Class Of Fractional-order Systems Research Articles

Fractional order PI observer for a class of fractional order systems

Currently the study of fractional order systems has become of great research interest, in particular the state estimation stands out within the lines of studies for this type of systems. Different methodologies have been proposed in order to solve this problem, however most of the techniques involve complete information of the system. Thus, this work presents a methodology for state estimation in a class of fractional order systems based on a fractional order observer which is constructed through an algebraic technique. This observer presents some significant properties, for instance, only variables of interest are estimated, i.e., it is a non redundant observer, it does not need complete information of the system and the initial conditions for the observer can be freely assigned. The stability of this observer is analyzed with the global Mittag-Leffler boundedness approach. Finally, the effectiveness and accuracy of the proposed methodology is verified through state estimation of a fractional-order Duffing chaotic system.

Fractional order output-feedback tube-MRAC design for a class of fractional order transfer functions with unknown parameters

In this paper, we address a class of fractional order systems defined by transfer function with unknown parameters and present a novel fractional order tube model reference adaptive control (FOTMRAC) design for their control despite the fact that the only available signals are the input and output of the system. This is the main difference with preceding authors' recent study, where the states of the system where available. Based on the output feedback, this strategy is a generalisation of the tube model reference adaptive control (TMRAC) to fractional order systems. The basic idea is to replace the single reference trajectory well known in reference model control theory by a set of admissible trajectories called tube reference. Also, an optimisation problem is resolved online in order to update the correction control signal minimising a control cost criterion. The asymptotic stability is proved using the Lyapunov extension theorem. Simulation results illustrate and confirm the effectiveness of the proposed control design.

Global Practical Conformable Stabilization by Output Feedback for a Class of Nonlinear Fractional‐Order Systems

Researchers have explored the concept of “practical stability” in the literature, pointing out that stability investigations always guarantee “practical stability” and the inverse is not true. The concept “practical stability” means that the origin is not an equilibrium point and the convergence of the system state is towards a ball centered at the origin. The primary purpose of this work is to investigate the notation of practical stability for a new class of fractional-order systems using the general conformable derivative. As a second objective, the nonlinear condition chosen is novel in that it is not Lipschitz as is customary, which is original in and of itself. In addition, some new analysis related to the LMI techniques was used to prove the main results. To begin, a method of stabilization is provided. Following that, the proposed system’s observer design is presented. Also, the principle of separation is described. Finally, a numerical example is offered to demonstrate the proposed methodology’s validity.

Closed-Form Oscillatory Condition in Electrical Circuits Containing Two Fractional Order Elements

Oscillatory condition in LTI dynamic systems is generally expressed as possessing purely imaginary solutions by their characteristic equations. Dealing with the class of fractional order systems, such a condition is equivalently restated as owing complex roots with specific arguments by a polynomial defined based on the system characteristic equation. The degree of this polynomial can be unboundedly high. Consequently, such statement for the oscillatory condition in fractional order systems, which is based on arguments of the roots of a polynomial with an unbounded degree, cannot be viewed as a closed-form expression. To tackle this challenge, this brief introduces an approach to obtain a closed-form oscillatory condition for electrical circuits containing two fractional order elements. It is shown that the obtained results are helpful in analytically finding the oscillatory curves in parametric regions. Moreover, it is discussed whether the existence of more than one frequency in the frequency content of steady-state oscillations produced by the considered class of fractional order circuits is possible.

Comments on ‘Less conservative conditions for robust LQR-state-derivative controller design: an LMI approach’ and new sufficient LMI conditions for invertibility of a convex combination of matrices

This note is concerned with conditions on a set of non-singular matrices , , so that any convex combination of these matrices is also non-singular. The first part of the note points out that Theorem 2.3 in a previous paper [Beteto et al. (2021). Less conservative conditions for robust LQR-state derivative controller design: An LMI approach. International Journal of Systems Science] provides only necessary conditions, which are not sufficient in the general case. In the second part, some stability results based on Linear Matrix Inequalities (LMIs) for a class of fractional order systems are used to establish new sufficient conditions. Numerical examples are presented for illustration. The results suggest that the new LMI conditions may be less conservative compared to a test proposed in the literature on P-matrices, and also to a positive-definiteness test based on matrix cross-products.

Formula omitted] adaptive controller design for a class of fractional order uncertain systems

ℒ1 adaptive control method is a new and effective technique to perform a robust tracking response with fast adaptation, its applications were generalized for both integer order affine and non-affine nonlinear systems. However, ℒ1 adaptive control scheme has not been yet investigated to control fractional order systems. In this work, an ℒ1 adaptive controller is designed based on a fractional order sliding surface to control a class of fractional order systems with matched uncertainties and external disturbances. The developed controller is able to achieve closed-loop stability with uniform performance bounds for system signals. Such that, increasing the adaptation gain can dramatically enhance the performances for both transient and steady state responses. Finally, the validity and the effectiveness of the controller is tested using a numerical simulation.

Control Techniques for a Class of Fractional Order Systems

The paper discusses several control techniques for a class of systems described by fractional order equations. The paper presents the unit frequency criteria that ensure the closed loop control for: Fractional Order Linear Systems, Fractional Order Linear Systems with nonlinear components, Time Delay Fractional Order Linear Systems, Time Delay Fractional Order Linear Systems with nonlinear components. The stability criterion is proposed for the systems composed of fractional order subsystems. These techniques are used in two applications: Soft Exoskeleton Glove Control, studied as a nonlinear model with time delay and Disabled Man-Wheelchair model, analysed as a fractional-order multi-system.

Stabilization of a Class of Fractional Order Systems With Both Uncertainty and Disturbance

This paper investigates the stabilization problem of fractional order systems with both model uncertainty and external disturbance. By combining the linear feedback control method, the dynamic feedback control method, and the uncertainty and disturbance estimator (UDE)-based control method, respectively, two new UDE-based control methods are developed. Using these methods, the fractional order systems can be stabilized by three steps. In the first step, the linear feedback and dynamic feedback controllers are designed to stabilize the nominal fractional order systems. The second step is to design a UDE-based fractional order controller to estimate the model uncertainty and external disturbance. In the third step, the two controllers are combined into a new controller to realize the stabilization of those fractional order systems. Finally, a numerical example is given to verify the correctness and validity of the proposed methods.

On Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative

In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.

Disturbance observer-based fractional-order nonlinear sliding mode control for a class of fractional-order systems with matched and mismatched disturbances

This study presents a novel fractional-order nonlinear sliding mode controller (FONSMC) based on an extended nonlinear disturbance observer (ENDOB) for a class of fractional order systems with matched and mismatched disturbances. Firstly, an ENDOB is introduced to estimate both the matched and mismatched disturbances. Then, the fractional-order nonlinear sliding surface is designed to satisfy the sliding condition in finite time. Accordingly, the corresponding FONSMC is proposed using the Lyapunov stability theorem. The proposed method shows an impressive disturbances rejection and also guarantees finite-time stability of closed-loop systems. Finally, the effectiveness of the proposed FONSMC-ENDOB structure is illustrated via numerical simulation. The simulation results exhibit the superiority of the proposed controlling method.

An adaptive fault-tolerant control scheme for a class of fractional-order systems with unknown input dead-zones

An adaptive backstepping fault-tolerant control scheme is presented for a class of fractional-order systems in the presence of unknown input dead-zones. The proposed fault-tolerant control scheme ensures all the closed-loop signals are bounded ultimately. Especially, the tracking error can be made as small as possible by choosing appropriate design parameters. Furthermore, this scheme not only works for fractional-order systems with unknown linear terms but also works for these with unknown nonlinear terms. Finally, several practical examples are simulated to verify the effectiveness of the proposed fault-tolerant control scheme.

Physical Significance Variable Control for a Class of Fractional-Order Systems

This paper studies a class of fractional-order systems (FOSs) and proposes control laws based on physical significance variables. Lyapunov techniques and the methods that derive from Yakubovici–Kalman–Popov Lemma are used, and the frequency criterions that ensure asymptotic stability of the physical significance variable closed-loop system are inferred. The asymptotic stability of the observer system is studied for a sector control law where the output is defined by the physical significance variables. Frequency criterions and conditions for asymptotic stability are determined. The control techniques are extended to a class of linear delay fractional-order systems and nonlinear FOS. Numerical simulations of a class of systems described by fractional-order models show the method efficiency.

A Unified Framework of Stability Theorems for LTI Fractional Order Systems With 0 < α < 2

This brief aims to provide a set of criteria to ensure the stability or stabilization of a class of fractional order systems (FOS) for a given order $\alpha $ that belongs to the interval (0, 2). These criteria are based on a unified structure of linear matrix inequalities (LMIs). Their add-value manifests in involving the least real decision variables of LMIs and presenting a unified LMI formulation with order between zero and two. Those criteria are necessary and sufficient conditions that can be straightforward used to solve the feasible solutions with LMI toolbox. Finally, it yields numerical examples to highlight the efficiency of the proposed pseudo-state state feedback controller, where some comparisons with the previous stabilization criteria for FOS are given to show its less conservatism.

Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation

The robust stability and stabilization of a class of fractional order systems under input saturation is studied using the Lyapunov method. In the problem formulation the Lipchitz and bounded conditions are adopted. Moreover, sufficient conditions, in the form of a Linear Matrix Inequality, for stabilizing the system via a state feedback control, are derived. Two numerical examples show the effectiveness of the proposed technique.

Fractional-disturbance-observer-based Sliding Mode Control for Fractional Order System with Matched and Mismatched Disturbances

This paper addresses the sliding mode control for a class of fractional order systems with matched and mismatched disturbances. Firstly, fractional disturbance observer is presented to estimate both the matched and mismatched disturbances, and the boundedness of the estimation error can be guaranteed. Secondly, sliding mode surface is constructed based on the output of the observer. The bounded stability of the closed-loop system under the designed controller is revealed by theoretical analysis. Finally, simulation results show that the proposed control strategy can effectively suppress the effect of the matched and mismatched disturbances on the system.

Direct Discretization Method for Realizing a Class of Fractional Order System in Delta Domain – a Unified Approach

Discretization is fundamental for digital realization of the fractional order system. The fractional order systems are digitally implemented by using either indirect or direct discretization methods. All these methods are typically developed in the conventional z domain in the form of s to z transforms. In this paper, a new direct discretization method is presented as a substitute of conventional z domain direct discretization method. The proposed approach capitalizes the feature of delta operator to unify both the discretized system and its continuous-time counterpart at the lower sampling time limit. The discretization is accomplished in two stages. At first, the delta operator is reframed by using the trapezoidal quadrature rule to obtain the suitable generating function. Then, the generating function is further expanded using the continued fraction expansion method to obtain the discrete-time approximation of the fractional order differentiator. Illustrative examples are presented to exhibit the efficacy of the proposed method with respect to two prevalent direct discretization methods of z domain on the basis of simulation results obtained using MATLAB.

Design of an integer order proportional–integral/proportional–integral–derivative controller based on model parameters of a certain class of fractional order systems

In this study, we deal with systems that can be represented by single fractional order pole models and propose an integer order proportional–integral/proportional–integral–derivative controller design methodology for this class. The basic principle or backbone of the design methodology of the proposed controller relies on using the inverse of the fractional model and then approximating this fractional controller transfer function by a low integer order model using Oustaloup filter. The emerging integer order controller reveals itself either in pre-filtered proportional–integral or proportional–integral–derivative form by emphasizing on the dominancy concept of pole-zero configuration. Parameters of the proposed controllers depend on the parameters of the single fractional order pole model and the only free design parameter left is the overall controller gain. This free design parameter is determined via some approximating functions relying on an optimization procedure. Simulation results show that the proposed controller exhibits either satisfactory or better results with respect to some performance indices and time domain criteria when they are compared to classical integer order proportional–integral–derivative and fractional order proportional–integral–derivative controllers. Moreover, the proposed controller is applied to real-time liquid level control system. The application results show that the proposed controller outperforms the other controllers.

Fractional order sliding mode control via disturbance observer for a class of fractional order systems with mismatched disturbance,

This article proposes a novel fractional order sliding mode control for a class of fractional order and integer order systems with mismatched disturbances. First, a new fractional order disturbance observer is designed to estimate the fractional order differential of the mismatched disturbance directly. Second, the fractional order sliding surface and controller are proposed based on the designed disturbance observer. Our method can deal with mismatched disturbance and has better control performance with faster response speed, lower overshoot, and less chattering effect. The simulations on Quad–Rotor UAV and Maglev suspension systems demonstrate the effectiveness of the proposed method.

Fractional Order Sliding Mode Control via Disturbance Observer for a Class of Fractional Order Systems With Mismatched Disturbance

This paper investigates a novel fractional order sliding mode control for a class of fractional order and integer order systems with mismatched disturbances. First, a new fractional order disturbance observer is designed to estimate the fractional order differential of the mismatched disturbance directly. Second, the fractional order sliding surface and controller are proposed based on the designed disturbance observer. This method can deal with mismatched disturbance and has better control performance with faster response speed, lower overshoot, and less chattering effect. The simulations on Maglev suspension system demonstrate the effectiveness of the proposed method.

A class of fractional order systems with not instantaneous impulses

A class of fractional order systems with not instantaneous impulses