Design of feedback for Inverted Pendulum System

Instructions:

Provide the original MATLAB code as well as the code on a word or pdf Explain your design procedures in detail on the same word or pdf Answer all three questions

System Description:

The inverted pendulum is hinged on a cart. The cart and the pendulum are assumed to be moving in one plane given as y. The problem is to maintain the pendulum at an angle θ with the vertical by designing appropriate feedback.

Two nonlinear equations describing the dynamics of the pendulum are given below:

Where θ is the angle in radians and u is the input

M = 0.5kg, m = 0.2kg, g = 9.8 m/sec2 and l = 0.3m

The linearized state space and output equations are obtained by assuming θ as small such that sin θ ≈ θ and by neglecting θ, as

X = AX + Bu

̇Here the output Y represents θ 

Question 1:

Design a PID controller to maintain the pendulum at vertical with rise time around 0.02 seconds, settling time around 0.24 seconds and overshoot is around 17%. (see pidTuner in MATLAB)

Question 2:

Design state feedback gains K such that u = r − KX where r is the reference and the output of the feedback system with r = 1 tends to 1 or -1 as time tends to infinity. (see function acker in MATLAB if place does not work)

Question 3: Numerically simulate the original two nonlinear equations with the state feedback using the K designed in Question 2 with u = r − KX . Here X is obtained from the simulation that represents  of the nonlinear equations.

First simulate with r = 0.1 , then with r = 0.5 and finally with r = 1. Compare the performance of the state feedback on the nonlinear equations. (for simulation refer to the attached Simulink)