Linear Dynamical Systems Research Articles

A novel non-intrusive ROM for randomly excited linear dynamical systems with high stochastic dimension using ANN

Analyzing large stochastic dynamical systems is computationally very expensive. A statistical simulation framework requires invoking the solver multiple times — ranging from thousands to millions. A non-intrusive reduced order model (ROM) serves as a computationally efficient alternative in this framework. Uncertainties in dynamical systems originate from two sources: system parameters and excitation. However, all the existing ROMs have been developed for uncertainty only in the system parameters. To the best of the authors’ knowledge, no ROM exists for dynamical systems under random excitation. A discretization of the input process increases the stochastic dimensionality, typically by a few hundred or thousand. This leads to a high dimensional interpolation or regression in the ROM, which is practically infeasible. This issue is addressed in this work by proposing a novel non-intrusive ROM that bypasses the need for such discretization. Accordingly, a regression is carried out directly on the random excitation using a neural network-based surrogate model. A principal component-based data compression is used in tandem to reduce the stochastic dimensionality of excitations. Detailed numerical studies are conducted to study the accuracy and efficiency of the proposed ROM using two examples: a mistuned bladed disk problem and a soil–structure interaction. The numerical results show that the proposed ROM is accurate and gains a significant speed-up of more than sixty for both examples. Using the proposed ROM, the cost of uncertainty quantification can be reduced significantly within the framework of Monte Carlo simulation.

On the integration of Physics-Based Machine Learning with hierarchical Bayesian modeling techniques

Machine Learning (ML) has widely been used for modeling and predicting physical systems. These techniques offer high expressive power and good generalizability for interpolation within observed data sets. However, the disadvantage of black-box models is that they underperform under blind conditions since no physical knowledge is incorporated. Physics-based ML aims to address this problem by retaining the mathematical flexibility of ML techniques while incorporating physics. In accord, this paper proposes to embed mechanics-based models into the mean function of a Gaussian Process (GP) model and characterize potential discrepancies through kernel machines. A specific class of kernel function is promoted, which has a connection with the gradient of the physics-based model with respect to the input and parameters and shares similarity with the exact Auto-covariance function of linear dynamical systems. The spectral properties of the kernel function enable considering dominant periodic processes originating from physics misspecification. Nevertheless, the stationarity of the kernel function is a difficult hurdle in the sequential processing of long data sets, resolved through hierarchical Bayesian techniques. This implementation is also advantageous to mitigate computational costs, alleviating the scalability of GPs when dealing with sequential data. Using numerical and experimental examples, potential applications of the proposed method to structural dynamics inverse problems are demonstrated.

Model order reduction for optimality systems through empirical gramians

In the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE) discretization. This leads to a specific linear but high-dimensional time variant (LTV) dynamical system. To reduce the size of the LTV system, we apply a tailored reduced order modeling technique based on empirical gramians and derived directly from the first-order optimality system. For testing purpose, we focus on two specific examples: a multiobjective optimization and a closed-loop optimal control problem. Our proposed methodology results to be better performing than a standard proper orthogonal decomposition (POD) approach for the above mentioned examples.

Global analysis on a continuous planar piecewise linear differential system with three zones

This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincare disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/83/abstr.html

On discrete-time dissipative port-Hamiltonian (descriptor) systems

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman–Yakubovich–Popov inequalities is also studied.

On the Dynamics of Two-Dimensional Fractional Linear Control Systems

On the Dynamics of Two-Dimensional Fractional Linear Control Systems

BLIND IDENTIFIABILITY OF A LINEAR DYNAMIC COMMUNICATION CHANNEL MODEL

The article describes the problem of developing a mathematical model of data transmission channel in communication systems in the case when signals at the channel input channel are unknown at the channel output. Development of mathematical models directly from observed data constitutes an identification problem, and the case under consideration belongs to the blind identification problem. The work consistently examines the concept of linear dynamic system identifiability in state space and shows the connection of this concept with the observability and controllability concepts. A number of statements about identifiability of a linear dynamic system describing communication channel are proved. In particular, it is shown that if the system is identifiable, it is sufficient to use the input-output model for its description, while the input-state-output model is redundant. Further, based on the proven statements, blind identifiability conditions are formulated for the case of a discrete system with one input and multiple outputs, which are imposed on the components of the system’s transfer matrix.

Exponential Consensus of Multiple Agents Over Dynamic Network Topology: Controllability, Connectivity, and Compactness

In this paper, we investigate the problem of securing exponentially fast consensus (exponential consensus for short) for identical agents with finite-dimensional linear system dynamics over dynamic network topology. Our aim is to find the weakest possible conditions that guarantee exponentially fast consensus using a Lyapunov function consisting of a sum of terms of the same functional form. We first investigate necessary conditions, starting by examining the system (both agent and network) parameters. It is found that controllability of the linear agents is necessary for reaching consensus. Then, to work out necessary conditions incorporating the network topology, we construct a set of Laplacian matrix-valued functions. The precompactness of this set of functions is shown to be a significant generalization of existing assumptions on network topology, including the common assumption that the edge weights are bounded piecewise constant functions or continuous functions. With the aid of such a precompactness assumption and restricting the Lyapunov function to one consisting of a sum of terms of the same functional form, we prove that a joint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\delta, T)$</tex-math></inline-formula> -connectivity condition on the network topology is necessary for exponential consensus. Finally, we investigate how the above two “necessities” work together to guarantee exponential consensus. To partially address this problem, we define a synchronization index to characterize the interplay between agent parameters and network topology. Based on this notion, it is shown that by designing a proper feedback matrix and under the precompactness assumption, exponential consensus can be reached globally and uniformly if the joint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\delta,T)$</tex-math></inline-formula> -connectivity and controllability conditions are satisfied, and the synchronization index is not less than one.

Characteristic curves and the exponentiation in the Riordan Lie group: A connection through examples

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Fréchet structure of the Riordan group described in [4]. After that, we describe some linear dynamical systems in K[[x]] with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to Pascal's triangle.

A convolutional neural network deep learning method for model class selection

AbstractThe response‐only model class selection capability of a novel deep convolutional neural network method is examined herein in a simple, yet effective, manner. Specifically, the responses from a unique degree of freedom along with their class information train and validate a one‐dimensional convolutional neural network. In doing so, the network selects the model class of new and unlabeled signals without the need of the system input information, or full system identification. An optional physics‐based algorithm enhancement is also examined using the Kalman filter to fuse the system response signals using the kinematics constraints of the acceleration and displacement data. Importantly, the method is shown to select the model class in slight signal variations attributed to the damping behavior or hysteresis behavior on both linear and nonlinear dynamic systems, as well as on a 3D building finite element model, providing a powerful tool for structural health monitoring applications.

Stages and main tasks of the century-long control theory and system identification development

The article provides a review of the mathematical description of the dynamics of continuous and discrete linear stationary systems and objects, used at the development stage of the classical theory of automatic control in the form of mathematical models of the «input-output» type. The time and frequency characteristics of continuous and discrete control systems are described, typical links of stationary systems are considered, parametric discrete models of objects as part of typical digital control loops are presented. Stochastic discrete autoregressive models of stationary time series used to describe the dynamic objects in the synthesis of digital control systems are considered. A review of standard control laws for the implementation of continuous and discrete controllers has been completed. A method for synthesizing discrete controllers for multidimensional controlled objects with different, unknown and changing delays is considered, through which variable delays are compensated in the characteristic equation of a closed-loop control system. A common technique for synthesizing one-dimensional and multidimensional controllers for stochastic objects with delays based on ARMAX models is considered. An analysis of approaches to identifying delays in controlled objects is carried out and a method for identifying delays when using input-output models is considered, based on the calculation and comparison of impulse responses for extended and non-extended models of the controlled object. An analysis of the advantages and disadvantages of «input-output» type models is given, as well as the possibilities of their application for solving various classes of control theory problems.

Synthesis of Robust Linear Stationary Dynamic Systems Based on Improvement of the Controllability and Observability of the Computational Model of the Object

An iterative method is developed for the synthesis of control systems of low parametric sensitivity with dynamic (polynomial) controllers, at each step of which singular numbers of controllability and observability gramians are intentionally changed, and a precontroller is formed that transforms the structure of the computational model of an object to increase its controllability and observability, taking into account the available opportunities and limitations. The method of polynomial modal control calculates the parameters of the main controller and evaluates the robust properties of the synthesized system in the given intervals of variation of the internal parameters.

A Novel Order Abatement Technique for Linear Dynamic Systems and Design of PID Controller

This article proposes a novel hybrid technique of order abatement for large-scale models that combines the Mihailov stability method (MSM) and the stability equation method (SEM). In this approach, the denominator coefficients of the higher-order system (HOS) are estimated using the MSM, while the numerator coefficients are computed using the SEM. The suggested approach is based on the MSM, which guarantees the stability of the estimated model if the actual model is stable. The MSM also makes sure that important factors of the original plant, such as dominant poles and stability, are retained in the reduced order system (ROS). The suggested approach is compared to several current conventional reduction methods using error indicators, and the smallest performance error indices values reflect the supremacy of the method. The transfer function (TF) of the ROS is then used to design controllers by employing the moment matching technique. When the controller designed with the approximated model is applied to the real HOS, it indicates that the response of the closed-loop system of the real model entirely overlaps with the response of the reference plant. To further demonstrate the efficiency of the proposed schemes, time-domain specifications are produced and time responses are plotted.

Combination of Optimal Three-Step Composite Time Integration Method with Multi-Point Iterative Methods for Geometric Nonlinear Structural Dynamics

This study focuses on solving the geometric nonlinear dynamic equations of structures using the multi-point iterative methods within the optimal three-step composite time integration method (OTCTIM). The OTCTIM, initially devised for linear dynamic systems, is now proposed to encompass nonlinear dynamic systems in such a way that the semi-static nonlinear equations in time sub-steps can be solved using multi-point methods. The Weerakoon–Fernando method (WFM), Homeier method (HM), Jarrat method (JM), and Darvishi–Barati method (DBM) have been extended as multi-point solvers for nonlinear equations in OTCTIM, which exhibit a higher convergence order than the Newton–Raphson method (NRM), without requiring the calculation of second and higher derivatives. Several structural examples were solved to examine the performance of these methods in the OTCTIM approach. The results demonstrated that the multi-point iterative methods outperform NRM (in terms of the number of iterations) within the OTCTIM for geometric nonlinear structural dynamics and, among the multi-point methods, the JM and DBM converged with fewer number of iterations and lower error levels. Furthermore, it has been observed that when solving nonlinear dynamic equations for structures with a high number of degrees of freedom, the incorporation of the DBM into the OTCTIM mitigates the convergence iterations and the average elapsed time for iterative sub-steps.

Bifurcation analysis of dynamical systems with fractional order differential equations via the modified Riemann-Liouville derivative

In this manuscript, the solutions of linear dynamical systems with fractional differential equations via themodified Riemann-Liouville derivative is derived. By using Jumarie type of derivative (JRL), we stated and provedthe Existence and uniqueness theorems of the dynamical systems with fractional order equations. Also a novel stability analysis of fractional dynamical systems by Jumarie type derivative is established and some important stability conditions are determined. The achieved results have various applications in mathematics, plasma physics and almost all branches of physics that have non-conservative forces. Finally, we investigated interesting application of nonlinear space-time fractional Korteweg-de Vries (STFKdV) equation in Saturn F-ring’s region. Moreover, our investigation could be basic interest to explain and interpret the effects of fractional and modification parameters on STFKdV equation. This is novel study on this model by dynamical system (DS) to describe the behavior of nonlinear waves without solve this system.

A robust and generalized effective DAE framework encompassing different methods, algorithms, and model order reduction for linear and nonlinear second order dynamical systems

For linear and nonlinear dynamical problems, we propose the novel development and advancement of the well-known Generalized Single Step Single Solve (GS4) family of second order time-accurate algorithms encompassing the entire class of LMS methods developed over the past 50 years or so with/without controllable numerical dissipation in conjunction with the Differential Algebraic Equation (DAE) framework and Proper Orthogonal Decomposition (POD). Unlike traditional practices which have severe limitations in the ability to provide generality and flexibility and a wide variety of choices to the analyst without losing order of time accuracy, the proposed implementation uniquely not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains in a single body, but also additionally allows the selection of different space discretized methods such as the Finite Element Method (FEM) and particle methods, and other spatial methods as well in a single analysis. Moreover, the addition of the POD further provides reduction in computational times. Furthermore, unlike existing state of the art, the present framework readily permits a wide array of implicit–implicit, implicit–explicit, and explicit–explicit couplings and the integration of such a technology is of interest here. Consequently, the present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration schemes in different subdomains in a selective manner together with Reduced Order Modeling (ROM) to optimize capturing the local and global features of the representative physics. The ROM also employs an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to the prescribed accuracy requirements. Such a novelty, and computational features, are not possible with existing state of art and numerical illustrations validate the claim.

Applications of the Linear Quadratic Regulator Optimal Control With Completely and Partially Observed Markovian Switching

Abstract We study optimal control problems where the dynamic system evolves according to a linear stochastic differential equation with (a) multiplicative and additive white noise, (b) a mixture of white noise, and (c) white noise and colored noise. The drift rate and the diffusion matrix of the linear dynamic system depend on a continuous-time Markov chain with finite state space that is (I) partially observed and (II) completely observed. Using the results of Wonham filter theory, we reduce the partially observed problems to one with complete observation. We solve the control optimal problems explicitly with the help of dynamic programing technique, and three applications are presented to illustrate our theoretical results.

Sequential stabilizing spline algorithm for linear systems: Eigenvalue approximation and polishing

The sequential stabilizing spline (SSS) algorithm is a remarkable algorithm for identifying linear dynamical systems. It can guarantee the stability of an estimated model by polishing the maximum modulus roots of an equation. Therefore, the root structure of an equation plays an important role in the SSS algorithm. Although the traditional power iterative method can be applied to approximate the extremal eigenvalues which have the largest modulus, it has two limitations: (1) the extremal eigenvalues must be real numbers, and (2) the structures of the extremal eigenvalues should be known a priori. In this paper, a novel power iterative method is proposed to approximate the extremal eigenvalues. Compared with the traditional power iterative method, the method in this paper can (1) determine the types of the extremal eigenvalues without prior knowledge of the matrix; (2) approximate the true values of the extremal eigenvalues regardless of their type; (3) become a worthy addition to SSS algorithm and gradient descent algorithm. Simulation examples demonstrate the effectiveness of the proposed method.

A New Scheme for the Approximation of Linear Dynamical Systems and Its Application to Controller Design

A New Scheme for the Approximation of Linear Dynamical Systems and Its Application to Controller Design

On the сontrollability, observability, and optimization of discrete nonstationary linear Volterra systems

In this article, we study discrete nonstationary linear dynamic systems of Volterra type. An essential feature of such kind of systems is that their current states depend on the previous states of this system. The formula Cauchy, which gives us the solution of linear Volterra systems with the control inputs, is obtained. The necessary and sufficient conditions of the pointwise controllability, pointwise output controllability, and observability are proven. Also the linear-quadratic optimization problem for the nonstationary Volterra control systems is studied.