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ÖgeController design methodologies for fractional order system models(Graduate School, 20220125)Fractional order calculation deals with cases where the derivative and integral order is noninteger. Although the notion of fractional order was introduced at the end of the 17th century, this concept in engineering was employed after the first quarter of the 19th century. Its first application to control engineering areas was made after the second quarter of the 20th century. Since fractional calculus is a generalized version of integer order calculus, it provides great flexibility in system modeling and controller design. In other words, fractional calculus offers three different combinations in terms of the controller and system types: Fractional order control for integer order system, Fractional order control for fractional order system, and Integer order control for fractional order system. In this respect, fractional calculus is an excellent tool to describe a control system compared to integer order calculus. Besides the flexibility, the notion brings more complexity to system modeling and controller tuning. Therefore, many studies over the last halfcentury have been trying to overcome these difficulties. Numerous realtime systems have nonlinear characteristics and highorder system dynamics. In literature, simple integerorder models, i.e. the first and second order with or without time delay, are used to represent system dynamics.

ÖgeDoğrusal olmayan sistemler için model öngörülü kontrol yöntemine ters optimal kontrol yapısının katılması(Lisansüstü Eğitim Enstitüsü, 20210802)Optimal kontrol probleminin amacı, bazı kontrol ve durum kısıtlamalarını sağlayacak ve bir başarım kriterini optimize edecek şekilde bir kontrol giriş fonksiyonu veya kontrol kuralı elde etmektir. Buna rağmen, optimal kontrol kuralı, kısıtsız ve doğrusal durumlarda bile oldukça kolay ve analitik olarak bulunamaz. Optimal kontrol kuralının çözümünün HamiltonJacobiBellman (HJB) denklemini çözmeyi gerektirdiği iyi bilinen bir gerçektir ki bu son derece zordur. Dahası, doğrusal olmayan sistemlerin çoğu için analitik bir HJB çözümü mevcut değildir. Sistem doğrusal olduğunda ve başarım kriteri ikinci dereceden olduğunda, HJB, belirli durumlarda analitik olarak çözülmesi zor olabilen bir Riccati denklemi olarak ortaya çıkar. Bu zorlukların üstesinden gelmek amacıyla önceden belirlenmiş bir sonlu ufuk için mevcut sistem durumunu, başlangıç durumu olarak atayarak, sistem modeli yardımıyla optimal kontrol problemini tekrar tekrar ve ardışıl olarak çözmek düşünülmüştür. Bu stratejiyi kullanan kontrol yaklaşımları, Model Öngörülü Kontrol (MÖK) olarak adlandırılır. Bu yaklaşımda, sistemin gelecekteki davranışı, sistem modeli kullanılarak tahmin edilir ve kontrol işareti, anlık sistem durumlarına göre her kontrol ufku için tekrar tekrar yenilenir. Öte yandan, HJB problemini çözmek yerine bize farklı bir bakış açısı sağlayan bir başka yaklaşım ise Ters Optimal Kontrol (TOK) teorisidir. TOK, HJB denklemini çözmenin zahmetli görevinden kaçınarak, doğrusal olmayan optimal kontrol problemini çözmek için alternatif bir yaklaşımdır. Son yıllarda, birçok gerçek zamanlı uygulamada doğrusal olmayan optimal kontrol problemlerini çözmek için ters optimizasyon yaklaşımı giderek daha fazla kullanılmaktadır. Tezde, ilk olarak model öngörülü kontrol yaklaşımının optimal kontrol problemini ele alış biçimi anlatılmıştır. Önerilecek yöntem ile karşılaştırabilmek amacıyla, klasik model öngörülü yaklaşımlarından, doğrusal sistem modelini kullanan gradyant tabanlı MÖK ve doğrusal olmayan sistem modeli RungeKutta tabanlı MÖK (RKMÖK) yaklaşımları verilmiştir. Daha sonra ters optimal kontrol (TOK) yaklaşımları incelenmiş ve ayrıkzamanlı girişteafin doğrusal olmayan sistemler için TOK problemini Kontrol Lyapunov Fonksiyonu (KLF) bulma problemine dönüştürerek çözen TOK yaklaşımı anlatılmıştır. TOK yaklaşımı için takip probleminde karşılaşılabilecek sorunlar üzerinde durulmuştur. Bu tezde ilk olarak, takip problemi sorunlarını çözebilmek amacıyla kontrol işareti ağırlık matrisinin her bir elemanı için sistem durum değişkenlerine bağlı bir sigmoid fonksiyon önerilmiştir. Önerilen yaklaşımın başarımını gösterebilmek için klasik TOK yaklaşımıyla karşılaştırma yapılmıştır. Bu tez çalışmasında, ayrıca girişteafin doğrusal olmayan sistemler için MÖK ve TOK yaklaşımları birleştirilerek yeni bir optimal kontrol yöntemi önerilmektedir. Gerçek hayatta ve literatürde karşılaşılan doğrusal olmayan sistemlerin ve sistem modellerinin çoğu, bazı doğrusal olmayan azaltma yöntemleri ile girişteafin biçime dönüştürülebilir. Önerilen yöntemin temel özelliği, her kayan ufuk ve sonuç olarak yeni bir başlangıç koşulu için çözülmesi gereken MÖK optimizasyon problemini TOK problemi olarak ele alıp, bu TOK problemini tekrar tekrar çözmesidir. Bu yaklaşımda, sistemin gelecekteki davranışının tahminini elde etmek için sistem modeli kullanılır ve önceden belirlenmiş bir kontrol ufku için TOK yönteminden elde edilen kontrol işareti sisteme uygulanır. TOK probleminin çözümü aşamasında, belirlenmesi gereken aday kontrol Lyapunov fonksiyon matrisinin parametreleri, evrimsel Büyük PatlamaBüyük Çöküş (BPBÇ) optimizasyon arama algoritması kullanılarak çevrim içi bir şekilde tahmin edilir. Önerilen kontrol yapısında, MÖK yaklaşımında her kontrol ufku için uygun bir KLF matrisinin aranması ile optimal kontrol problemi çözülmektedir. Diğer bir bakış açısından ise, MÖK yapısı TOK problemine dahil edilerek TOK problemi, her kayan ufkun başlangıcındaki farklı başlangıç koşulları kullanılarak tekrar tekrar çözülmekte ve böylece, TOK için çevrim içi bir düzeltme mekanizması elde edilmektedir. Bu yaklaşım ve literatürdeki diğer yöntemler kullanılarak top ve çubuk kontrol sistemi üzerinde benzetim çalışmaları ve gerçek zamanlı uygulama yapılmıştır. Elde edilen sonuçlar bazı kontrol başarım ölçütlerine karşılaştırılmış ve önerilen yaklaşımın başarımı değerlendirilmiştir.

ÖgeA stable, energy and time efficient biped locomotion(Lisansüstü Eğitim Enstitüsü, 2021)This thesis presents two different walking strategies for biped robots while ensuring energy efficiency. The first strategy is a closedloop walking controller based on the most used 3Dimensional (3D) Linear Inverted Pendulum Model (LIPM) which is used to calculate the Zero Moment Point (ZMP) approximately. The closedloop Proportional Integral (PI) controller's coefficients are searched by the Genetic Algorithm (GA), which is developed to overcome the 3D LIPM's dynamical insufficiency. Because of its ease of modeling, the key concept is to continue to use the 3D LIPM with a closedloop controller. For this purpose, the biped is modeled using the 3D LIPM, which is one of the most wellknown modeling approaches for humanoid robots due to its ease of use and quick computations during trajectory planning. Model Predictive Control (MPC) is applied to the 3D LIPM once the simple model is obtained to search the reference trajectories for the biped while meeting the ZMP criteria. The second strategy is to express the ZMP in a detailed model instead of an approximate model. For this purpose, the biped is modeled with the conventional robot modeling methods and the detailed expression of the ZMP is obtained. Then the problem is redefined as a Nonlinear MPC problem. The highly complicated biped model is implemented in Matlab with the use of CasADi Library which is a symbolic library and used on large symbolic solutions. The optimal control problem is solved with the Interior Point Optimizer (IPOPT), which is an optimization solver for large equations. With the solution of the optimal control problem, reference trajectories are found for the biped while satisfying the ZMP criteria. Both strategies suggested in this thesis are studied and implemented on a biped robot which means the robot has no upper body elements. The main idea is that if the dynamic flaws are suppressed without any upper body elements, this study will open a way to work on more modular robots. After obtaining two different walking strategies, the energyefficient trajectory for the swing leg is searched to have longer working durations on the field. The Big Bang Big Crunch with Local Search (BBBCLS) global optimization algorithm is used for energy efficiency. With the newly defined trajectory there became nearly 10% energy consumption reduction compared to the sinusoidal trajectory. To implement the algorithms to the real biped, a new communication library is written to meet the desired communication speed. But with the increased speed in communication, there became random packet losses on the feedback from the motors. These packet losses are examined and it is observed that these random packet losses may make the system unstable, so to suppress the effects of packet losses the problem is redefined as a time delay problem. With the redefinition of the problem, the wellknown Smith Predictor method is used to overcome the packet losses and from the results, it can be seen that with this redefinition the instability risk because of the packet losses has disappeared. In a short summary, a twolegged robot has been modeled using conventional methods in the literature. First, the dynamic defects of the simple model are eliminated with a conventional controller. Secondly, a more detailed dynamic model is obtained. Walking planning is done with both methods and comparisons are made with the method commonly used in the literature. The success of the proposed methods has been demonstrated in both simulations and experimental results. With the two methods proposed in this thesis, the oscillation problem encountered by one of the most widely used walking models in the literature has been resolved. After obtaining stable walking, energy optimization is studied so that the robot could work longer in the outdoor environment and trajectory improvement is made to reduce energy consumption during the robot's movement. Finally, a faster communication library is written to apply the designed algorithms to the real system and to solve the problems caused by communication speeds, the problem is redefined with a different approach and the traditional method, Smith Predictor, is used. Packet losses that are random thanks to the communication interfaces prepared for the mechanism; become predictable and the effects of packet losses are eliminated with Smith Predictor. Finally, all these control methods are applied to the system and used in experimental studies.

ÖgeDiscretetime adaptive control of port controlled hamiltonian systems(Fen Bilimleri Enstitüsü, 2020)In control theory, the design of the adaptive controllers in the discretetime setting for nonlinear systems has been an interesting area of research. The adaptive controller deals with the problem of finding an appropriate and efficient control structure with an adaptation mechanism to preserve stability and an acceptable closedloop performance in the existence of a considerable amount of uncertainties or timevarying parameters. It is well known that nonlinear systems are sensitive to disturbances, unknown noises, and parameter perturbations. For these kinds of perturbed systems, adaptive control theory is a powerful tool to establish compensation procedures in an effective way that automatically updates the controller to improve the performance of the controlled systems. This thesis study considers adaptive control of an important class of nonlinear systems socalled Portcontrolled Hamiltonian systems (PCH) with uncertainty in their energy function and proposes adaptive discretetime controllers with novel construction of parameter estimators for the multiplicative uncertainty case, the linearly parametrized case, and the nonlinearly parameterized case. The proposed method adopts the Interconnection and Damping Assignment Passivitybased control (IDAPBC) as the control design method and the Immersion and Invariance (I&I) for parameter(s) estimation. Therefore, the two approaches, namely, the IDAPBC and I&I techniques, are combined in a discretetime framework such that all the trajectories of the closedloop system are bounded, and system states successfully converge to the stable desired equilibrium points, namely the minimum of the desired energy function. As mentioned previously, the Immersion and Invariance (I&I) approach is considered to develop an automatic tuning mechanism for the adaptive IDAPBC controller. To comply with I&I conditions, for each case, the estimation error dynamic is defined such that it includes a free design function of the system states, and then the parameter estimator is constructed by establishing a parameter update rule and by presenting a novel function for the mentioned free design function such that Lyapunov stability of the estimator error dynamics is ensured. This novel design function includes some parameters, that can vary in a determined range, to provide the ability to assign desired dynamics to the estimator error system. By replacing the uncertain terms with the values obtained by the I&I estimator, the closedloop system is immersed in the desired closedloop system which would be obtained with the IDAPBC controller with true parameters. In the multiplicative uncertainty case, and as an initial formulation of this study, the uncertainties in energy function appear as multiplicative uncertainties to the gradient of the Hamiltonian function. Unlike the other two formulation cases, no specific perturbation is considered in the system parameters and instead, a general multiplicative uncertainty is presented to the gradient of the Hamiltonian function and thus the adaptive IDAPBC controller is constructed considering this multiplicative uncertainty formulation. The I&I based estimator is designed by selecting an update rule and presenting a general structure for the free design function such that the estimator error dynamics are Lyapunov asymptotically stable. The proposed general structure includes a free parameter that enables to assign different desired dynamics to the estimator. By including the proposed estimator in the constructed adaptive IDAPBC controller, the local asymptotic stability of the obtained closedloop system is shown in a sufficiently large set. One underactuated Hamiltonian system example is considered. In the linear parameterized case, the uncertainties of system parameters appear linearly in the energy function and thus the uncertain system dynamics are formulated such that these uncertainties appear in linearly parameterized form in the gradient of the Hamiltonian function. By considering this formulation of the linear parameterization of the uncertain system parameters, the adaptive IDAPBC controller is constructed. Since PCH is linearly parameterized in the proposed formulation, the gradient of the Hamiltonian function could be factorized in two terms such as one of the terms becomes a matrix that includes all the known terms of system states and system parameters while the other term is a vector of unknown parameters. The mentioned matrix can be a full column rank or not. In the case where this matrix is full rank, the Lyapunov asymptotic stability of the estimator is proved while the Lyapunov stability of the estimator is shown for the case when it is not full rank. It is also shown that, for the case of having not full rank matrix, the term representing the effect of uncertainties in the closedloop system dynamics obtained with the IDAPBC controller that uses the estimated parameters approaches to zero. Furthermore, the Lyapunov asymptotic stability of the obtained closedloop system is shown in a sufficiently large local set either the matrix is full rank or not. For the I&I based estimator design, a general structure for the free design function that includes some free parameters is presented that makes the estimator error dynamics Lyapunov stable where these free parameters are in a determined specific range. So that, by selecting different values for these free parameters in the determined range, different desired dynamics can be assigned to the estimation of each unknown parameter. Three linearly parameterized examples are considered; two fully actuated systems (One has a formulation with a full rank matrix while the other has a formulation with a not full rank matrix), and one underactuated system. In the nonlinear parametrized case, the parameter uncertainties that appear nonlinearly in the energy function are considered. A proper formulation for uncertain system dynamics is presented such that the uncertainties appear in nonlinearly parameterized form in the gradient of the Hamiltonian function and the adaptive IDAPBC controller is constructed considering this formulation. The conditions on the Lyapunov asymptotic stability of the estimator dynamics are derived. Namely, it is proved that if these conditions are satisfied, the estimator error dynamics become asymptotically stable. Assuming these conditions are satisfied, local asymptotic stability of the closedloop system, which is obtained when the proposed estimator is used with the adaptive IDAPBC controller, in a sufficiently large set is proved. For the I&I based estimator design, a structure for the free design function of the estimator is proposed including some other free design functions to satisfy these conditions however it is seen that it is not easy to give general suggestions for these last free functions. It is concluded that for each example, a special selection of these functions is needed. Two nonlinearly parameterized examples are considered and proper selections of the free design functions in the proposed structure is performed. One of the example is a fully actuated mechanical system while the other one is underactuated. The simulation results for each of the previously mentioned systems illustrated the effectiveness of the proposed adaptive controller in comparison to the nonadaptive controller for the same test conditions. The estimator successfully estimates the uncertain parameters and the adaptive IDAPBC controller that utilizing these parameters stabilizes the closedloop system and preserves the performance of the stable desired Hamiltonian systems.

ÖgeAnalysis and design of general type2 fuzzy logic controllers(Fen Bilimleri Enstitüsü, 2020)This thesis presents new interpretations on the design parameters of the general type2 fuzzy logic controllers by investigating their internal structures, proposes novel systematic design approaches for the general type2 fuzzy logic controllers based on comprehensive and comparative analyses, and validates theoretical findings as well as proposed tuning methods via simulation and realtime experiments. The fuzzy systems have been successfully realized in a wide variety of engineering areas such as controls, image processing, data processing, decision making, estimation, modeling, and robotics. The fuzzy logic systems provide complex mappings from inputs to outputs, and this benefit usually results in better performances in comparison to nonfuzzy counterparts. Due to this, the fuzzy logic controllers have been applied to numerous challenging control problems for decades. Nowadays, more attention has been given to a new research direction of the fuzzy sets and systems, the general type2 fuzzy logic controllers, which is the main motivation of this thesis. The internal structures of a class of TakagiSugenoKang type fuzzy logic controllers are first examined in detail. In this context, three fuzzy logic controller types (type1, interval type2, and general type 2) and two kinds of controller configurations (singleinput and doubleinput) are considered. The baseline controllers, i.e. type1 and interval type2 fuzzy logic controllers, are presented in the preliminaries section. The fuzzy sets, fuzzy relations, fuzzy rules, fuzzy operators, and PID forms of these fuzzy logic controllers are explained in detail. The design assumptions and design parameters are given, also the most common design approaches are listed. Afterward, the general type2 fuzzy sets and the general type2 fuzzy logic controllers are presented. The general type2 fuzzy logic controllers are described with αplane associated horizontal slices because the αplane representation provides useful advantages on the handling of the secondary membership function of the general type2 fuzzy sets and the calculation of the general type2 fuzzy logic controller output. It is shown that the αplane based general type2 fuzzy logic controller output calculation is accomplished through the wellknown interval type2 fuzzy logic computations. The secondary membership functions are further detailed in terms of their mathematical definitions and design options. The structure analysis on the general type2 fuzzy sets shows the interactions between nonfuzzy, type1 fuzzy, interval type2 fuzzy, and general type2 fuzzy sets happen in the secondary membership function. It is shown that the general type2 fuzzy logic controller can easily transform into interval type2 fuzzy, or type1 fuzzy counterparts based on the secondary membership function definitions. As an outcome of this structural analysis, a new representation of the trapezoid secondary membership function is proposed based on a novel parameterization of the parameters that form the trapezoid shape. It is shown that the parameterized trapezoid secondary membership function is capable to construct trapezoid, triangle, interval, and singleton shapes so that the general type2 fuzzy logic controllers are further capable to transform into interval type2 fuzzy, or type1 fuzzy counterparts. It is also shown that the proposed parameterization of the trapezoid secondary membership functions allows designing the control curves/surfaces of the general type2 fuzzy logic controllers with a single tuning parameter. Moreover, the structural design suggestions are presented not only to construct fuzzy controllers in a straightforward manner but also to ease the design of the controllers with few design parameters. The design parameters of the general type2 fuzzy logic controllers are grouped as the shape and the sensitivity design parameters with respect to their effects on the accuracy and the shape of the resulting fuzzy mapping. Accordingly, the tuning parameter of the secondary membership functions and the total number of αplanes are interpreted and as the sensitivity and shape design parameters, respectively. The shape analyses of the general type2 fuzzy logic controllers show the effects of the proposed shape design parameter on the control curves/surfaces. In this context, the resulting fuzzy mappings of single input and double input general type2 fuzzy logic controller structures are compared for various design settings of the shape design parameter. The comparative analyses provide interpretable and practical explanations on the potential advances of the shape design parameter. Based on the shape analyses, novel design approaches are proposed to tune the shape design parameter in a systematic way. In this context, it is suggested constructing the general type2 fuzzy logic controllers over their type1 and interval type2 baselines and tuning them via the shape design parameter by providing a tunable tradeoff between robustness and performance. Therefore, it is aimed to combine benefits of baseline type1 (relatively more aggressive control curves/surfaces better performance measures) and interval type 2 (relatively smoother control curves/surfaces, better robustness measures) fuzzy logic controllers. To enhance the control performance, two scheduling mechanisms are also proposed for onlinetuning of the shape design parameter with respect to the steadystate operating points as well as transientstate dynamics. The sensitivity analyses of the general type2 fuzzy logic controllers show the effects of the proposed sensitivity design parameter on the accuracy of the control curves/ surfaces. In this context, the resulting fuzzy mappings of single input and double input general type2 fuzzy logic controller structures are also compared for various design settings of the sensitivity design parameter. The comparative sensitivity analyses show interpretable and practical explanations of the sensitivity design parameter in terms of calculation accuracy and computation burden. Therefore, it is suggested tuning the sensitivity design parameter by considering the limitations of hardware components such as resolution and processing speed. To accomplish the design in accordance with a tradeoff between sensitivity and computational time, a novel iterative algorithm is proposed to tune the sensitivity design parameter. The simulation and realtime experimental control studies validate the proposed design recommendations, systematic design approaches, and tuning methods for the general type2 fuzzy logic controllers on benchmark control systems. In these control studies, the general type2 fuzzy logic controllers are designed based on the proposed design methods. In order to show the performance improvements on the control systems, the general type2 fuzzy logic controllers (tuned either online or offline) are compared with type1 fuzzy and interval type2 fuzzy counterparts. The performance measures clearly show that the onlinetuned general type2 fuzzy logic controllers outperform all general type2, interval type2, and type1 counterparts on account of the proposed scheduling mechanisms over the proposed systematic design rules. The results also show that the systematic design of the general type2 fuzzy logic controllers is simply accomplished by following the proposed tuning steps of the shape and sensitivity design parameters.