Time-invariant Fractional-order Systems Research Articles

Supplemental Stability Criteria with New Formulation for Linear Time-Invariant Fractional-Order Systems

In this paper, new stability criteria for linear time-invariant fractional-order systems (LTIFOSs) based on linear matrix inequalities (LMIs) are derived. The solved variable of the existing LMI formulations is generalized to a complex one. In addition, based on the congruent transformation, a new LMI formulation is obtained, which is different from those in the existing literature. To deal with the above LMIs more conveniently with simulation software, the complex LMIs are converted to equivalent real LMIs. Finally, numerical examples are presented to validate the effectiveness of our theoretical results.

On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena.This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore, periodic solutions can be expected in fractional-order systems in terms of the new fractional derivative operator. To confirm this assertion, one investigates two examples, one linear system for which one gives an exact periodic solution by its analytical expression and another nonlinear system for which one provides exact periodic solutions using qualitative and numerical methods.

An LMI approach to Mixed H_∞/H_- fault detection observer design for linear fractional-order systems

This study deals with the problem of robust fault detection for linear time-invariant fractional-order systems (FOSs) assumed to be affected by sensor, actuator and process faults as well as disturbances. The observer-based method was employed to solve the problem, where the detector is an observer. The problem was transformed into the mixed robust optimization problem to make the system disturbance-resistant on one hand and fault-sensitive on the other hand. Then, sufficient conditions were obtained to solve the problem in the linear matrix inequality (LMI) mode. Finally, the effectiveness and superiority of the method were demonstrated by simulating the solutions on a single-input multi-output thermal testing bench.

Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems

This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems. It is proved that given any stable linear time invariant fractional order system, there exists a positive definite functional with respect to the system state, and the first order time derivative of that functional is negative definite. A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.

Optimal approximation of fractional‐order systems with model validation using CFOA

Optimal integer-order transfer function approximation of linear time-invariant fractional-order systems (FOS) using a straight-forward approach is proposed here. The error between the rational approximant and the FOS (commensurate/non-commensurate/oscillatory types etc.) is minimised directly in the frequency domain, subject to design stability constraints, by using the colliding bodies optimisation (CBO) algorithm. Comparisons with other well-known metaheuristics regarding the computational time and the design effort expended in parameter tuning justify the efficiency of the proposed method. The CBO-optimised models achieve a superior solution quality regarding both the magnitude and phase responses in comparison to the reported literature. Analogue circuit implementations of the models using the current feedback operational amplifier are also presented. Simulations carried out in OrCAD PSPICE environment validate the practical efficacy of the proposed models.

Geometrical design of fractional PDμ controllers for linear time-invariant fractional-order systems with time delay

This article presents a simple procedure that allows a practical design of fractional –[Formula: see text] controllers for single-input single-output linear time-invariant fractional-order systems subject to a constant time delay. The methodology is based on a geometric approach, which provides practical guidelines to design stabilizing and non-fragile PDμ controllers. The simplicity of the proposed approach is illustrated by considering several numerical examples encountered in the control literature. Moreover, with the aim of showing the performance of the PDμ over a classical PD controller, both controllers were implemented at the end of the article in an experimental test-bench consisting a teleoperated robotic system.

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.

H∞ output feedback control of linear time-invariant fractional-order systems over finite frequency range

This paper focuses on the H∞ output feedback control problem of linear time-invariant fractional-order systems over finite frequency range. Based on the generalized Kalman-Yakubovic-Popov (KYP) Lemma and a key projection lemma, a necessary and sufficient condition is established to ensure the existence of the H∞ output feedback controller over finite frequency range, a desirable property in control engineering practice. By using the matrix congruence transformation, the feedback control gain matrix is decoupled and further parameterized by a scalar matrix. Two iterative linear matrix inequality algorithms are developed to solve this problem. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.

The Grünwald–Letnikov Fractional-Order Derivative with Fixed Memory Length

Contrary to integer-order derivative, the fractional-order derivative of a non-constant periodic function is not a periodic function with the same period. As a consequence of this property, the time-invariant fractional-order systems do not have any non-constant periodic solution unless the lower terminal of the derivative is ±∞, which is not practical. This property limits the applicability of the fractional derivative and makes it unfavorable, for a wide range of periodic real phenomena. Therefore, enlarging the applicability of fractional systems to such periodic real phenomena is an important research topic. In this paper, we give a solution for the above problem by imposing a simple modification on the Grünwald–Letnikov definition of fractional derivative. This modification consists of fixing the memory length and varying the lower terminal of the derivative. It is shown that the new proposed definition of fractional derivative preserves the periodicity.

LMI-based sufficient conditions for robust stability and stabilization of LTI-fractional-order systems subjected to interval and polytopic uncertainties

In this paper, by introducing a new general state-space form for uncertain linear time-invariant fractional-order systems subjected to interval and polytopic uncertainties, two problems including robust stability analysis and robust stabilization of the presented systems are investigated. Subsequently, two sufficient conditions in terms of several linear matrix inequalities for the problems mentioned are concluded as two separate theorems. It is assumed that the fractional order [Formula: see text] is a known constant belonging to [Formula: see text]. Simulation results of two different numerical examples demonstrate that the provided sufficient conditions are applicable and effective for tackling robust stability and stabilization problems.

Effect of Initialization on a Class of Fractional Order Systems: Experimental Verification and Dependence on Nature of Past History and System Parameters

The objectives of this work are (i) to verify experimentally the theoretical claim that neither Riemann–Liouville nor Caputo fractional derivative can be used to predict the time response of fractional order systems without proper correction for the system’s past history in terms of an initialization function and (ii) to study quantitatively how the error incurred due to ignoring initialization depends on the nature of the past history and the system parameters. The entire analysis is restricted to a special class of single input single output linear time invariant fractional order system which can be realized by a simple electrical circuit consisting of a resistance and a fractance in series. This work involves two different realizations of fractances or constant phase elements whose characteristic parameters are first determined based on their respective impedance frequency responses and then used to simulate the time responses of the circuit with the input same as the one used for experimentation using a numerical method for two cases: (i) taking past history into account and (ii) without taking past history into account. Thereafter, an integral square error criterion is presented and variation of the same is studied with respect to system parameters and nature of the history function to have a relative idea of how much partial past history in the absence of a complete one should suffice in practical applications.

State feedback H ∞ control of commensurate fractional-order systems

This article focuses on the state feedback H ∞ control problem for commensurate fractional-order systems with a prescribed H ∞ performance. For linear time-invariant fractional-order systems, a sufficient condition to guarantee stability with H ∞ performance is firstly presented. Then, by introducing a new flexible real matrix variable, the feedback gain is decoupled with complex matrix variables and further parametrised by the new flexible matrix. Moreover, iterative linear matrix inequality algorithms with initial optimisation are developed to solve the state feedback H ∞ suboptimal control problem for fractional-order systems. Finally, illustrative examples are given to show the effectiveness of the proposed approaches.

On the existence of periodic solutions in time-invariant fractional order systems

Periodic solutions and their existence are one of the most important subjects in dynamical systems. Fractional order systems like integer ones are no exception to this rule. Tavazoei and Haeri (2009) have shown that a time-invariant fractional order system does not have any periodic solution. In this article, this claim has been investigated and it is shown that although in any finite interval of time the solutions do not show any periodic behavior, when the steady state responses of fractional order systems are considered, periodic orbits can be detected.

Variable structure control of linear time invariant fractional order systems using a finite number of state feedback law

In this paper, an approach based on the variable structure control is proposed for stabilization of linear time invariant fractional order systems (LTI-FOS) using a finite number of available state feedback controls, none of which is capable of stabilizing the LTI-FOS by itself. First, a system with integer order derivatives is defined and its existence is proved, which has stability equivalent properties with respect to the fractional system. This makes it possible to use Lyapunov function and convex analysis in order to define the sliding sector and develop a variable structure control which enables the switching between available control gains and stabilizing the fractional order system.

Robustness Margin in Linear Time Invariant Fractional Order Systems

In this paper, the computation of robustness margin for linear time invariant fractional order systems is studied. For the definition of robustness margin, we employ the one introduced for polynomials (i.e. integer order) and extend it to fractional order functions. Using the well known concept of the value set and knowing its shape for the intended functions, this paper presents an easy way to obtain the robustness margin for fractional order systems. To illustrate the results, a numerical example is provided.

A proof for non existence of periodic solutions in time invariant fractional order systems

The aim of this note is to highlight one of the basic differences between fractional order and integer order systems. It is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals. As a result, a limit cycle cannot be expected in the solution of these systems. Our investigation is based on Caputo’s definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems.

FINITE TIME STABILITY ANALYSIS OF LINEAR AUTONOMOUS FRACTIONAL ORDER SYSTEMS WITH DELAYED STATE

For the first time, in this paper, a stability test procedure is proposed for linear time-invariant fractional order systems (LTI FOS). Paper extends some basic results from the area of finite time and practical stability to linear, continuous, fractional order time invariant time-delay systems given in state space form. Sufficient conditions of this kind of stability, for particular class of fractional time-delay systems is derived.