Stability analysis and control synthesis of hybrid time-varying linear systems using a discretization-based approach

This paper introduces a novel and rapid approach for the stability and performance analysis of linear time-varying (LTV) systems, which has a very wide application to the evaluation of control system of Aerospace vehicles. By introducing the concept of integral function for the LTV systems and showing its prosperities, a sufficient and necessary condition for the exponential stability of LTV systems is derived. Furthermore, by computing the radii of convergence of the integral function, the exponential decay rate of system trajectories of LTV systems can be obtained exactly, which provides a computable way for the analysis of system performance. Finally, the algorithm for computing the integral function is developed and the proposed approach is applied to the stability and performance analysis of the control system of international space station.

The problem of linear time-varying(LTV) system modal analysis is considered based on time-dependent state space representations, as classical modal analysis of linear time-invariant systems and current LTV system modal analysis under the “frozen-time” assumption are not able to determine the dynamic stability of LTV systems. Time-dependent state space representations of LTV systems are first introduced, and the corresponding modal analysis theories are subsequently presented via a stability-preserving state transformation. The time-varying modes of LTV systems are extended in terms of uniqueness, and are further interpreted to determine the system’s stability. An extended modal identification is proposed to estimate the time-varying modes, consisting of the estimation of the state transition matrix via a subspace-based method and the extraction of the time-varying modes by the QR decomposition. The proposed approach is numerically validated by three numerical cases, and is experimentally validated by a coupled moving-mass simply supported beam experimental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides a new way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

This paper proposes detailed stability analysis of linear time-varying (LTV) systems. In this sense, a survey of published approaches is presented and a numerical tool that allows the LTV systems stability analysis is developed. The analytic methods that are based on different concept as state matrix, state transition matrix or eigenvalues gives a sufficient conditions for LTV systems stability study. The developed numerical tool is then based on the presentation of different sufficiency criteria for the asymptotic stability of linear time-varying systems.

Linear time-varying (LTV) systems naturally arise when one linearizes nonlinear systems about a trajectory. In contrast the linear time-invariant (LTI) cases which have been thoroughly understood in the analysis and synthesis technologies, many features of the LTV systems are still limited and not clear. This paper addresses the problems of solution and stability of a general unforced LTV differential state space system. Unlike most of the work based on the Lyapunov theory, numerical simulations, or specific constraint systems, the paper proposes the spectral decompositions of the LTV systems by employing extended eigenpairs and with simple mathematical derivation. The spectral decompositions reveal the mechanisms of inherent characterization in general LTV systems, rather than a particular class. Moreover, a novel set of auxiliary equations is developed for guiding and obtaining the extended eigenpairs of its system matrix which completely characterize the LTV systems. The solutions to perform the commutative systems and the second-order systems with companion form are straightforward. The proposed innovative thinking provides a novel guided way to analyze the LTV systems. These findings are easily extended to LTI cases. Examples from the literature demonstrate the effectiveness and the superiority of the proposed approaches when compared with other methods. The proposed results may be of great interest in both for scientific research and application.

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). The proposed method allows that the systems may have higher harmonic frequencies. An example shows the effectiveness of the proposed analysis method.

Many engineering structures, such as cranes, traffic-excited bridges, flexible mechanisms and robotic devices exhibit characteristics that vary with time and are referred to as time-varying or nonstationary. In particular, linear time-varying (LTV) systems have been often dealt with on a case-by-case basis. Many concepts and analytic methods of linear time-invariant (LTI) systems cannot be applied to LTV systems, as for example the conventional definition of modal parameters. In fact, LTV systems violate one of the assumptions of the conventional modal analysis, which is stationarity.Subspace-based identification methods, proposed in the 1970s, have been attracting much attention due to their affinity to the modern control theory, which is based on the state space model. These methods are now successfully applied to many industrial cases and may be considered reference methods for identifying LTI systems.In this paper the use of a subspace-based method for identifying LTV systems is discussed and applied to both numerical and experimental systems. More precisely a modified version of the SSI method, referred to here as ST-SSI (Short Time Stochastic Subspace Identification) is introduced as well as a method for predicting time-varying stochastic systems using the angle variation between the subspaces; the latter is able to predict the system parameter in the “near” future.

Stability analysis of linear time-varying time-delay systems by non-quadratic Lyapunov functions with indefinite derivatives

Most of linear time-varying (LTV) systems except special cases have no general solution for the dynamic equations. Thus, it is difficult to design time-varying controllers in analytic ways, and other control design approaches such as robust control and gain-scheduling have been applied to control design for the LTV systems. A robust control method such as quantitative feedback theory (QFT) has an advantage of guaranteeing the stability and the performance specification in frozen time sense. However, if these methods are applied to the approximated linear time-invariant (LTI) plants with large uncertainty, the designed control will be constructed in complicated forms and usually not suitable for fast dynamic performance. In this paper, as a method to enhance the fast dynamic performance, the approximated uncertainty of time-varying parameters are reduced by the proposed gain-scheduling control design based on QFT for LTV systems with bounded time-varying parameters. To generate a continuous and smooth gain-scheduling function, multilayer neural network is used.

This work deals with the stability analysis of linear discrete-time time-variant systems, focusing on the most restrictive stability measures for free respectively forced systems, the exponential stability respectively BIBS (bounded-input bounded-state) stability. Additionally the connection of these stability concepts with the so-called Bohl exponent is in focus of this work. The Bohl exponent can be seen as a generalization of eigenvalues for the stability analysis of time-variant systems. The contribution of this work is the combination of the exponential and BIBS stability analysis with the numerical computation of the Bohl exponent, leading to a numerical stability analysis for linear discrete-time time-variant systems. This stability analysis is applied at a time-variant drivetrain control loop.

An analytical framework is developed that permits the input-output representations of discrete-time linear time-varying (LTV) systems in terms of biorthogonal bases on compact time intervals. Using these representations, the companion paper, Part II develops computational procedures for rapid identification of fast nonsmooth LTV systems based on short data records. One of the representations proposed is also used in H. Zhao and J. Bentsman, “Block Diagram Reduction of the Interconnected Linear Time-Varying Systems in the Time Frequency Domain,” accepted for publication by Multidimensional Systems and Signal Processing to form system interconnections, or wavelet networks, and develop subsystem connectibility conditions and reduction rules. Under the assumption that the inputs and the outputs of the plants considered in the present work belong to lp spaces, where p=2 or p=∞, their impulse responses are shown to belong to Banach spaces. Further on, by demonstrating that the set of all bounded-input bounded-output (BIBO) stable discrete-time LTV systems is a Banach space, the system representation problem is shown to be reducible to the linear approximation problem in the Banach space setting, with the approximation errors converging to zero as the number of terms in the representation increases. Three types of LTV system representation, based on the input-side, the output-side, and the input-output transformations, are developed and the suitability of each representation for matching a particular type of the LTV system behavior is indicated.

A method for determining whether a linear time-varying (LTV) system is exponentially stable has been proposed based on differential Lyapunov inequalities (DLIs). However, there are few established systematic methods to find solutions of DLIs. In this paper, we focus on a class of periodic LTV systems and propose a method to find solutions of DLIs by using linear matrix inequalities (LMIs). Moreover, we propose a method to design state observers, state feedback controllers and output feedback controllers for periodic LTV systems by utilizing the proposed DLI solutions search method. Some examples show the effectiveness of the proposed analysis and design method.

On asymptotic stability of linear time-varying systems

For most of linear time-varying (LTV) systems, it is difficult to design time-varying controllers in analytic way. Accordingly, by approximating LTV systems as uncertain linear time-invariant, control design approaches such as robust control have been applied to the resulting uncertain LTI systems. In particular, a robust control method such as quantitative feedback theory (QFT) has an advantage of guaranteeing the frozen-time stability and the performance specification against plant parameter uncertainties. However, if these methods are applied to the approximated linear time-invariant (LTI) plants with large uncertainty, the resulting control law becomes complicated and also may not become ineffective with faster dynamic behavior. In this paper, as a method to enhance the fast dynamic performance of LTV systems with bounded time-varying parameters, the approximated uncertainty of time-varying parameters are reduced by the proposed QFT parameter-scheduling control design based on radial basis function (RBF) networks.

This paper concerns passivity and stability analysis of linear time varying (LTV) systems characterized by parametric time variations. The first set of results quantify a trade off between the degree of passivity of the frozen systems and the rate of parameter variations, so that passivity of certain classes of such LTV systems is preserved. One class of systems examined has frozen systems whose transfer functions are multiaffinely parametrized. Another class considered includes linear circuits with time varying resistors, inductors capacitors and mutual inductors. The second set of results exposes the utility of these passivity results in stability 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.