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fractional-order-system
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Robust D-stability Analysis of a Class of Interval Fractional Order Systems
,
Ph.D. Dissertation
Sharif University of Technology
;
Fathi Jegarkandi, Mohsen
(Supervisor)
Abstract
Because of advancing fractional calculus and modeling physical phenomena by using fractional calculus more accurately than that by using integer calculus, and also existing uncertainties in models of real world systems, robust stability and performance analysis of fractional order systems are necessary. This thesis deals with the robust -stability analysis of LTI fractional order systems from kind of uncertain typical fractional order systems (UTFOS) and the robust stability analysis of LTI fractional order systems with delays from kind of uncertain retarded type systems (URTS) and uncertain neutral type systems (UNTS) with interval uncertainties. The coefficients of the numerator and the...
Conditions on Fractional-order LTI Systems to be Negative Imaginary
, M.Sc. Thesis Sharif University of Technology ; Tavazoei, Mohammad Saleh (Supervisor)
Abstract
In this thesis, we derive the analytic condition that linear time invariant fractional order system being positive real. Also, by inspiration of positive real condition, we derive the analytic condition for linear time invariant fractional order system that their imaginary part of frequency response being negative. Intended conditions for single input single output systems are derived in the form of necessary conditions in one way and derived in the form of sufficient conditions in another way. However, intended conditions in multiple input multiple output are just derived in the form of sufficient conditions. Furthermore, an upper band for phase of oustaloup approximation frequency response...
Control of Fractional Order Systems
, M.Sc. Thesis Sharif University of Technology ; Shahrokhi, Mohammad (Supervisor)
Abstract
This thesis addresses design of an adaptive tracking control of input-quantized strict-feedback fractional-order nonlinear systems with unknown control directions. The control design is achieved by using a hysteretic quantizer to avoid chattering. The main advantages of the proposed controller are: the fractional order is available for both commensurate and non-commensurate cases, the¬¬ controller design does not depend on the quantization parameters and some restrictive assumptions are relaxed. Additionally only one adaptive law has been used for the unknown control directions which leads to reduction of computational load. By utilizing the backstepping approach and based on the frequency...
Control of Fractional Order Systems with Input Constraints
, M.Sc. Thesis Sharif University of Technology ; Shahrokhi, Mohammad (Supervisor)
Abstract
Considering input constraints is an essential task in the controller design. In this thesis, a controller has been designed for incommensurate fractional order nonlinear systems in the nonstrict feedback form subject to unknown dynamics, input nonlinearity and actuator failures. The Lyapunov direct method and the backstepping technique have been used to design the controller and stability analysis. The number of actuator faults can be infinite. In addition, the proposed control algorithm can cope with different types of input nonlinearities namely, saturation, dead zone, dead zone-saturation, backlash and hysteresis. To estimate the system uncertainties, neural networks have been employed...
Design of an Adaptive Controller for Uncertain Fractional-order Systems Subject to Actuator Failure
, M.Sc. Thesis Sharif University of Technology ; Shahrokhi, Mohammad (Supervisor)
Abstract
The objective of this research is to design an adaptive controller for a class of fractional-order nonlinear systems in the strict-feedback form with unmodeled dynamics. Actuator saturation and actuator fault are also considered. All of the system states are assumed to be measurable, and all the sensors can be faulty. Fractional-order systems are chosen because, in the modeling of physical systems, the fractional-order calculus is often preferable to the classical integer-order calculus. The controller is designed by using the backstepping design technique. The fuzzy logic systems are used to eliminate the problem of "explosion of complexity" in the conventional backstepping method and also...
Controller Design for Nonlinear fractional Order Systems in the Presence of Input and Output Constraints
, M.Sc. Thesis Sharif University of Technology ; Shahrokhi, Mohammad (Supervisor)
Abstract
The purpose of this thesis is design of an adaptive tracking control for input-quantized strict-feedback fractional order nonlinear systems with unknown dynamics and asymmetric time-varying output constraints. The controller design is achieved by using a hysteretic quantizer to avoid chattering and not needing the quantization parameters. The fuzzy logic method has been used to solve the problem of unknown dynamics. Also, due to the asymmetric time-varying output constraints, the Barrier Lyapunov function has been used. In this thesis, less restrictive assumptions have been considered than the work performed in previous researches.By utilizing the adaptive backstepping approach and based on...
Robust non-fragile fractional order PID controller for linear time invariant fractional delay systems
, Article Journal of Process Control ; Vol. 24, issue. 9 , 2014 , pp. 1489-1494 ; Haeri, M ; Sharif University of Technology
Abstract
A fractional order PID controller is designed to stabilize fractional delay systems with commensurate orders and multiple commensurate delays, where the time delays in the system may belong to several distinct intervals. Moreover, the controller parameters should belong to given intervals. In order to stabilize the system, the D-subdivision method is employed to choose the stabilizing set of the controller parameters from their available values. Furthermore, the nearest values of the obtained stabilizing set to their mean values are selected as the controller parameters so that a non-fragile controller is concluded. Two numerical examples evaluate the proposed control design method
Fractional/distributed-order systems and irrational transfer functions with monotonic step responses
, Article JVC/Journal of Vibration and Control ; Vol. 20, issue. 11 , 2014 , pp. 1697-1706 ; Sharif University of Technology
Abstract
This paper deals with irrational transfer functions having monotonic nondecreasing step responses. Firstly, some results on the monotonicity of step responses in irrational transfer functions describing fractional- or distributed-order systems are presented. Then, some conditions guaranteeing the existence of monotonic nondecreasing step responses in more general forms of irrational transfer functions are found. Various examples are brought to show the usefulness of the obtained results in time response analysis of fractional/distributed-order systems. The achievements of the paper can be used in the design of control systems having monotonic step responses
Non-fragile control and synchronization of a new fractional order chaotic system
, Article Applied Mathematics and Computation ; Volume 222 , 2013 , Pages 712-721 ; 00963003 (ISSN) ; Delshad, S. S ; Hamidi Beheshti, M. T ; Tavazoei, M. S ; Sharif University of Technology
2013
Abstract
In this paper, we address global non-fragile control and synchronization of a new fractional order chaotic system. First we inspect the chaotic behavior of the fractional order system under study and also find the lowest order (2.49) for the introduced dynamics to remain chaotic. Then, a necessary and sufficient condition which can be easily extended to other fractional-order systems is proposed in terms of Linear Matrix Inequality (LMI) to check whether the candidate state feedback controller with parameter uncertainty can guarantee zero convergence of error or not. In addition, the proposed method provides a global zero attraction of error that guarantees stability around all existing...
Sensitivity analysis of CRA based controllers in fractional order systems
, Article Signal Processing ; Volume 92, Issue 9 , September , 2012 , Pages 2040-2055 ; 01651684 (ISSN) ; Haeri, M ; Sharif University of Technology
Elsevier
2012
Abstract
This paper focuses on robust performance analysis of a closed loop fractional order system through a sensitivity approach. The characteristic ratio assignment method is selected to attain a desired closed loop transient response. Then, we compute the sensitivity of such a desired transfer function with respect to its characteristic ratio and we explore its specifications. The relation between the coefficient diagram shape and the relative stability of the closed loop system is discussed. Also, the closed loop poles variations due to the changes in the characteristic ratios are investigated. Finally, we study a pseudo second order process to verify the robust performance of the characteristic...
Temperature control of a cutting process using fractional order proportional-integral-derivative controller
, Article Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME ; Volume 133, Issue 5 , March , 2011 ; 00220434 (ISSN) ; Haeri, M ; Sharif University of Technology
2011
Abstract
In this paper, the fractionalized differentiating method is implemented to reduce commensurate fractional order models complexity. The prominent properties of this method are its simplicity and guarantee of preserving the stability of a specific class of fractional order models in their reduced counterparts. The presented reduction method is employed in simplifying complicated fractional order controllers to a fractional order PID (FOPID) controller and proposing tuning rules for its parameters adjustment. Finally, the efficiency of the FOPID tuning rule obtained based on the proposed reduction method is shown in the temperature control of a cutting process
Maximal bound for output feedback gain in stabilization of fixed points of fractional-order chaotic systems
, Article Journal of Computational and Nonlinear Dynamics ; Volume 6, Issue 3 , February , 2011 ; 15551415 (ISSN) ; Sharif University of Technology
2011
Abstract
This paper deals with the problem of stabilizing the unstable fixed points of a class of fractional-order chaotic systems via using static output feedback. At first, a static output feedback controller designed to stabilize a fixed point of a fractional-order chaotic system is considered. Then, the maximal allowable perturbation bound around the nominal value of the output feedback gain of the designed controller, such that the stability of the intended fixed point in the closed-loop system is guaranteed, is analytically determined. Also, some numerical examples are presented to confirm the validity of the analytical results of the paper
Necessary and sufficient conditions for BIBO-stability of some fractional delay systems of neutral type
, Article IEEE Transactions on Automatic Control ; Volume 56, Issue 1 , 2011 , Pages 125-128 ; 00189286 (ISSN) ; Haeri, M ; Sharif University of Technology
2011
Abstract
In this note, bounded-input bounded-output (BIBO)-stability of a large class of neutral type fractional delay systems is investigated. Necessary and sufficient conditions of BIBO-stability are presented for the intended class of systems (the sufficient conditions have been provided for a more general case in the previous studies). Two lemmas are provided for checking a prerequisite imposed on the considered class of systems. Finally, two numerical examples are given to illustrate the obtained results
Maximum number of frequencies in oscillations generated by fractional order LTI systems
, Article IEEE Transactions on Signal Processing ; Volume 58, Issue 8 , May , 2010 , Pages 4003-4012 ; 1053587X (ISSN) ; Haeri, M ; Siami, M ; Bolouki, S ; Sharif University of Technology
2010
Abstract
In this paper, relation between the inner dimension of a fractional order LTI system and the maximum number of frequencies which exist in oscillations generated by the system is investigated. The considered system is defined in pseudo state space form and the orders of its involved fractional derivatives are rational numbers between zero and one. First, an upper bound is derived for the maximum number of frequencies. Then, using the restricted difference bases concept, a new method is introduced to design a multifrequency oscillatory fractional order system. Finally, based on the proposed method some lower bounds are derived for the maximum number of frequencies obtainable in solutions of a...
On robust stability of LTI fractional-order delay systems of retarded and neutral type
, Article Automatica ; Volume 46, Issue 2 , 2010 , Pages 362-368 ; 00051098 (ISSN) ; Haeri, M ; Sharif University of Technology
2010
Abstract
This paper deals with the analysis of robust BIBO-stability of LTI fractional order delay systems in the presence of real parametric uncertainties. Two large classes of these systems, namely retarded and neutral types, are considered. Two theorems are given to check the robust BIBO-stability of these two families of fractional order systems. One of these theorems provides necessary and sufficient conditions for the case of retarded type and another one presents only sufficient conditions for the case of neutral type. Furthermore, upper and lower bounds (cutoff frequencies) are provided for drawing the value sets. To illustrate the results, two numerical examples are presented
Stability analysis of distributed-order nonlinear dynamic systems
, Article International Journal of Systems Science ; Volume 49, Issue 3 , 2018 , Pages 523-536 ; 00207721 (ISSN) ; Tavazoei, M. S ; Sharif University of Technology
Taylor and Francis Ltd
2018
Abstract
The problem of asymptotic stability analysis of equilibrium points in nonlinear distributed-order dynamic systems with non-negative weight functions is considered in this paper. The Lyapunov direct method is extended to be used for this stability analysis. To this end, at first, a discretisation scheme with convergence property is introduced for distributed-order dynamic systems. Then, on the basis of this tool, Lyapunov theorems are proved for asymptotic stability analysis of equilibrium points in distributed-order systems. As the order weight function assumed for the distributed-order systems is general enough, the results are applicable to a wide range of nonlinear distributed-order...
Taming single input chaotic systems by fractional differentiator-based controller: theoretical and experimental study
, Article Circuits, Systems, and Signal Processing ; Volume 28, Issue 5 , 2009 , Pages 625-647 ; 0278081X (ISSN) ; Haeri, M ; Jafari, S ; Sharif University of Technology
2009
Abstract
A simple fractional differentiator-based controller is proposed to suppress chaos in a 3D single input chaotic system by stabilizing some of the fixed points. The tuning procedure for the proposed controller is based on the stability concepts in the incommensurate fractional order systems. To show the efficiency of the controller, some numerical simulations are given. Also, to evaluate the practical capability of the proposed controller, we experimentally apply it to control chaos in a chaotic circuit. Moreover, some mathematical analyses are presented to show the applicability of the proposed controller, when its structure is not exactly implementable. © Birkhäuser Boston 2009
Model reduction in commensurate fractional-order linear systems
, Article Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control Engineering ; Volume 223, Issue 4 , 2009 , Pages 493-505 ; 09596518 (ISSN) ; Haeri, M ; Sharif University of Technology
2009
Abstract
In this paper, some commonly used model reduction methods for integer-order systems are employed to approximate commensurate fractional-order linear systems. In comparison with the original system, the approximating model possesses a smaller inner dimension, while its fractional order is the same as that of the original system. The applied methods fall into the global reduction category, such as direct truncation and singular perturbation methods, and into the local reduction category, such as Pade approximation, partial realization, shifted Pade approximation, and rational interpolation methods. The applicability of these methods is illustrated by approximating a sample high-dimensional,...
Limitations of frequency domain approximation for detecting chaos in fractional order systems
, Article Nonlinear Analysis, Theory, Methods and Applications ; Volume 69, Issue 4 , 15 August , 2008 , Pages 1299-1320 ; 0362546X (ISSN) ; Haeri, M ; Sharif University of Technology
2008
Abstract
In this paper, we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases. For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in...
Estimating the fractional order of orthogonal rational functions used in the identification
, Article 2008 International Conference on Control, Automation and Systems, ICCAS 2008, Seoul, 14 October 2008 through 17 October 2008 ; December , 2008 , Pages 1130-1134 ; 9788995003893 (ISBN) ; Haeri, M ; Tavazoei, M.S ; Sharif University of Technology
2008
Abstract
This paper deals with the identification of fractional order systems via orthogonal rational functions. These functions have widely been used in system identification of classical integer order systems. It has been shown that due to some properties such as the presence of non-exponentional aperiodic multimodes in the fractional order systems, it is much better to use fractional orthogonal rational functions in approximation of these systems. One problem which arises in this area is the estimation of fractional order of these orthogonal rational functions. In the existing methods, these parameters have been found by trial and error which requires a large amount of calculations. To reduce the...