Problem 3 (restated for convenience): A causal system with input x(t) and output y(t) obeys the differential equation d²y dy dt² dt - 2y(t) = dx dt +3x(t) = 3(b) 10 points Assume that the output obeys the initial condition y(0) = 3, and the input is x(t) eu(t) as in 5(a). Can you find an initial condition for the derivative of the output, y'(0-), such that y(t) does not blow up as t gets large? If so, specify the value of the initial condition. If not, say why not. Zero input response 5(64(5) -4(0))-(0)-(sy (5) - 4(0))-2Y(=0 -ylo 5²-s-2) Y(St) = sy(o-) -y (σ`¯)+y' (0¯ ) Y(5) 2 35 - 3+ y' (0°) 35-3+4(0) (5-2) (5+1) A = B + 5-2 5+1 A = 6-3+y' (0-) = 1+5607/3 3 라 Need A = -1/3 to cancel out e²t term in zero state response | + 4 (0¯ ) = -1/2 =)) y ( 0 ) == 4 3

0.02 (e) The responses in the figure below are for a unit step change in the disturbance with just P-control. One of the responses used a controller gain of Ke = 5 and the other used a controller gain of Ke = 8. Match the controller gains with the responses. Response to a Unit Step Change in Disturbance With P-Control 0.07 Response 1 --- Response 2 0.06 0.05 H 2 0.04H S 0.03 0.01 5 10 15 20 Time (min) (f) The responses in the figure below are for a unit step change in set point with a PI controller. The gain is the same for both responses but one response uses an integral time of ti = 1 min and the other uses an integral time of t = 2 min. Match the integral times for the PI controllers with the responses. r Response to a Unit Step Change in Set Point With Pl-Control 1. Respanse 1 --- Respanse 2 1.4 1.2 0.8 O 0.6- 0.4 0.2H 10 15 20 Time (min) (h) Determine the missing information for the sensor system in the following table. The output for the sensor system is a mA signal. Range Zero…

Given the system in figure Realize using state space. 0.5 -1.16 2-1 ri(n) Z-1 r2(n) -1.25

4. A laser guidance system was used to control the boring machines drilling the channel tunnel to ensure that the machines, which were drilling from either end, met in the middle. The desired angle of heading of the boring machine y,, compared with the measured heading angle y and the error is used in a proportional controller of gain Kp, Figure Q4. Encountering non-uniform layers of earth or rock can cause disturbances in the heading of the boring machine denoted by D(s). (a) Derive the transfer function relating the heading ụ to the reference heading , (assuming no dis- turbance) and hence show that the steady state error due to a step change in the reference heading is zero irrespective of the value of Kp. (b) Derive the transfer function relating the heading y to the disturbance D assuming a zero reference value. (c) Calculate the natural frequency, damping, and overshoot of the system when i) K, = 50, ii) K, = 100. 1 Control 3 Section 3 Tutorial: Feedback Control Systems D(s)…