Find the closed loop transfer function gcls thetals

Question

1. Shown in is the control system for one joint of a robot arm. The controller is a PD compensator, given by the transfer function below.

The block diagram describing the system is shown in Figure 2

Find the plant transfer function of the motor and arm: G(s) = θ(s)/E_a(s), as in Figure 3.

1. Find the closed loop transfer function G_cl(s) = θ_L(s)/ θ_s(s)

 

2. Determine the initial and final values of θ_L(t) for a step input θc(t).

3. Assume K_D = 1. Find Kp such that the closed loop system will have a damping ratio of 0.8 and a critically damped response, respectively.

4. Then, with K_D =1 and the values of K_p found in 4), sketch the responses of the system θ_L(t) to a unit step input.

5. Sketch the step response of the closed-loop system manually and comment on upon the effect of proportional action (Provide the details of calculation of time response)

(1 ) K_D = 1, K_P= 2

(2) K_D = 1, K_P= 6

(3) K_D = 1, K_p= 18

(4) K_D = 1, K_p= 28

6. For different values of KID and Kp, analyse the stability of the closed loop system using Routh Hurwitz.