Chapter 12, Robust Control Systems Video Solutions, Modern Control Systems

Problem 1

Interest in unmanned underwater vehicles (UUVs) has been increasing recently, with a large number of possible applications being considered. These include intelligence-gathering, mine detection, and surveillance applications. Regardless of the intended mission, a strong need exists for reliable and robust control of the vehicle. The proposed vehicle is shown in Figure P12.1(a) [13].
We want to control the vehicle through a range of operating conditions. The vehicle is 30 feet long with a vertical sail near the front. The control inputs are sternplane, rudder, and shaft speed commands. In this case, we wish to control the vehicle roll by using the stern planes. The control system is shown in Figure P12.1(b), where $R(s)=0$, the desired roll angle, and $T_d(s)=1 / s$. We select $G_c(s)=K(s+2)$, where $K=4$. (a) Plot $20 \log |T|$ and $20 \log \left|S_K^T\right|$ on a Bode diagram. (b) Evaluate $\left|S_K^T\right|$ at $\omega_B, \omega_{B / 2}$, and $\omega_{B / 4}(T(s)=Y(s) / R(s))$.

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Problem 2

A new suspended, mobile, remote-controlled videocamera system to bring three-dimensional mobility to professional NFL football is shown in Figure P12.2(a) [24]. The camera can be moved over the field, as well as up and down. The motor control on each pulley is represented by the system in Figure P12.2(b), where $\tau_1=20 \mathrm{~ms}$ and $\tau_2=2 \mathrm{~ms}$.
(a) Select $K$ so that $M_{p \omega}=1.84$. (b) Plot $20 \log |T|$ and $20 \log \left|S_K^T\right|$ on one Bode diagram. (c) Evaluate $\left|S_K^T\right|$ at $\omega_B, \omega_{B / 2}$, and $\omega_{B / 4 \cdot}$ (d) Let $R(s)=0$ and determine the effect of $T_d(s)=1 / s$ for the gain $K$ of part (a) by plotting $y(t)$.

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01:23

Problem 3

Magnetic levitation (maglev) trains may replace airplanes on routes shorter than 200 miles. The maglev train developed by a German firm uses electromagnetic attraction to propel and levitate heavy vehicles, carrying up to 400 passengers at $300-\mathrm{mph}$ speeds. But the $\frac{1}{4}$-inch gap between car and track is difficult to maintain $[7,12,17]$.
The air-gap control system is shown in Figure P12.3(a). The block diagram of the air-gap control system is shown in Figure P12.3(b). The compensator is
$$
G_c(s)=\frac{K(s+5)}{(s+10)} .
$$
(a) Find the range of $K$ for a stable system. (b) Select a gain so that the steady-state error of the system is zero for a step input command. (c) Find $y(t)$ for the gain of part (b). (d) Find $y(t)$ when $K$ varies $\pm 15 \%$ from the gain of part (b).

Carson Merrill

Numerade Educator

Problem 4

Computer control of a robot to spray-paint an automobile is accomplished by the system shown in
Figure P12.4(a) [1]. We wish to investigate the system when $K=1,10$, and 20 . (a) For the three values of $K$, determine $\zeta, \omega_n$, percent overshoot, settling time (with a $2 \%$ criterion), and steady-state error for a step input. Record your results in a table. (b) Determine the sensitivity $\left|S_K^r\right|$ for the three values of $K$. (c) Select the best of the three values of $K$. (d) For the value selected in part (c), determine $y(t)$ for a disturbance $T_d(s)=1 / s$ when $R(s)=0$.

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Problem 5

An automatically guided vehicle is shown in Figure P12.5(a) and its control system is shown in Figure P12.5(b). The goal is to track the guide wire accurately, to be insensitive to changes in the gain $K_1$, and to reduce the effect of the disturbance $[15,25]$. The gain $K_1$ is normally equal to 1 and $\tau_1=1 / 25$ second.
(a) Select a compensator $G_c(s)$ so that the percent overshoot to a step input is less than or equal to $10 \%$, the settling time (with a $2 \%$ criterion) is less than $100 \mathrm{~ms}$, and the velocity constant $K_v$ for a ramp input is 100 .
(b) For the compensator selected in part (a), determine the sensitivity of the system to small changes in $K_1$ by determining $S_{K_1}^r$ or $S_{K_1}^T$.
(c) If $K_1$ changes to 2 while $G_c(s)$ of part (a) remains unchanged, find the step response of the system and compare selected performance figures with those obtained in part (a).
(d) Determine the effect of $T_d(s)=1 / s$ by plotting $y(t)$ when $R(s)=0$.

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11:24

Problem 6

A roll-wrapping machine (RWM) receives, wraps, and labels large paper rolls produced in a paper mill $[9,16]$. The RWM consists of several major stations: positioning station, waiting station, wrapping station, and so forth. We will focus on the positioning station shown in Figure P12.6(a). The positioning station is the first station that sees a paper roll. This station is responsible for receiving and weighing the roll, measuring its diameter and width, determining the desired wrap for the roll, positioning it for downstream processing, and finally ejecting it from the station.
Functionally, the RWM can be categorized as a complex operation because each functional step (e.g., measuring the width) involves a large number of field device actions and relies upon a number of accompanying sensors.
The control system for accurately positioning the width-measuring arm is shown in Figure P12.6(b). The negative pole $p$ of the positioning arm is normally equal to 2 , but it is subject to change because of loading and misalignment of the machine. (a) For $p=2$, design a
compensator so that the complex roots are $s=-2 \pm$ $j 2 \sqrt{3}$. (b) Plot $y(t)$ for a step input $R(s)=1 / s$. (c) Plot $y(t)$ for a disturbance $T_d(s)=1 / s$, with $R(s)=0$. (d) Repeat parts (b) and (c) when $p$ changes to 1 and $G_c(s)$ remains as designed in part (a). Compare the results for the two values of the negative pole $p$.

Nathan Prins

Numerade Educator

Problem 7

The function of a steel plate mill is to roll reheated slabs into plates of scheduled thickness and dimension $[5,10]$. The final products are of rectangular plane view shapes having a width of up to $3300 \mathrm{~mm}$ and a thickness of $180 \mathrm{~mm}$.
A schematic layout of the mill is shown in Figure P12.7(a). The mill has two major rolling stands, denoted No. 1 and No. 2. These are equipped with large rolls (up to $508 \mathrm{~mm}$ in diameter), which are driven by high-power electric motors (up to 4470 $\mathrm{kW}$ ). Roll gaps and forces are maintained by large hydraulic cylinders.
Typical operation of the mill can be described as follows. Slabs coming from the reheating furnace initially go through the No. 1 stand, whose function is to reduce the slabs to the required width. The slabs proceed through the No. 2 stand, where finishing passes are carried out to produce the required slab thickness. Finally, they go through the hot plate leveller, which gives each plate a smooth finish.
One of the key systems controls the thickness of the plates by adjusting the rolls. The block diagram of this control system is shown in Figure P12.7(b). The plant is represented by
$$
G(s)=\frac{1}{s\left(s^2+4 s+5\right)} .
$$
The controller is a PID with two equal real zeros. (a) Select the PID zeros and the gains so that the closedloop system has two pairs of equal roots. (b) For the design of part (a), obtain the step response without a prefilter $\left(G_p(s)=1\right)$. (c) Repeat part (b) for an appropriate prefilter. (d) For the system, determine the effect of a unit step disturbance by evaluating $y(t)$ with $r(t)=0$.

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04:07

Problem 8

A motor and load with negligible friction and a voltage-to-current amplifier $K_a$ is used in the feedback control system, shown in Figure P12.8. A designer selects a PID controller
$$
G_c(s)=K_P+\frac{K_I}{s}+K_D s,
$$
where $K_P=5, K_l=500$, and $K_D=0.0475$.
(a) Determine the appropriate value of $K_a$ so that the phase margin of the system is $30^{\circ}$. (b) For the gain $K_a$, plot the root locus of the system and determine the roots of the system for the $K_a$ of part (a). (c) Determine the maximum value of $y(t)$ when $T_d(s)=1 / \mathrm{s}$ and $R(s)=0$ for the $K_a$ of part (a). (d) Determine the response to a step input $r(t)$, with and without a prefilter.

Khoobchandra Agrawal

Numerade Educator

04:23

Problem 9

A unity feedback system has a nominal characteristic equation
$$
q(s)=s^3+3 s^2+3 s+6=0 .
$$

The coefficients vary as follows:
$$
\begin{aligned}
& 2 \leq a_2 \leq 4, \quad 1 \leq a_1 \leq 4, \\
& 4 \leq a_0 \leq 5 .
\end{aligned}
$$

Determine whether the system is stable for these uncertain coefficients.

James Kiss

Numerade Educator

01:59

Problem 10

Future astronauts may drive on the moon in a pressurized vehicle, shown in Figure P12.10(a), that would have a range of 620 miles and could be used for missions of up to six months. Boeing Company engineers first analyzed the Apollo-era Lunar Roving Vehicle, then designed the new vehicle, incorporating improvements in radiation and thermal protection, shock and vibration control, and lubrication and sealants.
The steering control of the moon buggy is shown in Figure P12.10(b). The objective of the control design is to achieve a step response to a steering command with zero steady-state error, an overshoot less than $20 \%$, and a peak time less than 0.3 second with a $|u(t)| \leq 50$. It is also necessary to determine the effect of a step disturbance $T_d(s)=1 / s$ when $R(s)=0$, in order to ensure the reduction of moon surface effects. Using (a) a PI controller and (b) a PID controller, design an acceptable controller. Record the results for each design in a table. Compare the performance of each design. Use a prefilter $G_p(s)$ if necessary.

Samantha Baker

Numerade Educator

Problem 11

A plant has a transfer function
$$
G(s)=\frac{25}{s^2} .
$$

We want to use a negative unity feedback with a PID controller and a prefilter. The goal is to achieve a peak time of 1 second with ITAE-type performance. Predict the system overshoot and settling time (with a $2 \%$ criterion) for a step input.

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Problem 12

A three-dimensional cam for generating a function of two variables is shown in Figure P12.12(a). Both $x$ and $\theta$ may be controlled using a position control system $[19]$. The control of $x$ may be achieved with a DC motor and position feedback of the form shown in Figure P12.12(b), with the DC motor and load represented by
$$
G(s)=\frac{K}{s(s+p)(s+4)},
$$
where $1 \leq K \leq 3$ and $1 \leq p \leq 3$. Normally $K=2.5$ and $p=2$. Design an ITAE system with a PID controller so that the peak time response to a step input is less than 3 seconds for the worst-case performance.

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Problem 13

Consider the closed-loop second-order system
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{rr}
0 & 3 \\
-5 & -K
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
0 \\
1
\end{array}\right] r \\
& y=\left[\begin{array}{ll}
2 & 0
\end{array}\right] \mathbf{x}+[0] u .
\end{aligned}
$$

Compute the sensitivity of the closed-loop system to variations in the parameter $K$.

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