Chapter 4, Feedback Control System Characteristics Video Solutions, Modern Control Systems

Problem 1

The open-loop transfer function of a fluid-flow system can be written as
$$
G(s)=\frac{\Delta Q_2(s)}{\Delta Q_1(s)}=\frac{1}{\tau s+1},
$$
where $\tau=R C, R$ is a constant equivalent to the resistance offered by the orifice so that $1 / R=1 / 2 \mathrm{kH}_0^{-1 / 2}$, and $C=$ the cross-sectional area of the tank. Since $\Delta H=R \Delta Q_2$, we have the following for the transfer function relating the head to the input change:
$$
G_1(s)=\frac{\Delta H(s)}{\Delta Q_1(s)}=\frac{R}{R C s+1} .
$$

For a closed-loop feedback system, a float-level sensor and valve may be used as shown in Figure P4.1. Assuming the float is a negligible mass, the valve is controlled so that a reduction in the flow rate, $\Delta Q_1$, is proportional to an increase in head, $\Delta H$, or $\Delta Q_1=-K \Delta H$. Draw a closed-loop flow graph or block diagram. Determine and compare the openloop and closed-loop systems for (a) sensitivity to changes in the equivalent coefficient $R$ and the feedback coefficient $K$, (b) the ability to reduce the effects of a disturbance in the level $\Delta H(s)$, and (c) the steady-state error of the level (head) for a step change of the input $\Delta Q_1(s)$.

Jack Chen

Numerade Educator

03:19

Problem 2

It is important to ensure passenger comfort on ships by stabilizing the ship's oscillations due to waves [13]. Most ship stabilization systems use fins or hydrofoils projecting into the water to generate a stabilization torque on the ship. A simple diagram of a ship stabilization system is shown in Figure P4.2. The rolling motion of a ship can be regarded as an oscillating pendulum with a deviation from the vertical of $\theta$ degrees and a typical period of 3 seconds. The transfer function of a typical ship is
$$
G(s)=\frac{\omega_n^2}{s^2+2 \zeta \omega_n s+\omega_n^2},
$$
where $\omega_n=3 \mathrm{rad} / \mathrm{s}$ and $\zeta=0.20$. With this low damping factor $\zeta$, the oscillations continue for several cycles, and the rolling amplitude can reach $18^{\circ}$ for the expected amplitude of waves in a normal sea. Determine and compare the open-loop and closedloop system for (a) sensitivity to changes in the actuator constant $K_a$ and the roll sensor $K_1$, and (b) the ability to reduce the effects of step disturbances of the waves. Note that the desired roll $\theta_d(s)$ is zero degrees.

Khoobchandra Agrawal

Numerade Educator

Problem 3

One of the most important variables that must be controlled in industrial and chemical systems is temperature. A simple representation of a thermal control system is shown in Figure P4.3 [14]. The temperature $\mathscr{T}$ of the process is controlled by the heater with a resistance $R$. An approximate representation of the dynamic linearly relates the heat loss from the process to the temperature difference $\mathscr{T}-\mathscr{T}_e$. This relation holds if the temperature difference is relatively small and the energy storage of the heater and the vessel walls is negligible. Also, it is assumed that the voltage $e_h$ applied to the heater is proportional to $e_{\text {desired }}$ or $e_h=k E_b=k_a E_b e(t)$, where $k_a$ is the constant of the actuator. Then the linearized open-loop response of the system is
$$
\mathscr{T}(s)=\frac{k_1 k_a E_b}{\tau s+1} E(s)+\frac{\mathcal{T}_e(s)}{\tau s+1},
$$
where
$$
\begin{aligned}
& \tau=M C /(\rho A), \\
& M=\text { mass in tank, } \\
& A=\text { surface area of tank, } \\
& \rho=\text { heat transfer constant, } \\
& C=\text { specific heat constant, } \\
& k_1=\text { a dimensionality constant, and } \\
& e_{t h}=\text { output voltage of thermocouple. }
\end{aligned}
$$

Determine and compare the open-loop and closedloop systems for (a) sensitivity to changes in the constant $K=k_1 k_a E_b$; (b) the ability to reduce the effects of a step disturbance in the environmental temperature $\Delta \mathscr{T}_e(s)$; and (c) the steady-state error of the temperature controller for a step change in the input, $e_{\text {desired. }}$

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02:45

Problem 4

Consider the feedback amplifier given in Figure P4.4.
(a) Find the overall gain if $K_a=1000$.
(b) Find the sensitivity of the amplifier to changes in $K_{u r}$
(c) Find the overall gain if $K_a$ decreases to 900.
(d) Find the overall gain if $K_a$ is increased to 2000.
(e) Compare answers obtained in (a), (c) and (d).

Khoobchandra Agrawal

Numerade Educator

05:25

Problem 5

Large microwave antennas have become increasingly important for radio astronomy and satellite tracking. A large antenna with a diameter of $60 \mathrm{ft}$, for example, is subject to large wind gust torques. A proposed antenna is required to have an error of less than $0.10^{\circ}$ in a $35 \mathrm{mph}$ wind. Experiments show that this wind force exerts a maximum disturbance at the antenna of $200,000 \mathrm{ft} \mathrm{lb}$ at $35 \mathrm{mph}$, or the equivalent to 10 volts at the input $T_d(s)$ to the amplidyne. One problem of driving large antennas is the form of the system transfer function that possesses a structural resonance. The antenna servosystem is shown in Figure P4.5. The transfer function of the antenna, drive motor, and amplidyne is approximated by
$$
G(s)=\frac{\omega_n^2}{s\left(s^2+2 \zeta \omega_n s+\omega_n^2\right)}
$$
where $\zeta=0.707$ and $\omega_n=15$. The transfer function of the power amplifier is approximately
$$
G_1(s)=\frac{k_a}{\tau s+1},
$$
where $\tau=0.15$ second. (a) Determine the sensitivity of the system to a change of the parameter $k_a$. (b) The system is subjected to a disturbance $T_d(s)=10 / \mathrm{s}$. Determine the required magnitude of $k_a$ in order to maintain the steady-state error of the system less than $0.10^{\circ}$ when the input $R(s)$ is zero. (c) Determine the error of the system when subjected to a disturbance $T_d(s)=10 / s$ when it is operating as an open-loop system $\left(k_s=0\right)$ with $R(s)=0$.

Shital Rijal

Numerade Educator

03:11

Problem 6

An automatic speed control system will be necessary for passenger cars traveling on the automatic highways of the future. A model of a feedback speed control system for a standard vehicle is shown in Figure P4.6. The load disturbance due to a percent grade $\Delta T_d(s)$ is also shown. The engine gain $K_e$ varies within the range of 10 to 1000 for various models of automobiles. The engine time constant $\tau_e$ is 20 seconds. (a) Determine the sensitivity of the system to changes in the engine gain $K_e$. (b) Determine the effect of the load torque on the speed. (c) Determine the constant percent grade $\Delta T_d(s)=\Delta d / s$ for which the vehicle stalls (velocity $V(s)=0$ ) in terms of the gain factors. Note that since the grade is constant, the steady-state solution is sufficient. Assume that $R(s)=30 / s \mathrm{~km} / \mathrm{hr}$ and that $K_e K_1 \gg 1$. When $K_{\mathrm{g}} / K_1=2$, what percent grade $\Delta d$ would cause the automobile to stall?

Shoukat Ali

Other Schools

12:28

Problem 7

A robot uses feedback to control the orientation of each joint axis. The load effect varies due to varying load objects and the extended position of the arm. The system will be deflected by the load carried in the gripper. Thus, the system may be represented by Figure P4.7, where the load torque is $T_d(s)=D / s$. Assume $R(s)=0$ at the index position. (a) What is the effect of $T_d(s)$ on $Y(s)$ ? (b) Determine the sensitivity of the closed loop to $k_2$. (c) What is the steady-state error when $R(s)=1 / s$ and $T_d(s)=0$ ?

Rory Naguib

Numerade Educator

04:10

Problem 8

Consider the feedback control system given in Figure P4.8. Find the value of gain $K$ such that the step response reaches $99 \%$ of the final value in $1 s$. Also find steady-state error to a unit step input.

James Kiss

Numerade Educator

03:52

Problem 9

A useful unidirectional sensing device is the photoemitter sensor [15]. A light source is sensitive to the emitter current flowing and alters the resistance of the photosensor. Both the light source and the photoconductor are packaged in a single four-terminal device. This device provides a large gain and total isolation. A feedback circuit utilizing this device is shown in Figure P4.9(a), and the nonlinear resistance-current characteristic is shown in Figure P4.9(b) for the Raytheon CK1116. The resistance curve can be represented by the equation
$$
\log _{10} R=\frac{0.175}{(i-0.005)^{1 / 2}},
$$
where $i$ is the lamp current. The normal operating point is obtained when $v_{\mathrm{o}}=35 \mathrm{~V}$, and $v_{\text {in }}=2.0 \mathrm{~V}$. (a) Determine the closed-loop transfer function of the system. (b) Determine the sensitivity of the system to changes in the gain, $K$.

Khoobchandra Agrawal

Numerade Educator

08:12

Problem 10

For a paper processing plant, it is important to maintain a constant tension on the continuous sheet of paper between the wind-off and wind-up rolls. The tension varies as the widths of the rolls change, and an adjustment in the take-up motor speed is necessary, as shown in Figure P4.10. If the wind-up motor speed is uncontrolled, as the paper transfers from the wind-off roll to the wind-up roll, the velocity $v_0$ decreases and the tension of the paper drops $[10,14]$. The threeroller and spring combination provides a measure of the tension of the paper. The spring force is equal to $k_1 y$, and the linear differential transformer, rectifier, and amplifier may be represented by $e_0=-k_2 y$. Therefore, the measure of the tension is described by the relation $2 T(s)=k_1 y$, where $y$ is the deviation from the equilibrium condition, and $T(s)$ is the vertical component of the deviation in tension from the equilibrium condition. The time constant of the motor is $\tau=L_a / R_a$, and the linear velocity of the wind-up roll is twice the angular velocity of the motor, that is, $v_0(t)=2 \omega_0(t)$. The equation of the motor is then
$$
E_0(s)=\frac{1}{K_m}\left[\tau s \omega_0(s)+\omega_0(s)\right]+k_3 \Delta T(s),
$$
where $\Delta T=$ a tension disturbance. (a) Draw the closed-loop block diagram for the system, including the disturbance $\Delta T(s)$. (b) Add the effect of a disturbance in the wind-off roll velocity $\Delta V_1(s)$ to the block diagram. (c) Determine the sensitivity of the system to the motor constant $K_m$. (d) Determine the steadystate error in the tension when a step disturbance in the input velocity, $\Delta V_1(s)=A / s$, occurs.

Keshav Singh

Numerade Educator

02:07

Problem 11

One important objective of the paper-making process is to maintain uniform consistency of the stock output as it progresses to drying and rolling. A diagram of the thick stock consistency dilution control system is shown in Figure P4.11(a). The amount of water added determines the consistency. The block diagram of the system is shown in Figure P4.11(b). Let $H(s)=1$ and
$$
G_c(s)=\frac{K}{10 s+1}, \quad G(s)=\frac{1}{2 s+1} .
$$

Determine (a) the closed-loop transfer function $T(s)=Y(s) / R(s)$, (b) the sensitivity $S_K^T$, and (c) the steady-state error for a step change in the desired consistency $R(s)=A / s$. (d) Calculate the value of $K$ required for an allowable steady-state error of $2 \%$.

Manik Pulyani

Numerade Educator

01:59

Problem 12

(a) Compute the overall transfer function $\frac{Y(s)}{R(s)}$ for the system given in Figure P4.12.
(b) Compute the transfer function $\frac{Y(s)}{D(s)}$ for the same system.
(c) Obtain an expression for $Y(s)$ when there is both reference input and disturbance.

James Kiss

Numerade Educator

Problem 13

One form of a closed-loop transfer function is
$$
T(s)=\frac{G_1(s)+k G_2(s)}{G_3(s)+k G_4(s)} .
$$
(a) Use Equation (4.16) to show that [1]
$$
S_k^T=\frac{k\left(G_2 G_3-G_1 G_4\right)}{\left(G_3+k G_4\right)\left(G_1+k G_2\right)} .
$$
(b) Determine the sensitivity of the system shown in Figure P4.13, using the equation verified in part (a).

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04:25

Problem 14

A proposed hypersonic plane would climb to 100,000 feet, fly 3800 miles per hour, and cross the $\mathrm{Pa}-$ cific in 2 hours. Control of the aircraft speed could be represented by the model in Figure P4.14. Find the sensitivity of the closed-loop transfer function $T(s)$ to a small change in the parameter $a$.

Zulfiqar Ali

Numerade Educator

08:50

Problem 15

The steering control of a modern ship may be represented by the system shown in Figure P4.15 [16, 20]. Find the steady-state effect of a constant wind force represented by $T_d(s)=1 / s$ for $K=5$ and $K=25$. (a) Assume that the rudder input $R(s)$ is zero, without any disturbance, and has not been adjusted. (b) Show that the rudder can then be used to bring the ship deviation back to zero.

Donald Albin

Numerade Educator

05:22

Problem 16

Figure P4.16 shows the model of a two-tank system containing a heated liquid, where $T_0$ is the temperature of the fluid flowing into the first tank and $T_2$ is the temperature of the liquid flowing out of the second tank. The system of two tanks has a heater in the first tank with a controllable heat input $Q$. The time constants are $\tau_1=10 \mathrm{~s}$ and $\tau_2=50 \mathrm{~s}$. (a) Determine $T_2(s)$ in terms of $T_0(s)$ and $T_{2 d}(s)$. (b) If $T_{2 d}(s)$, the desired output temperature, is changed instantaneously from $T_{2 d}(s)=A / s$ to $T_{2 d}(s)=2 A / s$, where $T_0(s)=A / s$, determine the transient response of $T_2(t)$ when $G_c(s)=K=500$. (c) Find the steadystate error $e_{s s}$ for the system of part (b), where $E(s)=T_{2 d}(s)-T_2(s)$.

Jack Chen

Numerade Educator

06:03

Problem 17

A robot gripper, shown in part (a) of Figure P4.17, is to be controlled so that it closes to an angle $\theta$ by using a DC motor control system, as shown in part (b). The model of the control system is shown in part (c), where $K_m=30, R_f=1 \Omega, K_f=K_i=1, J=0.1$, and $b=1$. (a) Determine the response $\theta(t)$ of the system to a step change in $\theta_d(t)$ when $K=20$. (b) Assuming $\theta_d(t)=0$, find the effect of a load disturbance $T_d(s)=A / s$. (c) Determine the steady-state error $e_{\mathrm{ss}}$ when the input is $r(t)=t, t>0$. (Assume that $\left.T_d(s)=0.\right)$

Rashmi Sinha

Numerade Educator