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Unformatted text preview: 862 AppendixF MATLAB Commands MATLAB Function (.m ﬁle) Description Page (s) pade Pade approximation for time delay 332 parallel Parallel connection of two systems 127 place Poleplacement 524, 547, 562, 568, 595
plot Plot function 34, 119, 122, 215 poly Form polynomial from its roots 115 printsys Print system in transfer function format 118 pzrriap ' Polezero map 137 residue Residues in partial fraction expansion 111, 115, 822 rlocfind Find root locus gain 304, 472 rlocus Root locus 276, 472, 534, 536
rltool Interactive root locus tool 350 roots Roots of a polynomial 162, 296, 514, 524
semilogx Semilog plot 102, 369, 572
semilogy Semilog plot 456 series Series connection of two systems 127, 572, 674 sqrt Square root 595 ss2tf State—space to transfer function 118, 121, 510, 513
ssZzp State—space to pole—zero conversion 121, 135 ss Conversion to statespace 44, 122, 507, 512, 515
ssdata Create a statespace model 681 step Step response 34, 44, 119, 162, 214, 682
thSS Transfer function to statespace 120, 497 tf22p Transfer function to polezero conversion 121, 135 tf Conversion to transfer function 34, 102, 127, 138, 154, 214, 534
tzero Transmission zeros 512, 514, 680 zeros Array of zeros 122 Solutions to Review
Questions CHAPTER 1 1. What are the main components of a feedback control system?
The process, the actuator, the sensor, and the controller. 2. What is the purpose of the sensor? .
To measure the output variable and, usually, to convert it to an electrical voltage. 3. Give three important properties of a good sensor.
A good sensor is linear (the output is proportional to the input signal) over
a large range of amplitudes and a large range of frequencies at its input, has
low noise, is unbiased, is easy to calibrate, and has low cost. The relative
values of these properties vary with the particular application. 4. What is the purpose of the actuator?
The actuator takes an input, usually electrical, and converts it to a signal such
as a force or torque that causes the process output to move or change over the required range. 863 64 Appendix G Solutions to Review Questions 5. Give three important properties of a good actuator. A good actuator has fast response and adequate power, energy, speed, torque,
and so on, to be able to cause the process output to meet the design speciﬁca
tions and is eﬁicient, light weight, small, cheap, and so on. As with sensors,
the relative value of these properties varies with the application. What is the purpose of the compensator? Give the input(s) and output(s)
of the compensator. The compensator is to take the sensor output (the input to the compensator)
and compute the control signal (the output of the compensator) to be sent to
the actuator. What physical variable(s) of a process can be directly measured by a Hall
effect sensor? A Hall effect device measures the strength of a magnetic ﬁeld and can be
most easily conﬁgured to measure relative positions of two bodies or relative
angles. What physical variable is measured by a tachometer?
A tachometer measures speed of rotation or angular velocity. Describe three different techniques for measuring temperature. In each case indicated below, it is important to realize that the devices men
tioned need to be calibrated and often corrected for nonlinearity in order to
give a reliable, accurate measure of temperature. (a) An early technique still used in many home thermostats is based on the
bimetallic strip composed of two strips of different metals that expand
with different coeﬁ‘icients with temperature. As a result, the strip bends
with temperature and the resulting motion can be used as a measure of
temperature. This principle was introduced in the eighteenth century
to maintain a constant length to a clock pendulum for precision time
keeping. (b) A technique related to the bimetallic strip is based on the fact that metals
with different work functions placed in contact will produce a voltage
that is proportional to temperature. Such a device is called a thermo
couple and is the basis of a standard laboratory technique for measuring
temperature. (c) A number of materials have electrical resistance that is dependent in a
monotonic way on temperature and a resistance bridge can be used with
one of these to indicate temperature. Such devices are called thermistors. (d) For high temperatures, it is well known that the color of the radiation
due to heat depends on temperature. A piece of iron placed in a ﬁre
will glow orange, and then red and will ﬁnally become white hot at
high temperatures. An instrument for measuring the frequency of the
radiation and thus the temperature is a pyrometer. (e) In ceramic kilns, cones of different materials that melt at different and
known temperatures are placed near the products in the kiln to indicate
when the design temperature has been reached. The potter watches until 10. AppendixG ChapterZ 865 the cone of importance begins to sag and then knows that the products
should be removed. These give a quantized measure of temperature. Why do most sensors have an electrical output, regardless of the physical
nature of the variable being measured? Electrical signals are the most easily manipulated and therefore most con~
trollers are electrical devices, either analog or digital. To provide the signal
input to such a device, the sensor needs to produce an electrical output. CHAPTER 2 1. What is a “freebody” diagram?
To write the equations of motion of a system of connected bodies, it is useful
to draw each body in turn with the inﬂuence of all other bodies represented by forces and torques on the body in question. A drawing of the collection
of such isolated bodies is called a “free—body diagram.” What are the two forms of Newton’s law? Translational motion is described by F 2 ma. Rotational motion is described
by M = Id. . Why is it convenient to write equations of motion in the statevariable form? It provides a standard way to describe the differential equations for any
dynamic system so that computeraided analysis can be carried out more
conveniently. It is also more convenient to analyze linear systems in terms
ofthe standard system description matrices. . For a structural process to be controlled such as a robot arm, what is the meaning of “collocated control”? “Noncollocated control”? When the actuator and the sensor are located on the same rigid body, the
control is said to be “collocated. ” When they are on different bodies that are
connected by springs, the control is “noncollocated. ” When, why, and by whom was the device named an “operational ampliﬁer”?
In a paper in 1947, Ragazzini, Randall, and Russell named the highgain,
widebandwidth ampliﬁer used in feedback to realize operational calculus
‘operations’ the operational ampliﬁer. . What is the major beneﬁt of having zero input current to an operational ampliﬁer?
With zero input current the ampliﬁer does not load the input circuit and thus the transfer function of the device is not dependent on the ampliﬁer
characteristics. Also, the analysis of the circuit is simpliﬁed in this case. . State Kirchhoff’s current law. The algebraic sum of all currents entering a junction or circuit is zero. State Kirchhoff’s voltage law. The algebraic sum of voltages around a closed path in an electric circuit is
zero. 866 Appendix G Solutions to Review Questions 9. 10. 11. 12. Why is it important to have a small value for the armature resistance, Ra, of an electric motor?
The armature resistance causes power loss when the armature current ﬂows and thus reduces the efficiency of the motor. What are the deﬁnition and units of the electric constant of a motor? A rotating motor produces a voltage (called the back emf) in its armature
proportional to the rotational speed. The electric constant K6 is the ratio of
this voltage to the speed so that e = K36. The units are voltsec/radian. What are the deﬁnition and units of the torque constant of an electric motor? When current, ia, ﬂows in the armature of an electrical motor, a torque, r,
is produced that is proportional to the current. The torque constant, K ,, IS
the constant of proportionality so that r = Ktia. The units are Newton meters/amp. Give the relationships for (a) heat ﬂow across a substance and (b) heat
storage in a substance. (a) Heat ﬂow is proportional to the temperature difference divided by the
thermal resistance, that is, 1
=—T—T.
q R(1 2) (b) The differential equation describing the heat storage is T—1
—Cq. where C is the thermal capacity of the material. 13. Name and give the equations for the three relationships governing ﬂuid ﬂow. Continuity: m 2 win — wom Force equilibrium: f = pA. 1
Resistance: w 2 E091 — p2)1/a 14. Why do we approximate a physical model of the plant (which is always nonlinear) with a linear model? Analysis and design of linear models is vastly simpler than with nonlinear
models. Furthermore, it has been shown (by Lyapunov) that if the linear
approximation is stable, then there is at least some region of stability for the nonlinear model. AppendixG Chapter3 867 CHAPTER 3 1. What is the deﬁnition of “transfer function”? The Laplace transform of the output of a linear, timeinvariant system, Y (s),
is proportional to the transform of its input, U (s). The function of pro
portionality is the transfer function, F(s), so that Y(s) = F (s)U (s). It is
assumed that all initial conditions are zero. 2. What are the properties of systems whose responses can be described by transfer functions? The system must be both linear (superposition applies) and timeinvariant
(the parameters do not vary with time). 3. What is the Laplace transform of f (t — A)1(t— A) if the transform of f (t) is F(s)?
£{f(t — x)1(t— M} = (“F(s). 4. State the Final Value Theorem (FVT). 10. If all the poles of sF(s) are in the left halfplane, then the ﬁnal value of f (t)
is given by [lim f(t) = lirr(1)sF(s).
A00 S—> . What is the most common use of the FVT in control? A standard test of a control system is the step response, and the FV T is used
to determine the steadystate error to such an input. Given a second—order transfer function with damping ratio g“ and natural
frequency a)", what is the estimate of the step response rise time? What
is the estimate of the percent overshoot in the step response? What is the
estimate of the settling time? These are given by t, E 1.8 / can ; M p is set by the damping ratio (see the curve
in Figure 3.28) and t, g 4.6/0. What is the major effect of an extra zero in the left halfplane on the second
order step response? Such a zero causes additional overshoot; and the closer the zero is to the
imaginary axis, the higher the overshoot. I f the zero is more than 6 times the
real part of the complex poles, the effect is negligible. . What is the most noticeable effect of a zero in the right halfplane on the step response of the second—order system?
Such a zero often causes an initial undershoot of the response. . What is the main effect of an extra real pole on the secondorder step response? A pole slows down the response and makes the rise time longer. The closer
the pole is to the imaginary axis, the more pronounced is the effect. I f the pole
is more than 6 times the real part of the complex poles, the effect is negligible. Why is stability an important consideration in control system design?
Almost any useful dynamic system must be stable to perform its function.
Feedback around a system that is normally stable can actually introduce instability so control designers must be able to assure the stability of their
designs. 868 Appendix G Solutions to Review Questions 11. 12. What is the main use of Routh’s criterion? With this method, we can ﬁnd (symbolically) the range of a parameter such
as the loop gain for which the system will be stable. Under what conditions might it be important to know how to estimate a
transfer function from experimental data? In many cases, the equations of motion are either extremely complex or not
known at all. Chemical processes such as a papermaking machine are often
of this kind. In these cases, if one wishes to build a good control, it is very
useful to be able to take transient data or steadystate frequency response
data and to estimate a transfer function from these. CHAPTER 4 1. Give three advantages of feedback in control.
(a) Feedback can reduce the steadystate error in response to disturbances.
(b) Feedback can reduce the steadystate error in tracking a reference. (C) Feedback can reduce the sensitivity of a transfer function to parameter
changes. (d) Feedback can stabilize an unstable process. Give two disadvantages of feedback in control. (a) Feedback requires a sensor that can be very expensive and may introduce
additional noise. (b) Feedback systems are often more difficult to design and operate than
open—loop systems. What is the main objective of introducing integral control?
Integral control will make the error to a constant input go to zero. It removes
the effects ofprocess noise bias. It cannot remove the effects of sensor bias. . What is the major objective of adding derivative control? Derivative control typically makes the system better damped and more stable. Why might a designer wish to put the derivative term in the feedback rather
than in the error path? When a reference input might include sudden changes, including it in the
derivative action might cause unnecessary large controls. What is integrator windup? If the plant actuator output signal saturates, then it may take a long time for
the error to be brought back to zero from an initial upset and during this
time the integrator output may grow or ‘wind up ’ much more than it would if
the system were linear. Special ‘antiwindup ’ circuits are designed to prevent
windup. A 7. 10. 11. 12. 13. 14. 15. AppendixG Chapter4 869 Why is an anti—windup circuit important?
When a control includes integral action and is subject to saturation, large inputs can cause large over shoots and slow recovering unless an antiwindup
circuit is included. Using the nonlinear saturation function having gain 1 and limits i1, sketch the block diagram of saturation for an actuator which has gain 7 and limits
$20. Ifthe output ofthe actuator is an“, and its input is u,,,, the control is given by 7 in
u,,,,, = 20sat < :0 > . A temperature control system is found to have zero error to a constant
tracking input and an error of 0.50 C to a tracking input that is linear in time,
rising at the rate of 400 C/sec. What is the system type of this control system
and what is the relevant error constant [K p or K U or etc.]? The system is type I and the K” is the ratio of input rate to error or K, =
40/5 = 80/ sec. What are the units of KP, K”, and K“ ?
Kp is dimensionless, KU is sec—1, and K, is sec—2. What is the deﬁnition of system type with respect to reference inputs? With only a polynomial of degree k reference input (no disturbances), the
type is the largest value of k for which the error is a constant. What is the deﬁnition of system type with respect to disturbance inputs? With only a polynomial of degree k disturbance input (no reference), the
type is the largest value of k for which the error is a constant. Why does system type depend on where the external signal enters the
system? Because the error depends on where the input enters, so does the value of
type. Give two reasons to use a digital controller rather than an analog controller.
(a) The control law is easier to change if the controller is digital. (b) A digital controller can perform logic and other nonlinear operations
much easier than an analog controller. (0) The hardware ofa digital controller can beﬁxed in the design before the
details of the actual control design is ﬁnished. Give two disadvantages to using a digital controller. (a) The bandwidth of a digital controller is limited by the possible sample
frequency. (b) The digital controller introduces noise by the quantization process. 870 Appendix G Solutions to Review Questions 16. Give the substitution in the discrete operator 2 for the Laplace operator s if the approximation to the integral in Eq. (4.108) is taken to be the rectangle
of height e(kT_,.) and base TS. CHAPTER 5 1. Give two deﬁnitions for the root locus. (a) The root locus is the locus of points in the splane where the equation
a(s) + Kb(s) = 0 has a solution. (b) The root locus is the locus of points in the s plane where the angle of
G(s) = b(s)/a(s) is 1800.
2. Deﬁne the negative root locus.
The negative root locus is the locus ofpoints where the equation a(s) —
Kb(s) = 0 has a solution or where the angle of G(s) = b(s)/a(s) is 00. 3. Where are the sections of the (positive) root locus on the real axis? Segments ofthe real axis to the left ofan odd number ofzeros and poles are
on the root locus. 4. What are the angles of departure from two coincident poles at s = —a on
the real axis. Assume there are no poles or zeros to the right of —a.
The loci depart at 390°. 5. What are the angles of departure from three coincident poles at s = —a on the real axis. Assume there are no poles or zeros to the right of —a.
The loci depart at i60O and 1800. 6. What is the requirement on the location of a lead compensation zero that
Wlll cause the locus to pass through a desired root location, r,, ? The angle from the lead zero to r,, must be such that the angle of the com
pensated transfer function at r,, is 1800. 7. What is the value of the compensator gain that will cause a closedloop pole
to be at r,, ? Ifthe compensated openloop transferfunction is KDG. then K = l/D(r(,)
G(m)l
8. What is the principal effect of a lead compensation on a root locus? The lead compensation generally causes the locus to bend toward the left half
plane, moving the dominant roots to a place ofhigher damping. 9. What is the principal effect of a lag compensation on a root locus in the
Vicinity of the dominant closedloop roots?
The lag compensation is normally placed so near the origin that it has neg ligible effect on the root locus in the vicinity of the dominant closedloop
roots. 10. 11. 12. 13. AppendixG Chapter6 871 What is the principal effect of a lag compensation on the steadystate error
to a polynomial reference input? A lag compensation normally raises the gain at s = 0 and thus increases
the velocity constant of a type I system and lowers the error to polynomial
inputs. Why is the angle of departure from a pole near the imaginary axis especially
important? If the locus starts toward the right half plane, then feedback will make the
system less stable. On the other hand if the locus departs toward the left
halfplane, then feedback is going to make the system more stable. Deﬁne a conditionally stable system. A system that becomes unstable as gain is reduced is considered to be con
ditionally stable. That is, its stability is conditioned on having an operating
compensator with at least a minimum value ofgain. Show, with a root locus argument. that a system having three poles at the
origin must be conditionally stable. With three poles at the origin, the angles of departure insure that two poles
leave the origin at 180““, i605, or, if there are poles on the real axis in the
right—halfplane, they may leave at 0"“, 1120‘“ which is to say that at least one
pole begins by moving into the right half—plane. As gain is reduced from the
operating level at least one root must pass into the right halfplane for gain
low enough and therefore the system must be conditionally stable. CHAPTER 6 Why did Bode suggest plotting the magnitude of a frequency response on
log—log coordinates? In log—log coordinates, the plotfor a rational transfer function can be well
guided by linear asymptotes and thus easily plotted and visualized. . Deﬁne a decibel. If a power ratio is P1/P2, then the measure in decibels is 1010g(P1/P2).
Because power is proportional to voltage squared, and a transfer function
would give a ratio ...
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