Discrete-time Singular Systems Research Articles

Peak-to-Peak Filtering for Discrete-Time Singular Systems

This brief on peak-to-peak filtering is carried out for discrete-time singular systems in this brief. For the sake of designing the required filter, it makes the filtering error system admissible (regular, causal, and stable), meanwhile, preserves the specified peak-to-peak performance level. By introducing a novel Lyapunov function, the peak-to-peak performance of the filtering error system is analyzed on the strength of matrix inequalities. Then, based on linear matrix inequalities (LMIs), significant design conditions of the peak-to-peak filter for discrete-time singular systems are proposed. In the end, a RC circuit is presented to verify the availability of the results.

Robust finite-time sliding mode control for discrete-time singular system with time-varying delays

This paper explores the finite-time bounded issue for discrete-time singular time-varying delay system via sliding mode control method. A suitable discrete-time sliding mode control law is constructed to drive the state trajectories onto the specified sliding surface in a given finite time interval. Meanwhile, sufficient conditions for finite-time bounded to the closed-loop delayed system are provided in both reaching phase and sliding motion phase. In addition, the finite-time sliding mode controller gain matrix can be solved by using the linear matrix inequalities approach. Finally, three numerical examples are illustrated to demonstrate the superiority and practicability of presented results.

Observer-Based Fuzzy Controller Design for Nonlinear Discrete-Time Singular Systems via Proportional Derivative Feedback Scheme

This paper investigates the observer-based fuzzy controller design method for nonlinear discrete-time singular systems that are represented by Takagi-Sugeno (T-S) fuzzy models. At first, the nonlinearity can be well-approximated with several local linear input-output relationships. The parallel distributed compensation (PDC) technology and the proportional derivative (PD) feedback scheme are then employed to construct the observer-based fuzzy controller. To solve the problem of unmeasured states, the impulsive phenomenon of singular systems, and the PD scheme’s reasonableness, a novel observer-based fuzzy controller is developed. By using the Lyapunov theory and projection lemma, the stability criteria are built in terms of linear matrix inequalities (LMI). Moreover, the gains of fuzzy controller and fuzzy observer can be calculated synchronously by using convex optimization algorithms. Finally, a biological economic system is provided to verify the effectiveness of the proposed fuzzy control method.

Linearly Monotonic Convergence and Robustness of P-Type Gain-Optimized Iterative Learning Control for Discrete-Time Singular Systems

In this article, the repetitive finite-length linear discrete-time singular system is formulated as an input-output equation by virtue of the lifted-vector method and a gain-optimized P-type iterative learning control profile is architected by sequentially arguing the learning-gain vector in minimizing the addition of the quadratic norm of the tracking-error vector and the weighed quadratic norm of the compensation vector. By virtue of the elementary permutation matrix and the property of the quadratic function, the optimized-gain vector is solved and explicitly expressed by the system Markov matrix and the iteration-wise tracking error. Then the linearly monotonic convergence of the tracking error is derived under the assumption that the initial state of the dynamic subsystem is resettable. Furthermore, for the circumstance that the system parameters uncertainties exist, the quasi scheme is established by replacing the exact system Markov matrix with the approximated one in the optimized gain. Rigorous analysis conveys that the proposed gain-optimized scheme is robust to the system internal disturbance within a suitable range. The validity and effectiveness are demonstrated numerically.

Convergence characteristics of iterative learning control for discrete-time singular systems

This paper investigates the convergence characteristics of the conventional P-type iterative learning control (ILC) scheme and exploits a gain-adaptive iterative learning control mechanism for a class of linear discrete-time singular systems. Based on the lifted vector technique, the paper reforms the discrete-time singular system as a kind of algebraic input-output transmission. For the conventional P-type ILC scheme, the asymptotical convergence in terms of the tracking-error vector is achieved and the monotonic convergence in the sense of 2-norm of the tracking-error vector is derived. Further, in order to improve the learning performance, a gain-adaptive iterative learning control (GAILC) strategy is developed, which argues the iteration-time-variable gain vector while minimising the increment of quadratic tracking-error vectors of two adjacent iterations. The existence of the optimal gain vector is explored through the optimisation criterion and the algebraic approach of the columns/ rows exchanging transformation of matrix. Then the non-conditionally strictly monotonic convergence of the GAILC is made by studying the eigenvalues of the quadratic function. Finally, the validity and the effectiveness of the P-type ILC are numerically demonstrated and the remarkable outcomes of the GAILC are illustrated.

Optimal control for uncertain discrete-time singular systems under expected value criterion

Optimal control problems governed by two different types of uncertain discrete-time singular systems are investigated under expected value criterion. The objective function including uncertain variables is optimized with the help of expected value method provided that the singular systems are both regular and impulse-free. At first, based on the principle of dynamic programming, a recurrence equation is derived to simplify an optimal control model for a class of uncertain discrete-time singular systems. After that, according to uncertainty theory and the recurrence equation, two kinds of optimal control problems subject to an uncertain linear singular system and an uncertain singular system with quadratic input variables are considered in order, and the optimal solutions are both presented by accurate expressions. A numerical example and a dynamic input-output model are settled to illustrate the effectiveness of the results obtained.

Stability analysis of discrete-time coupled systems with delays

Addressed in this paper is the stability issue of discrete-time delayed coupled systems consisting of a difference equation and an algebraic equation in coupled states. First, a necessary and sufficient condition of exponential stability is proposed for coupled systems with constant delays. Second, periodicity and stability are investigated for periodic coupled systems. Third, based on the Lyapunov inequality, a necessary and sufficient condition is presented for exponential stability of general time-varying coupled systems. Finally, as an application of these results, stability of discrete-time singular systems is discussed.

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.

State estimation and sliding mode control for non-linear singular systems with time-varying delay

Abstract This paper deals with the sliding mode control (SMC) problem for a class of nonlinear discrete-time singular systems with external disturbances and time-varying delay. The main contribution of this paper is to design an observer-based sliding mode control scheme for the system under consideration. First, a delay dependent criterion is built to guarantee the closed-loop system to be robustly admissible. Then, a (SMC) law is synthesized to guarantee the state trajectories of the closed-loop systems to be stable, to and ensure the reachability of the specified sliding surface in a short time interval. An illustrative example is given to numerically demonstrate the effectiveness of the proposed control scheme.

Robust Admissibilization for Discrete-Time Singular Systems with Time-Varying Delay

The problems of the admissibility and state feedback stabilization for discrete-time singular systems with interval time-varying delay and norm-bounded uncertainty are studied. The system is equivalently transformed into a new comparison form by decomposition. By taking advantage of the Seuret summation inequality, the reciprocally convex inequality, and some relaxation techniques, a delay-dependent criterion that ensures the admissibility of the concerned systems is established. The result on robust stabilization is also obtained by fixing some parameters. It should be pointed out that the results are less dependent on the parameters so that some conservatism is reduced. A numerical example is included to illustrate the effectiveness and improvement of the proposed methods.

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.

A fixed-time observer for discrete-time singular systems with unknown inputs

This paper is concerned with the fixed-time observer (FTO) design for discrete-time singular systems subject to unknown inputs. Firstly, the observer matching condition (OMC) and the strongly observability condition (SOC) are given. Then, a new FTO design framework with a reduced-order is given which is free from the influences of the unknown inputs. It is proven that the SOC together with the OMC guarantees the existence of a FTO which is able to provide an exact state estimation in an arbitrarily pre-defined time step. Another advantage of the FTO is that both the estimation accuracy and the convergence time step can be guaranteed regardless of whatever the initial values of the observer are. Finally, a numerical simulation validates the effectiveness of the proposed result.

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.

Robust Output Regulation in Discrete-Time Singular Systems With Actuator Saturation and Uncertainties

This brief studies robust output regulation in discrete-time singular systems with actuator saturation and matched uncertainties. The singularity of dynamical equations makes the design procedure more complex due to the appearance of both algebraic and difference equations in such systems. In addition to this, the actuator saturation constraint along with uncertain terms complicates the design problem. The idea for robust output regulation is to add an additional control term to the composite nonlinear feedback (CNF) controller which guarantees the output regulation of the uncertain closed-loop singular system with an ultimately bounded tracking error. In this regard, a theorem is given which guarantees the output regulation even if the saturation takes place. The accuracy of the theoretical results is tested through simulations for a practical example.

A State Space Decomposition Filtering Method for a Class of Discrete-Time Singular Systems

This paper develops a state space decomposition method for a class of discrete-time singular systems. The singular matrix in this system is allowed to vary not only in the value but also in the dimension. An orthogonal factorization is used to rewrite the original space as a direct sum of its two subspaces. Correspondingly, the singular system is replaced by two reduced-order systems, whose states are then calculated respectively. Based on this construction, the prediction of part of the state, optimal filtering result, an optimal one step smoothing result is proposed. Lastly, a numerical example is presented to demonstrate the effectiveness of the new method.

$\ell_\infty$ -Gain Analysis of Discrete-Time Positive Singular Systems With Time-Varying Delays

This paper is concerned with the l ∞ -gain analysis problem of discrete-time positive singular systems with time-varying delays. By introducing an auxiliary system, a necessary and sufficient positivity condition is proposed for systems. Then, by using of the positivity and linearity, we investigate the monotonic and asymptotic property of the system with time-varying delays. Then, by resorting to the comparing system with constant delays, an explicit expression of the l ∞ -gain of discrete-time singular systems with time-varying delays is given in terms of system matrices. The result shows that l ∞ -gain of discrete-time positive singular systems with bounded time-varying delays is insensitive to the magnitude of delays.

Event-based mixed ℋ ∞ and passive filtering for discrete singular stochastic systems

ABSTRACT This work devotes to the event-based mixed and passive filter analysis and design problems of discrete-time singular stochastic network systems with Markovian jump parameters. Using the event-triggered communication approach, we first derive a network-induced delay discrete error singular stochastic network model. With the help of event-triggered method and stochastic Lyapunov function scheme, sufficient conditions are then established such that the resulting error stochastic model is regular, causal, stochastically mean-square stable with a mixed and passivity performance index. By applying slack matrix variables and filter transfer function equivalent method, an event-triggered mixed and passive filter is designed such that the resulting error singular stochastic network system satisfies certain mixed and passive filter performance index, and the derived conditions could be solved in terms of linear matrix inequality method. Finally, the effectiveness of the proposed methods is illustrated by a numerical example.

Event‐triggered sliding mode control for discrete‐time singular system

This study focuses on the discrete-time sliding mode control (SMC) based on event-triggered approach for discrete-time singular system. The purpose is to design a SMC law under event-triggered to ensure that the discrete-time singular system is admissible. A novel event-triggered sliding mode surface is constructed, and a sufficient condition, in the form of linear matrix inequality (LMI), is investigated to guarantee the admissibility of the discrete-time singular system. One can verify this condition by utilising the toolbox YALMIP of MATLAB. In addition, the expected controller gain matrix can be represented by a feasible solution of LMI. Then, based on the above solved sliding mode controller gain, a new discrete-time event-triggered SMC law is constructed to ensure that the discrete-time singular system can be driven onto the sliding surface in finite time and then is maintained there for all subsequent time. Finally, three examples are presented to illustrate the effectiveness of the theoretical results.

Dynamic behaviors of a discrete-time predator\u2013prey bioeconomic system

Bifurcation and chaotic behavior of a discrete-time singular bioeconomic system are investigated. First, the traditional catch equation is modified after accounting for the handling time of the catch in a singular bioeconomic system. To discover the richer dynamics compared with the continuous form, the proposed system is considered difference scheme. Specially, the tangent space local parameterization condensed method for DAEs is generalized. The new local parameterization method is sufficiently general to be applicable to this type of discrete singular system. Also the dynamic behaviors of the system are investigated, by using normal form theory, center manifold theorem and bifurcation theory, it is shown that the system undergoes a Neimark–Sacker bifurcation and a flip bifurcation, on varying step-size in some range. In addition, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors.

Exponentially practical stability of discrete time singular system with delay and disturbance

In this paper, we consider exponentially practical stability of a discrete time singular system with disturbance. By using Lyapunov–Krasovskii stability theory, some criteria for exponentially practical stability of such a system are derived. Moreover, by using a Razumikhin-type technique, the criteria for exponentially practical stability of a discrete time singular system with delay and disturbance are also obtained. Some numerical examples are given to show the success of our theoretical results.