Consider the robot arm illustrated in the following figure, with a mass m and moment of inertia I.

1.

Derive the equation of motion for the arm and linearize the equation about .

2.

Via numerical simulation, (pick your own values for I, m, l and , and use any programming language and/or environment you want) verify the following ``rules of thumb'' for PID control in light of the definitions of rise time, peak time, overshoot, settling time and steady state error from the following figure:

(a)

For proportional control, i.e., k_p > 0, k_d = 0 and k_I=0, the solutions are oscillatory, and increasing k_p increases the frequency of oscillation (which decreases the rise time and peak time) but decreases the mean steady state error. The settling time is infinite.

(b)

If derivative control is added to the proportional controller, i.e., k_p > 0, k_d > 0 and k_I=0, then

i.

for small k_d the solutions are decaying oscillations;

ii.

increasing k_d decreases the settling time;

iii.

increasing k_d sufficiently eliminates the oscillatory behavior completely, resulting in an solution which exponentially decays to the final, steady state value;

iv.

increasing k_p decreases the final steady state error;

v.

increasing k_p decreases the rise time.

(c)

Adding integral control (PID control)

i.

eliminates the steady state error, even for small values of k_p,

ii.

increasing k_I generally increases the overshoot and settling time;

iii.

increasing k_p decreases rise time, but may increase overshoot;

iv.

increasing k_d increases damping and stability.

3.

Extra credit: verify the above rules of thumb analytically by determining analytically the solution to the linearized equation of motion for the system.