Linear Discrete Time-varying Systems Research Articles

Strict proof for the state transition matrix of linear discrete time‐varying stochastic systems

This note presents a strict proof for the state transition matrix of linear discrete time-varying stochastic systems, which corrects the errors of the previous references.

Assignability of dichotomy spectra for discrete time-varying linear control systems

In this paper, we show that for discrete time-varying linear control systems uniform complete controllability implies arbitrary assignability of dichotomy spectrum of closed-loop systems. This result significantly strengthens the result in [ 5 ] about arbitrary assignability of Lyapunov spectrum of discrete time-varying linear control systems.

Covert Attack Detection Based on Hi/H∞ Optimization for Cyber-Physical Systems

In this paper, a new detection scheme is proposed to detect covert attack in the framework of Hi/H∞ index optimization for cyber-physical system (CPS) which is modeled as a linear discrete time-varying (LDTV) system. First, a random modulation matrix that the attacker cannot know is inserted into the path of the control variables to destroy the stealthiness of covert attacks. Second, a detection filter is constructed which transforms the detection problem into an H-/H∞ or H∞/H∞ index optimization problem. The optimal solution is obtained by solving the Riccati equation. Third, a decision making mechanism is presented to trigger an alarm and further determine whether the cause of alarm is a covert attack or a fault. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

Online sensor performance monitoring and fault detection for discrete linear parameter-varying systems

This paper proposes a fault detection algorithm to identify online sensor performance degradation and failure, where the sensor faults are characterised by variations of the sensor measurement noise covariance matrix. That is, the proposed algorithm estimates the slowly-varying sensor measurement noise covariance and detects the abrupt and/or intermittent change of sensor measurement noise covariance. A memory-based technique is used to detect the abrupt (or intermittent) change of sensor noise covariance matrix. The memory-based technique is adopted due to its simplicity and online applicability. The proposed algorithm originally is designed for discrete linear time-varying (DLTV) systems and applied to discrete linear parameter-varying (DLPV) systems. Simulation results show that the proposed algorithm is capable of estimating the slowly-varying and detecting the abrupt (or intermittent) change of sensor measurement noise covariance for multiple-input and multiple-output discrete linear parameter-varying systems, where the scheduling parameters lie within a compact set. Furthermore, the proposed estimation algorithm shows a reasonable rate of convergence.

Finite-horizon Anisotropy-based Estimation with Packet Dropouts

In this paper, the estimation problem is studied for a class of linear discrete time-varying system with packet dropout in the framework of anisotropy-based theory. The extended vector of fragment of the disturbance sequence is from the set of random vectors with bounded anisotropy. The packet dropout effect is considered to be random and described by a binary switching sequence with Bernoulli distribution. The input-to-error dynamics is obtained for multiplicative noise system with mutually independent noises and input disturbance. By using anisotropy-based approach, the estimation problem is reduced to optimization one with convex constraints. The developed method provides the (sub)optimal estimator ensuring the boundedness of anisotropic norm for input-to-output error system. Numerical example is provided to demonstrate efficiency of proposed approach.

Ellipsoid bundle and its application to set-membership estimation

This paper studies set-membership estimation for discrete linear time-varying systems subject to unknown disturbance and noise, which are bounded by ellipsoids. To improve the existing ellipsoid-based set-membership estimation methods, we propose a new set representation tool, called ellipsoid bundle, which combines the advantages of ellipsoids and zonotopes for uncertainty set representation and computation. Then, ellipsoidal bundles are used to design a new set-membership estimation method.

Quadratic estimation for discrete time-varying non-Gaussian systems with multiplicative noises and quantization effects

This paper is concerned with the remote state estimation problem for a class of linear discrete time-varying non-Gaussian systems with multiplicative noises. Due to bandwidth constraints in digital communication networks, the measured outputs are quantized before transmission by a probabilistic uniform quantizer. Our attention is focused on the design of a recursive quadratic estimator that exploits the quadratic functions of the measurements. By introducing a proper augmented system which aggregates the original state vector and its second-order Kronecker power, we are able to transfer the quadratic estimation problem into a corresponding linear estimation problem of the augmented state vector. An upper bound is first established for the covariance of the estimation error that is expressed in terms of the solutions to certain matrix difference equations, and such an upper bound is then minimized by designing the filter parameters in an iterative manner. Subsequently, we discuss the monotonicity of the optimized upper bound with respect to the quantization accuracy. A numerical example is provided to verify the effectiveness of the proposed filtering algorithm.

Minimax Control of Deviations for the Outputs of a Linear Discrete Time-Varying System

It is demonstrated how to design optimal digital controllers on a finite horizon for the linear time-varying objects with uncertain initial conditions and external disturbances. The square of an optimality criterion, called the maximum output deviation, represents the exact guaranteed maximum value of the squared Euclidean norm of the system output normalized by the sum of the squared Euclidean norms of disturbances and a quadratic form of the initial state of a system. The maximum output deviations and the worst-case disturbances and/or initial conditions causing them, as well as the minimax controllers (including the multiobjective ones that minimize the maximum deviations of several outputs) are calculated as solutions to a semidefinite programming problem. The necessary and sufficient conditions for the finite-time stability and boundedness of a system are established; they can be used for controller design.

On Lyapunov and Upper Bohl Exponents of Diagonal Discrete Linear Time-Varying Systems

In this paper, we present necessary and sufficient conditions for two given functions to be the Lyapunov and the upper Bohl exponent of a certain discrete linear system with diagonal coefficients. The obtained conditions have a form of easily verifiable algebraic conditions.

Fault estimation of linear discrete time-varying systems with multiplicative noise based on finite impulse response filter

This paper aims to construct a finite impulse response (FIR) based fault estimator for a class of linear discrete time-varying systems (LDTV) with multiplicative noise. Drawing support of intensive stochastic analyses and matrix manipulations, a novel performance index is proposed such that the fault estimation error is minimized in stochastic sense. A necessary and sufficient condition is established to guarantee the existence of the FIR-based fault estimator with satisfied estimation accuracy. The optimal gain of the desired fault estimator is calculated in an analytical way by minimizing the aforementioned performance index. Several examples are presented to demonstrate the effectiveness and superiority of the proposed methods.

Optimal Gains of Iterative Learning Control with Forgetting Factor

In order to solve the optimization problems of convergence characteristics of a class of single-input single-output (SISO) discrete linear time-varying systems (LTI) with time-iteration-varying disturbances, an optimal control gain design method of PID type iterative learning control (ILC) algorithm with forgetting factor was presented. The necessary and sufficient condition for the ILC system convergence was obtained based on iterative matrix theory. The convergence of the learning algorithm was proved based on operator theory. According to optimization theory and Toeplitz matrix characteristics, the monotonic convergence condition of the system was established. The accurate solution of the optimal control gain and the relationship equation between the forgetting factor and the optimal control gains were obtained according to the optimal theory which ensures the fastest system convergence speed, thereby reaching the end of the system convergence improvement. The convergence condition is weaker than the known results. The proposed method overcomes the shortcomings of traditional optimal control gain in ILC algorithm with forgetting factor, effectively accelerates the system convergence speed, suppresses the system output track error fluctuation, and provides a better solution for LTI system with time-iteration-varying disturbances. Simulation verifies the effectiveness of the control algorithm.

Discrete Double-Sided Quaternionic Fourier Transform and Application

In the present article, we first introduce the definition of the discrete double-sided quaternionic Fourier transform (DQFT) and obtain its inverse. We give examples how to compute the DQFT. We derive a discrete version of the duality property of the DQFT. We finally present an application of the DQFT to study the two-dimensional discrete linear time-varying systems.

Bicoprime factor stability criteria for linear time-varying systems in the nest algebra

ABSTRACT This paper presents a bicoprime factorisation approach to deal with various stability problems for linear discrete time-varying systems within the framework of nest algebra. First, we show that a system has a bicoprime factorisation if and only if it has a right and left coprime factorisation by using the complete finiteness of the nest algebra. Based on the bicoprime factorisations, some stability criteria for internal stability and strong stabilisability are obtained, which generalises similar results for linear time-invariant systems. Finally, we derive a necessary and sufficient condition for simultaneous stabilisability of two systems, Additionally, the condition given in this paper lead to a constructive controller design to stabilise simultaneously two systems.

Fault Identifiability Analysis of Linear Discrete Time-Varying Systems

The objective of this paper is to propose a method for evaluating fault identifiability in linear discrete time-varying systems. The basic idea of this method is to quantify the difficulty in estimating an unknown value from an observation for the constructed moving window time-varying model. The analysis of the connection among fault detectability, fault isolability, and fault identifiability is presented in terms of the corresponding measures, distinguishability, and estimability, respectively. Furthermore, factors that influence the performances are analyzed, and an approach for selecting a better system configuration is proposed based on the Fisher information matrix. Finally, the effectiveness of the proposed method in terms of qualitative and quantitative analyses is illustrated through two examples.

Relations between Bohl exponents and general exponent of discrete linear time-varying systems

ABSTRACTIn the paper, properties of the upper Bohl exponents and senior upper general exponent of discrete linear time-varying systems are investigated. The relation of these exponents to uniform exponential stability is discussed. Moreover, an example of system, which is not uniformly exponentially stable but each trajectory tends uniformly and exponentially to zero is provided.

Discrete Fourier transform based frequency characteristics of iterative learning control for linear discrete-time systems

For discrete-time iterative learning control systems, the discrete Fourier transform (DFT) is a powerful technique for frequency analysis, and Toeplitz matrices are a typical tool for the system input–output transmission. This paper first exploits z-transform and DFT-based frequency properties for iterative learning control systems and studies the convergence property of a Toeplitz matrix to the power of iteration index. The exploitation exhibits that for the finite-length discrete-time iterative learning control systems, the time-domain convolution theorem for the z-transform and DFT is no longer true, and the Toeplitz matrix to the power of iteration index converges if and only if the identical diagonal element lies in the unit circle. Then, by considering the DFT to a finite-length sequence as a linear transform, it is easy to equivalently reform the input–output equation of linear discrete time-invariant and time-varying ILC systems as an algebraic discrete-frequency equation. Thus the derivative-type (D-type) iterative learning control (ILC) converges in a discrete-frequency domain if and only if it converges in a discrete-time domain. Numerical simulations are carried out to exhibit the validity and effectiveness.

Coprime Representations and Feedback Stabilization of Discrete Timevarying Linear Systems

Within the framework of nest algebra, this paper deals with the stabilization problems of causal, discrete time, time-varying linear systems based on the coprime representations. It is shown that a linear system is stabilizable if and only if it admits a left coprime representation or a right coprime representation. The parametrization of stabilizing controllers, strong stabilizability and simultaneous stabilizability criteria are characterized in terms of left coprime representations alone, or analogously right coprime representations alone.

Proportional Local Assignability of Lyapunov Spectrum of Linear Discrete Time-Varying Systems

We consider a local version of the pole assignment problem for linear discrete time-varying systems. Our aim is to obtain sufficient conditions to place the Lyapunov spectrum of the closed-loop system in an arbitrary position within some neighborhood of the Lyapunov spectrum of the free system using an appropriate time-varying linear feedback. Moreover, we assume that the norm of the matrix of linear feedback should be bounded from above by the distance between these two spectra with some constant multiplier. We prove that diagonalizability, Lyapunov regularity, and stability of the Lyapunov spectrum each separately are the required sufficient conditions provided that the open-loop system is uniformly completely controllable.

On the assignability of regularity coefficients and central exponents of discrete linear time-varying systems

In this paper we investigate the problem of assignability of the so-called regularity coefficients and central exponents of discrete linear time-varying systems. The main result presents a possibility of assignability of Lyapunov, Perron, Grobman regularity coefficients and central exponents by a linear time-varying feedback under the assumptions of uniform complete controllability.

H∞ Containment Control of Multiagent Systems Under Event-Triggered Communication Scheduling: The Finite-Horizon Case.

This paper investigates the finite-horizon H∞ containment control issue for a general discrete time-varying linear multiagent systems with multileaders. All followers in such a system are driven into a convex hull spanned by multiple leaders, which can be transformed into a problem of tracking a virtual trajectory generated by these leaders. For this purpose, a local state observer is put forward to estimate the state of each agent itself. Then, the estimated state is transmitted to corresponding neighbors governing by an innovation-based event-triggered scheduling protocol. The purpose of the addressed problem is to design both an event-based distributed controller and a state observer such that a prescribed H∞ containment index can be achieved over a given finite horizon. First, with the help of the completing the square method, a sufficient condition is established to ensure the desired H∞ containment performance. Then, by resort to a novel nominal energy cost index combined with Moore-Penrose pseudoinverse method, the desired controller and observer parameters are obtained by solving two coupled backward recursive Riccati difference equations. Two positive scalars in proposed nominal energy cost index provide a tradeoff among the controlled tracking errors, the energy of transformed control inputs, and the precision of estimated states. Finally, a simulation example is given to illustrate the usefulness of the proposed theoretical results.