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Unformatted text preview: OPTIMAL CONTROL OF AN INVERTED PENDULUM USING A DIGITAL DEADBEAT RESPONSE PREDICTION OBSERVER S. S. Mirsaid Ghazi Iran University of Science and Technology [email protected] Abstract: This paper presents an optimal control algorithm of an inverted pendulum with Linear Quadratic Regulator in which a digital prediction observer with deadbeat response is used to estimate the unmeasured state variables. Optimal control is a powerful algorithm considers the limitations in state variables and system actuators. The main weak point of optimal control in a practical view is its complete dependence in sense or estimation of state variables. A digital prediction observer offers the fastest way of estimation of unmeasured state variables in a digital- based control system. The successful results of implementing of the proposed algorithm in this paper show the effectiveness of the algorithm.. Keywords: Deadbeat response, Digital observer, Inverted Pendulum (IP), Optimal Control. 1 Introduction The inverted pendulum is a classical control problem, which involves developing a system to balance a pendulum and belongs to the class of under-actuated mechanical systems having fewer control inputs than degrees of freedom (A detailed description of The Inverted Pendulum system is considered in [1]). This renders the control task more challenging making the inverted pendulum system a classical benchmark for testing different control techniques (Useful techniques related with the proposed algorithm in this paper could be found in [3,4,6,7,9]). There are a number of different versions of the IP system offering a variety of interesting control challenges. A single rod on a cart IP is a rod mounted on a moving cart that can rotate on its pivot as shown in figure 1. This system has tow equilibrium points that one of them is stable (fig. 1.b) and another one is unstable (fig. 1.a). Stability and un-stability of these equilibrium points is simply provable by Liapunov stability theory with different methods as in [7,8,11]. With the rod exactly centered above the motionless cart, there are no sidelong resultant forces on the rod and it remains balanced as shown in Figure 1.1a. In principle it can stay this way indefinitely, but in practice it never does. Any disturbance that shifts the rod away from equilibrium, gives rise to forces that push the rod farther from this equilibrium point. Fig. 1: Equilibrium points Optimal control is a good choice of existing control algorithms for the systems in which state variables should be kept in complete restricted bounds and also actuators of the system are in danger of being saturated or damaged (design of an algorithm regardless of energy shaping of the system functions in nonlinear systems is frequently not useful, [7]-[9]). In order to implement the algorithm on a system, all state variables should be accessible simultaneously, therefore the estimation of some variables is vital in controlling the complex nonlinear variables. complex nonlinear variables....
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This note was uploaded on 08/19/2011 for the course ELECTRICAL 136 taught by Professor Haeri during the Spring '11 term at Amirkabir University of Technology.

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