Fractional Linear Systems Research Articles

Controllability and Observability Criteria for Systems Described by Fractional Differential Equations

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay. By using new conformable delayed matrix functions and the method of variation, we obtain a representation of their solutions. As an application, we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayed matrix functions. The obtained results are new, and they extend and improve some existing ones. Finally, an example is presented to illustrate the validity of our theoretical results.

Условие оптимальности типа принципа максимума Понтрягина в задаче управления линейными разностными уравнениями дробного порядка

The optimal control problem of fractional order linear two-dimensional difference equations sys-tems is considered. It is assumed that the control function is included in the boundary condition, and the functional is linear. A necessary and sufficient optimality condition is proved in the discrete maximum principal form. A sufficient optimality condition is proved in the nonlinear but convex cost functional case.

Fractional time-invariant compartmental linear systems

Fractional time-invariant compartmental linear systems

A New Method for Controlling Fractional Linear Systems

The study of fractional linear time-invariant (LTI) systems has been an area of active research over the past thirty years. Results indicate that such systems are becoming important in the representation of certain types of dynamical behavior in biology and engineering. Several methods have been developed for the representation, identification, and control of fractional LTI systems. The primary purpose of this paper is to introduce a new approach to their control. Its principal contribution is the transformation of a fractional LTI system into one without fractional terms (standard LTI system) using feedback and feedforward compensation. While the former may be quite difficult to control, tools widely available in linear systems theory may be used to control the latter. Transformation of a fractional system into a standard one requires compensators with fractional transfer functions. It is shown, using research results from the past two decades, that such functions can be realized using hardware and software subsystems. The authors demonstrate, using an example, that better control performance is possible with the new method when compared to current ones that are most common in the research literature. Robustness in the presence of noise and model uncertainties is also studied for the new method and currently available ones. Simulation results presented illustrate improvements achievable in control robustness with the new method.

Non-asymptotic and robust estimation for fractional order pseudo-state space model using an algebraic parametric method

This paper mainly investigates the derivative estimation problem for a class of fractional order linear systems. After applying the Laplace transform to the considered pseudo-state space representation model, a series of multiplications and derivations are applied to eliminate unknown initial conditions, which define an annihilator. Then, through strict mathematical derivation, an input-output integral equation is deduced when returning into the time domain. Based on the obtained equation and thanks to the fractional Leibniz formulas, algebraic integral formulas are provided, which can non-asymptotically and robustly estimate the fractional derivatives of the system output. Moreover, error analysis in discrete noisy cases is given to study the performance of the proposed differentiator. Finally, numerical simulations are provided to show the effectiveness of the proposed method.

About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays

The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under some natural assumptions of boundedness type. In the case when, in the system, the term which describes the outer forces is a locally Lebesgue integrable function and the initial function is continuous, it is proved that the studied IP has a unique solution, which has an integral representation via the corresponding resolvent kernel. Applying the obtained results, we establish that, from the existence and uniqueness of a resolvent kernel, the existence and uniqueness of a fundamental matrix of the homogeneous system and vice versa follows. An explicit formula describing the relationship between the resolvent kernel and the fundamental matrix is proved as well.

The Exact Solutions of Fractional Differential Systems with n Sinusoidal Terms under Physical Conditions

This paper considers the classes of the first-order fractional differential systems containing a finite number n of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool to solve linear fractional differential equations. However, this method requires to express the initial conditions in certain fractional forms which have no physical meaning currently. This issue formulated a challenge to solve fractional systems under real/physical conditions when applying the Riemann–Liouville fractional definition. The principal incentive of this work is to overcome such difficulties via presenting a simple but effective approach. The proposed approach is successfully applied in this paper to solve linear fractional systems of an oscillatory nature. The exact solutions of the present fractional systems under physical initial conditions are derived in a straightforward manner. In addition, the obtained solutions are given in terms of the entire exponential and periodic functions with arguments of a fractional order. The symmetric/asymmetric behaviors/properties of the obtained solutions are illustrated. Moreover, the exact solutions of the classical/ordinary versions of the undertaken fractional systems are determined smoothly. In addition, the properties and the behaviors of the present solutions are discussed and interpreted.

An effective adaptive algorithm for linear fractional dynamical systems

This study proposes a time-stepping [Formula: see text] scheme to approximate the linear fractional dynamical systems based on nonuniform mesh. The developed numerical scheme is unconditionally stable, and exhibits second-order accuracy when the suitable graded mesh is used. A posteriori error estimation is derived for the obtained numerical scheme and the corresponding adaptive algorithm is devised. Finally, two numerical examples are provided to demonstrate the effectiveness of our approach and verify the theoretical results.

Uniform asymptotic stability of q -deformed conformable fractional systems with delay and application

In this article, we initiate the study of new concepts of conformable \(q\)-fractional calculus. The conformable fractional \(q\)-derivative and \(q\)-integral are defined and their fundamental theorems are also proved. The uniform asymptotic stability of the \(q\)-deformed conformable fractional system with constant delay is investigated by using the Lyapunov-Razumikhin method. For application, a new asymptotic stability necessary condition for the conformable \(q\)-fractional linear system with constant delay is obtained in term of linear matrix inequality (LMI). A numerical example is demonstrated for the results given to illustrate the effectiveness.

Controllability of Fractional Stochastic Delay Systems Driven by the Rosenblatt Process

In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By introducing a fractional delayed Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear fractional stochastic delay systems. Furthermore, by employing Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional stochastic delay systems. Finally, an example is given to illustrate the main results.

Stabilization for a General Class of Fractional-Order Systems: A Sampled-Data Control Method

In this paper, based on sampled-data control method, we address the stabilization problem for a general class of linear continuous-time fractional systems whose solution contains the Mittag-Leffler function that does not obey the basic exponentiation identity. By considering the infinite memory and hereditary characteristics of fractional-order calculus, we propose a sampled-data controller that guarantees the resulting closed-loop system to be asymptotically stable. Simulation examples are presented to demonstrate the effectiveness of the proposed controller and verify the established results.

Reachability of fractional dynamical systems with multiple delays in control using ψ-Hilfer pseudo-fractional derivative

In this article, we study the reachability of linear and non-linear fractional dynamical systems with multiple delays in control in the sense of the ψ-Hilfer pseudo-fractional derivative. The necessary and sufficient conditions for the reachability of linear fractional dynamical systems are obtained using the Gramian matrix, which is expressed by Mittag–Leffler functions. Sufficient conditions for the reachability of nonlinear fractional dynamical systems are obtained by using Schauder’s fixed point theorem. Two numerical examples are offered to help better understand the theoretical results.

Analytical Solution of the Fractional Linear Time-Delay Systems and their Ulam-Hyers Stability

We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, and delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay systemD0,tμ,νzt+Azt+Ωzt−h=ftof order1<μ<2and type0≤ν≤1, with nonpermutable matricesAandΩ. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear time-delay systems with permutable matrices and new even for these fractional delay systems.

Controllability of linear convex combination of linear discrete-time fractional systems

Controllability of linear convex combination of linear discrete-time fractional systems

Distinguishability criteria of conformable hybrid linear systems

AbstractWe relate this article to the emerging idea of distinguishability of conformable linear hybrid time-invariant control systems. To obtain the necessary and sufficient conditions ofα\alpha-distinguishability for fractional cases, we develop the Leibnitz rule for conformable derivatives. Furthermore, with the help of a study of Laplace techniques, a more simple criterion ofα\alpha-distinguishability for the fractional linear system is developed.

Double-Preconditioning Techniques for Fractional Partial Differential Equation Solvers

This paper is devoted to the numerical computation of the solution to stationary space fractional Partial Differential Equations (PDEs). To this end, we derive some algorithms for solving fractional algebraic linear systems resulting from the discretization of these space fractional PDEs. The proposed approach is based on an efficient computation of Cauchy’s integrals allowing to estimate positive real powers of a (sparse) matrix A. A first preconditioner M is used to reduce the length of the Cauchy integral contour enclosing the spectrum of MA, hence allowing in some cases for a large reduction of the number of quadrature nodes along the integral contour. Next, ILU-factorizations are used to efficiently solve the linear systems involved in the computation of approximate Cauchy’s integrals. Therefore, the strategy is independent of the way the Cauchy integrals are computed, although in this paper, for the sake of simplicity, we use standard quadrature rules. Based on this approach, we derive some solvers for stationary deterministic or stochastic space fractional Poisson equations which are tested in a series of numerical experiments.

Trajectory controllability of nonlinear fractional Langevin systems

Abstract In this paper, we discuss the trajectory controllability of linear and nonlinear fractional Langevin dynamical systems represented by the Caputo fractional derivative by using the Mittag–Leffler function and Gronwall–Bellman inequality. For the nonlinear system, we assume Lipschitz-type conditions on the nonlinearity. Examples are given to illustrate the theoretical results.

Sampled-Data Stabilization of Fractional Linear System under Arbitrary Sampling Periods

The sampled-data stabilization of a fractional continuous linear system under arbitrary sampling periods was first investigated in this paper wherein novel co-designed sampled-data controllers were constructed based on the compensation of scaling gains. With the help of fractional difference approximation, sufficient and necessary conditions for global asymptotic stability were first presented in the discrete-time domain, and then co-designed sampled-data controllers were constructed with only the “newest” or “oldest” state information available for controller design. Due to the compensation scheme between scaling gains and sampling periods, much more flexibility on selecting different sampling periods was provided in the sampled-data stabilization of the fractional continuous linear system which is significantly preferred for digital implementation. Numerical studies are also presented to illustrate the effectiveness of our co-designed sampled-data controllers under different sampling periods.

Controllability of Linear Fractional Systems with Delay in Control

This paper discusses the controllability of continuous-time linear fractional systems with control delay. The Atangana-Baleanu fractional derivative with the Caputo approach is used. First, the solution expression for a linear fractional system is obtained. Then, the corresponding fractional delay controllability Gramian matrix is defined, and its non-singularity as necessary and sufficient conditions for the controllability is proved. Finally, another equivalent condition based on the matrix rank formed by the coefficients matrices of the original system is provided that is much easier to check.