A plant has the transfer function given by

$$G(s)=\frac{1}{(1+s)(1+0.5 s)}$$

and is controlled by a proportional controller $G_c(s)=K$, as shown in the block diagram in Figure 5.42. The value of $K$ that yields a steady-state error $E(s)=Y(s)-R(s)$ with a magnitude equal to $0.01$ for a unit step input is: a. $K=49$ b. $K=99$ c. $K=169$ d. None of the above In Problems 14 and 15 , consider the control system in Figure 5.42, where

$$G(s)=\frac{6}{(s+5)(s+2)} \text { and } G_c(s)=\frac{K}{s+50}$$

The Ferris wheel is often featured at state fairs and carnivals. George Ferris was born in Galesburg, Illinois, in 1859; he later moved to Nevada and then graduated from Rensselaer Polytechnic Institute in 1881. By 1891 , Ferris had considerable experience with iron, steel, and bridge construction. He conceived and constructed his famous wheel for the 1893 Columbian Exposition in Chicago [8]. Consider the requirement that the steady-state speed must be controlled to within $5 \%$ of the desired speed for the Ferris wheel speed control system shown in Figure (a) Determine the required gain $K$ to achieve the steady-state requirement. (b) For the gain of part (a), determine and plot the tracking error for a unit step disturbance. Does the speed change more than $5 \%$ ? (Set $R(s)=0$ and recall that the tracking error $E(s)=R(s)$ $T(s)$.

A unity feedback control system has a process transfer function

$$\frac{Y(s)}{E(s)}=G(s)=\frac{K}{s} .$$

The system input is a step function with an amplitude $A$. The initial condition of the system at time $t_0$ is $y\left(t_0\right)=Q$, where $y(t)$ is the output of the system. The performance index is defined as

$$I=\int_0^{\infty} e^2(t) d t .$$

(a) Show that $I=(A-Q)^2 /(2 K)$. (b) Determine the gain $K$ that will minimize the performance index I. Is this gain a practical value? (c) Select a practical value of gain and determine the resulting value of the performance index.

Consider the closed-loop system in Figure Develop an m-file to accomplish the following task: (a) Determine the closed-loop transfer function $T(s)=Y(s) / R(s)$.(b) Plot the closed-loop system response to an impulse input $R(s)=1$, a unit step input $R(s)=1 / s$, and a unit ramp input $R(s)=1 / s^2$. Use the subplot function to display the three system responses.

Consider the unity feedback control system in Figure where

$$L(s)=G_c(s) G(s)=\frac{K}{s(s+5)} .$$

The design specifications are: i. Peak time $T_p \leq 1.0$ ii. Percent overshoot $P . O$. S $10 \%$. With $K$ as the design parameter, it follows that a. Both specifications can be satisfied. b. Only the first specification $T_p \leq 1.0$ can be satisfied. c. Only the second specification $P . O . \leq 10 \%$ can be satisfied. d. Neither specification can be satisfied.

A system with unity feedback is shown in Figure . Determine the steady-state error for a step and a ramp input when

$$G(s)=\frac{16(s+2)}{s(s+1)(s+4)(s+8)} .$$

Develop an m-file to simulate the response of the system in Figure to a ramp input $R(s)=1 / \mathrm{s}^2$. What is the steady-state error? Display the output on an $x-y$ graph.

A speed control system of an armature-controlled DC motor uses the back emf voltage of the motor as a feedback signal. (a) Draw the block diagram of this system (see Example 2.5). (b) Calculate the steady-state error of this system to a step input command setting the speed to a new level. Assume that $R_a=L_a=J=b=1$, the motor constant is $K_m=1$, and $K_b=1$. (c) Select a feedback gain for the back emf signal to yield a step response with a percent overshoot of $P . O .=15 \%$

A system has the closed-loop transfer function $T(s)$ given by

$$T(s)=\frac{Y(s)}{R(s)}=\frac{2500}{(s+20)\left(s^2+10 s+125\right)} .$$

Using the notion of dominant poles, estimate the expected percent overshoot. a. P.O. $\approx 5 \%$ b. $P . O . \approx 20 \%$ c. P.O. $\approx 50 \%$ d. No overshoot expected

A closed-loop transfer function is

$$T(s)=\frac{Y(s)}{R(s)}=\frac{108(s+3)}{(s+9)\left(s^2+8 s+36\right)} .$$

(a) Determine the steady-state error for a unit step input. (b) Assume that the complex poles dominate, and determine the percent overshoot and settling time to within $2 \%$ of the final value. (c) Plot the actual system response, and compare it with the estimates of part (b).

In general, a third-order system can be approximated by a second-order system's dominant roots if the real part of the dominant roots is less than $1 / 10$ of the real part of the third root. True or False

A motor control system for a computer disk drive must reduce the effect of disturbances and parameter variations, as well as reduce the steady-state error. We want to have no steady-state error for the headpositioning control system. (a) What type number is required? (How many integrations?) (b) If the input is a ramp signal, and we want to achieve a zero steady state error, what type number is required?

Consider the closed-loop transfer function

$$T(s)=\frac{15}{s^2+8 s+15} .$$

Obtain the impulse response analytically and compare the result to one obtained using the impulse function.

An important problem for television systems is the jumping or wobbling of the picture due to the movement of the camera. This effect occurs when the camera is mounted in a moving truck or airplane. The Dynalens system has been designed to reduce the effect of rapid scanning motion; see Figure. A maximum scanning motion of $25^{\circ} / \mathrm{s}$ is expected. Let $K_g=K_{\mathrm{t}}=1$ and assume that $\tau_g$ is negligible. (a) Determine the error of the system $E(s)$. (b) Determine the necessary loop gain $K_a K_m K_t$ when a $1 \%$ s steady-state error is allowable. (c) The motor time constant is $\tau_m=0.40 \mathrm{~s}$. Determine the necessary loop gain so that the settling time (to within $2 \%$ of the final value of $v_b$ ) is $T_s \leq 0.03 \mathrm{~s}$.