(a) The first-order system has the transfer function T(s)=\frac{A}{1+s T} Find the system output response in time domain when a unit ramp signal, r(t)=t, is applied to its input. (b) The transfer function of a second-order system is: T(s)=\frac{4}{s^{2}+4 s+4} Find the system output response in time domain when a unit ramp signal, r(t)=t, is applied to its input. \text { (Note: The inverse Laplace transform of } F(s)=\frac{1}{(s+a)^{2}} \text { is } f(t)=t \cdot e^{-a t} \text {.) }

E1.13 Consider the inverted pendulum shown in Figure E1.13. Sketch the block diagram of a feedback control system. Identify the process, sensor, actuator, and controller. The objective is keep the pendulum in the upright position, that is to keep 0 = 0, in the pres- ence of disturbances. Optical encoder to measure angle 7/ 7, torque m, mass

1. Determine whether the following systems are linear systems or not, and show (or explain) why [note that x represents the input and y represents the output]: \text { System 1: } y=x+1 \text { System 2: } y=x+x^{3}

Problem 2. The sine input u = v2 sin 2t and the corresponding output of the stable linear dynamic system are shown in Fig. 1. During steady-state, they have the same frequency, but different amplitudes (the output amplitude is 1) and phase angles. The phase delay turns out to be one eighth (1/8) of the input period. Based on this observation, determine the frequency response function value at the input frequency, in other words G(2j).

E1.4 An automobile driver uses a control system to main- tain the speed of the car at a prescribed level. Sketch a block diagram to illustrate this feedback system.

P1.8 The student-teacher learning process is inherently a feedback process intended to reduce the system error to a minimum. Construct a feedback model of the learning process and identify each block of the system.

4. Let I=c=4 for the PI controller shown. The performance specifications require that τ= 0.2 (a) Compute the required gain values for each case, 1) <=0.707 2) <=1 (b) Use matlab to plot the unit-step command responses for each of the cases in (a). Compare.

1. Reduce the block diagram shown in Figure P5.1 to a single transfer function, T(s) = C(s)/R(s). Use the following methods: R(s) a. Block diagram reduction [Section: 5.2] b. MATLAB 50 s+1 2 S FIGURE P5.1 S 2 MATLAB ML C(s)

P1.5 A light-seeking control system, used to track the sun, is shown in Figure P1.5. The output shaft, driven by the motor through a worm reduction gear, has a bracket attached on which are mounted two photocells. Complete the closed-loop system so that the system follows the light source./nMotor S Gears € Photocell tubes * Light source FIGURE P1.5 A photocell is mounted in each tube. The light reaching each cell is the same in both only when the light source is exactly in the middle as shown.

B. The transfer function of a system is: T(s)=\frac{s(s+2)}{s^{2}+3 s+1} Find the steady-state response (output) of this system for: (a) Unit step input, (b) Ramp input with a slope of 2 units.