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Using fractional order calculus to model complex phenomena with combined behavior (i.e., dissipative, capacitive and inertia behavior) is one of the most recent research fields in the world. These accurate models lead to better understanding of the phenomena. For example, consider viscoelastic materials. In early models, dissipative behavior is separated from capacitive behavior but by using fractional order models, these two behaviors are considered simultaneously thus the complexity of the model would be reduced. To use fractional order models in real-life engineering applications, it is important to investigate the dynamics of these systems. Unfortunately, there are a few proper... 

The purpose of this thesis is presenting some decoupling methods for fractional-order two input-two outputs (TITO) systems in order to using in control applications. To achieve this purpose, some methods which have been proposed for decoupling of integer-order systems have been studied and then generalized to be used in decoupling of the fractional-order systems. These methods are classified in two classes: state space based methods and transfer function based methods. Also, in this thesis the proposed decoupling methods have been used in control of some fractional-order TITO systems  

Most of the real-world systems can be modeled as closed loop systems which are made up of a LTI subsystem in the forward path and a memoryless nonlinear subsystem in the feedback path. Such systems are called “Lur’e systems”, the name of the first who introduced these systems. Because of the generality and the wide applicability of this type of systems, studying the stability of them has been notable and interesting to many scientists for a long time. Meanwhile, in recent decades, many scientists in physics and engineering sciences have been interested in applications of the fractional order systems and as a result focused on studying the features of the fractional order systems. An... 

An algebraic method is presented for assessing the bounded input bounded output (BIBO) stability of a class of fractional delay systems with commensurate orders and multiple commensurate delays of retarded type. In the proposed method, first, by mapping the principal sheet of the Riemann surface and a pseudo-delay transformation, an auxiliary polynomial is generated. Then, this auxiliary polynomial is employed to find roots of the characteristic equation on the imaginary axis. The properties of the root path close to these roots are used to identify intervals of delay values, in which the system is stable. Moreover, a delay-independent BIBO-stability criterion is provided. Based on the... 

In this thesis, a new general state space form for uncertain LTI-fractional order system subjected to interval and polytopic uncertainties is introduced. Robust stability analysis and robust stabilization of presented system are investigated. Fractional derivative order assumed to be a known constant. Sufficient conditions in terms linear matrix inequalities (LMIs) are concluded to check the robust stability of the presented system. Then, these results are extended to derive some sufficient LMI conditions to design a state feedback controller in order to stabilize the mentioned uncertain system robustly. Effectiveness and correctness of presented sufficient conditions and application of... 

Time delay generates exponential transcendental terms in characteristic equations. Subsequently, the applied methods are narrow to assess the stability map of delayed fractional-order systems with vanishing fractions. In this case, it is convenient to approximate these equations with its integer-order counterparts to design controllers or investigating features of the systems. But, delay can cause enormous differences between characteristic of these two equations. This study offers a method to survey analytically the stability of fractional-order time delayed linear time invariant systems with infinitesimal fractions. In this method, equations are transferred to an explicit expression which... 

Stabilizing periodic orbit of fractional order chaotic system via linear state feedback is the subject of this research. Firstly, for detection of the unstable periodic orbits (UPO) in fractional order chaotic systems, the famous shooting method for integer order systems is extended to fractional order systems. After that, for stabilizing the fractional order system a hypothesis is stated and a theorem is proved in a specific condition. Besides the proven theorem, some examples are presented as evidences that verify the hypothesis. So, through the hypothesis, common methods for stabilizing integer order systems can be extended to the fractional order systems. Finally, for stabilizing the... 

The PID controller has been used for a long history in control engineering and is acceptable for many real applications due to its simplicity in architecture. Hence, in many real industrial applications, the PID controller is still widely used even though lots of new control techniques have been proposed. In recent years, by developing fractional calculus in control applications, using fractional order PID controller —which is simply called — was proposed by Podlubny. Up to now, several methods of tuning fractional order PID controller has been established for time invariant systems, but dynamics of real systems are mostly time varying. Because of this fact, using adaptive control and... 

In this thesis, some basic concepts in distributed order systems are rewieved. BIBO stability condition of these systems is introduced and stability boundary of distributed order systems in eigenvalue plane of system dynamic matrix is found. In next section, some basic properties of stability boundary are investigated. Stability boundary changes due to changes in distribution function discussed in next part. Furthermore, finding stability areas in cases which stability boundary divides plane of system dynamic matrix to several separated areas is done. A new method for identifying order distribution function by information obtained from stability boundary presented and after that, an... 

In this thesis, stability and stabilization of fractional-order linear time invariant swarm systems are studied. In recent years it has been proved that the exact model of dynamic agents in most of the swarm systems can be modeled more accurate by fractional order differential equations. Stating the motivation of choosing this subject, the achievements of the thesis can be divided into two general parts: Investigating the stability of fractional-order swarm systems and how to stabilize such systems. After introducing the fractional-order systems and fractional-order model of swarm systems in the introduction part, the literature review is presented in Chapter 2. In Chapter 3 the results... 

In this thesis, two essential issues are investigated. The first issue is, when a general incommensurate fractional-order controller would be a stabilizing controller for an incommensurate fractional-order system in the standard feedback structure, and how one can possibly construct such closed-loop stabilizing controller. The second issue is to synthesize internal model principle based fractional-order controllers. These controllers consist of two main parts: the internal model corresponding to exogenous signals and an arbitrary fractional order controller. The main objective in the internal model principle based controllers, is to synthesize the second part of the controller such that the... 

In this thesis, Model reference adaptive control in fractional order systems was discussed. Various types of this problem can be introduced based on fractional order dynamics in system equations, reference model equations or adaptation laws, which some of them play more important role in real-world applications, also some of them have more complexity in governing mathematical equations and in stability analysis. Fractional order dynamics were recently used in many real-world applications like model reference control, Chaos synchronization and Chaos control, so studying adaptive control in this fields, could be important. Nowadays, Model reference adaptive control was used in many industrial... 

In this thesis, first the fundamental theory of stochastic fractional systems has been introduced and the basic of this theorem in control system consisting the controllability and stability has been introduced. It is shown that these basics are playing enormous role in define and understanding such system. Also, Linear Quadratic Regulators has been introduced and appropriate steps to define such controller in the systems. Simulation results shows the effectiveness of the proposed method. On the other hand, the problem of filtering and estimation has been presented with two different method. The first method is to use the idea of optimality and Kalman filter as optimal filter and the second... 

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... 

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... 

The maximum number of oscillation frequencies which can be generated in a linear integer order system at the steady state is the half of its inner dimension. In this thesis, we seek the using of fractional order delayed systems with minimum order to produce maximum number of oscillation frequencies. We found a relation between the system parameters and the oscillation frequency and then adjust the system’s coefficients so that all the roots of the characteristic equation, except the roots on the imaginary axis, lie on the left-hand side of the imaginary axis. Taking into account the diagonal and triangular lower for the state space matrices of fractional order systems with one delay, we... 

This thesis studies undamped oscillations generated by marginally stable fractional order linear time invariant (LTI) systems. In this study, the concept of integral curve is developed for fractional order LTI systems. The proposed concept is obtained based on equivalent integer order linear time varying (LTV) systems. Then, an analytical approach is proposed to be used for harmonic analysis of such oscillations in marginally stable commensurate order LTI systems. Also, it is shown that the Q-semi norm of the limit sets for a trajectory of these systems can be analytically determined based on the Q-semi norm of the initial condition, where Q is a specific matrix. Moreover, this result is... 

Transfer functions of some important classes of infinite-dimensional LTI systems contain semi-polynomials in fractional powers of Laplace variable , possibly in combination with delay terms or exponentials of fractional powers of . They can be observed in many systems and subjects such as biological systems, distributed parameter systems and heat flowing phenomena. Some appropriate theorems relevant to the subject of BIBO-stability are available for a large class of such systems, though many difficulties arise in applying them analytically. In this work, some parameters of the transfer functions of such systems (e.g., the coefficients of the numerators and denominators) are considered as... 

This thesis, argues on recently published methods for tuning of fractional PID controllers. Proposed algorithms provided tuning rules to achieve design objective such as, phase margin, gain cross-over frequency and zero derivative of the phase versus frequency at cross-over frequency. Third constraint imposed to reach high robustness to gain loop variation for controller. Tuning rules attained by consideration on the insensitivity of both stability and desired performance to the system parameter variations, but not on own controller. At the present paper, based on geometric approach, fragility analysis of the fractional-order of some types of PID setting for considering the robustness of the... 

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...