart, and each developed its own tools and problems. My dissertation takes a fresh look at nonlinear input/output systems theory. I generalize two of the most successful tools in linear systems theory and linear network theory—the Nyquist diagram and the frequency response of a transfer function—to nonlinear systems using tools from monotone operator theory.

Data-Driven Simulation of Generalized Bilinear Systems via Linear Time-Invariant Embedding

Nonparameteric representations of linear time-invariant systems that use Hankel matrices constructed from data are the basis for data-driven simulation and control. This article extends the approach to data-driven simulation of a class of nonlinear systems, called generalized bilinear. The generalized bilinear class includes Hammerstein, finite-lag Volterra, and bilinear systems. The key step of the generalization is an embedding result that is of independent interest. The behavior of a nonlinear system is included into the behavior of a linear time-invariant system. The method proposed is illustrated and compared with a model-based method on simulation examples and real-life data.

Block Factor-Width-Two Matrices and Their Applications to Semidefinite and Sum-of-Squares Optimization

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this paper, we introduce a new notion of block factor-width-two matrices and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS optimization, this leads to a block extension of the scaled diagonally dominant sum-of-squares (SDSOS) polynomials. By varying a matrix partition, the notion of block factor-width-two matrices can balance a trade-off between the computation scalability and solution quality for solving semidefinite and SOS optimization problems. Numerical experiments on a range of large-scale instances confirm our theoretical findings.

Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again

The fundamental lemma by Jan C. Willems and co-workers is deeply rooted in behavioral systems theory and it has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions – in particular Polynomial Chaos Expansions (PCE) of L2-random variables, which date back to Norbert Wiener’s seminal work – enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the L2-random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear time-invariant systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts.