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ÖgeDiscretetime adaptive control of port controlled hamiltonian systems(Fen Bilimleri Enstitüsü, 2021) Alkrunz, Mohammed ; Yalçın, Yaprak ; 657354 ; Kontrol ve Otomasyon Mühendisliği Ana Bilim DalıIn 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.