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

In this paper the extremum periodic trajectory of the two-dimensional selector-linear differential inclusion (SLDI) is used to estimate boundary of the invariant set of the nonlinear time-varying system arising in the stability analysis of the wheeled robot control. The motion is supposed to be planar without a lateral slippage. The control goal is to drive the target point of the robot plaform to the specified trajectory and to stabilize the motion along it. The trajectory consists of line segments and circular arcs. The current curvature of the trajectory of the target point is taken as control. The control must satisfy two-sided constraints. Given control law, the attraction domain estimation problem is considered. The attraction domain must be inscribed into the certain parallelepiped of the 'distance to the trajectory-orientation' phase space. Time-varying curvature of the target trajectory is considered as arbitrary varying function which takes values from the specified interval. The feedback linearization scheme is used for synthesis of the control low. The 'saturation function' is then used to take into account control constraints. The closed loop system takes form of the nonlinear system with parametric disturbances. The absolute stability approach is explored for stability analysis. Some nonlinearities take values from the interval. Other nonlinearities satisfy sector constraints. Along with the nonlinear time-varying system the uncertain linear time varying system is considered. Every solution of the nonlinear system is also solution of the time varying system for certain set of time-varying disturbances. To estimate the attraction domain of the nonlinear closed loop system, the Lyapunov function for SLDI is constructed. A convex invariant function is known to exist at the boundary of the absolute stability of SLDI. The extremum trajectory, corresponding to the boundary of the absolute stability in the second order case belongs to the level set of the invariant function and is the periodic solution. The periodic solution has finite number of switches on the period. It circumscribes the boundary of the attraction domain estimate. The illustrative example is given.

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.

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.

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.

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.

In this article, the problem of intermittent fault (IF) detection is investigated for stochastic linear time-varying (LTV) systems using dynamic event-triggered methods. Using the nonuniform sampling approach, the event-triggered system is transformed into a time-varying system with varying sampling periods. Using the moving horizon estimation strategy, a new IF detection filter is designed to generate residual signals, which can be decoupled from event-triggered transmission errors and estimation errors. Moreover, an event-triggered IF detection algorithm is proposed such that the appearance time and disappearance time of IFs can be detected quickly for stochastic LTV systems. In order to analyze the detectability of IFs for systems with/without event-triggered cases, the concept of distinguishability is introduced for IFs. Sufficient conditions are derived to guarantee the detectability of IFs for LTV systems. Finally, an experiment concerning the rotary steerable drilling tool system is provided to illustrate the effectiveness of the proposed IF detection method.

In this paper, we introduce two new types of local instabilities which exclusively pertain to a linear time-varying (LTV) system. Investigation of these local instabilities has lead to new necessary and sufficient conditions for a LTV system to be globally stable. It is shown that a direct extension of the stability conditions of a linear time-invariant (LTI) system is not applicable to a LTV system. The admissible area for a set of coefficients of a globally stable LTV system has been established and this area is found to be much larger than its LTI system counterpart.

For output-feedback control of linear time-varying (LTV) and nonlinear systems, this paper focuses on control based on the forward propagating Riccati equation (FPRE). FPRE control uses dual difference (or differential) Riccati equations that are solved forward in time. Unlike the standard regulator Riccati equation, which propagates backward in time, forward propagation facilitates output-feedback control of both LTV and nonlinear systems expressed in terms of a state-dependent coefficient (SDC). To investigate the strengths and weaknesses of this approach, this paper considers several nonlinear systems under full-state-feedback and output-feedback control. The internal model principle is used to follow and reject step, ramp, and harmonic commands and disturbances. The Mathieu equation, Van der Pol oscillator, rotational-translational actuator, and ball and beam are considered. All examples are considered in discrete time in order to remove the effect of integration accuracy. The performance of FPRE is investigated numerically by considering the effect of state and control weightings, the initial conditions of the difference Riccati equations, the domain of attraction, and the choice of SDC.

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.

In this paper, two algorithms are proposed to identify linear, time-varying (LTV) systems and model nonstationary signals. The first one is based on the time-varying cumulants (TVC); and the second one applies the time-varying sum-of-pseudo cumulants (TVSPC). We also have shown that if the output of the LTV system is corrupted by stationary/nonstationary noise with symmetric distribution, the time-varying coefficients of the system can be identified using the time-varying sum-of-cumulants and sum-of-pseudo cumulants algorithms. Some LTV system identification and nonstationary signal modeling examples are given, using the above algorithms.

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.

In this paper some previously developed series and parallel differential spectral concepts for scalar linear time-varying (LTV) systems are extended to some subclasses of multi-input-multi-output (MIMO) LTV systems. The extension is facilitated by the new concepts of differential determinant and differential adjoint matrix introduced herein, which are natural extensions of the familiar concepts of determinant and adjoint matrix to a noncommutative differential ring. Explicit matrix fractional representations are obtained for the subclasses of MIMO LTV systems for which the new results are applicable. The new results have important applications in the analysis and control of MIMO LTV systems.

Signal analysis and system synthesis are essential tasks in linear time-varying (LTV) systems electrical and control engineers are called upon to perform. This paper emphasizes the analysis of LTV systems in the frequency domain by the aid of the two-dimensional Laplace transform (2DLT)techniques. Specifically, the bifrequency-domain models of basic LTV network elements are determined. Applications of the bifrequency characterization of LTV systems are illustrated in detail to demonstrate the merit of the proposed convenient 2DLT approach.

The objective of this paper is to extend application of Laplace-Carson transform (LCT) to the realization of linear time-varying (LTV) systems. We propose using the unilateral and bilateral two-dimensional Laplace transform (2DLT) for realization of LTV systems, which exhibit a circular symmetry. Application of two-dimensional Laplace and Fourier transforms to LTV systems results in a bifrequency realization. In particular, it is shown that the 2DLT of LTV systems is, in fact, a Hankel transform of order zero.