Commutativity of time-varying systems

A high order multiplication perturbation method for linear time-varying dynamic system is presented. Firstly the time domain is dispersed with small time interval and the coefficient matrix of the dynamic system is decomposed into a “large amount” and a “small amount” in each time interval. One order perturbation system is then presented from the original dynamic system by a variable transformation which is also a linear time-varying dynamic system. The coefficient matrix of the new system is also decomposed into a “large amount” and a “small amount” and then higher order perturbation system can be obtained by another variable transformation. The final perturbation system can be solved exactly after abandoning the “small amount” of the coefficient matrix in the final system, and then the answer to the original problem can be determined through a series of inverse transform. Since the transfer matrix is the product of a series of exponential matrix which can be calculated accurately by the precise time integration method, so the proposed method has fine accuracy, efficiency and stability. The proposed method actually is a high order symplectic conservative perturbation method for the Hamiltonian system. The examples show that the proposed method can also give good results even though a large time step selected, and with the increase of the perturbation number, the perturbation solutions can tend to exact solutions rapidly.

A robust tracking control algorithm based mainly on the theory of variable structure systems and Lyapunov's direct method is developed in this paper, to treat the tracking control problem of a class of non-linear time-varying systems with uncertain dynamics. A nominal model is used instead of an exact one. together with a description of the bounds of plant uncertainty. The objective is to design a control law such that all the plants lying within a specified band of uncertainly will be driven to reach the goal. The modelling error, time-varying system dynamics, system uncertainties, and disturbances are all treated as general disturbances to the nominal system. This algorithm greatly eases the difficulties present in requiring a precise model of the physical system for the control purpose and for handling the non-linear time-varying system. The non-linear time-varying system is first linearized to a linear time-varying system (with time-varying coefficients completely defined) via the method of variable s...

The purpose of this paper is to derive necessary and sufficient conditions for target path controllability (TPC) of time-varying dynamical systems, and to relate the results to those of Brockett and Mesarović in 1965. It is shown that conditions for TPC in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W^{p}_{1}</tex> by means of L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> -controls are easy to verify, but are very strong. TPC implies that the number of target variables is less than or equal to the number of control variables.

This paper presents a new on-line procedure for identifying MIMO time-varying stochastic dynamic systems. It is based on a decomposition model derived from the state-space representation. This scheme gives an estimated model with a minimal number of parameters. The parameters of the MIMO system are identified from short records with unknown dead times. Extended recursive least-squares and fixed-lag smoothing identification algorithms are applied to identify the validity of the model structure. Artificial and hydrological records are used to illustrate the efficiency of the proposed procedure.

A technique for determining the dynamic response characteristics of time-varying linear systems is presented. A model composed of a parallel connection of filters whose impulse responses are orthogonalized exponential functions is used. The input to the system is fed to each of the filters. The coefficients of linear regression of the system output on each of the filter outputs are determined. Time-varying characteristics are measured by determining successively the coefficients from short samples of the input and output signals of the system. Many of the statistical properties of regression coefficients are known and can be used to estimate the length of sample of input and output signals required to determine the coefficients with given variance. The measurement technique has been implemented on a high speed digital computer. Results obtained by applying the technique to measurement of a variety of digitally simulated filters and to measurement of human pilot control characteristics are presented.

Motivated by applications to automatic control of aircraft, we consider the problem of output tracking for time-varying linear systems. The signals to be tracked are bounded on (-/spl infin/, /spl infin/), and our goal is to compute bounded controls and bounded state trajectories that result in the desired tracking. A convolution integral representation for the desired solution is given in the literature, and we show that certain integrability assumptions can be eliminated from the associated results. Despite the fact that our system is time varying and our signals do not necessarily have Fourier transforms in the classical sense, we show that computations can be carried out using generalized Fourier transforms. We also consider an output tracking problem in which the driven dynamical equation is an ordinary differential equation.

In this paper, an effective method is proposed called the “Dynamic Routh’s stability Criterion (DRSC)”, which is developed using the Dynamic Pole Motion (DPM) approach (DPMA). This innovative technique extends the classical Routh’s stability criterion—traditionally limited to linear time-invariant (LTI) systems—to encompass a broader range of systems, including linear time-varying and nonlinear dynamic systems. By incorporating the behavior of pole trajectories over time, the proposed method offers a more comprehensive framework for analyzing system stability in more complex and realistic scenarios. DPM approach describes the notion of dynamic poles (DP) in a three dimensional ‘g-plane’, which is an extension to the two dimensional ‘s-plane’. The s-domain approach is based upon Laplace transform and has been used for linear time-invariant (LTI) system only. This novel g-plane framework is versatile enough to be applied to both linear and nonlinear dynamic systems. For the stability of dynamic systems, the DP must present in the left-hand side (LHS) of the g-plane. For nonlinear systems, the locations of DP are the function of system states; and these systems states are the function of initial conditions with amplitude and frequency (AnF) of input signals. Therefore, in nonlinear systems, stability is influenced not only by the initial conditions but also by the AnF of the input signals. For example, for given amplitude of the input signal, system may be unstable at low frequency; however, it may become stable at high frequency or vice-versa. These stability conditions are illustrated by several examples.

The objective of the paper is to explain a modern Hilbert transform method for analysis and identification of mechanical non-linear vibration structures in the case of quasiperiodic signals. This special kind of periodicity arises in experimental vibration signals. The method is based on the Hilbert transform of input and output signals in a time domain to extract the instantaneous dynamic structure characteristics. The paper focuses on the dynamic analysis and identification of three groups of dynamics systems: • Forced vibrations of linear and non-linear SDOF systems excited with quasiperiodic force signal. • Combined forced vibrations of quasiperiodic time varying linear and non-linear SDOF systems excited with harmonic signal. • Combined self-excited and forced vibrations of non-linear SDOF systems excited with harmonic signal. The study focuses on signal processing techniques for nonlinear system investigation, which enable us to estimate instantaneous system dynamic parameters (natural frequencies, damping characteristics and their dependencies on a vibration amplitude and frequency) for different kinds of system excitation.

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh −. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.

Decoupling by state-variable feedback and forming of the inverse has been done for multivariable systems with constant coefficients. In this letter, the theory is extended to linear time-varying systems. For this, general conditions are formulated when a time-varying multivariable systems can be decoupled. Further, the matrices are given for the decoupling. The inverse of a linear time-varying multivariable systems is also determined.

Fuzzy dynamic Bayesian network based on a discrete aggregation method for risk assessment of marine nuclear power platform hinge joints accidents

Mathematical modeling of many engineering systems yields coupled linear or nonlinear ordinary differential equations which contain periodic parametric excitation terms. Such systems, in general, do not possess exact solutions and one must resort to some approximate analytical or numerical techniques. For weakly excited systems, it has been possible to apply averaging and perturbation methods and obtain meaningful equivalent time-invariant systems. In the event when the parametric excitation is strong, such techniques can not yield meaningful results. In this paper, a new approach for constructing equivalent approximate time-invariant forms for general multi-dimensional dynamical systems subjected to periodic parametric excitations is presented. The technique is based on the application of the well-known Liapunov-Floquet (L-F) transformation and normal form theory. The proposed approach is illustrated through examples of linear as well as nonlinear time-periodic systems. It is shown that a dynamical system described by a set of second-order differential equations with periodic coefficients can be transformed to a similar set of equations which is time-invariant. The transformed form can then be used to study the response of the original time-varying system when forced with deterministic or stochastic excitations. Further, the time-invariant form also permits to obtain solution of the order-reduction problem associated with a large-scale periodic time-varying systems. In case of nonlinear problems, the application of L-F transformation yields similar dynamical systems whose linear part are time-invariant. The solutions of such systems can be obtained through an application of the time-dependent normal form theory. When dynamical systems with critical eigenvalues are considered, both time-dependent center manifold reduction and normal form theory can be employed in the study of local stability of such systems. It is shown that the solutions of time-invariant dynamical systems obtained via time-independent normal form theory provide excellent results when compared with numerical solutions. Some engineering problems are included as illustrative examples.

The feasibility of using an Artificial Neural Network (ANN) for controlling time- varying dynamical system is presented. The direct adjusting of neural controller by direct adaptive control (DAC) is available, by using the error between output of plant and desired input. The finite recurrent back propagation (FRBP) is used in the learning process, because the ability of this method to capture the nonlinearly and overcome the problem of time varying system. Hybrid controller structure used in this paper, where the parameters of classical controller are adjusted with time at specified freezing points for time varying dynamical system, and summed the outputs of two controllers and enter to the plant, identify of system by ANN to get the optimal initial condition for neuro controller. A single channel for Spacecraft model is used as an example in this paper, satisfactory results are obtained, which explain the ability of recurrent neural network (RNN) to identify time varying dynamical system and overcome for all its problem and explain the ability of this structure of hybrid neuro controller to use with time varying dynamical system.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

Observability of Time-Varying Fractional Dynamical Systems with Caputo Fractional Derivative