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13FR-02-30 - 130 Asian Journal of Control Vol 6 No 1 pp...

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130 Asian Journal of Control, Vol. 6, No. 1, pp. 130-135, March 2004 INPUT-STATE LINEARIZATION OF A ROTARY INVERTED PENDULUM Chih-Keng Chen, Chih-Jer Lin, and Liang-Chun Yao ABSTRACT The aim of this paper is to design a nonlinear controller for the rotary inverted pendulum system using the input-state linearization method. The system is linearized, and the conditions necessary for the system to be lin- earizable are discussed. The range of the equilibriums of the system is also investigated. Further, after the system is linearized, the linear servo control- lers are designed based on the pole-placement scheme to control the output tracking problem. The performance of the controller is studied with different system parameters. The computer simulations demonstrate that the controller can effectively track the reference inputs. KeyWords: Input-state linearization, nonlinear control, rotary inverted pen- dulum, pole-placement method. I. INTRODUCTION The rotary inverted pendulum is a widely investi- gated nonlinear system due to its static instability. This paper deals with a rotary inverted pendulum system (see Fig. 1), which is composed of the following components: a rotating disk, which is driven by a motor with a rod mounted on its rim. The rod moves as an inverted pen- dulum in a plane perpendicular to the rotating disk. The system discussed in this paper is not the same as the normal inverted pendulum or Furuta’s pendulum. In contrast to the later two systems, in which the equilibria are at two points (with the pendulum upright or down vertically), the equilibria of this system are important and will be investigated in the paper. The system is used as a simplified model for the control of rider-motorcycle systems in circular motion on paths of different radii. The results of this research can be used to study how the riding speeds of the motorcycle and the leaning angles of the rider affect the system dynamics and its motion. Wu and Liu [1] used genetic algorithm and auto-tuning to improve the performance of a fuzzy controller for their system. Yurkovich and Widjaja [2] dealt with the control problem using two strategies. For swing-up control, a fuzzy supervisory mechanism was used. For balancing Manuscript received August 8, 2002; revised March 5, 2003; accepted April 8, 2003. The authors are with Department of Mechanical and Automation Engineering, Da-Yeh University, Changhua, Taiwan 51505, R.O.C. b θ m θ l = 0.3 m m = 0.04 kg Pendulum Mass Inverted 6.3kg M Disk Mass = Motor r = 0.21 m Fig. 1. A schematic representation of the rotary inverted pendulum. control, a direct fuzzy controller based on the LQG lin- ear control was utilized. These methods do not directly use the nonlinear mathematical model in the controller design. Feedback linearization methods can be viewed as ways of algebraically transforming a nonlinear system dynamic fully or partially into a simple linear one. In the standard approach to exact input-state linearization, one uses coordinate transformation and static state feedback such that the closed-loop system, in the defined region, Brief Paper
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C.K. Chen et al.: Input-State Linearization of a Rotary Inverted Pendulum
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