New results in realization theory for linear time-varying analytic systems

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.

ABSTRACTIn this paper, we describe a constructive test to decide whether a given linear time-varying (LTV) differential system admits a stabilising compensator for the control tasks of tracking, disturbance rejection or model matching and construct and parametrise all of them if at least one exists. In analogy to the linear time-invariant (LTI) case, the ring of stable rational functions, noncommutative in the LTV situation, and the Kučera–Youla parametrisation play prominent parts in the theory. We transfer Blumthaler's thesis from the LTI to the LTV case and sharpen, complete and simplify the corresponding results in the book ‘Linear Time-Varying Systems’ by Bourlès and Marinescu.

The aim of this paper is to present an algebraic theory of the realization of linear and nonlinear time-varying systems. In the linear case, the correspondence between Kalman's factorization criterion and the representative functions leads to a very simple and transparent formulation of the realization theory. In the nonlinear case, the time-varying realization is related to the strong controllability of the system. The connection between linear and nonlinear time-varying systems is investigated by using noncommutative generating series.

Polynomial matrix-fraction representations for linear time-varying systems

An identification approach for linear and nonlinear time-variant structural systems via harmonic wavelets

A kernel functional representation of linear time-varying systems is employed in conjunction with B-spline functional approximation techniques to construct non-asymptotic state and parameter estimators for LTV systems. Total observability of the estimated system must be assumed. Practical identifiability conditions for parametric estimation are also stated. In the absence of output measurement noise the observer provides almost exact reconstruction of the system state and delivers high fidelity functional estimates of the time varying system parameters. It also shares the usual superior features of algebraic observers such as independence of the initial conditions of the system and good noise attenuation properties. Other advantages of the kernel and B-spline based identification of linear time-varying systems are elucidated.

In the case of multiple nonstationary independent source signals and linear instantaneous time-varying mixing systems, it is difficult to adaptively separate the multiple source signals. Therefore, the adaptive blind source separation (BSS) problem is firstly formally expressed and compared with tradition BSS problem. Then, we propose an adaptive blind identification and separation method based on the variable learning rate equivariant adaptive source separation via independence (EASI) algorithm. Furthermore, we analyze the scope and conditions of variable-learning rate EASI algorithm. The adaptive BSS simulation results also show that the variable learning rate EASI algorithm provides better separation effect than the fixed learning rate EASI and recursive least-squares algorithms.

For linear time-varying single-input single-output systems, the paper treats the problem of designing an output feedback controller resulting in an (internally) stable closed-loop system that tracks a given reference signal. The design is accomplished by augmenting the given system to include an internal model of the reference signal. The internal model reduces to the usual form when the given plant is time invariant. An explicit construction for the internal model is given in the case of a step reference.

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 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.

Minimal realizations for discrete-time linear periodic 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.

This paper discusses the time-varying characteristics of the output signals resulted from linear time variant (LTV) systems using level crossing properties. These systems which are used for generating synthetic records based on a target record contain two identification processes: 1) amplitude modulating function, 2) time-varying impulse response function (IRF) parameters. For efficient tracking of the IRF parameters, the zero-crossing and positive minima/negative maxima cumulative curves are usually utilized as the measures of the instantaneous spectral power. Using spectral moments, analytic relations for zero-crossing and positive minima/negative maxima cumulative count are developed with respect to the IRF’s evolutionary power spectrum. In order to efficiently investigate the low-frequency components of the output records, the zero-crossing rate of the velocity and the displacement records are tracked and rigorous analytic equations for their descriptions are developed. Some of the issues concerning application of the developed equations are explored using two LVT models, and the limitations of the proposed procedures are explained. Through introduction of new compatibility measures, the necessity of implementing Monte Carlo simulations would no longer be needed and it would be possible to generate the synthetic acceleration records with desirable evolutionary characteristics with much less computation time.

Statistical detection and isolation of additive faults in linear time-varying systems