Answer Happy

simulate using matlab/simulink using equations (3) and
(4).
Ball on Beam (BoB) Project Description of System The balancing of a ball on a beam is a classical demonstration of the application of feedback control. The schematic diagram of a typical system is shown below: мотор Ball on Beam The actuator is a d-c motor that can provide the torque to rotate the beam upon which the ball rolls (theoretically) without slipping. Under ideal conditions, the dynamics of the system are governed by the following nonlinear equations: (I + mx²)ē + 2mxzė +mgr cos = T 5 ễ+ (gsin Đ –zỞ*) – 0 (1) (2) where e is the shaft (beam) angle z is the ball displacement from the pivot (the motor shaft) (In the diagram it's labeled r) and I is the inertia of the beam about the motor shaft, m is the ball mass, g is the acceleration of gravity 7 is the torque supplied by the motor The coriolis and centrifugal accelerations are typically negligible; also if the angle is small, we can use the approximations: sin 0; cos021 in which case (1) and (2) reduce to (I + mx²)ő + +mgz - 5 7+790=0 (4) For an ideal d-c motor, the torque is proportional to the motor current i which is given by T=K(e-KB)/R 6
Numerical data for an existing system are given in the following table: Symbol Value Units I 0.02 Kg m² Kg m 0.05 9 9.81 m sec-2 Nm/amp K .01 R 10. ohm
For the "design model", you can use (3) and (4) with the control torque 7 given by the following equa- tion, and with a assumed negligible. This will result in a linear model. The design can then be accomplished using state-variable feedback.