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Ball-and-Beam Balancer

System ModelingFig 1 shows schematic representations of the ball on beam balancer system. The lagrangian approach was used to model the system. Because the actuator in this system is a DC motor, there is no elastance, and hence, no potential energy in the system. Thelagrangian, therefore, is the kinetic energy, U, given by

Figure1: Schematic for the ball on beam balancer system

Where m is the ball mass, v is the translational velocity of the ball,

is the angular velocity of the ball,

is the angle of the beam shaft, Ia

is the beam inertia, and Ib

is the ball inertia.

Because the sensors in this system measure ball translational position, x, and shaft angular position, , we want to write the above equation in terms of just these two variables. Therefore, we now nd relationships between r and , and x. We rst note that thedistance traveled by the ball is given by

Where is the rotational angle of the ball with respect to the shaft with respect to the shaft and r is the rolling, or effective, radius of the ball. The total angle of the ball is the sum of the angle of the ball with respect to the shaft, , and the angle of the shaft, . Therotational velocity of the ball, therefore, is given by

The translational velocity of the ball, v, is given by (See Figure 3-1b)

Using the equations for rotational and translational velocities in the Lagrangian gives us

Lagranges equation for the ball on beam system is:

Performing the indicated operation on L results in the non-linear differential equation

To linearize this equation, we assume that the control system, for small disturbances, will act to return the ball to rest with minimal shaft movement. Therefore, the shaft angle and its derivatives are assumed to be small, and the dynamic equation becomes

The plant model is found by taking the Laplace transform of the preceding

We can simplify this transfer function by substituting the following inertia relationship into the above equation

This gives us

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System Simulation and Controller Design

Fig 2 shows the Simulink model of the system. The rst part (with the feedback loop) is representative of the DC motor dynamics and the next transfer function corresponds to the ball-on-beam balancer transfer function. Fig 3 is the system response of theaforementioned system.

Figure 2: Simulink Model of the open loop system

Figure 3: Step response of the open loop system

To show that the system is unstable, the root locus of the system is drawn in g 3. To make the system stable, the poles in the root locus are moved to the left half plane by adding two zeros in the form of K(1+as)(1+bs) to the system transfer function. Theresults of this is shown in g 4. The step response of t he new system is plott ed in g 5 (after choosing values for K , a and b based on the new root locus).

Figure 4: Root Locus of the BOBB System

Figure 5: Root locus of the ball on beam balancer system and its two-zero PD compensator

Figure: Step response of the ball on beam balancer with two-zero PD compensator