Linear Fractional-order Systems Research Articles

Observer-based model reference control for linear fractional-order systems

In the literature, the issue of designing model-reference controllers for integer-order systems has been extensively addressed. The extension of the existing works to the fractional-order calculus is still a fertile field of research. It is within this context that the present paper is established. Herein, a new observer-based model-reference control scheme is proposed for linear fractional-order systems. The developed methodology is based on the Caputo definition of fractional derivatives, the Lyapunov theory and the linear matrix inequality (LMI) technique. Finally, a simulation study is given in order to support the theoretical findings of the paper.

A Collocation Method for Solving Fractional Order Linear System

In this paper we propose the collocation method to achieve the numerical solution of fractional order linearsystems where a Fractional Derivative is defined in the Caputo sense.We use Taylor collocation method,which is based on collocation method for solving fractional differential equation. This method is based onfirst taking the truncated Taylor expansions of the vector function’s solution in the Fractional order linearsystem and then substituting their matrix forms into the system. Using collocation points, we have a systemof linear algebraic equation.The method has been tested by some numerical examples.DOI : http://dx.doi.org/10.22342/jims.23.1.257.

Global stability analysis of delayed complex-valued fractional-order coupled neural networks with nodes of different dimensions

In this paper, the delayed fractional-order complex-valued coupled neural networks (FCCNNs) with nodes of different dimensions are investigated. Firstly, stability theorems for linear fractional-order systems with multiple delays are presented. Secondly, by using the homeomorphism theory, the existence and uniqueness of the equilibrium point for delayed FCCNNs are proved. Then, the global stability criteria for delayed FCCNNs are derived by comparison theorem. Finally, numerical examples are given to illustrate the effectiveness of the presented results.

Design of fractional-order hyper-chaotic multi-scroll systems based on hysteresis series

This paper develops a novel fractional-order (FO) hyper-chaotic multi-scroll (FOHMS) system derived from a 4-dimensional FO linear system. Hysteresis function series are employed and added to the original system to extend the number of equilibrium points with index 2. The hyper-chaotic dynamics of the new FOHMS system is explored through phase portraits, largest Lyapunov exponents and Poincare maps. A circuit for the 0.96-order FOHMS is implemented and several FOHMS attractors are observed. Numerical and circuit-based simulations show the feasibility of the theoretical formulation.

Stability and Stabilization of Linear Fractional Order Systems with Input Delay Using Linear Matrix Inequalities

Stability and stabilization of delayed linear fractional order systems with input delay using linear matrix inequalities were considered in the present study. At first, the input delay fractional order system was changed into a free delay fractional order system using an alternate variable. Thereafter, using a state feedback, the systems were turned into a closed loop. Later on, the system stabilization was examined by applying linear matrix inequality theorems and, finally, an example was used to demonstrate the efficiency of the proposed method.

Optimal control problem for linear fractional-order systems, described by equations with Hadamard-type derivative

Two kinds of optimal control problem are investigated for linear time-invariant fractional-order systems with lumped parameters which dynamics described by equations with Hadamard-type derivative: the problem of control with minimal norm and the problem of control with minimal time at given restriction on control norm. The problem setting with nonlocal initial conditions studied. Admissible controls allowed to be the p-integrable functions (p > 1) at half-interval. The optimal control problem studied by moment method. The correctness and solvability conditions for the corresponding moment problem are derived. For several special cases the optimal control problems stated are solved analytically. Some analogies pointed for results obtained with the results which are known for integer-order systems and fractional-order systems describing by equations with Caputo- and Riemann-Liouville-type derivatives.

A novel chaotification scheme for fractional system and its application

Little seems to be known about the chaotification control of fractional order linear and nonlinear systems. This paper proposes a novel chaotification method for fractional order nonlinear systems based on the negative damping instability mechanism and fractional calculus technique. We then apply it to chaotify the fractional order Lorenz system with order lying in ( 1 , 2 ) , which is stable originally with specific parameters. Moreover, we introduce three critical effective orders to distinguish different four dynamics: singleton sets attractor, self-excited attractor, coexisting attractors, and blow up behavior. Many simulations are carried out to illustrate the effectiveness of the results. • A new method for chaotifying the fractional order nonlinear systems is proposed. • Fractional operator lying in (1, 2) has the negative damping instability effect. • Three critical orders are introduced to distinguish different complex dynamics.

New class of conformable derivatives and applications to differential impulsive systems

In this paper, first we introduce new definition of conformable fractional derivatives and study their algebraic properties. As application, we consider certain classes of conformable differential linear systems subject to impulsive effects and establish qualitative behavior of the nontrivial solutions such as stability, disconjugacy, nonexistence, upper bound estimation for maximum number of zeros of the nontrivial solutions and some oscillatory properties. The main tool of this establishment concerns with Lyapunov inequalities of under study fractional order linear systems.

Numerical solution of fractional variable order linear control system in state-space form

Abstract The aim of this paper is to introduce a matrix approach for approximate solving of non-commensurate fractional variable order linear control systems in state-space form. The approach is based on switching schemes that realize variable order derivatives. The obtained numerical solution is compared with simulation and analog model results.

Fractional-order Kalman filters for continuous-time linear and nonlinear fractional-order systems using Tustin generating function

ABSTRACTThis paper presents the fractional-order Kalman filters using Tustin generating function for linear and nonlinear fractional-order systems involving process noise and measurement noise. By using the Tustin generating function, the differential equation model is obtained by discretising the investigated continuous-time fractional-order system. The two kinds of fractional-order Kalman filters are given for the correlated and uncorrelated cases in terms of the process noise and measurement noise for linear fractional-order system, respectively. In addition, based on the first-order Taylor expansion formula, the extended fractional-order Kalman filter using Tustin generating function is proposed to improve the accuracy of state estimation. Finally, three examples are illustrated to verify the effectiveness of the Tustion fractional-order Kalman filters for linear and nonlinear fractional-order systems.

Complex‐domain stability criteria for fractional‐order linear dynamical systems

Complex‐domain stability criteria for fractional‐order linear dynamical systems

Optimal control problems for linear fractional-order systems defined by equations with Hadamard derivative

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.

Sufficient conditions for asymptotic stability and stabilization of autonomous fractional order systems

We discuss the asymptotic stability of autonomous linear and nonlinear fractional order systems where the state equations contain same or different fractional orders which lie between 0 and 2. First, we use the Laplace transform method to derive some sufficient conditions which ensure asymptotic stability of linear fractional order systems. Then by using the obtained results and linearization technique, a stability theorem is presented for autonomous nonlinear fractional order system. Finally, we design a control strategy for stabilization of autonomous nonlinear fractional order systems, and apply the results to the chaotic fractional order Lorenz system in order to verify its effectiveness.

Parameter estimation of the fractional‐order Hammerstein–Wiener model using simplified refined instrumental variable fractional‐order continuous time

This study proposes a direct parameter estimation approach from observed input–output data of a stochastic single-input–single-output fractional-order continuous-time Hammerstein–Wiener model by extending a well known iterative simplified refined instrumental variable method. The method is an extension of the simplified refined instrumental variable method developed for the linear fractional-order continuous-time system, denoted. The advantage of this novel extension, compared with published methods, is that the static output non-linearity of the Wiener model part does not need to be invertible. The input and output static non-linear functions are represented by a sum of the known basis functions. The proposed approach estimates the parameters of the linear fractional-order continuous-time subsystem and the input and output static non-linear functions from the sampled input–output data by considering the system to be a multi-input–single-output linear fractional-order continuous-time model. These extra inputs represent the basis functions of the static input and output non-linearity, where the output basis functions are simulated according to the previous estimates of the fractional-order linear subsystem and the static input non-linear function at every iteration. It is also possible to estimate the classical integer-order model counterparts as a special case. Subsequently, the proposed extension to the simplified refined instrumental variable method is considered in the classical integer-order continuous-time Hammerstein–Wiener case. In this paper, a Monte Carlo simulation analysis is applied for demonstrating the performance of the proposed approach to estimate the parameters of a fractional-order Hammerstein–Wiener output model.

Stability and synchronization of fractional-order memristive neural networks with multiple delays

The paper presents theoretical results on the global asymptotic stability and synchronization of a class of fractional-order memristor-based neural networks (FMNN) with multiple delays. First, the asymptotic stability of fractional-order (FO) linear systems with single or multiple delays is discussed. Delay-independent stability criteria for the two types of systems are established by using the maximum modulus principle and the spectral radii of matrices. Second, new testable algebraic criteria for ensuring the existence and global asymptotic stability of the system equilibrium point are obtained by employing the Kakutani’s fixed point theorem of set-valued maps, the comparison theorem, and the stability criterion for FO linear systems with multiple delays. Third, the synchronization criterion for FMNN is presented based on the linear error feedback control. Finally, numerical examples are given demonstrating the effectiveness of the proposed results.

An innovative parameter estimation for fractional order systems with impulse noise

This paper investigates the problem of parameter estimation for fractional order linear systems when the output signal is polluted by impulse noise. The conventional least square error objective function is replaced by a new approximate least absolute error (ALAE) function to restrain the influence of impulse noise. Then a novel parameter estimation approach is designed based on the stochastic gradient method, the fractional order parameter update law and the ALAE criterion, which improves estimation accuracy and enhances robustness at the same time. It is shown that the adoption of the fractional order parameter update law ensures larger selection range of update order and smoother convergence of the algorithm. The effectiveness and superiority of the proposed method are verified by strict mathematical analysis and detailed numerical examples.

Sufficient conditions for domain stabilisability of uncertain fractional‐order systems under static‐output feedbacks

Extended linear matrix inequality (LMI) conditions, ensuring the stability of commensurate fractional-order linear systems by static-output feedbacks, are given. It is assumed that the system uncertainties are constant and possibly present in all the system matrices. The stabilising static-output feedback is conceived to overcome the system uncertainty and place the poles of the closed-loop system in a well-defined domain that is formed by the intersection of many regions in the complex plane. The control design is formulated as the solution of a set of linear matrix inequality conditions. The validity of the obtained results is testified through an example of a fractional-order system with polytopic uncertainties.

Modulating functions-based fractional order differentiator for fractional order linear systems with a biased output

This paper aims at designing a non-asymptotic fractional order differentiator for a class of fractional order linear systems with zero initial conditions, where the fractional orders can be commensurate or non-commensurate, and the output is corrupted by a non zero-mean noise. Firstly, a set of fractional differential equations are constructed, based on which the modulating functions method is applied, such that the fractional derivative with an arbitrary order of the output can be exactly given by an algebraic formula using a recursive way. In particular, this arbitrary order can be different to the fractional orders defining the considered system. Unlike the improper integral in the definition of the fractional derivatives, the obtained formula can be given by proper integrals by choosing appropriate modulating functions. Moreover, by taking an additional condition on the modulating functions, the term biasing the output can be eliminated in the obtained formula. After constructing the needed modulating functions, a digital fractional order differentiator is proposed in discrete noisy case with some error analysis. Finally, the efficiency and the robustness of the proposed fractional order differentiator is shown in numerical results.

Stability and Stabilization Analysis of Fractional-Order Linear Systems Subject to Actuator Saturation and Disturbance

This paper focuses on the fractional order linear systems subject to actuator saturation and external disturbance, and proposes method to estimate the domain of attraction. The quadratic Lyapunov functions are adopted to obtain the invariance of considered systems. The convex combination form of quadratic Lyapunov function is utilized to obtain a large domain of attraction. A continuous feedback law is derived to achieve disturbance rejection for closed-loop systems. Numerical example is provided to illustrate the effectiveness of the developed method.

Lebesgue‐p NORM Convergence OF Fractional‐Order PID‐Type Iterative Learning Control for Linear Systems

AbstractThis paper discusses first‐ and second‐order fractional‐order PID‐type iterative learning control strategies for a class of Caputo‐type fractional‐order linear time‐invariant system. First, the additivity of the fractional‐order derivative operators is exploited by the property of Laplace transform of the convolution integral, whilst the absolute convergence of the Mittag‐Leffler function on the infinite time interval is induced and some properties of the state transmit function of the fractional‐order system are achieved via the Gamma and Bata function characteristics. Second, by using the above properties and the generalized Young inequality of the convolution integral, the monotone convergence of the developed first‐order learning strategy is analyzed and the monotone convergence of the second‐order learning scheme is derived after finite iterations, when the tracking errors are assessed in the form of the Lebesgue‐p norm. The resultant convergences exhibit that not only the fractional‐order system input and output matrices and the fractional‐order derivative learning gain, but also the system state matrix and the proportional learning gain, and fractional‐order integral learning gain dominate the convergence. Numerical simulations illustrate the validity and the effectiveness of the results.