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In this study, some of robust and exact tuning methods for fractional order PD and PI controllers are investigated in order to extract their solution existence conditions. Using aforementioned methods needs solving a set of nonlinear equations to obtain desired controller parameters which satisfy the control objectives. Hence, one of the aims of this study is to find necessary and sufficient condition on tuning specifications which guarantee the applicability of the aforementioned methods. The achievable performance region of the methods is obtained through the extracted necessary and sufficient conditions. The other aim of this work is to investigate the uniqueness of the methods... 

In this thesis, various fractional order controllers (FOCs), e.g., FO-PID, FO-TID, CRONE and FO-Lead/Lag, are implemented using PLC. For this purpose first, several methods are used to approximate the differentiator and integral operators. Based on these methods, the equivalent rational transfer functions of the fractional order controllers are obtained. Then, using STEP7 software, the FOCs are implemented in Siemens PLC. Moreover, to facilitate the use of the implemented FOCs in the industry and to take full advantage of the controllers, WinCC software as an interface between human and machine (HMI) is used. The WinCC software enables the operator to access the PLC through a computer (PC)... 

In this thesis, a new method has been developed for tuning of fractional PI ( ) and fractional [PI] ( ) controllers for the first order plus dead-time ( ) systems. By using the performance-map method ( ) and based on phase and gain margins' criteria, a formula which is function of fractional order ( ) and normalized dead-time ( ) has been proposed for the proportional and integral coefficients. In addition, an equation has been given for the fractional order of controller to minimize the integrated absolute error ( ) index. Moreover, overshoot and settling-time of the step response of the system have been improved by dynamic setpoint weighting. Simulation results have shown that proposed... 

Synchronous machines is one of the most popular machines in industry and in power plants. Because the dynamic of this machine is nonlinear, it can exhibit complex behaviors. In this project we have studied a special case in Permanent Magnet Synchronous Machines. A case in which speed of rotor, frequency, currents amplitude and voltages amplitude change in an irregular manner. This special case is called chaos. Chaos occurs for some specific values of parameters and inputs. In this study in the first step, we have tried to find the chaotic area of parameters of system (Prediction of chaos). Then in the next step, we have designed some controllers which can eliminate chaos in the system. The... 

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... 

Today, the fractional-order systems are widely used in the system identification and design and implementation of control systems. By production of electrical elements with fractional-order relationship between voltage and current of their terminals, implementation of fractional-order functions with passive elements is possible. Fractional order capacitors are one of these elements. It was shown that necessary and sufficient condition for the implementation of an integer order function by passive elements is positive realness of such a function.
Like systems with integer-order, in this thesis the aim is to find corresponding conditions for implementation of fractional-order functions... 

In the present thesis, first of all we consider a particular class of fractional order plants which have uncertainty in their time constant.In order to make the closed loop system robust against variations of the aforementioned parameter, a robust fractional order controller will be designed and the corresponding tuning rule will be expressed. By using the proposed controller, phase margin of the system will be preserved on a desired value for any value of the uncertain parameter and also gain crossover frequency of the system, in the nominal case, will be adjustable.Also another class of fractional order system will be considered. For these systems, the problem is the same as the former... 

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, variable order operators as a generalization for fractional operators are introduced and their characteristics are investigated. Then fractional variable order systems are defined whose their corresponding differential equations are stated by variable order operators. Most of this research delineate some beneficial control characteristics of fractional variable order systems. The first is type number subject. In this section two important theorems are stated which are useful in determining type number in an intensive range of systems. The other important subject is norm of functions. In this part a useful theorem is presented that states a sufficient and necessary condition... 

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... 

Benefiting from the great potential of the mathematical tool of fractional calculus, fractional order proportional integral (FOPI) and proportional integral derivative (FOPID) controllers have been proposed as the next generations for the popular PI and PID controllers. On the other hand, the integral performance indices such as the integral square error (ISE), integral square time error (ISTE), integral absolute error (IAE) and integral time absolute error (ITAE) are commonly used in design and evaluate of the performance of practical control systems. Considering these points, the present thesis deals with optimal tuning of the free parameters of fractional order PI and PID controllers, to... 

Due to the simplicity and generality, design of feedback control systems in the frequency domain is among the most prominent topics in the control systems engineering. The problem of compensation means properly adjusting the loop frequency response at critical frequencies. Considering the point that the traditional methods usually use trial and error procedures for compensation, analytical design of fixed-structure controllers for exactly adjusting such values has great importance. Benefitting from the capability of fractional-order dynamics, the compensation problem is studied from three aspects in this thesis. At first according to the importance of adjusting the system frequency response... 

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... 

Distributed order dynamic systems are studied in this thesis. These systems incorporate distributed order operators to describe physical phenomena that naturally occur in diffusion, relaxation and viscoelasticity. Hence, the exact solutions of distributed order system of differential equations are derived analytically which are described by what we later call distributed order mittag-leffler functions. Then robust stability of distributed order LTI continuous time systems with uncertain weight functions and dynamic matrices is investigated and some sufficient conditions for stability/instability are introduced in this matter. Also, after discussing some of the properties of stability... 

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... 

The present thesis introduces a new method to design non-fragile controllers. The idea of design is to set the centroid of the stability region as the tuning point of two-parameter controllers. This ideahas been motivated by the fact that the centroid of a convex region belongs to a set of pointsthat have farthest distance from its boundary. In case that stability region is not convex, a correction may be needed because the centroid of non-convex region may be very near to or out of its boundary. One modified version of design method when stability region is not convex, is to find maximum convex sub-area inside stability region and set its centroid as the tuning point of controller... 

The main effort of this thesis is to present a numerical method to simulate and implement a special type of controllers, called proportional-weighted integral (PWI) controllers. For this purpose, at first mathematical equations of the fractional order proportional-integral controllers have been investigated in the time domain, and then by generalization of these controllers, the PWI controllers are achieved. Next the discretization of these controllers are carried out and the performance of the controllers are simulated by means of this discretization and comparison between these controllers and the conventional PI controllers has been held. After that a method for estimation of these... 

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... 

A servomechanism is a system with the objective of to controlling the position or velocity of a mechanical device. In many servomechanisms, one of the major performance measures is tracking the reference input in the shortest time possible. In practice, due to reasons such as modeling error and existence of disturbances, time-optimal control strategy, which is obtained using the classical calculus of variations methods, leads to the undesirable chattering phenomenon. The Proximate Time-Optimal Servomechanism control method is among the practical approaches proposed to obtain responses arbitrarily close to time-optimal, and at the same time avoid chattering. This methodology has found... 

Adaptation is a recurring phenomenon in biology by which biochemical systems can robustly- and in some cases completely- adapt their regulated outputs to en- vironmental disturbances and uncertainties. In nature, such a feature has been evolved in biological circuits through the construction of integral feedback control structures. Three types of biologically plausible controllers have been recently in- troduced which can ensure the existence of robust perfect adaptation for arbitrary plants. In this thesis, the main purpose is to have a comparison between them. To this end, we first find a stability criterion to relate the system’s stability with its internal parameters. We then calculate a...