Linear Time-varying Discrete-time Systems Research Articles

Stability analysis of LTV systems via gain and phase

Abstract This paper considers the feedback stability analysis of infinite-dimensional discrete-time linear time-varying systems with combined gain and phase information. The framework considers a system to be a causal linear operator described by a lower triangular infinite-dimensional complex matrix. For $r$-sectorial operators, we first derive a sufficient condition for the invertibility of $I+AB$ using the gain and phase information. Finally, we use this invertibility criterion to give a sufficient condition for feedback stability of $r$-sectorial systems.

Invariant properties of linear-iterative distributed averaging algorithms and application to error detection

We consider the problem of average consensus in a distributed system comprising a set of nodes that can exchange information among themselves. We focus on a class of algorithms for solving such a problem whereby each node maintains a state and updates it iteratively as a linear combination of the states maintained by its in-neighbors, i.e., nodes from which it receives information directly. Averaging algorithms within this class can be thought of as discrete-time linear time-varying systems without external driving inputs and whose state matrix is column stochastic. As a result, the algorithms exhibit a global invariance property in that the sum of the state variables remains constant at all times. In this paper, we report on another invariance property for the aforementioned class of averaging algorithms. This property is local to each node and reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors (i.e., nodes to which it sends information directly) throughout the execution of the averaging algorithm. We show how this newly-discovered invariant can be leveraged for detecting errors while executing the averaging algorithm.

On-policy and off-policy Q-learning strategies for spacecraft systems: An approach for time-varying discrete-time without controllability assumption of augmented system

This article investigates two On-policy and Off-policy Q-learning algorithms for time-varying linear discrete-time systems (DTSs) in the presence of complete dynamic uncertainties. To handle the challenge of time-varying description, the lifting method is employed to transform the original time-varying linear DTS into time-invariant linear DTS in the absence of the conventional controllability condition, which affects to the convergence of traditional Q-learning algorithms. Based on theoretical analysis of the structure in the obtained time-invariant linear DTS, On-policy and Off-policy algorithms are proposed to guarantee the convergence of Q-learning algorithms. Both On-policy and Off-policy Q-learning algorithms guarantee model-free consideration under the data collection. Especially, the Off-policy technique is able to develop the algorithm with high data efficiency because the collected data can be utilized again after each iteration. Finally, the simulation results of two-dimensional systems and spacecraft control systems are presented to validate the effectiveness of the two proposed control schemes.

Ellipsoidal Set-Membership Filtering for Discrete-Time Linear Time-Varying Systems

This paper studies set-membership estimation for discrete-time linear time-varying systems with unknown-but-bounded uncertainties. An ellipsoidal set-membership filtering method with a new estimator is proposed. Based on the S-procedure, the proposed set-membership filtering is formulated as an optimization problem in terms of linear matrix inequalities. Moreover, it is shown that the optimal ellipsoids estimated by two existing set-membership filtering approaches are equivalent to the optimal estimation results obtained by two special cases of the proposed method, respectively. The proposed estimator has more design parameters which can be effectively optimized to improve the estimation accuracy. Simulation results are given to show the performance of the proposed method.

Direct Data-Driven Control of Linear Time-Varying Systems

Considering discrete-time linear time-varying systems with unknown dynamics, controllers guaranteeing bounded closed-loop trajectories, optimal performance and robustness to process and measurement noise are designed via convex feasibility and optimisation problems involving purely data-dependent linear matrix inequalities. For the special case of periodically time-varying systems, infinite-horizon guarantees are achieved based on finite-length data sequences.

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

Iterative Learning Identification and Control for Discrete-time Linear Time-varying Systems with Iteration Varying Lengths

Iterative Learning Identification and Control for Discrete-time Linear Time-varying Systems with Iteration Varying Lengths

On consistency and stability of distributed Kalman filter under mismatched noise covariance and uncertain dynamics

This paper considers the state estimation problem for a class of discrete-time linear time-varying systems over a peer-to-peer sensor network under mismatched noise covariance and uncertain dynamics. To deal with the inconsistency and instability of distributed Kalman filter for the problem, we propose a general consistent distributed Kalman filter framework by considering compensation to the mismatched noise covariance and the uncertain dynamics. The estimated error covariance of the distributed filter is proven to be monotonic with respect to compensation parameters. Futhermore, we show the design methods of compensation matrices to ensure consistency of the proposed filter in each sensor. Then, stability of the proposed distributed Kalman filter is demonstrated. Simulation examples show the effectiveness of our methods.

Decomposition of Second-Order Discrete-Time Linear Time-Varying Systems into First-Order Commutative Pairs

Decomposition of Second-Order Discrete-Time Linear Time-Varying Systems into First-Order Commutative Pairs

An Optimization Framework for Resilient Batch Estimation in Cyber-Physical Systems

This article proposes a class of resilient state estimators for linear time-varying discrete-time systems. The dynamic equation of the system is assumed to be affected by a bounded process noise. As to the available measurements, they are potentially corrupted by a noise of both dense and impulsive natures. The latter, in addition to being arbitrary in its form, need not be strictly bounded. In this setting, we construct the estimator as the set-valued map, which associates with the measurements the minimizing set of some appropriate performance functions. We consider a family of such performance functions, each of which yielding a specific instance of the proposed general estimation framework. It is then shown that the proposed class of estimators enjoys the property of resilience, i.e., it induces an estimation error, which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise. Hence, the estimation error may be bounded, while the measurement noise is virtually unbounded. Moreover, we provide several error bounds (in different configurations), whose expressions depend explicitly on the degree of observability of the system being observed and on the considered performance function. Finally, a few simulation results are provided to illustrate the resilience property.

Input–Output Finite-Time Stabilization of Linear Time-Varying Discrete-Time Systems

This article deals with the input–output finite-time stability (IO-FTS) of linear discrete-time systems. In current control science, discrete-time systems play a very important role in many engineering contexts. Moreover, many discrete-time control problems are defined over a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite</i> -interval of time; therefore, the development and application of finite-time control methodologies is of particular relevance. The first contribution of this article is a pair of necessary and sufficient conditions for IO-FTS. The former involves the solution of a set of generalized difference Lyapunov equations; the latter allows one to establish the feasibility of an optimization problem by solving a set of difference linear matrix inequalities (DLMIs). The second contribution of this article is a theorem for IO finite-time stabilization via state feedback, followed by a more general one for stabilization via output feedback. Both conditions are necessary and sufficient, and lead to optimization problems cast in the form of DLMIs. The applicability of the devised results is illustrated through a numerical example.

Dynamic Learning From Adaptive Neural Control for Discrete-Time Strict-Feedback Systems.

This article first investigates the issue on dynamic learning from adaptive neural network (NN) control of discrete-time strict-feedback nonlinear systems. To verify the exponential convergence of estimated NN weights, an extended stability result is presented for a class of discrete-time linear time-varying systems with time delays. Subsequently, by combining the n -step-ahead predictor technology and backstepping, an adaptive NN controller is constructed, which integrates the novel weight updating laws with time delays and without the σ modification. After ensuring the convergence of system output to a recurrent reference signal, the radial basis function (RBF) NN is verified to satisfy the partial persistent excitation condition. By the combination of the extended stability result, the estimated NN weights can be verified to exponentially converge to their ideal values. The convergent weight sequences are comprehensively represented and stored by constructing some elegant learning rules with some novel sequences and the mod function. The stored knowledge is used again to develop a neural learning control scheme. Compared with the traditional adaptive NN control, the proposed scheme can not only accomplish the same or similar tracking tasks but also greatly improve the transient control performance and alleviate the online computation. Finally, the validity of the presented scheme is illustrated by numerical and practical examples.

Input-to-state stability of discrete-time time-varying impulsive delay systems

This paper investigates the input-to-state stability (ISS) of nonlinear discrete-time time-varying impulsive delay systems with the coexistence of stabilising and destabilising impulses. Some ISS criteria are established for discrete-time impulsive delay systems by employing the Lyapunov functional where the coefficient of the estimated upper bound of the time difference of the Lyapunov functional is allowed to be a time-varying function which can take both negative and positive values. In addition, the ISS of a class of discrete-time linear time-varying impulsive delay systems is discussed by utilising a non-quadratic Lyapunov functional. Numerical examples are presented to verify the effectiveness of the main results.

Sufficient conditions for assignability of nonuniform dichotomy spectrum of discrete time-varying linear systems

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds.

On Sufficient Optimality Conditions for a Guaranteed Control in the Speed Problem for a Linear Time-Varying Discrete-Time System with Bounded Control

On Sufficient Optimality Conditions for a Guaranteed Control in the Speed Problem for a Linear Time-Varying Discrete-Time System with Bounded Control

Exponential estimation of the unknown frequencies of discrete-time multi-tone sinusoidal signals

While the estimation problem of the unknown frequencies for a continuous-time multi-tone sinusoidal signal was thoroughly solved in the early 2000s, the discrete-time version of this problem was studied only recently via a nonlinear control approach. In this paper, we present a simpler approach to the estimation problem of the unknown frequencies for a discrete-time multi-tone sinusoidal signal. Our approach only involves the design of two linear filters together with a linear adaptation law. As a result, the implementation of our two linear filters is easier than that of nonlinear ones. Moreover, the linear form of our filters lends itself to the establishment of the exponential convergence of the estimated frequencies to their actual values. In addition, we obtain an interesting exponential stability result for a class of discrete-time linear time-varying systems that can be viewed as a discrete-time counterpart of a well-known stability result arising from continuous-time adaptive identification.

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

Networked Robust Stability for LTV Systems with Simultaneous Uncertainties in Plant, Controller, and Communication Channels

In this paper, we study the robust stability of a networked control system (NCS) under the framework of infinite-dimensional discrete-time linear time-varying (LTV) systems. The NCS consists of a pair of uncertain plant and controller, as well as an uncertain bilateral communication channel in between. The uncertainties in the plant and controller are measured by the gap metric. The communication channel between the plant and controller is described by a cascade of two-port networks whose transmission matrices are subject to norm-bounded additive uncertainties. Such an uncertain two-port network can model distortions and interferences occurring during control and measurement signal transmissions. The causality of the LTV subsystems is characterized by using nest algebras. A necessary and sufficient condition for the robust stability of the NCS, with the causality of all system components explicitly considered, is established in the form of an arcsine inequality which generalizes a similar result for linear time-invariant NCSs.

시변 이산 선형 시스템에 대한 지수가중 유한 임펄스 응답 필터

In this paper, an exponentially weighted Finite Impulse Response (FIR) filter is proposed for discrete-time linear time-varying systems. The weighting method of measurements was first introduced for the FIR structured estimators, then an optimal design of the exponentially weighted FIR filter was developed with respect to a minimum variance criterion for discrete-time linear time-varying systems. The proposed filter is designed to be unbiased and independent of initial state information. Due to the inclusion of the weighting factor, it is expected to be less prone to model uncertainty and numerical errors than the conventional optimal FIR filter. Moreover, the proposed FIR filter is reduced to the optimal time-varying FIR filter with an equal weight, and can thereby provide a more general solution than the conventional optimal FIR filter. Through a numerical study, the performance of the proposed FIR filter was shown by comparing it with the exponentially weighted Kalman filter and the conventional optimal FIR filter when the modeling uncertainties exist.

Sequential predictors for delay compensation for discrete time systems with time-varying delays

We study time-varying linear discrete time systems with uncertainties and time-varying measurement delays, whose outputs are perturbed by uncertainty. We build sequential predictors, which ensure input-to-state stability with respect to the uncertainties and which can be constructed using output values under arbitrarily long delays. The number of required sequential predictors is any upper bound for the delay in our feedback stabilized closed loop systems. We illustrate our work in a digital control problem for a continuous time system that is discretized through sampling.