Singular Discrete-time Research Articles

Kalman Filter for Linear Discrete-Time Rectangular Singular Systems Considering Causality

This paper proposes a Kalman filter for linear rectangular singular discrete-time systems, where the singular matrix in the system is a rectangular matrix without full column rank. By using two different restricted equivalent transformation methods and adding the measurement equation to the state equation, the system is transformed into a square singular system satisfying regularity and observability. During this process, the causality of the system is taken into account, and multiple matrix transformations are applied accordingly. Based on these modifications, state estimation results are obtained using the Kalman filter. Finally, a numerical example is employed to demonstrate the effectiveness of our approach.

Suboptimal finite-time control of singular large scale discrete-time equations with delayed interconnections

The problem of suboptimal finite-time control for a class of large scale singular discrete-time equations with delayed interconnections is studied in this paper. Making use of singular value theory and Lyapunov function method, new delay-dependent conditions are established for solving the finite-time H∞ control and guaranteed cost control problems of such equations. The conditions are presented in the form of tractable linear matrix inequalities (LMIs). Illustrated examples with simulations show the feasibility and effectiveness of the proposed results.

State bounding estimation of positive singular discrete-time systems with unbounded time-varying delays

This paper investigates the problem of state bounding for linear positive singular discrete-time systems with unbounded time-varying delay. Our approach is based on the comparison principle and the spectral analysis of matrices. Using the proposed approach, we provide new sufficient conditions for the regularity, causality, positivity and the existence of the smallest ultimate upper bound of such systems. The conditions are given in terms of linear programming problems, which can be solved by LP optimal toolbox. Numerical examples with simulation are given to demonstrate the advantage and validity of the proposed theoretical results. The proposed method is the first trial in the state bounding problem of linear singular discrete-time systems with unbounded time-varying delays.

Finite-Time H∞ Robust Controller Design for a Class of Singular Discrete-Time Markov Jump Delay Systems With Packet Loss Compensation and Input Saturation

Finite-Time H_∞ Robust Controller Design for a Class of Singular Discrete-Time Markov Jump Delay Systems With Packet Loss Compensation and Input Saturation

The Samuelson macroeconomic model as a singular linear matrix difference equation

In this paper, we revisit the famous classical Samuelson’s multiplier–accelerator model for national economy. We reform this model into a singular discrete time system and study its solutions. The advantage of this study gives a better understanding of the structure of the model and more deep and elegant results.

Finite-time stability and stabilisation of singular discrete-time linear positive systems with time-varying delay

ABSTRACTIn this paper, the problem of finite-time stability and stabilisation for positive singular discrete-time linear systems with time-varying delay is investigated. We first present novel delay-dependent sufficient conditions for positivity and finite-time stability of the unforced systems. We then apply the obtained results to solve finite-time stabilisation problem of the considered systems. The sufficient conditions for the positivity and finite-time stabilisable of such systems are formulated in terms of a standard linear programming (LP) problem. Numerical examples are provided to illustrate the effectiveness and advantages of our results.

State bounding for positive singular discrete‐time systems with time‐varying delay and bounded disturbances

This study deals with the problem of state bounding for positive singular discrete-time systems with time-varying delay and bounded disturbances. First, the authors derive some new conditions for the existence of componentwise bounds of the state vector of the systems without disturbances. Then, in the case of systems with bounded disturbances, they obtain a sufficient condition for the existence of componentwise ultimate bounds. Since the conditions are given in terms of the spectral radius of the system matrices, they can check them easily and compute directly the smallest componentwise ultimate bound. In addition, the problem of state bounding for positive time-varying singular systems is also investigated in this study. Finally, numerical examples are provided to illustrate the effectiveness and advantages of their results.

On dissipativity‐based filtering for discrete‐time switched singular systems with sensor failures: a persistent dwell‐time scheme

The work is interested in the extended dissipative filtering issue for switched singular discrete-time systems with sensor failures. The persistent dwell-time switching regularity is utilised to describe the successive occurrence of slow switching and fast switching in a switched singular system. A set of variables, which take values in a fixed interval, are utilised for the quantitative analysis of the sensor failures. The main objective is centred on designing a reliable mode-dependent filter which can ensure that the resulting filtering error system is exponentially admissible and simultaneously satisfies the extended dissipative property in the presence of sensor failures. By adopting suitable decoupling methods and introducing reasonable matrix transformation techniques, some relaxed sufficient conditions are derived. The filter gains can be obtained by solving a convex optimisation problem in terms of linear matrix inequalities. Finally, an illustrative example is presented to verify the applicability of the designed filter.

Existence of Solutions and Finite-Time Stability for Nonlinear Singular Discrete-Time Neural Networks

This paper investigates the problem of finite-time stability and control for a class of nonlinear singular discrete-time neural networks with time-varying delays and disturbances. First, based on the implicit function theorem and singular value decomposition method, a sufficient condition for the existence of the solution of such systems is established in terms of a linear matrix inequality (LMI). Then, using the Lyapunov functional approach combined with LMI technique we provide new delay-dependent sufficient conditions for robust $$H_{\infty }$$ finite-time stability and control. Finally, some numerical examples are given to illustrate the efficiency of the proposed results.

An efficient observer design method for singular discrete-time systems with time-delay and nonlinearity: LMI approach

In this paper, the observer design method for linear and nonlinear singular discrete-time systems with constant time-delays is proposed. By constructing appropriate Lyapunov–Krasovskii functional and using linear matrix inequality (LMI) technique, the asymptotic convergence criterion is developed in terms of LMIs, which can be solved numerically using MATLAB® LMI® toolbox. The sufficient condition for the existence of a full-order observer is obtained and the states are estimated using Schur complement and S-procedure lemma very well. Moreover, an extension procedure for observer design of singular linear system with time-varying delay is presented. Simulation results are included to prove the efficiency of the suggested approach.

Punktowa zupełność oraz punktowa degeneracja wybranej klasy układów dyskretnych singularnych niecałkowitego rzędu

Punktowa zupełność oraz punktowa degeneracja wybranej klasy układów dyskretnych singularnych niecałkowitego rzędu

Fault Detection for Singular Discrete-Time Linear Systems with Signal-to-Noise Ratio Constrained Channels

This paper deals with the fault detection problem for singular discrete-time linear systems with additive white Gaussian noise channels subject to signal-to-noise ratio constraints. An observer-based fault detection scheme which consists of residual generator, threshold and false alarm rate is constructed. In the proposed scheme, the design of residual generator is realized in the context of $${H_\infty }$$Hź-filtering problem when the fault and unknown disturbance are present, and the estimations of both system state and fault are obtained in the design procedure. Then, in order to detect the fault, an online threshold based on the stochastic characteristics of channel noise is established. For the purpose of evaluating the fault detection system performance, the false alarm rate is guaranteed to be below a certain level. Finally, a simulation example is exploited to illustrate the application and effectiveness of the proposed method.

On a Non-homogeneous Singular Linear Discrete Time System with a Singular Matrix Pencil

In this article, we study the initial value problem of a non-homogeneous singular linear discrete time system whose coefficients are either non-square constant matrices or square with an identically zero matrix pencil. By taking into consideration that the relevant pencil is singular, we provide necessary and sufficient conditions for existence and uniqueness of solutions. More analytically we study the conditions under which the system has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally we provide some numerical examples based on a singular discrete time real dynamical system to justify our theory.

Design of reduced-order 2–∞ filter design for singular discrete-time systems using strict linear matrix inequalities

In this study, the problem of designing regular robust 2–∞ filter for a class of linear singular discrete-time systems with norm-bounded uncertainties is investigated. A class of regular linear filters is fully analysed, then necessary and sufficient conditions of robust admissibility are established. Guaranteed 2–∞ performance of the filtered system is provided and expressed in terms of the solution of strict linear matrix inequalities (LMIs). The cases of full- and zeroth-order realisable filter are derived as limiting cases of design. Numerical examples are worked out to illustrate the theoretical development.

Delay‐dependent robust control for singular discrete‐time Markovian jump systems with time‐varying delay

AbstractThe problem of delay‐dependent robust stabilization for uncertain singular discrete‐time systems with Markovian jumping parameters and time‐varying delay is investigated. In terms of free‐weighting‐matrix approach and linear matrix inequalities, a delay‐dependent condition is presented to ensure a singular discrete‐time system to be regular, causal and stochastically stable based on which the stability analysis and robust stabilization problem are studied. An explicit expression for the desired state‐feedback controller is also given. Some numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.

Regularization and robust control of uncertain singular discrete-time linear systems

New sufficient linear matrix inequality (LMI) condition for regularization of singular discrete-time systems subject to norm-bounded uncertainties is given. Then a new class of feedback is proposed to stabilize singular uncertain discrete-time systems with unknown time delays. The regularization and the stabilizability condition of this class of systems is given in terms of one strict LMI. A numerical example is given to show the novelty of the control design.

Optimal and Self-Tuning State Estimation for Singular Stochastic Systems: A Polynomial Equation Approach

This paper is concerned with the optimal steady-state estimation for singular stochastic discrete-time systems using a polynomial equation approach. The key to the optimal estimation is the calculation of an optimal estimator gain matrix. The main contribution of the paper is the derivation of a simple method for computing the gain matrix. Our method involves solving one simple polynomial equation which is derived from the uniqueness of the autoregressive moving average (ARMA) innovation model. The approach covers prediction, filtering, and smoothing problems. Further, when the noise statistics of the model are not available, self-tuning estimation is performed by identifying one ARMA innovation model.

Balancing singular discrete-time systems

In this work, we deal with normalizable-balanced singular systems, that is, singular systems where the obtained closed-loop system is balanced by an appropriate feedback. In the invariant case, we use a proportional and derivative feedback to obtain systems without infinite poles. In addition, the existence and construction of state-feedbacks to obtain normalizable-balanced N-periodic singular systems is analyzed.

How to Solve Singular Discrete-Time Riccati Equations

There exists a solution to the discrete time algebraic Riccati equation giving closed loop eigenvalues inside or on the unit circle, assuming the system is stabilizable. This solution is always unique. Numerical methods, like the sorted generalized Schur form method and the Kleinman iteration, often fail in case of zeros on the unit circle or lack of left invertibility. It is here suggested how such singular cases could be made regular by reduction operations on the system matrix pencil [−zI + A, B; C, D]. The solution may be discontinuous with respect to parameter variation, but the reduction approach seems numerically appealing. This is demonstrated using simple illustrative examples. The solution is the largest symmetrical matrix satisfying a corresponding LMI. To obtain a feasible solution for interior-point methods it is also necessary to do a reduction, for some problems even further than what is described in this paper.

Sampled-Data and Discrete-Time $H_2$ Optimal Control

This paper deals with the sampled-data $H_2$ optimal control problem. Given a linear time-invariant continuous-time system, the problem of minimizing the $H_2$ performance over all sampled-data controllers with a fixed sampling period can be reduced to a pure discrete-time $H_2$ optimal control problem. This discrete-time $H_2$ problem is always singular. Motivated by this, in this paper we give a treatment of the discrete-time $H_2$ optimal control problem in its full generality. The results we obtain are then applied to the singular discrete-time $H_2$ problem arising from the sampled-data $H_2$ problem. In particular, we give conditions for the existence of optimal sampled data controllers. We also show that the $H_2$ performance of a continuous-time controller can always be recovered asymptotically by choosing the sampling period sufficiently small. Finally, we show that the optimal sampled-data $H_2$ performance converges to the continuous-time optimal $H_2$ performance as the sampling period converges to zero.