Linear Discrete Time-varying Systems Research Articles

On robustness in the gap metric and coprime factor uncertainty for LTV systems

In this paper, we study the problem of robust stabilization for discrete linear time-varying (LTV) systems subject to time-varying normalized coprime factor uncertainty. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, we compute an upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure. We point to a necessary and sufficient condition which guarantees compactness of the TV Hankel operator, and in which case singular values and vectors can be used to compute the time-varying stability margin and TV controller.

On the number of upper Bohl exponents for diagonal discrete time-varying linear system

The Bohl exponents, similarly as the Lyapunov exponents, are one of the most important numerical characteristics of dynamical systems used in control theory. By contrast with properties of the Lyapunov characteristics, properties of the Bohl exponents are much less investigated. In this paper we present some new properties of the upper Bohl exponents of the diagonal discrete time-varying linear system.

Transitivity in simultaneous stabilization for a family of time-varying systems

This paper develops necessary and sufficient conditions for the so-called transitivity and strong transitivity (precise definitions later) for a family of discrete time-varying linear systems, which when satisfied lead to characterizations for simultaneous stabilizability and simultaneously stabilizing feedback controllers. Some criteria for the strong transitivity are established in terms of single coprime factorizations of only one plant and one controller, the resulting characterization involves coprime factorizations of fewer systems compared to previous work.

H∞Fault Detection for Linear Discrete Time-Varying Descriptor Systems with Missing Measurements

This paper deals with the problem ofH∞fault detection for a class of linear discrete time-varying descriptor systems with missing measurements, and the missing measurements are described by a Bernoulli random binary switching sequence. We first translate theH∞fault detection problem into an indefinite quadratic form problem. Then, a sufficient and necessary condition on the existence of the minimum is derived. Finally, an observer-basedH∞fault detection filter is obtained such that the minimum is positive and its parameter matrices are calculated recursively by solving a matrix differential equation. A numerical example is given to demonstrate the efficiency of the proposed method.

Finite-Horizon Distributed H∞ Fault Estimation for Time-Varying Systems in Sensor Networks: A Krein-Space Approach

In this paper, the finite-horizon distributed H∞ fault estimation problem is investigated for a class of linear discrete time-varying systems in sensor networks. Sensor nodes are distributed according to a given network topology and each sensor receives the information from both the system and its neighbors. A set of measurement models is established for each sensor node that describes all the available information from the node and its neighbors. The distributed H∞ fault estimation problem addressed is first transformed into an optimization issue of a certain indefinite quadratic form. Then, by employing innovation analysis and projection theory in Krein spaces, the desired distributed H∞ fault estimators are designed in terms of the solution to a set of recursive equations. Finally, an illustrative example is presented to show the effectiveness of the proposed distributed fault estimation scheme.

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.

HiH∞ optimization based fault detection for linear discrete time-varying systems with delayed state

This paper investigates the problem of fault detection for linear discrete time-varying systems with delayed state by using Hi/H∞ and/or Hi/H∞ optimization. A residual generator, which is the so-called Hi/H∞ fault detection filter (FDF), is designed by finding filter gain matrices and a post-filter. By utilizing projection and innovation re-organization technique, an optimal Hi/H∞ -FDF can be given in terms of recursions. Moreover, the obtained evaluation function is a unified solution and, in order to detect the presence of a fault, the selection of a threshold is dependent on the characteristics of unknown input and a prescribed false alarm rate. A numerical example is used to demonstrate the effectiveness of the proposed method.

Alternative formulae for lower general exponent of discrete linear time-varying systems

The Bohl exponents, similarly as Lyapunov exponents, are one of the most important numerical characteristics of dynamical systems used in control theory. Properties of the Lyapunov characteristics are well described in the literature. Properties of the Bohl exponents are much less investigated. In this paper we consider the so-called junior lower general exponent of discrete linear time-varying system and present some alternative formulae for it. We also discuss relations between lower Bohl exponents of the perturbed system and junior lower general exponent of the unperturbed system.

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.

Iterative learning based fault diagnosis for discrete linear uncertain systems

In order to detect and estimate faults in discrete linear time-varying uncertain systems, the discrete iterative learning strategy is applied in fault diagnosis, and a novel fault detection and estimation algorithm is proposed. And the threshold limited technology is adopted in the proposed algorithm. Within the chosen optimal time region, residual signals are used in the proposed algorithm to correct the introduced virtual faults with iterative learning rules, making the virtual faults close to these occurred in practical systems. And the same method is repeated in the rest optimal time regions, thereby reaching the aim of fault diagnosis. The proposed algorithm not only completes fault detection and estimation for discrete linear time-varying uncertain systems, but also improves the reliability of fault detection and decreases the false alarm rate. The final simulation results verify the validity of the proposed algorithm.

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.

A MIMO Sampling Rate Dependent Controller

A sampling-rate-dependent (SRD) controller is proposed for the uniform output tracking problem of multiinput multioutput discrete time-varying linear systems. A proportional-integral-derivative (PID) controller is a special case of the proposed controller. The SRD controller aims at achieving uniform output tracking in the sense of attaining arbitrary small steady-state errors as well as arbitrary small settling time. The SRD controller makes use of higher sampling rates for larger variations in the desired trajectories. An analytic approach for finding the parameters of the SRD controller that guarantees reference output trajectory tracking is presented. Four different numerical examples and two different experimental applications are provided in order to illustrate the output tracking capability of the proposed SRD controller. Performance is compared to a PID controller with fuzzy gain scheduling, four multivariable PID controllers, an H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> optimal controller, and an iterative-learning-control algorithm.

Finite-Horizon $H_{\infty}$ Fault Estimation for Uncertain Linear Discrete Time-Varying Systems With Known Inputs

In this brief, the finite-horizon H∞ fault estimation problem is investigated for a class of uncertain linear discrete time-varying systems with known inputs. A new H∞ performance index including the known inputs is put forward in order to better reflect the effect of the known input on the whole fault estimation systems. To cope with the uncertainties, an auxiliary system is constructed with a certain indefinite quadratic form. By recurring to the Krein-space theory, the optimization problem of the associated indefinite quadratic form is solved, and a sufficient condition with much less conservativeness is established for the existence of the desired fault estimator. Then, all the estimator parameters are derived simultaneously in terms of an explicit solution to a matrix equation. Finally, an illustrative numerical example is employed to demonstrate the effectiveness of the proposed fault estimation scheme.

Formula omitted] fixed-lag smoothing for linear discrete time-varying systems with uncertain observations

This paper deals with the problem of H∞ fixed-lag smoothing for linear discrete time-varying systems with uncertain observations and l2-norm bounded noise. First, the design of H∞ smoother is equivalent to the problem that a certain indefinite quadratic form with respect to a new state-space constraints has a minimum and the smoother is such that the minimum is positive. Then, by introducing a Krein space stochastic system and applying re-organized innovation analysis approach, the minimum of indefinite quadratic form and its existence condition are derived. Finally, through analyzing the existence condition of the minimum and guaranteeing the positivity of the minimum, a sufficient and necessary condition for the existence of an H∞ smoother is proposed and the smoother is obtained in terms of two standard Riccati difference equations. A numerical example is given to show the effectiveness of the proposed method.

A projection-based method of fault detection for linear discrete time-varying systems

This article is aimed at the development of a residual generator by making use of an arbitrary linear combination of measurement output estimation error sequence. First, Gramian matrix-based criteria are proposed to measure the influences of an unknown input and a fault on the residual. Then, the design of the residual generator is formulated into a sensitivity/robustness ratio maximisation problem. It is shown that the optimal solution is not unique and one of them can be derived by directly applying an orthogonal projection of the measurement output and, with the aid of an innovation analysis, a more generalised residual generator is also obtained. Furthermore, it is demonstrated that the obtained residual generators in state space description also provide optimal observer-based fault detection filters for the linear discrete time-varying systems subject to l 2-norm bounded unknown inputs or stochastic noise sequences. To show the effectiveness of the proposed method, a numerical example is given.

Stabilization of slowly time-varying discrete systems with state delays

This paper studies the stabilization of the infinite-dimensional linear discrete time-varying systems with state delays and slowly varying coefficients , , where A(k) are bounded linear operators acting on a Banach space X, B(k) are X-valued bounded linear operators defined on a Banach space U and A 1(k) are bounded linear operators acting on a Banach space X. Assuming appropriate conditions, we will show that the stabilization of the ‘frozen’ (time invariant) system , for each fixed integer m, implies the stabilization of the corresponding time-varying system. For such systems, explicit conditions for the feedback exponential stabilizability are established. As an application of the main results, we study the exponential stabilizability of a general nonlinear control system with several state delays.

On the spectrum of discrete time-varying linear systems

In this paper, we investigate the influence of small perturbations of the coefficients of discrete time-varying linear systems on the Lyapunov exponents. For that purpose we introduce the concepts of central exponents of the system and we show that these exponents describe the possible changes in the Lyapunov exponents under small perturbations. Finally, we present several formulas for the central exponents in terms of the transition matrix of the system and the so-called upper sequences. The results are illustrated by numerical examples.

Integrated trade‐off design of fault detection system for linear discrete time‐varying systems

In this study, the problem of fault detection (FD) system design for linear discrete time-varying systems is addressed. Different from most existing methods which separately handle residual generator and evaluator, this study focuses on integrated design of residual generator and evaluator. Instead of traditional trade-off design between sensitivity to faults and robustness to disturbances, this study aims at maximising FD rate given a predefined false alarm rate. The proposed approach in this study follows two steps: first, in the norm-based framework, the parity relation-based offline FD system achieves maximal FD rate by minimising the set of undetectable faults; then the optimal offline FD system is equivalently transformed into a recursive FD algorithm for online use. Relationship between the obtained solution in this study and the existing ones is analysed. The performance improvement of the proposed approach is illustrated by comparing against existing methods through Monte Carlo simulations.

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.

Minimum System Sensitivity Study of Linear Discrete Time Systems for Fault Detection

Fault detection is a critical step in the fault diagnosis of modern complex systems. An important notion in fault detection is the smallest gain of system sensitivity, denoted asℋ−index, which measures the worst fault sensitivity. This paper is concerned with characterizingℋ−index for linear discrete time systems. First, a necessary and sufficient condition on the lower bound ofℋ−index in finite time horizon for linear discrete time-varying systems is developed. It is characterized in terms of the existence of solution to a backward difference Riccati equation with an inequality constraint. The result is further extended to systems with unknown initial condition based on a modifiedℋ−index. In addition, for linear time-invariant systems in infinite time horizon, based on the definition of theℋ−index in frequency domain, a condition in terms of algebraic Riccati equation is developed. In comparison with the well-known bounded real lemma, it is found thatℋ−index is not completely dual toℋ∞norm. Finally, several numerical examples are given to illustrate the main results.