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Deep brain stimulation (DBS) is an efficient therapy to control movement disorders of Parkinson's tremor. Stimulation of one area of basal ganglia (BG) by DBS with no feedback is the prevalent opinion. Reduction of additional stimulatory signal delivered to the brain is the advantage of using feedback. This results in reduction of side effects caused by the excessive stimulation intensity. In fact, the stimulatory intensity of controllers is decreased proportional to reduction of hand tremor. The objective of this study is to design a new controller structure to decrease three indicators: (i) the hand tremor; (ii) the level of delivered stimulation in disease condition; and (iii) the ratio of the level of delivered stimulation in health condition to disease condition. For this purpose, the authors offer a new closed-loop control structure to stimulate two areas of BG simultaneously. One area (STN: subthalamic nucleus) is stimulated by an adaptive controller with feedback error learning. The other area (GPi: globus pallidus internal) is stimulated by a partial state feedback (PSF) controller. Considering the three indicators, the results show that, stimulating two areas simultaneously leads to better performance compared with stimulating one area only. It is shown that both PSF and adaptive controllers are robust regarding system parameter uncertainties. In addition, a method is proposed to update the parameters of the BG model in real time. As a result, the parameters of the controllers can be updated based on the new parameters of the BG model.

In this study, a model of basal ganglia (BG) is applied to develop a deep brain stimulation controller to reduce Parkinson's tremor. Conventionally, one area in BG is stimulated, with no feedback, to control Parkinson's tremor. In this study, a new architecture is proposed to develop feedback controller as well as to stimulate two areas of BG simultaneously. To this end, two controllers are designed and implemented in globus pallidus internal (GPi) and subthalamic nucleus (STN) in the brain. A proportional controller and a backstepping controller are designed and implemented in GPi and STN, respectively. The proposed controllers deliver suitable stimulatory control signals to GPi and STN based on hand tremor amplitude (as the feedback). When tremor reduces, these controllers decrease the stimulatory energy intensity proportionally. Therefore, additional stimulatory signal is not delivered to the brain. Subsequently, the side effects from the excessive stimulation intensity become much less. Comparing with one area stimulation, the results show that stimulating two areas of BG results in reduction of the level of the stimulation intensity. It is observed that these two controllers are both robust in terms of changing the system parameters.

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In this paper, we present a computational method for solving optimal control problems and the controlled Duffing oscillator. This method is based on state parametrization. In fact, the state variable is approximated by Boubaker polynomials with unknown coefficients. The equation of motion, performance index and boundary conditions are converted into some algebraic equations. Thus, an optimal control problem converts to a optimization problem, which can then be solved easily. By this method, the numerical value of the performance index is obtained. Also, the control and state variables can be approximated as functions of time. Convergence of the algorithms is proved. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.