Linear Time-varying Discrete-time Research Articles

Stability Analysis of Discrete-Time Switched Linear Time-Varying Systems Based on Function-Dependent LMIs

This paper considers the problem of uniformly asymptotic stability (UAS) for discrete-time switched linear time-varying (DSLTV) systems. Starting with discrete-time linear time-varying (DLTV) systems, some stability conditions are given by using function-dependent linear matrix inequalities (LMIs). Comparing with the existing results, the conditions obtained allow the norm and rate of variation of system matrix are unbounded. Furthermore, the obtained results are extended to study the UAS of DSLTV systems, the stability condition is given by combining the methods of function-dependent LMIs with average dwell time. Finally, some further discussions about the relaxation of conditions obtained are also proposed. Three numerical examples are given to illustrate the theoretical results.

Deterministic learning from sampling data

In this paper, based on the deterministic learning mechanism, we present an alternative systematic scheme for dynamics identification from sampling data sequences. The proposed scheme belongs to a dynamical machine learning framework based on Lyapunov stability theory rather than optimization estimation approaches. Given a sampling data sequence collected from an unknown deterministic nonlinear dynamical system, we show that the inherent dynamics of the sampling data sequence can be locally accurately captured and represented as a converged constant neural network. This accurate identification result can be derived from the exponential stability of a class of linear time-varying (LTV) error systems usually appearing in adaptive control and identification fields. However, there are few existing results concerning the exponential stability problem of the discrete-time LTV case. To this end, by using the small gain theorem of input-to-state stability (ISS), we provide a rigorous proof for establishing the exponential stability of the LTV system under the persistent excitation (PE) condition. Based on the exponential stability results, the exponential convergence of neural weights to their optimal values can be analytically guaranteed, which leads to the achievement of the accurate dynamics identification. The significance of the paper is that we preliminarily explore the essential problem of machine learning of dynamical systems by innovatively transforming tools and methodologies from dynamic system stability and control theory. Numerical simulation results are provided to validate the effectiveness of the proposed scheme with comparison to previously mentioned approaches.

Blind identification of fully observed linear time-varying systems via sparse recovery

Discrete-time linear time-varying (LTV) systems form a powerful class of models to approximate complex dynamical systems with nonlinear dynamics for the purpose of analysis, design and control. Motivated by inference of spatio-temporal dynamics in breast cancer research, we propose a method to efficiently solve an identification problem for a specific class of discrete-time LTV systems, in which the states are fully observed and there is no access to system inputs. In addition, it is assumed that we do not know on which states the inputs act, which can change between time steps, and that the total number of inputs is sparse over all states and over time. The problem is formulated as a compressive sensing problem, which incorporates the effect of measurement noise and which has a solution with a partially sparse support. We derive sufficient conditions for the unique recovery of the system model and input values, which lead to practical conditions on the number of experiments and rank conditions on system outputs. Synthetic experiments analyze the method’s sensitivity to noise for randomly generated models.

On Asymptotic Stability of Discrete-Time Linear Time-Varying Systems

This note is concerned with asymptotic stability analysis of discrete-time linear time-varying (DLTV) systems. The contribution can be summarized as follows. First, the difference between nonuniformly exponential stability (ES), which was not well recognized in the literature, and uniformly ES (UES) was revealed and illustrated by numerical examples. Second, some state transition matrices and difference Lyapunov inequalities (DLIs) based necessary and sufficient conditions were derived for testing different stability concepts including asymptotic stability (AS), ES, and UES of general DLTV systems. The main feature of the DLIs-based criteria is that the time-difference of the Lyapunov function is not required to be negative definite. Finally, the ES concept was utilized for analysis stability of a class of upper triangular DLTV systems and perturbated DLTV systems. The effectiveness of the proposed results was demonstrated by some examples.

A constructive test for exponential stability of linear time-varying discrete-time systems

We complete the stability results of the paper Bourlès et al. (SIAM J Control Optim 53:2725–2761, 2015), and for this purpose use the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.) of this paper. In the main theorem we characterize the e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. We extend the theory of discrete-time LTV behaviors, developed in the quoted publication, from the coefficient field of rational functions to that of locally convergent Laurent series or even of Puiseux series. The stability test can and has to be applied in connection with the construction of stabilizing compensators.

Constrained control for linear time-varying systems in the presence of disturbances

The aim of this paper is twofold. In the first part, we revisit the robust invariance for ellipsoidal sets with respect to discrete-time linear time-varying and uncertain systems with bounded additive disturbances. In this context we provide an extension of the invariance condition of the previously published results in the literature. In the second part a novel robust interpolation based control method involving several local unconstrained robust optimal controls is proposed. Among them, one controller plays the role of a performance candidate, while the remaining controllers are used in the interpolation scheme for enlarging the domain of attraction. By using an appropriate objective function, recursive feasibility and a robustly asymptotically stable closed loop behavior are guaranteed.

Interpolation based control for constrained linear time-varying or uncertain systems in the presence of disturbances

In this paper, the robust control of constrained discrete-time linear time-varying or uncertain systems affected by bounded additive disturbance is considered. The main idea is to use an interpolation technique between several local unconstrained robust optimal controls. From the implementation point of view, the scheme is shown to be computationally attractive, at each time instant a quadratic programming (QP) problem being solved on-line. The proofs of recursive feasibility of the optimization procedure and ISS stability of the closed loop system are given.

A structured matrix approach to efficient calculation of LQG repetitive learning controllers in the lifted setting

This article discusses the use of repetitive control for output reference tracking in linear time-varying discrete time systems with both repetitive and non-repetitive noise components. The design of such controllers is formulated as a lifted linear stochastic output feedback problem on which the mature techniques of discrete linear control may be applied. In many modern applications, the large size of the system matrices in such a control problem inhibits the application of standard solvers and optimisation techniques. For linear quadratic Gaussian (LQG) problems, the matrices of the lifted feedback problem can be fitted into the recently developed sequentially semi-separable structure. Innovative numerical solutions are developed that have 𝒪(N) computational complexity (where N is the trial length) in both controller synthesis and implementation, comparable to that of many non-lifted and Fourier transform based learning control methods. Moreover, within this formulation, the system is allowed to vary over the learning cycle, closed-loop stability is guaranteed, and stochastic noise and disturbances are handled in an LQG sense.

Biorthogonal Wavelet Based Identification of Fast Linear Time-Varying Systems—Part I: System Representations12

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.