Fractional-order Time-delay Systems Research Articles

An analytical criterion for stability testing of delayed Bode’s transfer function with unity feedback

The unity feedback where delayed Bode’s transfer function is placed in forward path is used as reference transfer function in controller design for fractional time delay systems. In this study, bounded input-bounded output stability of this reference transfer function is investigated. This investigation is based on a theorem for fractional order time delay systems using Hassard’s techniques. Sufficient and necessary conditions are obtained for testing the stability of this reference transfer function. Moreover, an explicit formula is given for calculating the number of unstable roots in case of instability. The illustrative examples show that this stability criterion works efficiently.

Wirtinger-based fractional summation inequality for stability analysis of nabla discrete fractional-order time-delay systems

Wirtinger-based fractional summation inequality for stability analysis of nabla discrete fractional-order time-delay systems

Event-Triggered State Observer Design for a Class of Nonlinear Time-Delay Fractional-Order Systems

Event-Triggered State Observer Design for a Class of Nonlinear Time-Delay Fractional-Order Systems

NOVEL UNIFIED STABILITY CRITERION FOR FRACTIONAL-ORDER TIME DELAY SYSTEMS WITH STRONG RESISTANCE TO FRACTIONAL ORDERS

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic stability, is only slightly affected by the change of the fractional order. In addition, the unified stability criterion is applied to general multi-dimensional nonlinear fractional-order systems with time delays, the corresponding asymptotic stability criterion is applied by combining the vector Lyapunov function with the M-matrix method. Compared with the traditional stability criterion, the unified stability criterion is slightly influenced by the changing fractional order and large time delays. The reliability and effectiveness of the novel uniform stability criterion were verified through three representative examples.

Stability Analysis of Finite Time for a Class of Nonlinear Time-Delay Fractional-Order Systems

In this study, we delve into the examination of Finite Time Stability (FTS) within a specific class of Fractional-Order Systems (FOS) with time delays. By applying a fixed-point theorem, we establish novel sufficient conditions to ensure FTS for time-delayed FOS within 1<σ<2. Moreover, we investigate the existence and uniqueness of global solutions for this particular system. To demonstrate the credibility of our results, we substantiate our findings through the presentation of two illustrative examples.

Anti-disturbance observer-based proportional-retarded control design for polytopic uncertain fractional-order systems

This study deals with the stabilisation problem of fractional-order time-delay systems with polytopic uncertainties and multiple disturbances with the use of Lyapunov stability theory. Notably, the multiple disturbances comprising both the known and unknown signals, are respectively characterised by the norm-bounded and exogenous system which signifies the harmonic signals with modelling perturbations. The control protocol is configured by the integration of proportional-retarded controller with gain perturbations and the output of an anti-disturbance observer. Precisely, the disturbance observer is taken into account for attenuating the influence of disturbance signals. Moreover, the implementation of a time-delay term in the feedback loop replicates the dynamic features of a derivative action that relies only on position measurements and its time differencing capabilities make it to be minimally sensitive against noise and provide a smoothing effect. By virtue of these circumstances, the asymptotic stability of the considered system is assured. With the aid of a suitable Lyapunov function, a set of robust order-dependent sufficient conditions is derived and then solved through the MATLAB LMI toolbox. As a result, the controller gain matrices can be computed and two numerical examples, including an electrical circuit model with simulation results are presented to certify the inherent potential of the theoretical outcomes.

Novel Stability and Stabilization Conditions of Linear Fractional-Order Time-Delay Systems Using Free Matrix Approach

The stability and stabilization problem of linear fractional-order time-delay systems is investigated. By introducing a free matrix, the linear fractional-order time-delay systems is transformed to a delay-free system interconnected with uncertainties. New stability and stabilization criteria are derived based on the small gain theorem. The criteria are formulated as linear matrix inequalities and are computationally efficient. Since the free matrix is utilized, the new conditions are less conservative than the existing ones. Two numerical examples indicate that the proposed criteria are less conservative than existing ones.

Quantized iterative learning control for singular nonlinear fractional-order time-delay multi-agent systems with iteration-varying reference trajectories and switching topologies

This paper addresses the consensus tracking of the leader-following singular nonlinear fractional-order system with state time-delay in face of iteration-varying communication topologies and reference trajectories via an iterative learning control scheme. Firstly, by means of the neighbor information, the closed-loop Dα-type quantized iterative learning protocol is constructed based on output tracking errors. Secondly, sufficient conditions of consensus errors between each follower agent and leader agent are given in a fixed time interval and convergence analysis is also presented under the fixed topology and reference trajectory. Thirdly, the extension to iteration-varying reference trajectory tracking and switching topologies cases are investigated. It is shown that the developed protocol can also effectively work along the iteration axis on a finite time interval, although the communication topologies and reference trajectories vary dynamically with respect to iteration. Finally, numerical examples are shown to verify the effectiveness of the obtained results.

Adaptive Neural Network Synchronization Control for Uncertain Fractional-Order Time-Delay Chaotic Systems

We propose an adaptive radial basis (RBF) neural network controller based on Lyapunov stability theory for uncertain fractional-order time-delay chaotic systems (FOTDCSs) with different time delays. The controller does not depend on the system model and can achieve synchronous control under the condition that nonlinear uncertainties and external disturbances are completely unknown. Stability analysis showed that the error system asymptotically tended to zero in combination with the relevant lemma. Numerical simulation results show the effectiveness of the controller.

Inite-Time Stability of Impulsive Fractional-Order Time Delay Systems with Damping Behavior

inite-Time Stability of Impulsive Fractional-Order Time Delay Systems with Damping Behavior

Stabilization of nonlinear fractional order faulty systems with time-varying delay

This paper focuses on the issue of stabilization of nonlinear fractional order time-delay systems in the presence of fault. A general class of nonlinear fractional order systems is considered in which fault causes variations in the system dynamic and actuators. Besides, time-varying delay is noticed in the system equations that is of high importance in control of real-life systems. In order to facilitate the problem of controller design for such systems, a precise method for modeling complex nonlinear systems, namely, Takagi–Sugeno fuzzy model, is adopted. Regarding a Caputo derivative–based Lyapunov–Krasovskii functional, new delay-dependent stabilization conditions are established in the form of linear matrix inequalities. Eventually, two examples including a truck–trailer and Lorenz system are simulated in order to evaluate the results of this research work.

Finite-time stability of nonlinear stochastic $ \psi $-Hilfer fractional systems with time delay

<abstract><p>In this paper, we study the finite time stability of stochastic $ \psi $-Hilfer fractional-order time-delay systems. Under the stochastic analysis techniques and the generalized Gronwall's inequality for $ \psi $-fractional derivative, the criterion of finite time stability of the solution for nonlinear stochastic $ \psi $-Hilfer fractional systems with time delay is obtained. An example is provided to illustrate the effectiveness of the proposed methods. Some known results in the literature are extended.</p></abstract>

Global output feedback stabilization for nonlinear fractional order time delay systems

Global output feedback stabilization for nonlinear fractional order time delay systems

State and disturbance simultaneous estimation for a class of nonlinear time-delay fractional-order systems

This article addresses the problem of estimating simultaneously the state and unknown disturbance of one-sided Lipschitz fractional-order systems with time-delay. The nominal models of nonlinearities are assumed to satisfy both the one-sided Lipschitz condition and the quadratically inner-bounded condition. Different from the state observer reported in the literature, which only dealt with one-sided Lipschitz integer-order time-delay systems or nonlinear fractional-order time-delay systems where the nonlinear function satisfying Lipschitz condition, the state observers in this article can be applied to a wide class of nonlinear time-delay systems (one-sided Lipschitz fractional-order time-delay systems and one-sided Lipschitz integer-order time-delay systems). We employ the Razumikhin stability theorem and a recent result on the Caputo fractional derivative of a quadratic function to derive a sufficient condition for the asymptotic stability of the observer error dynamic system. The stability condition is obtained in terms of linear matrix inequalities, which can be effectively solved using the MATLAB LMI Control Toolbox. Two examples are provided to show the effectiveness of the proposed design approach.

Parameter identification of fractional-order time delay system based on Legendre wavelet

This paper proposes a parameter identification method of fractional-order time delay system based on Legendre wavelet. An integration operational matrix and delay operational matrix of a Legendre wavelet are constructed. The process is derived to convert a fractional-order time delay system based on a Legendre wavelet operational matrix to algebraic equation. The least squares method is used to simultaneously estimate the order, model parameters and time delay coefficient of a system. In addition, wavelet decomposition and reconstruction algorithm are used to pre-process the output signal, which separate the noise from signal. The proposed method overcomes the initial-state dependence and randomness of the identification result in the algebraic solution of the traditional operational matrix, and reduces the effect of noise on the accuracy of parameter identification. It is especially suitable for engineering application. The effectiveness is verified by numerical simulation and a temperature process control system.

Stability Regions of Fractional First Order Controllers Applied to Fractional Order Delay Systems

This paper focuses on the problem of stabilizing fractional order time delay systems by fractional first order controllers. A solution is proposed to find the set of all stability regions in the controller’s parameter space. The D-decomposition method is employed to find the real root boundary and complex root boundaries which are used to identify the stability regions. Illustrative examples are given to show the effectiveness of the proposed approach, and it is remarked that the stability region obtained for the fractional order controller is larger than the non-fractional controller.

On calculation of ISE performance indices for fractional-order time-delay systems

In this paper, we apply the product-to-sum decomposition (PSD) E(s)E( − s) = Z(s) + Z( − s) to investigate the integral-square-error (ISE) computation for fractional-order time-delay (FOTD) feedback control systems. First, we show that the PSD approach to ISE evaluation is inapplicable to delay-free fractional systems. Then we show that, based on performing operations on polynomials in the delay operator z = e−τs, the PSD approach can be extended to FO systems with commensurate orders and delays. A key advantage of the PSD approach is that the Ip part of ISE contributed by the dynamics of infinite number of poles of E(s) can be computed from the finite number of poles of the decomposed function Z(s). Also, we present PSD-based formulae for ISE calculation. To demonstrate the PSD approach to ISE calculation for FOTD systems, illustrative examples are provided.

T-S Fuzzy Control of Uncertain Fractional-Order Systems with Time Delay

In this article, the adaptive control of uncertain fractional-order time-delay systems (FOTDSs) with external disturbances is discussed. A Takagi-Sugenu (T-S) fuzzy model with if-then rules is adopted to characterize the dynamic equation of the FOTDS. Besides, a fuzzy adaptive method is proposed to stabilize the model. By utilizing the Lyapunov functions, a robust controller is constructed to stabilize the FOTDS. Due to the uncertainty of system parameters, some fractional-order adaptation laws are designed to update these parameters. At the same time, some if-then rules with linear structure based on the fuzzy T-S adaption concept are established. The designed method not only guarantees that the state of closed-loop system asymptotically converges to origin but also keeps the signal in the FOTDS bounded. Finally, the applicability of the control method is proved by simulation examples.

Synthesised fractional‐order PD controller design for fractional‐order time‐delay systems based on improved robust stability surface analysis

An improved robust stability surface analysis method is proposed in this study for fractional-order time-delay systems. Through the stabilisation process, a synthesised fractional-order PD controller can be designed with guaranteed robustness specifications. Firstly, the specification for improved robustness requirements is proposed. In order to find selectable robust controller design parameter combinations for the controlled system, stability region is discussed based on the stability boundary locus, and the robust stability surface is derived straight after. Secondly, the selectable parameter combinations are checked to find the one that best fulfils all the proposed robustness specifications. Finally, numerical simulations are given to demonstrate the effectiveness and flexibility of the presented control algorithm.

Functional interval observers for nonlinear fractional-order systems with time-varying delays and disturbances

This article considers the estimation problem for a class of uncertain fractional-order systems with time-varying delays and disturbances, in which the time-varying delays are present in the state, while the disturbances are present in the output. A new design method of functional interval observers is proposed. In contrast to interval observers currently available in the literature, which dealt with integer-order systems or fractional-order systems without time delays, the functional interval observers with fractional-order [Formula: see text] in this article can deal with fractional-order time-delay systems and integer-order time-delay systems (when [Formula: see text]). Existence conditions of functional interval observers for non-linear fractional-order systems with time-varying delays are established and they are formulated in terms of linear programmings. Two algorithms are provided for determining the functional interval observers. Illustrative examples of the functional interval observer application are presented.