A new interpretation of adjoint method in linear time-varying system analysis

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 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.

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.

Control of time-varying systems has become increasingly popular in recent years, as various domains such as aviation, automobile and manufacturing industries require a quick and precise response in the presence of uncertainties. Adaptive control, a popular strategy is used to control systems with unknown or time-varying properties. Classic adaptive control is insufficient to cope with time-varying parameters to produce desired results with minimal parametric error. Furthermore, for these strategies to work, identification model must be initialised closer to the plant. Else the transient response of the plant is likely to grow unbounded. This paper mainly concentrates on employing adaptive control for a type of linear time-invariant (LTI) and linear time-varying (LTV) systems. The time variations in LTV plant parameters are either slow, rapid or fast. This research in turn proposes a two-stage adaptive scheme which swiftly determines the compact region in which plant parameters reside. This method in turn ensures atleast one model to be initialised closer to the plant which significantly reduces parametric estimation error. The uniform boundedness of identification error, tracking error as well as the parametric error of LTI and LTV systems, are deduced. Sufficient examples are also considered to demonstrate the efficacy of the proposed two-stage scheme.

This paper investigates model reference control for linear time-varying (LTV) systems. It is shown that the problem can be decomposed into two sub-problems: a state feedback stabilization problem of LTV systems and a feed-forward compensation problem. Firstly, the sufficient condition for the existence of the model reference control is deduced. The condition concerns two coefficient matrices such that two matrix equations are met simultaneously, and the state feedback gain matrix stabilizes LTV systems according to the unified spectral theory. Secondly, generally complete parametrization of the model reference tracking controller including a state feedback controller and a feed-forward compensator is proposed based on the solution to a type of generalized Sylvester matrix equations. Furthermore, a general procedure for solving the model reference control problem is also provided. Finally, a numerical example shows the feasibility and effectiveness of the parametric design approach to the model reference control in LTV systems.

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

The stability and behavior of continuous-time nonlinear time-varying (NTV), nonlinear time-invariant (NTI) and linear time-varying (LTV) systems are generally very difficult to ascertain using current analytical tools. This paper presents sufficient conditions under which a NTV system with a bounded system matrix, A(t,X), is globally exponentially stable (GES). The continuous-time NTV system is modeled as an uncertain polytopic system even though A(t,X) may be reasonably known. It is shown that if all polytope-vertex matrices are Hurwitz, then the system is GES. Also, a method for determining the boundary estimate of a NTV system state-trajectory norm is developed. Presented is a unified stability analysis approach for a large class of NTV, NTI, and LTV systems. Even for large systems, application of the GES theorem presented is straight forward using readily available software and computational power. Examples are given that illustrate the application of the theorem offered.

A necessary and sufficient condition for linear time-varying (LTV) systems to be exponentially stable has been given based on differential Lyapunov inequalities (DLIs). The exponential stability of an LTV system is guaranteed if the DLI for the system has a solution, however, few systematic methods have been reported for finding a solution of DLIs. We focus on a class of periodic LTV systems and propose a sufficient condition to find solutions of DLIs for the class by using linear matrix inequalities (LMIs). Some examples show the effectiveness of the proposed analysis method.

We extend the approximate transfer function calculus of "underspread" linear time-varying (LTV) systems introduced by W. Kozek (1992). Our extension is based on a new, generalized definition of underspread LTV systems that does not assume finite support of the systems' spreading function. We establish explicit bounds on various error quantities associated with the transfer function approximation. Our results yield a simple and convenient transfer function calculus for a significantly wider and practically more relevant class of LTV systems than that previously considered.

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

Takage-Sugeno (T-S) fuzzy systems are equivalent to linear time-varying (LTV) systems, the theory of LTV systems can therefore be used to design global feedback controllers to stabilize T-S fuzzy systems. It is seen that the closed-loop T-S fuzzy system is first transformed into the canonical form by using a Lyapunov transform, a global feedback controller is then designed to assign all Parallel D-spectrum eigenvalues (PD-eigenvalues) of the closed-loop T-S fuzzy system to the desired locations whose real parts have negative extended means. The proposed control scheme not only guarantees asymptotic stability of the closed-loop T-S fuzzy systems, but also allows the designers to directly design a simple global controller instead of deriving a complex global controller by aggregating all local controllers using fuzzy membership functions. A simulation example is given in support of the proposed scheme.

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.

Properties of double (or two-dimensional) Laplace transform (2DLT) for analysis of linear time-varying (LTV) systems are explored. The one-dimensional to two-dimensional expansion formulas, as well as the two-dimensional to one-dimensional reduction formulas, are outlined. The merit of the proposed technique for frequency analysis of LTV systems is demonstrated.

This study presents an eigenvalue assignment control architecture for a class of linear time-varying (LTV) systems in the presence of time-varying disturbances. Eigenvalue assignment architecture is presented as a way to achieve feedback stabilisation for LTV disturbed systems. With an input compensator design, the output of slowly LTV systems with disturbances can practically track the reference. Moreover, the implementation of the proposed eigenvalue assignment technique in practice has been demonstrated by using adaptive estimation of time-varying disturbances.