Controller Design using Homotopy Singular Perturbation Method in Nonlinear Fractional order Systems

نوع: Type: thesis

مقطع: Segment: PHD

عنوان: Title: Controller Design using Homotopy Singular Perturbation Method in Nonlinear Fractional order Systems

ارائه دهنده: Provider: Mahnaz Abolvafaei

اساتید راهنما: Supervisors: Dr. Soheil Ganjefar

اساتید مشاور: Advisory Professors:

اساتید ممتحن یا داور: Examining professors or referees: Dr. Mohammad Hasan Moradi, Dr. Mohammad Farrokhi, Dr. Hamid Reza Momeni

زمان و تاریخ ارائه: Time and date of presentation: 2021-12-05, PM 14-17

مکان ارائه: Place of presentation: online

چکیده: Abstract: Modeling and controlling are two essential issues in describing the behavior of nonlinear industrial systems. Investigating the dynamic demeanor of many systems, it has been found that many industrial systems have inherently fractional-order behavior. Hence, fractional calculus theory is an effective tool for modeling and behavioral justification of industrial systems, which provides a more accurate description of the system performance than integer-order modeling. Research shows that most industrial and natural systems and phenomena have complex nonlinear structures. These structures include severe nonlinear couplings between system components, parameter uncertainties, unmodeled dynamics, external disturbances, and unknown perturbations. As a result, modeling and studying such dynamics have complicated the controller design, the performance evaluation, and system stability analysis. By describing the system behavior using fractional calculus, because the derivative of a variable is dependent on all its previous values, the fractional-order model behaves like an infinite memory. The memory feature leads to the fact that fractional-order modeling, despite its desirable properties, adds to the complexity of modeling and makes analysis and design more difficult than integer-order systems. Now, what is clear is that the simpler system dynamics may cause an easier process in analysis and design. Therefore, simplifying the dynamic model requires the use of powerful methods. So that by applying these methods, in addition to reducing the computational load and complexity of the model, the main characteristics of the system maintain, such as stability, the number of equilibrium points, location of equilibrium points, and sometimes system memory property. Controller design based on a simplified dynamic model can play an important role in forcing the system to achieve the control aims. Due to the importance of these two issues, i.e., modeling and controller design, in two different steps, various methods are presented in this thesis for simplifying fractional-order models and controller design. In the first step, the problem of model simplification in the field of fractional-order systems is investigated by presenting two various proposed methods. These two approaches are system dynamics simplification based on maintaining some features of fractional order modeling and simplification based on achieving the integer-order approximation of fractional order dynamics. In the first strategy, the issue under consideration is the simplification of system dynamics into two simpler subsystems such as fractional-order and integer-order subsystems. In such a way that the preservation of properties, such as stability, desirable fractional-order characteristics, and the number and location of equilibrium points are evaluated along with reducing the complexity of analysis, design, and computational load. Hence, in this method, the proposed homotopy singular perturbation method is offered by presenting the fractional-order singular perturbation method and the modified homotopy perturbation method. In the second approach, achieving an integer-order approximation for the fractional-order system is evaluated. Therefore, to present the proposed method, it is assumed that the orders of all fractional derivatives of the system have a small deviation from the nearest integers. By defining this small deviation as a perturbation parameter, an integer order approximation method is presented for approximating the fractional-order system into a perturbed integer order system in the time domain. This proposed approach can be implemented on linear or nonlinear fractional-order systems in the time domain. Also, by applying the proposed strategy to linear systems, the system eigenvalues don't change and have a lower computational load than the previously presented methods in this field. In the second step, the utilization of the proposed modeling methods is evaluated in the control systems design. Hence, to evaluate the various modeling and present proposed control approaches, the wind turbine system is used. A wind turbine, as a renewable energy source, plays an essential role in providing clean energy. The inability of a wind turbine to perform desired control can cause to incur higher operating costs, reduce wind power extraction, and even damage system components. Therefore, it would implicate designing an advanced control approach. The control techniques have a fundamental role in preventing and maintaining the system from the adverse operating situation by monitoring control input, blade pitch angle, and rotor speed. Therefore, applying effective control approaches and accurate system modeling can improve the system behavior and raise the number of operational hours of wind turbine systems. Hence, in this thesis, various modeling methods and control approaches by applying to wind turbine systems are evaluated from different perspectives, such as maximum power extraction and mechanical load reduction. Therefore, after presenting the proposed modeling methods and applying them to the wind turbine system, various controllers based on nonlinear optimal control method, sliding mode control method from different orders, and adaptive control method in the presence of uncertainty and unknown disturbances are designed to achieve the control objectives. A fractional optimal control is an optimal control problem, in which the performance index and/or the model dynamics can be described by the fractional-order derivative. Therefore, the optimal control design for fractional-order systems can be classified into four categories, such as fractional dynamics with an integer-order performance index, fractional dynamics with a fractional-order performance index, mixed fractional-integer dynamics with an integer-order performance index, and mixed fractional-integer dynamics with a fractional-order performance index. As a result, according to this classification, for the first three cases, three nonlinear control approaches using optimal fractional high-order fast terminal sliding mode control are proposed. Generally, the design procedure of these proposed approaches for each performance index includes two-stages process. In the first phase, an equivalent control law is presented by the optimal controller approach. Then a switching controller is offered in the second phase that is obtained by defining the fractional high-order sliding manifold and fractional nonsingular fast terminal sliding manifold to overcome the unknown disturbance. In the proposed combined strategies, the optimal control and sliding mode control approaches are combined to minimize the control input and tackle the effects of parametric uncertainties, unmodeled dynamics, and external disturbances. In these methods, the high order sliding mode control scheme is used to attenuate the chattering phenomenon whereas the fast terminal sliding surface is applied to achieve better tracking, obtain the maximum wind power extraction, reduce mechanical loads on the drive train, and attain fast finite-time convergence by using a small control input. In the proposed controller design for fractional-order systems with different performance indexes such as integer and fractional order performance indexes, a state-space model of the wind turbine system error dynamics is firstly obtained by defining the desired trajectory. Then, proposed approaches are designed using the defined error dynamics. Hence, in both cases, the optimal control technique is designed using the nominal error dynamics of the system. Therefore, the tracking control problem is converted into a regulatory control problem by defining the nominal error dynamics of the system. However, in the fractional-order system with a fractional-order performance index, the fractional-order performance index is first converted to an integer-order performance index to simplify the controller design process. Therefore, the system error dynamics and the controller design process are changed by applying this transformation. Then, using the new error dynamics, the proposed controller is designed. Therefore, for the first time, in this thesis, two optimal fractional high-order fast terminal sliding mode controllers are proposed that the order of the controllers is smaller than the second order. In the third case, i.e. mixed fractional-integer dynamics with an integer-order performance index, by applying proposed homotopy singular perturbation method, fractional-order nonlinear DFIG based wind turbine dynamics is firstly divided into nonlinear integer order subsystem and linear fractional-order subsystem. Then, a nonlinear optimal second-order fast terminal sliding mode control is developed for nonlinear integer-order subsystem using the Lyapunov-based approach with negative definite solution. Also, a fractional optimal second-order fast terminal sliding mode control is proposed by defining the subsystem error dynamics for linear fractional order subsystem. In the continuation of this thesis, an adaptive second-order sliding mode control based on the sign TPID sliding surface is designed for the integer-order approximation method obtained for fractional-order wind turbine system. PID sliding surface is applied to ensure better tracking, achieve a zero steady-state error, reduce the mechanical loads, and attenuate the chattering phenomenon, whereas describing the PID sliding surface as a sign TPID sliding manifold helps have maximum wind power extraction and establish fast finite-time convergence by applying a small control input. The use of the sign function in describing the sliding surface increases the convergence speed of the tracking error to zero. Finally, the effectiveness of the proposed methods in the control systems design of the wind turbine and wind turbine based on DFIG is evaluated using simulation results. For this purpose, the proposed modeling methods and controllers are compared with some existing modeling methods and control laws. Hence, in some cases, the performance of the proposed methods is verified through various scenarios, i.e., random variation of wind speed, step change of wind speed, and robustness against parameter uncertainties. Based on the obtained results, various proposed modeling and control approaches provide better performance and an almost perfect behavior in tracking the desired trajectory and achieving maximum wind power through different scenarios, fast finite-time convergence, and reducing mechanical load and chattering phenomenon. Also, using the obtained simulation results, the advantages and disadvantages of various modeling methods are specified. Therefore, the simulation results confirm the effectiveness of the proposed controllers. The issues explained, the methods presented, and the analysis performed of this research can use for developing fractional-order systems, especially the industrial wind turbine system. Key Words: Model simplification, Homotopy singular perturbation method, Integer-order approximation, Nonlinear optimal control, High order terminal sliding mode control, Maximum power extraction

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