Essentials of Process Control phần 7 pps

CIIAPTIIK ~1: Frequency-Domain Dynamics 365MATLAB program for frequency response plots % Program “tempb0de.m ” uses Matlab to plot Bode, Nyquist % and Nichols plots for three heated tank

arguments of the individual components This is particularly useful when there is adeadtime element in the transfer function If you try to calculate the phase angle fromthe final total complex number, the FORTRAN subroutines ATAN and ATAN can-not determine what quadrant you are in ATAN has only one argument and thereforecan track the complex number only in the first or fourth quadrant So phase anglesbetween -90” and - 180” will be reported as +90” to 0” The subroutine ATAN2, since it has two arguments (the imaginary and the real part of the number), can accu-rately track the phase angle between + 180” and - 180°, but not beyond Getting thephase angle by summing the angles of the components eliminates all these problems.

A complex number G always has two parts: real and imaginary These parts can

be specified using the statement

G= CMPLX(X, Y)

where G = complex number

X = real part of G

Y = imaginary part of GComplex numbers can be added (G=GI +G2), multiplied (G=GI*G2), and divided(C=GZ/G2) The magnitude of a complex number can be found by using the state-ment

XX=CARS(G)

364 PAW 7‘1 IREE: Frequency-Domain Dynamics and Control

where XX = the real number that is the magnitude of G Knowing the compk~~ number G, we can find its real and imaginary parts by using the statements

A deadtime ,element Gtsj = eens

Gtiw, = e- iD” = cos(Dw) - isin ( 10.55)can be calculated using the statement

G=FMPLX(COS(D”W),-SIN(D*W)) where D = deadtime

W = frequency, in radians per minute in the FORTRAN program

The program in Table 10.1 illustrates the use of some of these complex FORTRANstatements

10.4.2 MATLAB Program for Plotting Frequency Response

Table 10.2 gives a MATLAB program that generates Nyquist, Bode, and Nichols

plots for the three-heated-tank process Figure 10.25 gives the plots The num and den polynomials are defined in the same way as in the root locus plots in Chapter 8 The frequency range of interest is specified by a logspace function from o = 0.01

to 10 radians/time The magnitudes and phase angles of Gfio> are found by using

the [magphase, w] = bode(num,den, w) statement The ultimate gain and frequency

are found by searching through the vector of phase angles until the - 180” point iscrossed

The real and the imaginary parts of Gfiwj for making.the Nyquist plot can be

found in two different ways The easiest is to use the {greaZ,gimag, w] = nyquist(num, den,w) statement Alternatively, the real part can be calculated from the product of

the magnitude and the cosine of the phase angle (in radians) at each frequency Thisterm-by-term multiplication is accomplished in MATLAB by using the * operation.Handling deadtime in MATLAB is not at all obvious Larry Ricker (private

communication, 1993) suggested a method for accomplishing it using the polyval

function As illustrated in the program given in Table 10.3, the numerator and nominator polynomials are evaluated at each frequency point Then these polynomi-als are divided at each frequency point by using the /division operation Finally,each of the resulting complex numbers is multiplied by the corresponding deadtime

de-exp(-d*w) at that frequency by the * multiplication operation.

Another problem encountered in systems with deadtime is large phase angles

As the curves wrap around the origin for higher frequencies, it becomes difficult

to track the phase angle MATLAB has a convenient solution to this problem: the

unwrap command As illustrated in Table 10.3, the phase angles are first calculated

for each frequency by using a “for” loop to run through all the frequency points

Then the unwrup(r-adiarzi) command is used to avoid the jumps in phase angle

CIIAPTIIK ~1: Frequency-Domain Dynamics 365

MATLAB program for frequency response plots

% Program “tempb0de.m ” uses Matlab to plot Bode, Nyquist

% and Nichols plots for three heated tank process

% Calculate magnitudes and phase angles at all frequencies

% using the “bode” function 70

ylabel(‘Log Modulus (dB) ‘) grid

text(2, - lO,[ ‘Ku= ‘,num2str(kult)]) text(2, -2O,[ ‘wu= ‘,num2str(wult)]) subplot(212)

semilogx( w,phase) xlabel( ‘Frequency (radians/hr) ‘) ylabel( ‘Phase Angle (degrees) ‘) grid

pause print -dps pjiglO2S.p~

70

% Make Nichols plot

%

elf plot(phase,db) title(‘Nicho1 Plot for Three Heated Tank Process’) rlabel( ‘Phase Angle (degrees) ‘)

.dahell ‘Log Modulus (dB) ‘) grid

pause

fl-ABLE 1 0 2 (CONTINUEU)

MATLAB program for frequency response plots

% Alternatively you can USE “nichols ” command

% [mugphase, w]=nichols(num.den, w);

print -dps -append pjiglO25

Of0

9’0 Make Nyquist plot

70

% Using the “nyquist” command

[grealgimag, w]=nyquist(num,den, w);

% Alternatively you can calculate the real and imaginary parts

70 from the magnitudes and phase angles

%radians=phase*3.1416/180;

%greal=mag.*cos(radians);

%gimag=mag.*sin(radians);

elf

axis( ‘square ‘)

plot(greal,gimag)

grid

title(‘Nyquist Plot for Three Heated Tank Process’)

xlabel( ‘Real(G)‘)

ylabel( ‘Imag(G) ‘)

pause

print -dps -append p$g1025

Bode Plot for Three Heated Tank Process

. .

I .

Frequency (radians/hr)

lo*

Frequency (radians/hr)

10*

FIGURE 10.25

Frequency response plots for three-heated-tank process.

Nichol Plot for Three Heated Tank Process 10

0

-10

5

9 - 2 0

2

=,

B

= - 3 0

g

-I

- 4 0

- 5 0

.y

0.2 0 - 0 2 6 2 - 0 4 E - 0 6 - 0 8 -1 Phase Angle (degrees) Nyquist Plot for Three Heated Tank Process I I I I I I I I ~ ~ , ~ ,

: I .

.

.~~~~ ~~.~~~~~ ~

Real(G)

FIGURE 10.25(CONTINUED)

Frequency response plots for three-heated-tank process

367

368 t+wrTtttwE: Frequency-Domain l!)ynamics and Control

TABI, IO.3

MATLAB program for deadtime Bode plots

70 Program “deadtime.m”

% Calculate frequency response of process with deadtime

% using the Larry Ricker method.

%

% Process is a first-order lag with time constant tau=lO minute,

70 a steady-state gain qf kp=l and a d=S minute deadtime.

% Evaluate numerator and denominator polynomials at all frequencies

% Note the “.I operator which does term by term division

title(‘Bode Plot for Deadtime Process’)

xlabel( ‘Frequency (radiandmin) ‘)

ylabel(‘Log Modulus (dB)‘)

text(0.02, - 10, ‘Tau=lO’)

text(0.02, - 15, ‘Deadtime=S’)

grid

CIIAITEK IO: Frequency-Domain Dynamics 369

‘rAIiI,1( 10.3 (CONTINUED)

MATLAB program For deadtime Bode plots

subplot{2 12)

semilogx(w,phase)

xlabel( ‘Frequency (rudiandmin) ‘)

ylabel( ‘Phase Angle (degrees) ‘)

grid

pause

print - dps pjig IO26.p.~

Bode Plot for Deadtime Process

:

10-l Frequency (radians/min)

.

.

.

.

.

.

.

.

.

.

.

.

.

10-l Frequency (radians/min)

loo

FIGURE 10.26

Figure

10.5

10.26 gives the resulting plots for a first-order lag and deadtime process

CONCLUSION

(10.56)

We have laid the foundation for our adventure into the China mainland We’ve learned the language,*and we have learned some useful graphical and computer soft-ware tools for working with it In the next chapter we apply all these to the problem

3 7 0 PARTTHREE: Frequency-Domain Dynamics and Control

of designing controllers for simple SlSO systems In later chapters tie use thesefrequency-domain methods to tackle some very complex and important problems:multivariable systems and system identification

PROBLEMS

10.1 Sketch Nyquist, Bode, and Nichols plots For the following transfer functions:

I(a) G(s) = (s + 1)3

1

@) G(s) = (s + l)(lOs + I)( loos 91)

Cc> G(s) = 1

sqs + 1)7-s + 1

if the frequency is changed and will be different for different kinds of processes.(a) How can the magnitude ratio MR and phase angle 8 be found from this cur\,e?(6) Sketch Lissajous curves for the following systems:

(9 (-4s) = K,,(ii) GC.7) = ;’ at o = 1 radian/time

1

at w = - radians/time

70

We show in Section 11.1 that closedloop stability can be determined from thefrequency response plot of the total openloop transfer function of the system (processopenloop transfer function and feedback controller G~~(i~jGc(i~)) This means that

a Bode plot of GM(~~~Gc(;~) is all we need As you remember from Chapter 10, thetotal frequency response cuive of a complex system is easily obtained on a Bode plot

by splitting the system into its simple elements, plotting each of these, and merelyadding log moduli and phase angles together Therefore, the graphical generation

of the required G M(;~)G~(~~) curve is relatively easy Of course, all this algebraicmanipulation of complex numbers can be even more easily performed on a digitalcomputer

11.1

NYQUIST STABILITY CRITERION

The Nyquist stability criterion is a method for determining the stability of systems inthe frequency domain It is almost always applied to closedloop systems A working,but not completely general, statement of the Nyquist stability criterion is:

!f (1 pd~r* plot qf’tht> total openloop transferfunction of the system GM(iwjGC(iw) wraps U~O~~IIC/ the (- 1, 0) point in the GMG~ plane as frequency w goes from zero to iilfirlit\: tilt’ s\~.vtcm is . closedloop unstable.

(a) Polar plots showing closedloop stability or instability (b) s plane location

of zeros and poles (c) Argument of (s - ~1)

The two polar plots sketched in Fig 11 la show that system A is closedloop unstablewhereas system B is closedloop stable

On the surface, the Nyquist stability criterion is quite remarkable We are able

to deduce something about the stability of the closedloop system by making a

fre-quency response plot of the c~perzlo~~p system! And the encirclement of the mystical,

374 PART THREE: Frequency-Domain Dynamics and Contd

magical ( I , 0) point somehow tells US that the system is closedloop unstable Thisall looks like blue smoke and mirrors However, as we will prove, it all goes back

to finding out if there are any roots of the closedloop characteristic equation in theRHP (positive real roots)

11.1.1 Proof

The Nyquist stability criterion is derived from a theorem of complex variables

If a complex function F,,, has Z zeros and P poles inside a certain area of the s

plane, the number N of encirclements of the origin that a mapping of a closed contour around the area makes in the F plane is equal to Z - P.

arg F(,) = =g(s - zl > + arg(s - z2) - arg(s - pl>

Remember, the argument of the product of two complex numbers zt and z2 is thesum of the arguments

~1~2 = (rle’Bi)(r2ei82) = rlr~ei(el+e2) q(z1z2) = 81 + 62

And the argument of the quotient of two complex numbers is the difference betweenthe arguments

arg 20

= 8, -02Z2

Let us pick an arbitrary point s on the contour and draw a line from the zero zt

to this point (see Fig Il lc) The angle between this line and the horizontal, 6,, , isequal to the argument of (s - ~1) Now let the point s move completely around thecontour The angle 6:, or arg(s - ~1) will increase by 2n radians Therefore, arg Fts)

will increase by 27r radians for each zero inside the contour.

’ ctiAi~n,K I I: Frequency-Domain Analysis of Closedloop Systems 3 7 5

A similar development shows that arg FQ, &creases by 27~ for each pole insidethe contour because of the negative sign in Eq ( 1 I 3) Two zeros and one pole meanthat arg Fc,, must show a net increase of +2+rr Thus, a plot of F&, in the complex Fplane (real part of F(,j versus imaginary part of Ftg)) must encircle the origin once

as s goes completely around the contour

InthissystemZ = 2andP = 1,andwehavefoundthatN = Z-P = 2 - l =

I Generalizing to a system with Z zeros and P poles gives the desired theorem[Eq ill]

If any of the zeros or poles are repeated, of order M, they contribute 27rM ans Thus, Z is the number of zeros inside the contour with Mth-order zeros counted

radi-M times And P is the number of poles inside the contour with Nth-order polescounted N times

B Application of theorem to closedloop stability

To check the stability of a system, we are interested in the roots or zeros of thecharacteristic equation If any of them lies in the right half of the s plane, the system

is unstable For a closedloop system, the characteristic equation is

So for a closedloop system, the function we are interested in is

If this function has any zeros in the RHP, the closedloop system is unstable

If we pick a contour that goes completely around the right half of the s planeand plot 1 + GM(~JGc(+ Eq (11.1) tells us that the number of encirclements of theorigin in this (1 + GMG~) plane will be equal to the difference between the zerosand poles of 1 + GMG~ that lie in the RHP Figure 11.2 shows a case where thereare two zeros in the RHP and no poles There are’two encirclements of the origin inthe (1 + GMG~) plane

We are familiar with making plots of complex functions like GM(iw)Gc(iw) inthe GMG~ plane It is therefore easier (but more confusing unless you are careful tokeep track of the “apples” and the “oranges”) to use the GMGC plane instead of the(1 + GMGc) plane The origin in the (1 + GMG~) plane maps into the (- 1,O) point

in the GMG~ plane since the real part of every point is moved to the left one unit

We therefore look at encirclements of the (- 1,O) point in the GMGC plane, instead

of encirclements of the origin in the (1 + GMGc) plane

After we map the contour into the G,+rGc plane and count the number N ofencirclements of the (- 1, 0) point, we know the difference between the number ofzeros 2 and the number of poles P that lie in the RHP We want to find out if thereare any zeros of the function Ftsj = 1 + GM(~,Gc(~) in the RHP Therefore, we mustfind the number of poles of Fts, in the RHP before we can determine the number ofzeros

The poles of the function F’(,s, = I + GMcsjGctsj are the same as the poles

of Gw(.s)Gc(s, It the process is openloop stable, there are no poles of GM(~,Gc(~) inthe RHF? Ai1 openloop-stable process means that P = 0 Therefore, the number N of

376 PARTTHREE: Frequency-Domain Dynamics and Control

FIGURE 11.2

-encirclements of the (- I, 0) point is equal to the number of zeros of I + C~M(,r~G~~s~

in the RHP for an openloop-stable process Any encirclement means the closedlqopsystem is unstable

If the process is openloop umtable, G,+,M(,rJ has one or more poles in the RHP, soF(s) = 1 + G~u(x)Gc(.s) also has one or more poles in the RHP We can find out howmany poles there are by solving for the roots of the openloop characteristic equation(the denominator of GM(.~J) Once the number of poles P is known, the number ofzeros can be found from Eq (I 1.6)

C, contour On the C, contour the variable s is a pure imaginary number Thus, s = iw

as o goes from 0 to +m Substituting io for s in the total openloop system transfer

function gives

We now let o take on values from 0 to +m and plot the real and imaginary parts ofG,+,M(;wjG~(iw) This, of course, is just a polar plot of GM~,~)Gc(~J, as sketched in Fig 11.3~.The plot starts (w = 0) at i K, on the positive real axis It ends at the origin, as o goes

to infinity, with a phase angle of -270”

CR contour On the CR contour,

R will go to infinity and 8 will take 011 values from +7r/2 through 0 to -7r/2 radians.Substituting Eq ( I I 9) into GM(,)Gc,(,,) gives

378 PAKTI’IIKEE: Frequency-Domain Dynamics and Control

As K becomes large, the + I term in the denominator can be neglected

lim GMG~ = lim

The magnitude of GMGc goes to zero as R goes to infinity Thus, the infinitely largesemicircle in the s plane maps into a point (the origin) in the GwGc plane (Fig 1 I 3b).The argument of G,+,Gc goes from -342 through 0 to +3~/2 radians

C- contour On the C- contour s is again equal to io, but now o takes on values from

co to 0 The GMG~ on this path is just the complex conjugate of the path with positivevalues of w See Fig 11.3~

The complete contour is shown in Fig 11.3d The bigger the value of K,, the fartherout on the positive real axis the GMG~ plot starts, and the farther out on the negative realaxis is the intersection with the GMGc plot.

If the GMGc plot crosses the negative real axis beyond (to the left of) the critical(- 1,O) point, the system is closedloop unstable There would then be two encirclements

of the (- I, 0) point, and therefore N = 2 Since we know P is zero, there must be two

zeros in the RHP

If the G,MGc plot crosses the negative real axis between the origin and the ( - 1.0)point, the system is closedloop stable Now N = 0, and therefore Z = N = 0 Thereare no zeros of the closedloop characteristic equation in the RHP

There is some critical value of gain K, at which the GMGC plot goes right throughthe (- I, 0) point This is the limit of closedloop stability See Fig 11.3e The \alue ofK, at this limit should be the ultimate gain K,, that we dealt with before in making rootlocus plots of this system We found in Chapter 8 that K, = 64 and o, = fi Let us

see if the frequency-domain Nyquist stability criterion studied in this chapter gives thesame results

380 f’AR7‘7‘tu<El:: Frequency-F)otnain Dynamics and Control

At the limit of closedloop stiihility

co= h=w,,This is exactly what we found from our root locus plot This is the value of frequency atthe intersection of the GMGc plot with the negative real axis

Equating the real part of Eq (11.13) to - 1 gives

(&)(I - 3w2)(1 - 3w2)” + (3w - LtJ3)2 = -l

($KJl - 3(3)1

[I - 3(3>]2 + (3 J? - 3 & =

-I

-Kc-= I 1$ J&=64=&

64This is the same ultimate gain that we found from the root locus plot

Remember also that for gains greater than the ultimate gain, the root locus plotshowed two roots of the closedloop characteristic equation in the RHP This is exactlythe result we get from the Nyquist stability criterion (N = 2 = 2) n

EXAMPLE 11.2 The system of Example 8.5 is second order

KC

It has two poles, both in the LHP: s = - 1 and s = - f Thus, the number of poles ofGMG~ in the RHP is zero: P = 0 Let us break up the contour around the entire RHP

into the same three parts used in the previous example

C, contour s = iw as o goes from 0 to +m This is just the polar plot of GM(iw)GC(iw).

CR contour s = Re” as R -+ ~0 and 8 goes from ~12 to -rf2,

GwGc(.,~ = (Re’” + I)(SR@ + 1)KC

382 PARTTHREE: Frequency-Domain Dynamics and Control

FIGURE 11.4 (CONTINUED)

(c) Nyquist plot of system with integrator

Thus, the infinite semicircle in the s plane again maps into the origin in the GrnGc plane,This happens for all transfer functions where the order of the denominator is greater thanthe order of the numerator

C- contour s = io as o goes from oo to 0 The GM(~~JGc(~~) curve for negative values

of w is the reflection over the real axis of the curve for positive values of o So wereally don’t need to plot the C- contour The C, contour gives us all the information weneed

The complete Nyquist plot is shown in Fig 11.4~ for several values of gain K,.Notice that the curves will never encircle the (- 1,O) point, even as the gain is madeinfinitely large This means that this second-order system can never be closedloop un-stable This is exactly what our root locus curves showed in Chapter 8

As the gain is increased, the GMG~ curve gets closer and closer to the (- 1,O) point.Later in this chapter we use the closeness to the (- 1,0) point as a specification for de-

E X A M P L E 1 1 3 If the openloop transfer function of the system has poles that lie on theimaginary axis, the s plane contour must be modified slightly to exclude these poles Asystem with an integrator is a common example

c*trAr~rt<r< I I: Frequency-Domain Analysis of Closedloop Systems 383

C, contour s = iw as o goes from rr) to K, with r() going to 0 and R going to +m.

iw( I + io7,,I)( I + io7,2) -K&T,,, + 7,~) - iK,( 1 - ~~1 TRAWL) (11.17)ZZ

W2(To, + To~)~ + @(I - 7,1TozW~)~

The polar plot is shown in Fig 11.4~

The CR contour maps into the origin in the GMG~ plane

C- contour The GMGc curve is the reflection over the real axis of the GMGC curve for

The maximum value of Kc for which the system is still closedloop stable can be

found by setting the real part of GM(iojGc(iw) equal to - 1 and the imaginary part equal

to 0 The results are

EXAMPLE I 1.4 Figure 11.5n shows the polar plot of an interesting system that hasconditional stability The system openloop transfer function has the form

Kc(7,,.s + I)

GMM(s’GC(s) = (T,,S + i)(T,2S + i)(T,j.S + l)(Td.S + 1) ( 11.23)

If the controller gain K, is such that the (- 1,O) point is in the stable region indicated inFig 1 I 5tr, the system is closedloop stable Let us define three values of controller gain:

System with conditional stability.

K1 = value of K, when 1 GMcjw, ,Gc(io,, 1 = 1

K2 = value of KC when IGM(iqjGC(iq)/ = 1

K3 = value of KC when [G~~~~,,G~~~~,,~ = 1The system is closedloop stable for two ranges of feedback controller gain:

K, < K, a n d K2 < K, < K3 (11.24)This conditional stability is shown on a root locus plot for this system in Fig 11.5b w

11.1.3 Representation

In Chapter 10 we presented three different kinds of graphs that were used to representthe frequency response of a system: Nyquist, Bode, and Nichols plots The Nyquiststability criterion was developed in the previous section for Nyquist or polar plots

<*IIAIWK I I: t;rcquency-uolnain Analysis (if Closedloop Systems 385

The critical point for closedloop stability was shown to be the (- I, 0) point on theNyquist plot

Naturally we also can show closedloop stability or instability on Bode andNichols plots The (- I, 0) point has a phase angle of - 180” and a magnitude ofunity or a log modulus of 0 decibels The stability limit on Bode and Nichols plots

(a) Nyquist plot

386 PA~TT~~REE: Frequency-Domain Dynamics and Control

is therefore the (0 dB, - 180”) point At the limit of closedloop stability

The system is closedloop stable if

L<OdB at8 = - 1 8 0 ”t9>-180” a t L = O d BFigure 11.6 illustrates stable and unstable closedloop systems on the three types ofplots

Keep in mind that we are talking about closedloop stability and that we arestudying it by making frequency response plots of the total openloop system trans-fer function These log modulus and phase angle plots are for the openloop system

So we could use the terminology h and 80 for our Bode and Nichols plots of theopenloop GM Gc frequency response plots

We consider openloop-stable systems most of the time We show how to dealwith openloop-unstable processes in Section 11.4

11.2

CLOSEDLOOP SPECIFICATIONS IN THE FREQUENCY DOMAIN

There are two basic types of specifications commonly used in the frequency main The first type, phase margin and gain margin, specifies how near the openloop

do-GM(iwJGc(iw) polar plot is to the critical (- 1,O) point The second type, maximum closedloop log modulus, specifies the height of the resonant peak on the log modu-lus Bode plot of the closedloop servo transfer function So keep the apples and theoranges straight We make openloop transfer function plots and look at the (- 1,O)point We make closedloop servo transfer function plots and look at the peak in thelog modulus curve (indicating an underdamped system) But in both cases we areconcerned with closedloop stability

These specifications are easy to use, as we show with some examples in tion 11.4 They can be related qualitatively to time-domain specifications such asdamping coefficient

Sec-11.2.1 Phase Margin

Phase margin (PM) is defined as the angle between the negative real axis and aradial line drawn from the origin to the point where the GMGc curve intersects theunit circle See Fig 11.7 The definition is more compact in equation form

If the GMGC polar plot goes through the (- 1,O) point, the phase margin is zero Ifthe GMGr polar nlot crosses the ncmtive renl auir tn the r;=ht nfthn /- 1 fi\ n-l-+ l hr.

phase margin is some positive angle The bigger the phase margin, the more stable

is the closedloop system A negative phase margin means an unstable closedloopsystem

Phase margins of around 45” are often used Figure 11.7 shows how phase gin is found on Bode and Nichols plots

mar-388 IMTTHKEB: Frequcl!lcy-Dornnit, Dynamics and Control

CIIA~TEK I I: Frequency-Domain Analysis of CIosedIoop Systems 3 8 9

Figure I 1.8 shows gain margins on Nyquist, Bode, and Nichols plots Gain marginsare sometimes reported in decibels

If the GMG~ curve goes through the critical (- 1,O) point, the gain margin isunity (0 dB) If the GMG~ curve crosses the negative real axis between the originand - 1, the gain margin is greater than 1 Therefore, the bigger the gain margin, themore stable the system is, i.e., the farther away from - 1 the curve crosses the realaxis Gain margins of around 2 are often used

A system must be third or higher order (or have deadtime) to have a ful gain margin Polar plots of first- and second-order systems do not intersect thenegative real axis

meaning-11.2.3 Maximum Closedloop Log Modulus (Ly)

The most useful frequency-domain specification is the maximum closedloop logmodulus The phase margin and gain margin specifications can sometimes give poorresults when the shape of the frequency response curve is unusual

For example, consider the Nyquist plot of a process sketched in Fig 11.9a, wherethe shape of the GMG~ curve gives a good phase margin but the curve still passesvery close to the (- 1,0) point The damping coefficient of this system would be quitelow This type of GMG~ curve is commonly encountered when the process has a largedeadtime Figure 11.9b shows a GMG~ curve that has a good gain margin but passestoo close to the (- 1,0) point These two cases illustrate that using phase or gainmargins does not necessarily give the desired degree of damping This is becauseeach of these criteria measures the closeness of the GMG~ curve to the (- LO) point

at only one particular spot

The maximum closedloop log modulus does not have this problem since it rectly measures the closeness of the GinGc curve to the (- 1,O) point at all frequen-cies The closedloop log modulus refers to the closedloop servo transfer function:

di-Y(s) _ G&c(s)

(s) 1 + G~(s,Gc(s)The feedback controller is designed to give a maximum resonant peak or hump inthe closedloop log modulus plot

All the Nyquist, Bode, and Nichols plots discussed in previous sections havebeen for openfoop system transfer functions G~~iw~Gc(iW) Frequency response plotscan be made for any type of system, openloop or closedloop The two closedlooptransfer functions that we derived in Chapter 8 show how the output Y(,) is affected

in a closedloop system by a setpoint input Yf$ and by a load Lt,, Equation (11.28)gives the closedloop servo transfer function Equation (11.29) gives the closedloopload transfer function

L(S) 1 + Gw.s)Gc(s)

390 PARTTHREE : Frequency-Domain Dynanlics and Control I

(b) Gain margin doesn’t work

FIGURE 11.9

Nyquist plots

1 -GM

Typical log modulus Bode plots of these two closedloop transfer functions are shown

in Fig 11.10~ If it were possible to achieve perfect or ideal control, the two idealclosedloop transfer functions would be

~IIAI~II:K I I : Frequency-Domain Analysis of Closedloop Systems 391

log w (a) Load and setpoint closedloop transfer functions

log w @r -Resonant frequency(c) Typical setpoint closedloop transfer functions

FIGURE 11.10

Closedloop log modulus curves.

In most systems, the closedloop servo log modulus curves move out to higherfrequencies as the gain of the feedback controller is increased The system has a

“wider bandwidth,” as the mechanical and electrical engineers say This is desirablesince it means a faster closedloop system Remember, the breakpoint frequency isthe reciprocal of the closedloop time constant

But the height of the resonant peak also increases as the controller gain is creased This means that the closedloop system becomes more underdamped Theeffects of increasing controller gain are sketched in Fig 11 lOc

in-3 9 2 r’A’AKrrriKr:E: Frequency-Domain Dynamics and Control

(a) Nichols chart

A commonly used maximum closedloop log modulus specification is +2 dB.The controller parameters are adjusted to give a maximum peak in the closedloopservo log modulus curve of +2 dB This corresponds to a magnitude ratio of 1.3 and

is approximately equivalent to an underdamped system with a damping coefficient

fre-A Nichols chart is a graph that shows what the closedlcop log modulus L,

and closedloop phase angle 8, are for any given openloop log modulus b andopenloop phase angle 130 See Fig I 1.1 la The graph is completely general and can