(a) Sketch the loci of the system poles for varying K. (6 %) (b) Find K for a damping ratio 0.707 of the dominating poles. (7 %)

1. For a fluid power position servo with loop gain function

GH

s(s2 t 6 s t 1 3 ) K .YTkc

(a) Sketch the loci of the system poles for varying K. (6 %) (b) Find K for a damping ratio 0.707 of the dominating poles. (7 %)

(c) Where is the third pole for this K? (7 ^%)

2. In the Figure shown, let G(s)=l/[(s+l)(s+4)] and let the compensator G, have the form G , =K(T,s + 1)/(T2s + I). This is called phase-lead compensation if TI > T2 and phase-lag compensation if TI < T,. Both are very common and their design will be discussed at length. One approach, considered here, is to choose the zero of G, to cancel one of the plant poles. If the system is to be designed for a damping ratio 0.5 using

1. Proportional control G, = K

2. Phase-lag compensation K(s+l)/(Ss+l)

3. Phase-lead compensation K(0.25~+1)/(0.05~+1) then:

(a) Determine the values of K required. _{( 5}

(b) Find and compare the steady-state errors following unit step inputs. (5 %) (c) Determine the closed-loop system time constants and compare the speeds of

response. (5 %)

3. In the Figure as shown below, the system is at rest initially. At t

0 a unit-step displacement input is applied to point A. Assuming that the system remains linear throughout the .response period and is underdamped, find the response x(t) as well as the values of x(O+), x(O+) , and x(w). (15 %)

Unit step

(

4. For the system shown below, a step torque T(t) is applied at B1(t) shaft.

(a) Find a state space representation of the system in vector-matrix form. (5%) (b) 1f ~ = l k ~ - m ~ , design D and K so that the percent overshoot is 16.3% and the peak

time is 1.2092 sec. (5%)

(c) If it is also desired that the settling time is less than 2.5sec, can this design meet the requirement? If not, how would you change your design? (5%) (d) Continued from (b), if it is desired to reduce the rise time by changing the values of J and K (with D fixed). How would you tune these two values, and which one is

more effective? (5%)

5. Derive the transfer function a, and use it to design m and k such that the step U s )

response of this system has a rise time about 2sec and percent overshoot about 16.3%. What is the smallest rise time if the overshoot is to be kept within 16.3%?

Without the requirement on the overshoot, what is the achievable range of the rise

time? (15%)

6. Find the transfer function T(s)=Y(s)/R(s), determine the system type and find the steady state error with respect to an input R(s)=lO/s. Is this system a minimum

phase one? Stable? (1 5%)