State Of Linear System Research Articles

TWO APPROACHES TO THE FORMATION OF A QUANTITATIVE MEASURE OF STABILITY BASED ON MULTIPLE ESTIMATES OF THE PARAMETERS OF AN ENSEMBLE OF TRANSIENT PROCESSES

The article is devoted to the further development of the theory of stability of dynamic systems, namely of quantitative methods of stability assessment. A review and critical analysis of various approaches, which allow to introduce a quantitative measure of stability of dynamic systems to one degree or another, is given. The limitations of the existing methods, which are primarily related to the assessment of the behavior of the transient processes of individual trajectories, as well as the difficulty of obtaining an assessment of the behavior of the ensemble of transient processes when trying to apply the methods of N. D. Moiseyev. are substantiated. A method of quantitative assessment of a dynamic system stability based on the numerical estimates of the behavior of the area of initial deviations from the equilibrium position on the trajectories of the dynamic system is substantiated. Based on the Liouville formula, it is shown that changes in the volume of the area of the initial deviations on the trajectories of the system does not depend on the form of the latter one. This allowed to limit the area of initial deviations in the shape of a hypersphere and to obtain a simple expression for a quantitative measure of the stability of a linear stationary dynamic system, the geometric sense of which is to estimate the rate of change of the volume of the control surface. The article proposes and substantiates the criterion of uniformity of deformation of the area of initial deviations. The essence of the problem is that in the transient process, the values of some components of the phase vector may reach unacceptable deviations from the equilibrium position. A theoretical estimate of deformation non-uniformity for linear systems is obtained, which is taken to be the deviation of the trace of the ellipsoid matrix from the deviations of the trace of the hypersphere matrix of the corresponding volume. A method for a quantitative measure of the stability based on an integral quadratic functional calculated on a set of transient processes of initial deviations in the form of a set of ellipsoids with a normalized volume is proposed and substantiated. Diagonal positive normalized matrices are considered as a set of matrices of the integral quadratic criterion. A simple algorithm for calculation of the multiple integral quadratic criterion is proposed.

On the External Estimation of Reachable and Null-Controllable Limit Sets for Linear Discrete-Time Systems with a Summary Constraint on the Scalar Control

The problem of constructing reachable and null-controllable sets for stationary linear discrete-time systems with a summary constraint on the scalar control is considered. For the case of quadratic constraints and a diagonalizable matrix of the system, these sets are built explicitly in the form of ellipsoids. In the general case, the limit reachable and null-controllable sets are represented as fixed points of a contraction mapping in the metric space of compact sets. On the basis of the method of simple iteration, a convergent procedure for constructing their external estimates with an indication of the a priori approximation error is proposed. Examples are given.

Synthesis of an Observer Providing a Finite-Time State Estimation Based on the Output

The paper proposes a new method for synthesizing a full-order state vector observer for a certain class of linear stationary systems with unknown input signal. Two approaches that ensure the convergence of the estimation error (i.e., the error between the state vector estimate and the true state vector) to zero in a finite time are considered. The convergence time is set by the designer in the process of observer construction. The system parameters are assumed to be known and stationary, but the relative degree (from the measured output and the unknown input) is equal to one. The synthesis process using the first approach consists of two stages. In the first stage two auxiliary observers are used to estimate the initial conditions of the object in finite time. In the second stage, a third observer is introduced into the loop to perform online estimation of the system state vector. All the above three observers have the same structure that provides their insensitivity to the input signal, but different parameters. The second approach uses aggregation of auxiliary observers and introduce a delay operator to estimate the state vector. This method provides convergence of the observation error to zero after a delay time. The paper contains a rigorous mathematical proofs of the performance of the obtained solutions. To demonstrate the effectiveness and operability of the proposed approach computer simulation was carried out in the MATLAB Simulink software environment. A two-channel fourth-order mechanical system is considered as an plant. The simulation results illustrate the convergence of the state vector estimation error to zero in a finite time specified by the developer.

Frequency criterion for stability «as a whole» of modulation-type pulse voltage converters according to Lyapunov

Background. The use of methods for researching and designing stable nonlinear dynamic systems by constructing the frequency characteristics of the loop gain of the negative feedback of a pulsed voltage converter through time characteristics obtained by numerical methods for solving nonlinear differential equations that describe electromagnetic processes in the converter takes a lot of time and has low accuracy in calculating optimal modes, when you need to calculate hundreds of options.
 Aim. Therefore, it seems necessary to further develop the theory of stability of modulation-type pulse voltage converters using frequency stability criteria, Lyapunov functions, amplitude and phase frequency characteristics of the converter loop gain.
 Methods. Stability in the «small» is solved using linearized models, which are constructed using the so-called first Lyapunov method - the «first approximation» method. The second method for studying the stability of nonlinear dynamic systems is the harmonic linearization method, when the nonlinear element is replaced by a harmonically linearized link with a transmission coefficient only for the first harmonic.
 Results. The Lyapunov function for a pulse voltage converter with nonlinear relay-type elements was found. Conclusion. The results obtained in the work make it possible to simplify the design of Lyapunov-stable pulse voltage converters through the use of the method of amplitude-frequency characteristics used for stationary linear systems.

Stability Analysis of Distributed-Order Systems

The paper deals with stability of distributed order systems. A recently proposed method for stability analysis of a wide class of linear stationary systems is shown to be applicable to distributed order systems as well. In addition, we demonstrate that the well-known distributed order PID control algorithm should be used with utmost care, since the corresponding transfer functions introduce dipoles in the open right-half complex plane. The analysis is illustrated by a numerical example.

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.

On Some Properties of Sets of Bounded Controllability for Stationary Linear Discrete Systems with Total Control Constraints

We consider the problem of constructing reachable sets, i.e., sets of terminal states into which a system can be transferred from the origin in a fixed time, and 0-controllability sets, i.e., sets of initial states, from which a system can be transferred to the origin in a fixed time, for stationary linear discrete systems with a total control constraint. The representation of reachable and 0-controllable sets as linear transformations of superellipsoidal sets of finite and infinite dimensions is proved. A constructive method for describing the desired sets based on the apparatus of supporting half-planes is proposed, including for the limit sets of reachability and controllability. In the case of Euclidean spaces, the description is obtained explicitly. Examples are given. For a three-dimensional satellite motion control system in a near-circular orbit, reachability sets are modeled.

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

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

Synthesis of Adaptive Observer of State Variables for a Linear Stationary Object in the Presence of Measurement Noise

The paper is devoted to the problem of state variables observers synthesis for linear stationary system operating under condition of noise or disturbances in the measurement channel. The paper considers a completely observable linear stationary system with known parameters. It is assumed that the state variables are not measured, and the measured output variable contains a small amplitude (in general, modulo less than one) additive noise or disturbance. It is also assumed that there is no a priori information about the disturbance or noise in the measurement channel (for example, frequency spectrum, covariance, etc.). It is well known that many observer synthesis methods have been obtained for this type of systems, including the Kalman filter, which has proven itself in practice. Under the condition of complete observability and the presence of some a priori information about a random process (which is typical for the case when a disturbance in the measurement channel can be represented as white noise), approaches based on Kalman filtering demonstrate the highest quality estimates of state variables convergence to true values. Without disputing the numerous results obtained using the application of the Kalman filter, an alternative idea of the state variables observer constructing is considered in this paper. The alternative of the new approach is primarily due to the fact that there is no need to use the usual approaches based on the Luenberger observer. The paper proposes an approach based on the estimation of unknown parameters (in this case, an unknown vector of initial conditions of the plant state variables) of a linear regression model. Within the framework of the proposed method, after a simple transformation, a transition is made from a dynamic system to a linear regression model with unknown constant parameters containing noise or disturbing effects. After that, a new nonlinear parametrization of the original regression model and an algorithm for identifying unknown constant parameters using the procedure of dynamic expansion of the regressor and mixing are proposed which ensure reduction the influence of noise. The article presents the results of computer simulations verifying the stated theoretical results.

Test Methods for Signal Reconstruction of Linear Stationary Systems

The test methods designed for mathematical reconstruction of input signals of linear stationary information processing systems from distorted and noisy output signals are proposed and analyzed. It is shown that the use of test data allows one to carry out a non-blind reconstruction without determining the hardware functions of processing systems, which, in general, can belong to the class of generalized functions. The features of the use of the regularization technique in test methods are considered when solving ill-posed and ill-conditioned problems of reconstruction of real non-deterministic signals. The criteria for selecting test signals are analyzed. The results of numerical experiments on the restoration of one-dimensional signals and two-dimensional images at different noise levels are presented.

Test Methods for Signal Reconstruction of Linear Stationary Systems

Test Methods for Signal Reconstruction of Linear Stationary Systems

Достаточные условия существования Н∝ -наблюдателя состояния линейных непрерывных динамических систем

<p>The article deals with the problem of finding the observer of the state vector of linear continuous non-stationary dynamical systems with uncertainty of the initial conditions, limited external influences and measurement errors over a finite time interval. Sufficient conditions for the existence of an observer are formulated and proved on the basis of the expansion principle. Relationships are obtained for finding the parameters of the observer and the worst laws of change in external influences and measurement errors. As a limiting case, the problem of observer synthesis for stationary linear dynamical systems on a semi-infinite time interval is considered. Two applied problems of estimating the aircraft state vector based on the results of incomplete and inaccurate measurements are solved.</p>

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.

QUASI-ANALYTIC METHOD OF LINEAR DYNAMIC SYSTEMS INVERSION

The problem of inversion of dynamic systems has become widespread while solving the problems of control, identification, and measurement problems arising during the design and research of electrical and mechanical dynamic systems. Inverting is an effective way of implementing disturbance control processes, as well as in combined control systems with a predictive model. The analysis of information sources showed that in the practical solution of most inversion problems, a number of difficulties arise, which are associated with the high sensitivity of the results in relation to the accuracy of the parameters of the mathematical model of the control object, the instability of the inverse model of non-minimum-phase objects, and the violation of the conditions of physical feasibility. The work offers an effective method of inverting linear stationary dynamic systems, free from the mentioned shortcomings in many respects. The basis of the method is the presentation of input and output signals in the form of infinite linear combinations of their derivatives. A method of determining the sequence of matrix coefficients of linear representations of input and output signals is proposed. The main theoretical result is obtaining relationships between matrix coefficients of input and output signals. The work considers mathematical models of linear dynamic systems in the form of differential equations in the state space and in the equivalent "input-output" form. The considered systems must meet the conditions of asymptotic stability, as well as the condition of equal dimensions of the input and output vectors. Requirements for mathematical models of input and output signals are given, the fulfillment of which allows, instead of infinite sums representing signals, to be limited to a finite number of terms.

Reduction of Band Matrices in Large Dynamic Systems Controllability and Observability

An approach is proposed for linear stationary dynamical system with controllability and observability band matrices making it possible to simplify procedures for evaluating controllability and observability of this system. The obtained results are based on the fact that the controllability and observability criteria of a dynamic system are equivalent due to their required and sufficient properties; therefore, any transformations of one criterion not violating the conditions of necessity and sufficiency could be reduced to transformations in a sense equivalent to the initial transformations. The Popov --- Belevich --- Hautus transformations of the controllability and observability criteria were taken as a basis, and then results of such transformations were correctly extended to the band criteria. It was proved that the controllability and observability analysis of a linear stationary system with a large number of the state dimensions was reduced to studying the matrices rank of a much smaller size. The proposed approach is based on the existence condition for a numerical matrix of a certain rank of the nondegenerate matrices that satisfy certain transformations. The corresponding controllability and observability theorems for the stationary dynamical systems were provided. It was shown that for systems with one input and one output, the controllability and observability analysis was reduced to the analysis of scalars

IDENTIFICATION OF LINEAR DYNAMIC SYSTEMS IN THE ENVIRONMENT OF POLYNOMYAL SIGNALS

A method for the structural and parametric identification of one-dimensional linear stationary dynamic systems, represented by differential "inputoutput" constraint equations, is proposed. The method is focused on both active and passive experiments. The method is based on a polynomialrepresentation of the input and output signals of the identified dynamic system. A compact vector-matrix representation of polynomials is proposed,which makes it possible to find the forced component of the solution of linear differential equations as a result of performing simple linear algebraicoperations. The vector-matrix representation of polynomials made it possible to quite simply solve the problem of inversion of linear dynamicalsystems and the problem of compensating the measured perturbation. The issues of representing time signals in polynomial form are not considered inthis paper. Based on the obtained linear representation of a one-dimensional dynamic system, which links the parameters of the input and outputsignals with the parameters of the differential equation of the identified dynamic system mathematical model, a linear system of algebraic equations forunknown coefficients of the differential process equation is obtained. In the general case, the resulting system belongs to the class of overdeterminedsystems, and therefore its solution can be obtained by the least-square technique and is reduced to finding a pseudoinverse matrix. A block diagram ofsoftware for solving the problem of structural and parametric identification in the environment of polynomial signals is proposed. The algorithmincludes the procedure of comparing the results of numerical simulation of the identified model with the output experimental signal and correcting thestructure of the model based on the results of the comparison.

Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation of the controllability and observability Gramians of continuous linear and bilinear stationary systems with many inputs and one output, based on the method of spectral expansion of the Gramians and the iterative method for computing the bilinear systems Gramians. An important feature of the concept is the idea of separability of the Gramians expansion: separate computation of the scalar and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation to calculation of numerical sequences of their elements. For the continuous linear systems with one output the method and the algorithm of the spectral decomposition of the controllability Gramian are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the Gramian matrices are obtained, and by making use of which the rest of the elements can be calculated. For continuous linear systems with many outputs the spectral decompositions of the Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly reduce the number of calculations. The obtained results are generalized for continuous bilinear systems with one output. Iterative spectral algorithms for computation of elements of Gramians of these systems are proposed. Examples are given that illustrate theoretical results.

Complex Frequency Domain Analysis of Memristor Based on Volterra Series

At present, the application of different types of memristors in electronics is being deeply studied. Given the nonlinearity characterizing memristors, a circuit with memristors cannot be treated by classical circuit analysis. In this paper, memristor is equivalent to a nonlinear dynamic system composed of linear dynamic system and nonlinear static system by Volterra series. The nonlinear transfer function of memristor is derived. In the complex frequency domain, the n-order complex frequency response of memristor is established by multiple Laplace transform, and the response of MLC parallel circuit is taken as an example to verify. Theoretical analysis shows that the complex frequency domain analysis method of memristor transforms the problem of solving nonlinear circuit in time domain into n times complex frequency domain analysis of linear circuit, which provides an idea for nonlinear dynamic system analysis.

New trends in control theory

The article outlines the conceptual foundations of new trends in control theorythat have been intensively developing in recent years. Unlike classical controltheory, which was formed in the last century and is based on well-known mathematical models of controlled processes in the form of local equations, new approaches to linear stationary systems use input-output relations that follow directly from the Cauchy formula for both continuous and discrete systems. On thebasis of the same description, it is possible to substantiate and obtain the socalled data-based models, which are directly linked to data that form, at the observation intervals, the trajectories of already implemented past processes andfuture ones, for which control is to be synthesized. This approach is focusedprimarily on finding control from the prediction model. At the same time, thecurrent measurements carried out at the plant make it possible to implementfeedback and, in case of discrepancies between the forecast and the real process,to correct the predictive control, i.e. such a way to stabilize it. Control by trajectory prediction model allows to exclude model identification by trajectory data,and control directly on their base. Since the data contain errors, the most important issue in the considered approach is the robustness of the chosen control.A large number of published works are dedicated to this problem, where theguaranteed approach, focused on the worst-case in the data, is the most in demand. In most cases, control synthesis is reduced to solving various optimizationproblems, mainly on the finite prediction horizon. Considerable attention in thearticle is paid to methods for solving synthesis problems based on SVD decomposition. To reduce the complexity of the tasks to be solved, it is proposed to reduce it to terminal control on the horizon of a short duration. Then an iterativecontrol strategy is implemented, which, due to feedback, ensures the feasibilityof the global control goal.