Hierarchical fuzzy controllers for an astronomical telescope tracking Abdel-Fattah Attia National Research Institute of Astronomy and Geophysics (NRIAG), 11421 Helwan, Cairo, Egypt 1. Introduction The mathematical model of the astronomical telescope driven by electrical motors is described through highly nonlinear-coupled differential equations. These equations contain a varying inertia term, a centrifugal and coriolis term, and gravity term. Meanwhile, the gravity term tends to be equal to zero for a well-balanced telescope. This raised difficulty to design an accurate conventional controller to cover a wide range of operating points in nonlinear- coupled differential equations [1,2]. Various modern control strategies have appeared in recent literature to deal with nonlinearity and strong coupling of the telescope dynamics. Many of these strategies generally neglect friction, backlash and other unmodeled dynamics. The controller designs are generally based on the assumption of exact knowledge about the model structure [2,8]. Recently, motivated by the rapidly developed advanced microelectronics and digital processors, conventional PID con- trollers have gone through a technological evolution, from pneumatic controllers via analog electronics to microprocessors via digital circuits [7]. However, it has been known that conventional PID controllers generally do not work well for nonlinear systems, higher order and time-delayed linear systems, and particularly complex and vague systems that have no precise mathematical models [16]. To overcome these difficulties, various types of modified conventional PID controllers such as auto tuning and adaptive PID controllers were developed lately [7,8]. Also, a class of non-conventional type of PID controller employing fuzzy logic has been designed and simulated for this purpose [4,9]. The fuzzy logic controller (FLC) has emerged as one of the most active and fruitful research areas [1]. It has been applied successfully in several applications such as control of astronomical telescope [2] and [15] electrical machines control [11]. This paper uses a hierarchical FLC (HFLC) controller for PID feedback control in order to focus on the proportional-plus- derivative feedback separately from the integral feedback. A hierarchical FLC controller has parallel and series stages where the output of some stages becomes the input to the others. Typically, pairs of inputs are fuzzified and applied to the rules of the preliminary stages [13,4]. The outputs of these stages are applied to the rules of subsequent stages of the fuzzy logic system until the result of the final stage gives the output of the complete fuzzy logic controller. The remainder of the paper is organized as follows. In Section 2, the dynamic model formulation of the 14 00 Celestron telescope is introduced. Then a detailed description of the PID computed- torque controller is given in Section 3. The structure of the hierarchical fuzzy PID controller is highlighted and discussed in Sections 4 and 5. The simulation results are given in Section 6 to demonstrate the improvement of the hierarchical fuzzy PID controller, as compared with PID control systems. Conclusion and future work are explained in Section 7. Applied Soft Computing 9 (2009) 135–141 ARTICLE INFO Article history: Received 2 May 2006 Received in revised form 21 February 2008 Accepted 21 March 2008 Available online 29 March 2008 Keywords: Hierarchical control: PDFLC PIDFLC Fuzzy switch ABSTRACT The paper presents an application of fuzzy logic controller (FLC) with hierarchically structured rule base for a two-link direct drive Celestron telescope. The hierarchical fuzzy logic controller (HFLC) is implemented, as nonlinear blocks used in a control system, for supervision of PID controller for position tracking of the telescope driven by electric motors. The HFLC is composed of two FLC stages connected in cascade. The input variables, for the first stage, of the HFLC are the position error and the rate of change in the position error. Then the output from the first stage and the position error integral are used as input variables for the second stage of the HFLC PID. The simulation results of the proposed HFLC PID, when the telescope is subjected to a unit step in reference position, are compared with the PID controller. The dynamic responses of the HFLC PID improve the rise time, damping factor and settling time compared with the PID controller. Also, the proposed techniques, hierarchical fuzzy PID controller, reduce the computation time due to reduction of rule base. The simulation results show the effectiveness of the proposed HFLC PID controller scheme as a promising technique. ß 2008 Elsevier B.V. All rights reserved. E-mail address: attia@nriag.sci.eg. Contents lists available at ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc 1568-4946/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2008.03.011