Analysis Of A Pendulum Problem
Jan Jantzen 	firstname.lastname@example.org:1
The ball-balancer, or cart-ball system, demonstrates some basic concepts in control be-
ing nonlinear, multivariable, and non-minimum phase. It is basically an inverted pendulum
problem, which is a much used benchmark problem. The objective here is to provide an
analysis of the system, which can be the basis for designing different kinds of controllers.
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Technical University of Denmark, Department of Automation, Bldg 326, DK-2800 Lyngby, DENMARK.
Tech. report no 98-E 863 (cartball), 19 Aug 1998.
The ball-balancing system in this paper consists of a cart with an arc made of two parallel
pipes on which a steel ball rolls. The cart moves on a pair of tracks horizontally mounted
on a heavy support (Fig. 1). The control objective is to balance the ball on the top of the
arc and at the same time place the cart in a desired position. The cart-ball system was built
for teaching electrical engineers about automatic control, originally with a focus on state-
space control theory. It is educational, because the laboratory rig is sufficiently slow for
visual inspection of different control strategies and the mathematical model is sufficiently
complex to be challenging.
Figure 1: Laboratory rig.
The system was built during an M.Sc. project, and later the mathematical model was
published in an educational journal (Jørgensen, 1974). A description, which includes equa-
tions, can be found in that paper, but it has a few errors, and much emphasis is on state-space
concepts. A simulator of the same system was built in Matlab much later for a course on the
Internet (Jantzen, 1996b; Jantzen & Dotoli, 1998). The simulator is based on a linearised
model of the system,
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The model is a linear state-space model, where [ is a vector of