The high order multiplication perturbation method for time-varying dynamic system

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.

Traffic Signal Control Algorithms of a Traffic Network

Since multivariate time series (MTS), which lie in a non-Euclidean space, exhibit temporal evolution and correlation characteristics, its classification is considered a non-trivial task. To mitigate the impact of the time-varying characteristics and thus enhance the classification accuracy, in this paper, we propose to model MTS data using a time-varying linear dynamical system followed by a neural network-based classification on the Grassmannian manifold. Our experiments on publicly available MTS datasets show promising classification results.

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.

A physical parameter identification method for the linear time-varying system is presented in this paper. For an arbitrarily time-varying system, a series of Hankel matrices are established directly from the combined measurements of displacement, velocity, and acceleration together with the input excitations from a single experiment with the system. The time-varying equivalent state-space system matrices of the structure at each time instance are then identified by singular value decomposition using the free or random responses. A time-varying transformation matrix for transforming the identified equivalent system matrices into the physical system matrices is developed to determine the mass, stiffness, and damping matrices. Details of the proposed approach are investigated with a two degrees-of-freedom spring-mass-damper system with three kinds of time-varying parameters. Numerical results show that the proposed method is accurate and effective in identifying time-varying physical parameters.

The paper considers non-local control and optimal control problems for a linear time-varying dynamic system described by an ordinary differential equation with undivided multipoint conditions and with an integral condition imposed on the components of the state vector in some part of the time interval of the system operation. For the problem of optimal control, it is assumed that the quality criterion is specified over the whole time interval of system operation and has the meaning of the norm of some normed space. Conditions for complete controllability of a dynamic system with imposed non-local conditions are formulated. Conditions are obtained under which a solution to the control problem with such nonlocal conditions exists, and this solution is constructed. A constructive approach to solving control problems and a method for constructing an optimal control function are proposed. Continuity and non-uniqueness of the control function are shown. A solution to a specific control problem is given as an illustration.

Chang transformation was introduced for decoupling the slow and fast dynamics of a singularly perturbed Linear Time Invariant (LTI) system, and it was subsequently extended to Linear Time-Varying (LTV) systems under the slowly-varying assumption by way of frozen-time eigenvalues. This paper extends Chang transformation from slowly-varying LTV systems to LTV systems, when the singularly perturbed system has a semi-proper coefficient matrix A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">22</sub> (t) and LTV system ẇ(t) = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">22</sub> (t)w(t) is exponentially stable. Instead of using frozen-time eigenvalues based on slowly time-varying conditions, PD-eigenvalues for LTV systems are employed to characterize the exponential stability, thereby circumventing the slowly-varying constraint. Our results provide a larger bound on epsilon, which is a gauge on the validity of the Chang transformation, in some situations than using previous techniques. We have also recast the original Chang transformation so as to provide more insight into the decoupled subsystems. The insight will be useful in subsequent investigation on estimate of the Singular Perturbation Margin (SPM) for LTV systems. An equivalence relationship has recently been established for LTI system between the SPM and the Phase Margin (PM). The new results in this paper will facilitate the development of a PM-type stability margin metric for LTV systems, and for nonlinear, time-varying systems in the future.

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.

Results are given concerning the commutativity of time-varying systems. It is shown that a constant-parameter dynamic system does not commute with a time-varying dynamic system. Conditions are derived for low-order time-varying systems to commute. A general condition for a set of time-varying systems to commute is the existence of a variable time scaling which reduces the time-varying systems to a set of constant-parameter systems.

In this paper, adaptive optimal control is proposed for linear discrete time-varying (LDTV) systems subject to unknown system dynamics. The idea of the method is a direct application of the Q-learning adaptive dynamic programming for time-varying systems. In order to derive the optimal control policy, an actor-critic structure is constructed and the time-varying least square method is adopted for parameter adaptation. The derived control policy robustly stabilizes the time-varying system and guarantees an optimal control performance. As no particular system information is required throughout the process, the proposed method provides a feasible solution to a large variety of applications. The validity of the proposed method is verified through simulation studies.

In this paper a new systematic switching control approach to adaptive stabilization of linear time-varying (LTV) discrete-time systems is presented.A feature of the localization-based method is its high model falsification capability, which in the case of LTI systems is manifested as the rapid convergence of the switching controller.We believe that the proposed method may help pave the way for design of practical adaptive switching controllers applicable to a wide range of linear time-invariant and timevarying systems.

The goal of this paper is to assess the robustness of an uncertain linear time-varying (LTV) system on a finite time horizon. The uncertain system is modeled as a connection of a known LTV system and a perturbation. The state matrices of the LTV system are assumed to be rational functions of time. This is used to model the uncertain LTV system as an connection of a time invariant system and an augmented perturbation that includes time. The input/output behavior of the perturbation is described by time-domain, integral quadratic constraints (IQCs). Static and dynamic IQCs are developed for the multiplication by time. A sufficient condition to bound the induced L2 gain is formulated using dissipation inequalities and IQCs. The approach is demonstrated with two simple examples.

4 Time-Varying Linear Dynamic System Model

Periodic Motions for Estimation of the Attraction Domain in the Wheeled Robot Stabilization Problem