Build the closed loop system using simulink-find poles

Question

Part -1:

Matlab Assignment

For the mathematical model expressed by the following governing equation:

x¨ + 8.684x = -0.3797u + 12.41∫∫u.dt

Write a program that performs the following using line commands on a script file (save as .m file)

1- Find transfer function 2- Find state space

3- Plot the open loop response due to the inputs: unit step, sin(t), over a range of time 0 < t < 200.

4- Add a negative feedback with H(s)=0.02 then construct a closed loop transfer function.

5- Plot the closed loop response due to the same inputs from part 3. 6- Obtain response information for both step and sin responses.

7- Find poles, zeros and pole-zero map for the closed loop system.

Build the closed loop system using Simulink.

Part -2:

Matlab Assignment

The linearized mathematical model of the shown inverted pendulum is:

x¨ + 8.684x = -0.3797u + 12.41∫∫u.dt

θ¨ = -8.684θ + 1.266f

The controller transfer function is:

G_c = k₁ + k₂/s + k₃.100s/(s+100) 

Where k₁ = 65, k₂ = 50, k₃ = 2

Inverted pendulum system and Controller Gc are connected in series → open loop, and then the open loop system connected in a negative feedback forming a closed loop system that's subjected to a unit step input:

Outputs needed to be measured are X and θ, the feedback signal is θ, the controlled signal is error = f - θ

Using Simulink:

1- Perform the closed loop system using block diagram.

2- Perform the closed loop system using state space.

3- Obtain open loop (without feedback) and closed loop (with feedback) step response.

4- Tune only k3 and find the optimum value of k3 that improves the response.

5- Tune k3 and k2 together and find the optimum values that improve the response.

6- Show at least 3 responses resulted from tuning for tuned values of K's for parts 4 & 5.