Fractional-order Linear Time-invariant 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.

A negative imaginary lemma and state-feedback negative imaginary synthesis for commensurate fractional-order systems

This article focuses on negative imaginariness of commensurate fractional-order linear time-invariant systems. Enlightened by the generalized Kalman–Yakubovic–Popov lemma, minimal state-space realization based necessary and sufficient conditions are developed to characterize negative imaginariness of commensurate fractional-order systems for both 0<α<1 and 1<α<2 cases. Meanwhile, the problem of negative-imaginary state-feedback controller synthesis for fractional-order systems is addressed. A fractional-order viscoelastic system, a fractional-order RLC circuit network and a numerical example are used as illustrative examples to demonstrate the main developed theory.

Handling a Commensurate, Incommensurate, and Singular Fractional-Order Linear Time-Invariant System

From the perspective of the importance of the fractional-order linear time-invariant (FoLTI) system in plenty of applied science fields, such as control theory, signal processing, and communications, this work aims to provide certain generic solutions for commensurate and incommensurate cases of these systems in light of the Adomian decomposition method. Accordingly, we also generate another general solution of the singular FoLTI system with the use of the same methodology. Several more numerical examples are given to illustrate the core points of the perturbations of the considered singular FoLTI systems that can ultimately generate a variety of corresponding solutions.

Dynamical system modelling to discriminate tissue types for bipolar electrosurgery

Electrosurgery uses electric current to heat, coagulate and ligate tissue. The current flow induces a voltage due to the bio-impedance of the tissue. The flow of the current should be controlled precisely to avoid tissue damage by overheating. The control requires the proper mathematical formulation of the current and voltage relationship. This mathematical description can be explained as a dynamical system. It is composed of linear time invariant (LTI) system and/or the static nonlinearity. The in-depth understanding of the system behaviour requires parameter estimation of the LTI block and the nonlinearity. It is often done stepwise by identifying the best linear approximation (BLA) to fit the whole combination of LTI block and nonlinearity. In this paper, we study the potential discrimination of a block-oriented model between femoral, mesentery, renal and tendon tissue. The estimated system fits up to 99.84% accurately with the measured signal after optimization for the electrosurgery data of Covidien Inc. The parameters exhibit that a weak nonlinearity exists in the measurement data. Finally, the fractional order LTI systems are explored to cope with the diffusion due to the generated heat.

Robust H∞output feedback control of uncertain fractional-order systems subject to input saturation

This article considers the problems of robust stability and stabilization for a class of fractional-order linear time-invariant systems via 0&lt;α&lt;1 subject to input saturation with convex polytopic uncertainties, by a kind of [Formula: see text]–proportional–integral–derivative control synthesis. A new saturation-based stability condition is suggested to estimate the stable region ([Formula: see text]) and region of attraction (ε) using the ellipsoid approach. The main concept of the presented strategy is based on transforming the proportional–integral–derivative controller design problem into the static output feedback control problem. Gronwall–Bellman lemma and sector bounded condition of the saturation function is used for stability investigation and stabilization of such systems. The static output feedback control laws are obtained with the aid of a non-iterative linear matrix inequality implication. From the static output feedback controller, the proportional–integral–derivative controller is then restored. Stability problem is addressed as an optimization problem to obtain the controller gains and the stable region, as well as the largest region of attraction. Some theorems and numerical examples are provided to illustrate the results of proposed mechanism.

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.

Positive real lemmas for singular fractional‐order systems: the case

This study investigates the positive real lemmas for singular fractional-order linear time-invariant systems with the fractional order α ∈ ( 0 , 1 ) . Firstly, a novel condition for the stability and extended strictly positive realness of fractional-order systems is derived in terms of linear matrix inequalities. Secondly, a novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is obtained with two complex variables. Then, another novel condition for the admissibility and extended strictly positive realness of singular fractional-order systems is established with only one complex variable. Thirdly, the positive real controller synthesis problems are analysed and the corresponding positive real controllers are designed based on the positive real lemmas. Finally, four numerical examples are provided to show the effectiveness of the proposed results in this study.

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.

Robust admissibility and stabilization of uncertain singular fractional-order linear time-invariant systems

This paper addresses the robust admissibility problem in singular fractional-order continuous time systems. It is based on new admissibility conditions of singular fractional-order systems expressed in a set of strict linear matrix inequalities &#x0028 LMIs &#x0029. Then, a static output feedback controller is designed for the uncertain closed-loop system to be admissible. Numerical examples are given to illustrate the proposed methods.

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.

Robust fixed‐order dynamic output feedback controller design for fractional‐order systems

This study deals with designing a robust fixed-order dynamic output feedback controller for uncertain fractional-order linear time-invariant systems by means of linear matrix inequalities (LMIs). The authors’ purpose is to design a low-order controller that stabilises the fractional-order linear system in the presence of model uncertainties. No limiting constraint on the state space matrices of the uncertain system is assumed in the design procedure. Furthermore, adopting the most complete model of linear controller, with direct feedthrough parameter, does not disturb the LMI-based approach of developing robust stabilising control. Eventually, the authors illustrate the advantages of the proposed method by some examples and their numerical simulation.

Rectified fractional order iterative learning control for linear system with initial state shift

In this paper, a new rectifying action is combined into different proportional-α-order-derivative-type iterative learning control algorithms for a class of fractional order linear time-invariant systems. Unlike the existing fractional order iterative learning control techniques, the proposed algorithms allow the initial state value of a fractional order iterative learning control system at each iteration to shift randomly. By introducing the Lebesgue-p norm and using the method of fractional integration by parts and the generalized Young inequality of convolution integral, the tracking performances with respect to the initial state shift under the proposed algorithms are analyzed. These analyses show that the tracking errors are incurred by such a shift and improved by tuning the rectifying gain. Numerical simulations are performed to demonstrate the effectiveness of the proposed algorithms.

Asymptotic Stabilization of Fractional Permanent Magnet Synchronous Motor

This paper is mainly concerned with asymptotic stability for a class of fractional-order (FO) nonlinear system with application to stabilization of a fractional permanent magnet synchronous motor (PMSM). First of all, we discuss the stability problem of a class of fractional time-varying systems with nonlinear dynamics. By employing Gronwall–Bellman's inequality, Laplace transform and its inverse transform, and estimate forms of Mittag–Leffler (ML) functions, when the FO belongs to the interval (0, 2), several stability criterions for fractional time-varying system described by Riemann–Liouville's definition is presented. Then, it is generalized to stabilize a FO nonlinear PMSM system. Furthermore, it should be emphasized here that the asymptotic stability and stabilization of Riemann–Liouville type FO linear time invariant system with nonlinear dynamics is proposed for the first time. Besides, some problems about the stability of fractional time-varying systems in existing literatures are pointed out. Finally, numerical simulations are given to show the validness and feasibleness of our obtained stability criterions.

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.

Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations

With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems.

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.

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.

Input–output finite time stability of fractional order linear systems with 0 &lt; α &lt; 1

The input–output finite time stability (IO-FTS) for a class of fractional order linear time-invariant systems with a fractional commensurate order 0 &lt; α &lt; 1 is addressed in this paper. In order to give the stability property, we first provide a new property for Caputo fractional derivatives of the Lyapunov function, which plays an important role in the main results. Then, the concepts of the IO-FTS for fractional order normal systems and fractional order singular systems are introduced, and some sufficient conditions are established to guarantee the IO-FTS for fractional order normal systems and fractional order singular systems, respectively. Finally, the state feedback controllers are designed to maintain the IO-FTS of the resultant fractional order closed-loop systems. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

Finite-time stability and finite-time boundedness of fractional order linear systems

In this paper, the finite-time stability (FTS) and the finite-time boundedness (FTB) for the fractional order linear time invariant (LTI) systems with 0<α<1 are studied. First, some conditions to guarantee the FTS and the FTB for a class of fractional order LTI systems are derived by combining a new property for Caputo fractional derivatives, the generalized Gronwall inequality and the Laplace transform. Moreover, based on these conditions, the method for the design of state feedback controllers is presented. Finally, illustrative examples are provided to show the effectiveness of the results.

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.