This computer project is to model and simulate a Ball-Beam balancing control system, as shown in Figure 1, which is also referred to P2.29 of the current textbook on Page 142. The detailed procedure is described as follows:

1. Derive a dynamic equation for this nonlinear system through the Lagrange Equation:

The torque to drive the beam is considered to be the only input u of the system. Then, the second and third terms of the above dynamic equation become

3. Design a state-feedback control u = Kx to place the new poles of the states into

   (0.8, 1.5, 2 2j);

4. All the parameters of the system are given as follows:

   I1 =0.2Kg-m2,I2 =0.05Kg-m2,m2 =2Kg.,r=0.11m.,a=0.1m.,andg=9.81 m./sec2 .

5. The initial conditions are suggested as (0) = 270, y(0) = 0.2 m., (0) = 0 and y (0) = 0, and set the sampling interval t = 0.02 sec. to run about 16 seconds duration;

6. Using the above results that you have just achieved, program your state-feedback control law and the real-plant into MatlabTM to run the simulation and/or graphics animation.

7. You can work on this project in a group of at most 3 people, and one report is required for each group, including an introduction, modeling and analysis, plots and discussion, as well as attaching your MatlabTM code. It is also recommended to animate this 2D system in MatlabT M , but is not required.

The nonlinear real-plant can be simulated in computer by using either the ode45( ) that is built-in MatlabT M , or the following Runge-Kutta Algorithm in solving x = f (x), x Rn with x(0) = x0. Namely, first,

w()+ (wz wa) = 79 = (). A where the position vector q = ( ), and the inertial matrix ly/ ( 11 + 12 + m2(y2 +r2) mar + 12 m2r + 12 m2 + ( 200 ) (wi - wa)- Tg = m2 ( 20 ) m2g **sin cos"). C = Ax + Bu, where x = | 9 is the 2. Locally linearize it to determine c = f(x, u) = 4-dimensional state vector; I g 21 = f(xx)At, 22 = f(*+ *)At, z3 = f(xx + 2)At, Z4 = f(3x + 23)At, and then, Xk+1 = 2x +=(21+222 +223 + za), k = 0,1, ..., 0 y 12 , m2 11 w()+ (wz wa) = 79 = (). A where the position vector q = ( ), and the inertial matrix ly/ ( 11 + 12 + m2(y2 +r2) mar + 12 m2r + 12 m2 + ( 200 ) (wi - wa)- Tg = m2 ( 20 ) m2g **sin cos"). C = Ax + Bu, where x = | 9 is the 2. Locally linearize it to determine c = f(x, u) = 4-dimensional state vector; I g 21 = f(xx)At, 22 = f(*+ *)At, z3 = f(xx + 2)At, Z4 = f(3x + 23)At, and then, Xk+1 = 2x +=(21+222 +223 + za), k = 0,1, ..., 0 y 12 , m2 11