Linear Discrete Time-varying Research Articles

Iterative learning control approach for linear discrete time-varying systems with different initial time points

Different from the existing iterative learning control (ILC) works for dynamical systems with iteration-varying trial lengths, in which the terminal time points are not identical, this paper presents an ILC approach for linear discrete time-varying (LDTV) systems with different initial time points. A piecewise ILC law is designed by using the segment compensation techniques in time axis. Through a rigorous mathematical analysis, it is theoretically proved that as the mathematical expectation of the iterative initial state equals to the desired initial state, the ILC tracking error at the maximum universal time interval can be driven into a bounded region in mathematical expectation sense. Particularly, the mathematical expectation of the ILC tracking error at the maximum universal time interval can even be driven to zero under a certain initial state condition. Numerical simulations are provided to verify the effectiveness of the proposed ILC scheme.

Stability analysis for linear time‐varying systems using bicoprime factorization

AbstractThis paper presents a bicoprime factorization (BCF) approach to deal with various stability problems for linear discrete time‐varying (LTV) systems within the framework of nest algebra. Based on the bicoprime factorization, we give necessary and sufficient conditions for (strongly) simultaneous stabilization of three systems, which generalize similar results for systems by coprime factorization (CF). Finally, when the bicoprime factorization for the plant and controller has additive perturbations, we derive sufficient conditions using the norm inequality for robust stability of the system.

Finite impulse response filter based fault estimation with computational efficiency for linear discrete time-varying systems subject to multiplicative noise

The robust fault estimation problem for linear discrete time-varying (LDTV) systems subject to multiplicative noise is investigated by means of finite impulse response (FIR) filter. A novel analytical redundancy, expressed via all states of the previous time window, is originally established to construct the fault estimator. To ensure the satisfactory fault estimation accuracy in stochastic sense under the interference of random uncertainty, a new performance index in forms of matrix trace function is proposed. An easy-to-check necessary and sufficient condition is presented to obtain the optimal filter gain via minimizing the performance index at each time instant. It is analytically demonstrated that, the newly proposed fault estimation algorithm enjoys obvious computational advantages in updating the filter gain, especially as the length of the time window increases for time-varying systems. Simulation results are finally provided to verify its feasibility and superiority.

A Krein space-based approach to event-triggered [formula omitted] filtering for linear discrete time-varying systems

This paper is concerned with the problem of event-triggered H∞ filtering for linear discrete time-varying (LDTV) systems. Using the lifting technique, we firstly establish an equivalent relationship with a certain equivalent minimum problem of indefinite quadratic form subject to LDTV systems with non-uniform sampling periods. Then, based on Krein space projection and innovation analysis, sufficient and necessary conditions for the existence of desired filter are derived and a feasible solution is obtained in terms of Riccati recursions. Thus, an algorithm based on the time-update and event-update recursions is given for the implementation of event-triggered H∞ filtering. Different from some existing results, a new event-triggered H∞ filtering scheme is provided so that the estimation error can be completely decoupled from the event-triggered transmission error. Moreover, the new proposed Krein space approach is less conservative and more computational attractive than the existing methods based on recursive linear inequality matrix. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

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.

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.

A survey on model-based fault diagnosis for linear discrete time-varying systems

Abstract To meet the rising demands for safety and reliability of modern industrial control systems, the model-based fault diagnosis problem has attracted much attention in the past few decades both from the academic communities and in practical applications. Among the rich body of literature, the majority of the results are proposed for linear time invariant systems despite the fact that most practical processes are inherently time-varying by nature and discretized for online implementations. Recently, with the ever-increasing system complexity and scale, the model-based fault diagnosis issue for linear discrete time-varying (LDTV) systems has become a hot research topic of both theoretical importance and practical significance, and a great number of results have been reported with many open problems deserving further investigation. To reflect the latest progress in this emerging research branch, this survey aims to provide a timely reference to readers interested in this topic. More specifically, in this survey, the techniques for model-based fault diagnosis for LDTV systems are classified into observer-based methods, parity space-based approaches and parameter estimation schemes, which constitute the three main parts of this survey. The background and the latest progress for each class of these techniques are comparatively discussed with particular focuses on the fault detection and the fault estimation issues. Additionally, an overview on their practical applications to specific industrial plants is provided. Furthermore, the possible future research directions concerning the model-based fault diagnosis for LDTV processes are pointed out, followed by comprehensive concluding remarks.

Stationary wavelet transform aided design of parity space vectors for fault detection in LDTV systems

This study deals with the problem of fault detection (FD) for linear discrete time-varying (LDTV) systems by combining the parity space-based method with the stationary wavelet transform (SWT). For the purpose of achieving good FD performance and simultaneously reducing the online computational burden resulted by high parity space order, SWT on residual signal is the first step in this study. This allows the formulation of residual generation as the selection of wavelet basis for SWT and the design of time-varying parity space vectors in a framework of multiscale optimisation. A recursive algorithm is developed for the computation of optimal parity space vectors and, based on this, a bank of residual signals can be obtained. Moreover, a weighted multiscale residual evaluation strategy is presented to detect the occurrence of a fault. The major contribution of the study lies in the development of a recursive design of SWT aided parity space vectors, which delivers an improvement of FD performance for a lower parity space order and an easy online implementation. A numerical example is demonstrated to illustrate the effectiveness of the proposed method.

Optimal residual generation for fault detection in linear discrete time-varying systems with uncertain observations

This work deals with the problem of optimal residual generation for fault detection (FD) in linear discrete time-varying (LDTV) systems subject to uncertain observations. By introducing a generalized fault detection filter (FDF) with four parameter matrices as the residual generator, a novel FDF design scheme is formulated as two bi-objective optimization problems such that the sensitivity of residual to fault is enhanced and the robustness of residual to unknown input is simultaneously strengthened. A generalized operator based optimization approach is proposed to deduce solutions to the corresponding optimization problems in operator forms, where the related H∞/H∞ or H−/H∞ FD performance index is maximized. With the aid of the addressed methods, the connections among the derived solutions are explicitly announced. The parameter matrices of the FDF are analytically derived via solving simple matrix equations recursively. It is revealed that our proposed results establish an operator-based framework of optimal residual generation for some kinds of linear discrete-time systems. Illustrative examples are given to show the applicability and effectiveness of the proposed methods.

A New Scheme of Fault Detection for Linear Discrete Time-Varying Systems

In this note, a new scheme of fault detection (FD) is proposed for linear discrete time-varying (LDTV) systems subject to $l_{2}$ -norm bounded unknown inputs. The basic idea is to find an optimal estimation of the $l_{2}$ -norm of the unknown inputs including the unknown initial state variables. This leads to a natural design of the FD system with the $l_{2}$ -norm boundedness of the unknown inputs as a threshold and the $l_{2}$ -norm of the unknown input estimate as the evaluation function. To avoid heavy computational burden, projection technique in Krein space is applied which allows a recursive computation of the evaluation function. It is shown that the achieved FD system satisfies both the worst case and best case sensitivity/robustness ratio criteria and further applications to observer-based FD lead to an alternative design of $H_{\infty}/H_{\infty}$ and/or $H_{-}/H_{\infty}$ fault detection filter.

Parity space-based fault detection for linear discrete time-varying systems with unknown input

In this paper, a new approach to parity space-based fault detection (FD) is put forward for linear discrete time-varying (LDTV) systems. The focus of our study is on solving the problem with heavy online computational burden. The main contributions are two folds. Firstly, the problem of parity space-based FD is formulated as finding a minimum of quadratic form of unknown input. Secondly, by applying projection in Krein space, a solution of optimal parity space-based FD is obtained via recursively computing innovations. It is shown that the reduction of computation cost is achievable, especially with the increase of parity space order. Finally, a numerical example is given to demonstrate the application of the proposed method.

An H∞ Disturbance Estimation Approach to Fault Detection for Linear Discrete Time-Varying Systems

This paper is concerned with a new fault detection scheme for linear discrete timevarying systems by using an H∞ disturbance estimation approach. A unified estimation problem is formulated for linear discrete time-varying systems in an H∞ framework. Such a formulation not only achieves a novel H∞ disturbance estimation problem but also includes the classical H∞ fault estimation as a special case. By using projection and innovation analysis in the Krein space, the H∞ disturbance and fault estimation problems are solved in a unified way, and their estimates are given in terms of the solution to a set of Riccati difference equations. Then, both the H∞ disturbance and fault estimates are used as residuals for the purpose of fault detection, and the detection performances are discussed based on these two residual generation strategies. It is an interesting finding that, in the case of sensor fault-free, the H∞ fault estimation approach is no longer available while the H∞ disturbance estimation approach is still effective. Finally, a simulation example is employed to illustrate the results derived in this paper.

Fault Detection for Linear Discrete Time‐varying Systems with Intermittent Observations and Quantization Errors

AbstractThis paper deals with the fault detection (FD) problem for linear discrete time‐varying (LDTV) systems subject to multiple intermittent observations and quantization errors. A set of independent identical distributed random variables are introduced as the indicators of the observation sequences to describe the multiple intermittent measurements. The measured observation is quantized by a logarithmic type quantizer. Our focus is to construct an observer‐based fault detection filter (FDF) to recognize the fault in spite of multiple measurement packet dropouts and quantization inaccuracy. By defining generalized input‐to‐output operators, the FD problem is formulated into a two‐objective optimization framework such that stochastic H∞/H∞ or H−/H∞ performance index is maximized. Probability/indicators‐dependent and probability‐dependent analytical solutions are respectively derived by virtue of an adjoint operator based optimization approach for two cases. One is that the indicators are on‐line known while the other one is that the indicators are not available at each time instant. An illustrative example is employed to demonstrate the effectiveness and applicability of the proposed approach.

Statistical χ2 Testing Based Fault Detection for Linear Discrete Time-delay Systems

This paper is concerned with statistical χ2 testing based fault detection (FD) for a class of linear discrete time-varying (LDTV) stochastic systems with delayed state. Different from the traditional residual based FD, we propose to construct the evaluation function by directly using measurement observations. Then an equivalent solution can be given in terms of Riccati recursion by utilizing projection and innovation analysis technique. Moreover, the fault free case evaluation function is with central χ2 distribution and the heavy computational burden is reduced. Furthermore, strategies of χ2 statistic testing on evaluation function are also discussed. Finally, a numerical example is given to illustrate the proposed method.

OnH∞Fault Estimator Design for Linear Discrete Time-Varying Systems under Unreliable Communication Link

This paper investigates theH∞fixed-lag fault estimator design for linear discrete time-varying (LDTV) systems with intermittent measurements, which is described by a Bernoulli distributed random variable. Through constructing a novel partially equivalent dynamic system, the fault estimator design is converted into a deterministic quadratic minimization problem. By applying the innovation reorganization technique and the projection formula in Krein space, a necessary and sufficient condition is obtained for the existence of the estimator. The parameter matrices of the estimator are derived by recursively solving two standard Riccati equations. An illustrative example is provided to show the effectiveness and applicability of the proposed algorithm.

Fault Detection for Linear Discrete Time-Varying Systems with Measurement Packet Dropping

The fault detection (FD) problem for linear discrete time-varying (LDTV) systems with measurement packet dropouts is considered. The objective is to design a new observer-based fault detection filter (FDF) as a residual generator through employing packet dropout information on the measurement sequence. Based on some new defined input-to-output operators, the FD problem is formulated in a framework of maximizing stochasticH−/H∞orH∞/H∞performance index. By introducing an adjoint-operator-based optimization method, the analytical optimal solution can be derived in terms of solving a modified Riccati equation. A numerical example is provided to demonstrate the effectiveness of the proposed approach.

Stability analysis for time-varying systems via quadratic constraints

In this paper, we introduce the concept of “quadratic constraint” (QC) to deal with various stability problems for infinite dimensional linear discrete time-varying (LTV) systems. First, we derive a necessary and sufficient condition for the close-loop stability based on quadratic constraints (QCs). An equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system. Moreover, applying this stability criterion and the complete finiteness of the nest algebra, we show that the plant is stabilizable if and only if it has a single strong representation. Finally, we use QCs to give a necessary and sufficient condition for the robust stabilization of connected uncertainty set defined by gap metric.

Optimal fault detection for linear discrete time-varying systems

This paper deals with the problem of observer-based fault detection for linear discrete time-varying (LDTV) systems. A problem formulation is first proposed to address the optimization of the fault detection filter (FDF) design, which is expressed in terms of maximizing a finite horizon H ∞ / H ∞ or H − / H ∞ performance index. This formulation can be applied to FDF design of LDTV systems subject to l 2 -norm bounded unknown inputs or stochastic noise sequences. It is shown that a unified optimal solution to the FDF can be obtained by solving the discrete time Riccati equation and the optimal FDF is not unique. A numerical example is given to illustrate the proposed method.

On the computation of the gap metric for LTV systems

In this paper, we consider the robust stabilization problem for linear discrete time-varying (LTV) systems using the gap metric. In particular, we show that the time-varying (TV) directed gap reduces to an operator with a TV Hankel plus Toeplitz structure. Computation of the norm of such an operator can be carried out using an iterative scheme involving a TV Hankel operator defined on a space of Hilbert–Schmidt causal operators. The “infimization” in the TV directed gap formula is shown to be, in fact, a minimum by using duality theory. The latter holds as well in the time-invariant case.