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4 Basic Properties

ofFeedback

A Perspective on the Properties of Feedback

A major goal of control design is to use the tools available to keep the error

small for any input and in the face of expected parameter changes. Al-

though in this book we will focus on the selection of the controller transfer

function, the control engineer must be aware that changes to the plant may

be possible that will greatly help control of the process. It is also the case

that the selection and location of a sensor can be very important. These

considerations illustrate the fact that control is a collaborative enterprise

and control objectives need to be considered at every step of the way from

concept to finished product. However, in this book, we consider mainlythe case of control of dynamic processes and begin with models that

can be approximated as linear, time-invariant, and described by transfer

functions. Discussion of the theoretical justification of this assumption is

deferred until Chapter 9, in which the theories of Lyapunov are introduced.

Given a model, the next step in the design is formulation of specifi-

cations of what it is that the control is required to do. While maintaining

the essential property of stability, the control specifications include both

static and dynamic requirements such as the following:

The permissible steady-state error in the presence of a constant orbias disturbance signal.

The permissible steady-state error while tracking a polynomial refer-

ence signal such as a step or a ramp.

The sensitivity of the system transfer function to changes in modelparameters.

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166

Feedback Control of Dynamic Systems, Fifth Edition,by Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini.

ISBN 0-13-149930-0. 2006 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.


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The permissible transient error in response to a step in either the ref-erence or the disturbance input.

The two fundamental structures for realizing controls are open-loop

control (Fig. 4.1) and closed-loop control, also known as feedback con-Open-loop and closed-loopcontrol trol (Fig. 4.3). Open-loop control is generally simpler, does not require a

sensor to measure the output, and does not, of itself, introduce stabil-

ity problems. Feedback control is more complex and may cause stabil-

ity problems but also has the potential to give much better performance

than is possible with open-loop control. If the process is naturally (open-

loop) unstable, feedback control is the only possibility to obtain a stable

system and meet any performance specifications at all. Before specific

design techniques such as the root locus are described, it is useful to

develop the equations of systems in general terms and to derive expres-

sions for the several specifications in order to have a language describing

the objectives toward which the design is directed. As part of this activity,

a comparison of open-loop to closed-loop control will expose both theadvantages and the challenges of feedback control.

Chapter Overview

This chapter begins with consideration of the basic equations of feed-

back and the comparison of a feedback structure with open-loop control.

In Section 4.1 the equations are presented first in general form and used

to discuss the effects of feedback on disturbance rejection, parame-

ter sensitivity, andcommand tracking. In Section 4.2 the steady-state

errors in response to polynomial inputs are analyzed in more detail. As

part of the language of steady-state performance, control systems are

frequently classified by type according to the maximum degree of the

input polynomial for which the steady-state error is a finite constant. InSection 4.3 the issue of dynamic tracking errors is introduced by con-

sidering a modification of the closed-loop characteristic equation using a

classical structure of proportional,integral, and differentialcontrol, the

PIDcontroller. This study will illustrate the interaction of steady-state with

transient performance and will set the tone for the more sophisticated de-

sign techniques to be described in later chapters. Finally, in Section 4.4,

several extensions of the material of the chapter are presented that are

interesting and important, but something of a distraction from the main

issues of the chapter. Issues discussed there are digital controllers, tuning

PID controllers, Truxels formula for error constants, and time-domain sen-

sitivity. The most important of these is the implementation of controllers

in digital form, introduced in Section 4.4.1. If time permits, considerationof this section is highly recommended because almost all modern con-

trollers are realized by digital logic. A more complete discussion of this

important issue is given in Chapter 8.

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168 Chapter 4 Basic Properties of Feedback

Figure 4.1

Open-loop control system

yController

Dol

Plant

G

Input shaping

Hr

U

W

R

4.1 The Basic Equations of Control

We begin by collecting the basic equations and transfer functions that will beused throughout the rest of the text. For the open-loop system of Fig. 4.1, if wetake the disturbance to be at the input of the plant, the output is given by

Yol = Hr DolGR + GW (4.1)

and the error, the difference between reference input and system output, isgiven by

Eol = R Yol (4.2)

= R [Hr DolGR + GW] (4.3)

= [1 Hr DolG]R GW (4.4)

= [1 Tol]R GW. (4.5)

The open-loop transfer function in this case is Hr DolG, for which we will usethe generic notation Tol(s) .

For feedback control, Figure 4.2 gives the basic structure of interest, butwith the disturbance and the sensor noise entering in unspecific ways. We willtake these signals to be at the inputs of the process and the sensor, respectively,as shown in Figure 4.3. The sensor transfer function is Hy and may show im-portant dynamics. However, the sensor can often be selected to be fast andaccurate. If this is the case, its transfer function can be taken to be a constantHy , with units ofvolts/unit-of-output. The reference input r has the same units

Figure 4.2

Feedback control system

YController

Dcl

Plant

G

SensorHy

Input shaping

Hr

W

V

R





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Section 4.1 The Basic Equations of Control 169

Figure 4.3

Basic feedback control

block diagram

Y

V

Controller

Dcl

Plant

G

Sensor

Hy

Input shaping

Hr

R

W

u

as the output, of course, and the input filters transfer function is Hr , also withunits ofvolts/unit-of-output. An equivalent block diagram is drawn in Fig. 4.4,with controller transfer function D(s) = Hr Dcland with the feedback transferfunction as the ratio H = Hy

Hr. It is standard practice, especially if Hy is con-

stant,toselectequalscalefactorssothat Hr = Hyand the block diagram can bedrawn as a unity feedback structure as shown in Figure 4.5. We will develop theequations and transfer functions for this standard structure. When we use these

equations later, it will be important to be sure that the preceding assumptionsactually apply. If the sensor has dynamics that cannot be ignored, for example,then the equations will need to be modified accordingly.

For the feedback block diagram of Figure 4.5, the equations for the outputand the control are

Ycl =DG

1 + DGR +

G

1 + DGW

DG

1 + DGV , (4.6)

U =D

1 + DGR

DG

1 + DGW

D

1 + DGV . (4.7)

Perhaps more important than these is the equation of the error,Ecl

= R Ycl

:

Ecl = R

DG

1 + DGR +

G

1 + DGW

DG

1 + DGV

(4.8)

=1

1 + DGR

G

1 + DGW+

DG

1 + DGV . (4.9)

Figure 4.4

Equivalent feedback block

diagram with Hr included

inside the loop

HrDclD G

W

R

Y

V

HHr

Hy





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Figure 4.5

Unity feedback system

when Hr = Hy and

letting D = Hr Dcl

W

R

Y

V

uController

D

Plant

G

This equation is simplified by the definition of the sensitivity function 1 andthe complementary sensitivity function Tas

=1

1 + DG(4.10)

and

T= 1 =DG

1 + DG. (4.11)

In terms of these definitions, the equation for the closed-loop error is

Ecl = R GW+ TV . (4.12)

For future reference, it is standard to define the transfer function around a loopas the loop gain, L(s). In the case of Fig. 4.4, we have L = DGH, for example.

4.1.1 Watts Problem of Disturbance Rejection

One of the early uses of the steam engine in Britain was in mining, to pumpwater out of mines and to haul wagons loaded with coal. In carrying out thesetasks, the steady-state speed of early engines would change substantially whenpresented with added torque caused by a new load. To correct the problem,Wattscompany introduced theflying ball governor shown in Fig. 1.11, wherebythe speed of the engine was fed back to the steam chest to change the torqueof the engine. We will illustrate the principles of operation of this feedbackinnovation through study of the simple equations of motion of an engine withspeedeand external load torque .

Equation (4.13)describes the dynamics of an engine with inertiaJ, viscousfrictionb , control u , and load torque (t) :

Je + be = A1u + A2. (4.13)

1 The reason for the name, coined by H. W. Bode, will be given shortly.





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If we take the Laplace transform of Eq. (4.13), let the velocity transform bee(s)and the transform of the load torque beT(s) , we obtain the transformedequations of open-loop speed control as

sJ e(s) + be(s) = A1U(s) + A2T(s), (4.14)

sJ e(s) + be(s) = A1[U(s) +A2

A1T(s)], (4.15)

s e(s) + e(s) = A[U+ W]. (4.16)

In deriving Eq. (4.16), we have defined the parameters = J/b,A = A1/b,and the disturbance variable to be W = A2

A1T . In transfer function form the

equation is

e(s) =A

( s + 1)U(s) +

A

( s + 1)W(s) (4.17)

= G(s)[U(s) + W(s)] (4.18)

= G(s)W(s) ifU(s) = 0. (4.19)

In the feedback case, with no reference input and with control proportional toerror as U = Kcle , the equations of proportional feedback control are

s e(s) + e(s) = A[Kcle + W], (4.20)

e(s) = G(s)Kcle(s) + G(s)W, (4.21)

[1 + GKcl]e(s) = GW, (4.22)

e(s) =G

1 + GKclW. (4.23)

In the open-loop case, if the control input is U(s) = 0 and W = wos , the finalvalue theorem gives2

ss = G(0)wo = Awo. (4.24)

To make the comparison with the closed-loop case, suppose that G(0) = 1,wo = 1, and justfor fun, we take the controller gainto beKcl = 99. The steady-state output in the open-loop case is ss =1, and in the closed-loop case it is

ss = 1

1+GKcl

s0= 11+99 =0.01. Thus the feedback system will have an error

to disturbance that is 100 times smaller than in the open-loop case. No wonder

Watts engine was a success!

2 We assume for the moment that G(0)is finite.





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This result is a particular case of application of the error equations. FromEq. (4.4) the error in the open-loop case is Eol = GW, and from Eq. (4.12)in the feedback case the error is Ecl = GW = Eol .Thus, in every case,the error due to disturbances is smaller by a factor in the closed-loop case

compared with the open-loop case.

Major advantage of feedback System errors to constant disturbances can be made smaller with feedbackthan they are in open-loop systems by a factor of = 11+DG(0) , where D G(0)is the loop gain at s = 0.

4.1.2 Blacks Problem: Sensitivity of System

Gain to Parameter Changes

During the 1920s, H. S. Black was working at Bell Laboratories to find a designfor an electronic amplifier suitable for use as a repeater on the long lines of thetelephone company. The basic problem was that electronic components driftedand he needed a design that maintained a gain with great precision in the face ofthese drifts. His solution was a feedback amplifier. To illustrate the advantageshe found, we compare the sensitivity of open-loop control with that of closed-loop control when a parameterchanges. The change might come about becauseof external effects such as temperature changes, because of aging, or simplyfrom an error in the value used for the parameter from the start. Suppose thatthe plant gain in operation differs from its original design value of A to beA + A , which represents a fractional change of A

A. The open-loop controller

gain is taken to be fixed at Dol(0) = Kol . In the open-loop case the nominaloverall gain is Tol = KolA,3 and the perturbed gain would be

Tol + Tol = Kol(A + A) = KolA + KolA = Tol + KolA.

Thus, Tol = KolA . To give a fair comparison, we compute the fractionalchangeinTol ,definedasTol/Tolfor a given fractional changeinA .Substitutingthe values, we find that

Tol

Tol=

KolA

KolA=

A

A. (4.25)

This means that a 10% error in A would yield a 10% error in Tol . H. W. BodeSensitivitycalled the ratio of T/T to A/Athe sensitivity of the gain with respect tothe parameter A . In the open-loop case, therefore, = 1.

The same change in A in the feedback case (Eq. (4.23)) yields the newsteady-state feedback gain

Tcl + Tcl =(A + A)Kcl

1 + (A + A)Kcl,

3 We use Tol and Tcl for the open-loop and closed-loop transfer functions, respectively. Theseare not to be confused with the transform of the disturbance torque Tol used earlier.





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whereTclis the closed-loop gain. We can compute the sensitivity of this closed-loopgaindirectlyusingdifferentialcalculus.Theclosed-loopsteady-stategainis

Tcl

=AKcl

1 + AKcl.

The first-order variation is proportional to the derivative and is given by

Tcl =dTcl

dAA.

The general expression for sensitivity of a transfer function Tto a parameterAis thus given by

Tcl

Tcl=

A

Tcl

dTcl

dA

A

A

= (sensitivity) AA .

From this formula the sensitivity is seen to be

TclA

= sensitivity ofTclwith respect toA

=

A

Tcl

dTcl

dA,

so

TclA =

A

AKcl/(1 + AKcl)

(1 + AKcl)Kcl Kcl(AKcl)

(1 + AKcl)2

=1

1 + AKcl. (4.26)

This result, which explains our use of the name sensitivity earlier, exhibits an-other major advantage of feedback:

Advantage of feedback In feedback control, the error in the overall transfer function gain is less sen-sitive to variations in the plant gain by a factor of = 11+DG compared witherrors in open-loop control.

As with the case of disturbance rejection, if the gain is such that 1 + DG = 100,a 10% change in plant gainAwill cause only a 0.1% change in the steady-stategain. The open-loop controller is 100 times more sensitive to gain changes thanthe closed-loop system with loop gain of 100.





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The results in this section so far have been computed for the steady-stateerror in the presence of constant inputs, either reference or disturbance. Verysimilar results can be obtained for the steady-state behavior in the presence ofsinusoidal reference and disturbance signals. This is important because there

are times when such signals naturally occur, as with a disturbance of 60 Hz dueto power-line interference in an electronic system, for example. The concept isalso important because more complex signals can be described as containingsinusoidal components over a band of frequencies and analyzed using super-position of one frequency at a time. For example, it is well known that humanhearing is restricted to signals in the frequency range of about 60 to 15,000 Hz.A feedback amplifier and loudspeaker system designed for high-fidelity soundmust accurately track any sinusoidal (pure tone) signal in this range. If we takethe controller in the feedback system shown in Fig. 4.5 to have the transferfunctionD(s)and we take the process to have the transfer functionG(s), thenthe steady-state open-loop gain at the sinusoidal signal of frequency owill be|D(jo)G(jo)|and the error of the feedback system will be

|E(jo)| = |R(jo)|

11 + D(jo)G(jo) .

Thus, to reduce errors to 1% of the input at the frequency o , we must make|1+DG| 100 or |D(jo)G(jo)|>100,andagoodaudioamplifiermusthavethisloopgainovertherange 260 215,000. We will revisit this conceptin Chapter 6 as part of the design based on frequency response techniques.

4.1.3 The Conflict with Sensor Noise

Finally, it must be noticed that the feedback system error has a term that is

missing from the open-loop case. This is due to the sensor, which is not neededin the open-loop case. The error due to this term is Ecl = TV and will be smallifTis small. Unfortunately, keeping both error due to Wand error due to Vsmall requires that in the one case be small and in the other case Tbe small.However, Eq. (4.11) shows that this is not possible.Thestandardsolutiontothisdilemma is frequency separation. The reference and the disturbance energiesare typically concentrated in a band of frequencies below some limitlets callit c . On the other hand, the sensor can usually be carefully designed so thatthe sensor noiseV is held small in the low-frequency band belowc , where theenergyin RandWare substantial.4 Thus thedesign shouldhavesmall whereRandWare large and whereV is small, and should then makeTbe small (andnecessarily larger) for higher frequencies, where sensor noise is unavoidable.It is compromises such as this that will occupy most of our attention in the

design of controllers in later chapters.

4 The moral of this is that money spent on a good sensor is usually money well spent.





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4.1.4 The RADAR Problem: Tracking a Time

Varying Reference

In addition to rejectingdisturbances, many systems arerequired to track a mov-

ing reference, for which the generic problem is that of a tracking RADAR. Ina typical system, electric pulses are sent from a parabolic antenna, the echoesfrom the target airplane are received, and an error between the axis of the an-tenna and the vector pointing to the target is computed. The control is requiredto command the antenna pointing angles in such a way as to keep these vectorsaligned. The dynamics of the systemare of central importance. A constant-gainopen-loop controller has no effect on the dynamics of the system for eitherreference or disturbance inputs. Only if an open-loop controller includes a dy-namic input filter, Hr (s) , can the dynamic response to the reference signal bechanged, but the plant dynamics will still determine the systems response todisturbances. On the other hand, feedback of any kind changes the dynamics ofthe system for both reference and disturbance inputs. In the case of open-loopspeed control, Eq. (4.17) shows that the plant dynamics are described by the

(open-loop) timeconstant. The dynamics with proportional feedback controlare described by Eq. (4.23), and the characteristic equation of this system is

1 + GKcl = 0, (4.27)

1 +AKcl

s + 1 = 0, (4.28)

s + 1 + AKcl = 0, (4.29)

s = 1 + AKcl

. (4.30)

Therefore, the closed-loop time constant, a function of the feedback gain Kcl ,is given by cl = 1 + AKcl

, and is decreased as compared with the open-loop

value. It is typically the case that closed-loop systems have a faster responseas the feedback gain is increased and, if there were no other effects, this isgenerally desirable. As we will see, however, the responses of higher ordersystems typically become lesswelldamped and eventuallywill become unstableas the gain is steadily increased. Thus a definite limit exists on how large wecan make the gain in our efforts to reduce the effects of disturbances andthe sensitivity to changes in plant parameters. Attempts to resolve the conflictbetween small steady-state errors and good dynamic response will characterizea large fraction of control design problems. The conclusion is as follows:

Property of feedback Feedback changes dynamicresponse and often makes a system both faster andless stable.





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4.2 Control of Steady-State Error: System Type

In the speed-control case study in Section 4.1 we assumed both reference anddisturbancesto be constants andalso tookD(0)andG(0)to be finite constants.

In this section we will consider the possibility that either or both ofD(s)andG(s)have poles at s = 0. For example, a well-known structure for the controlequation of the form

u(t) = kp + kI

te()d+ kD

de(t)

dt(4.31)

is called proportional plus integral plus derivative (PID) control, and the cor-responding transfer function is

D(s) = kp +kI

s+ kDs. (4.32)

In a number of important cases, the reference input will not be constant but

can be approximated as a polynomial in time long enough for the system toeffectively reach steady-state. For example, when an antenna is tracking the el-evation angle to a satellite, thetime history as thesatellite approaches overheadis an S-shaped curve as sketched in Fig. 4.6. This signal may be approximatedby a linear function of time (called a ramp function or velocity input) for asignificant time relative to the speed of response of the servomechanism. Inthe position control of an elevator, a ramp function reference input will directthe elevator to move with constant speed until it comes near the next floor. Inrare cases, the input can be approximated over a substantial period as having aconstant acceleration. In this section we consider steady-state errors in stablesystems with such polynomial inputs.

The general method is to represent the input as a polynomial in time and toconsider the resulting steady-state tracking errors for polynomials of differentdegrees. As we will see, the error will be zero for input polynomials below acertain degree, and will be unbounded for inputs of higher degrees. A stablesystem can be classified as asystem type,defined to be the degree of the poly-Definition of system typenomial for which the steady-state system error is a nonzero finite constant. Inthe speed-control example, proportional control was used and the systemhad aconstantfiniteerrortoastepinput,whichisaninputpolynomialofzerodegree;therefore this system is called a type zero (type 0) system. If the error to a rampor first-degree polynomial is a finite nonzero constant, such a system is calledtype one (type 1), and so on. System types can be defined with regard to eitherreference inputs or disturbance inputs, and in this section we will consider both

Figure 4.6

Signal for satellite tracking

Time (sec)

us





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Section 4.2 Control of Steady-State Error: System Type 177

classifications. Determining the system type involves calculating the transformof the system error and then applying the Final Value Theorem. As we will see,a determination of system type is easiest for the case of unity feedback, so wewill begin with that case.

4.2.1 System Type for Reference Tracking:

The Unity Feedback Case

In the unity feedback case drawn in Fig. 4.5, the system error is given byEq. (4.9). If we consider only the reference input alone and set W = V =0 ,then, using the symbol for loop gain, the equation is simply

E =1

1 + LR = R. (4.33)

To consider polynomial inputs, we letr(t) = tk1(t) ,forwhichthetransform

is R = 1sk+1. As a generic reference nomenclature, step inputs for which k = 0are called position inputs, ramp inputs for which k = 1 are called velocityinputs, and if k =2, the inputs are called acceleration inputs, regardless ofthe units of the actual signals. Application of the Final Value Theorem to theerror gives the formula

limt

e(t) = ess = lims0

E(s) (4.34)

=lims0

s1

1 + LR(s) (4.35)

=lims0

s1

1 + L1

sk+1. (4.36)

We consider first a system for whichLhas no poleat the originand a step inputfor which R(s) = 1

s. In this case, Eq. (4.36) reduces to

ess = lims0

s1

1 + L1s

(4.37)

=1

1 + L(0). (4.38)

We define such a system to betype 0and we define the constant L(0) = Kpas

the position error constant. IfLhas one pole at the origin, we could consider

both step and ramp inputs, but it is quite straightforward to evaluate Eq. (4.36)in a general setting. For this case, it is useful to be able to describe the behaviorof the controller and plant as s approaches 0. For this purpose, we collect allthe terms except the pole(s) at the origin into a function L(s), which is thus





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finite ats = 0 so that we can define the constantLo(0) = Knand write the looptransfer function as

L(s) =Lo(s)

sn . (4.39)

Forexample,ifLhasno integrator, thenn = 0.Ifthesystemhasoneintegrator,thenn = 1, and so forth. Substituting this expression into Eq. (4.36), we have

ess =lims0

s1

1 +Lo(s)

sn

1

sk+1 (4.40)

=lims0

sn

sn + Kn

1

sk. (4.41)

From this equation we can see at once that ifn > k , then e =0, and ifn < k ,thene . Ifn = k = 0, theness = 11+K0, and ifn = k = 0, theness =

1Kn

. Ifn = k = 0, the input is a zero-degree polynomial otherwise known as a step orposition, the constant Ko is called the position constant, written as Kp , and

the system is classified as type 0, as we saw before. Ifn = k = 1, the input isa first-degree polynomial, otherwise known as a ramp or velocity, the constantK1is called the velocity constant, written as Kv , and the system is classifiedtype 1. In a similar way, systems of type 2 and higher types may be defined.The type information can be usefully gathered in a table of errors as follows:

TABLE 4.1 Errors as a Function of System Type

Input

Type Step (Position) Ramp (Velocity) Parabola (Acceleration)

Type 0 11 + Kp

Type 1 0 1Kv Type 2 0 0 1

Ka

The most common case is that of simple integral control leading to a type1 system. In this case, the relationship betweenKvand the steady-state errorType 2 systemsto a ramp input is shown in Fig. 4.7. Looking back at the expression givenfor DcGin Eq. (4.39), we can readily see that the several error constants canbe calculated by counting the degree n of the poles of L at the origin (thenumber of integrators in the loop with unity gain feedback) and applying theappropriate one of the following simple formulas

Kp =lims0

L(s ), n = 0, (4.42)

Kv =lims0

s L(s ), n = 1, (4.43)

Ka =lims0

s2L(s ), n = 2. (4.44)





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Figure 4.7

Relationship between ramp

response and Kv

0 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

Time (sec)

r,yr y

1K

ess

EXAMPLE 4.1 System Type for Speed Control

Determine the system type and the relevant error constant for the speed-control exam-ple shown in Fig. 4.4, withproportional feedback given byD(s) = kp . The plant transferfunction is G = A

s +1 .

Solution. In this case,L = kp As +1 , and applying Eq. (4.42), we see that n = 0, as there is

no pole at s = 0. Thus the system istype 0, and the error constant is a position constantgiven by Kp = kpA.

EXAMPLE 4.2 System Type Using Integral Control

Determine the system type and the relevant error constant for the speed-control exam-ple shown in Fig. 4.4, with PI feedback. The plant transfer function is G = A

s +1 , and in

this case the controller transfer function is Dc = kp + kIs .

Solution. In this case,the transfer function isL(s) = A(kp s+kI)

s(s+1) ,andasaunityfeedbacksystem with a single pole ats = 0, the system is immediatelyseenas type 1. The velocityconstant is given by Eq. (4.43) to be Kv = lim

s0sL(s) = AkI.

The definition of system type helps us to identify quickly the ability ofa system to track polynomials. In the unity feedback structure, if the processparameters change without removing the pole at the origin in a type 1 system,the velocity constant will change, but the systemwill still have zero steady-stateerror in response to a constant input and will still be type 1. Similar statements

can be made for systems of type 2 or higher. Thus, we can say that system typeis a robust propertywith respect to parameter changes in the unity feedbackRobustness of system typestructure. Robustness is the major reason for preferring unity feedback overother kinds of control structure.





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Figure 4.8

Block diagram reduction

to an equivalent unity

feedback system

YD

H1

GUE

R

4.2.2 System Type for Reference Tracking:

The General Case

If the feedbackH = HyHr

in Fig. 4.4 is different from unity, the formulas given inthe unity feedback case do not apply, and a more general approach is needed.There are two immediate possibilities. In the first instance, if one adds andsubtracts 1.0 from H, as shown by block diagram manipulation in Fig. 4.8, thegeneral case is reduced to the unity feedback case and the formulas can beapplied to the redefined loop transfer function L = DG1+(H1)DG , for which the

error equation is again E = 11+L R = R .Another possibility is to develop formulas directly in terms of the closed-

loop transfer function, which we call the complementary sensitivity functionT(s) . From Fig. 4.4, the transfer function is

Y(s)

R(s)= T(s) =

DG

1 + H DG, (4.45)

and therefore the error is

E(s) = R(s) Y(s) = R(s) T(s)R(s).

The reference-to-error transfer function is thus

E(s)

R(s)= 1 T(s),

and the system error transform is

E(s) = [1 T(s)]R(s) = R.

We assumethat the conditions of theFinalValue Theorem aresatisfied, namelythat all poles ofsE(s)are in the left half plane. In that case the steady-state

error is given by applying the Final Value Theorem to get

ess = limt

e(t) =lims0

sE(s) =lims0

s[1 T(s)]R(s). (4.46)





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With a polynomial test input, the error transform becomes

E(s) =1

sk+1[1 T(s)],

and the steady-state error is given again by the Final Value Theorem:

ess = lims0

s1 T(s)

sk+1 =lim

s0

1 T(s)sk

. (4.47)

The result of evaluating the limit in Eq. (4.47) can be zero, a nonzero constant,or infinite. If the solution to Eq. (4.47) is a nonzero constant, the system isreferred to as type k . For example, if k =0 and the solution to Eq. (4.47) isa nonzero constant equal, by definition, to 11+Kp , then the system is type 0.Similarly, if k =1 and the solution to Eq. (4.47) is a nonzero constant, thenthe system is type 1 and has a zero steady-state error to a position input and

a constant steady-state error equal, by definition, to 1/Kv to a unit velocityreference input. Type 1 systems are by far the most common in practice. Asystem of type 1 or higher has a closed-loop DC gain of 1.0, which means thatT (0) = 1.

EXAMPLE 4.3 System Type for a Servo with Tachometer Feedback

Consider an electric motor position-control problem, including a nonunity feedbacksystem caused by having a tachometer fixed to the motor shaft and its voltage (whichis proportional to shaft speed) is fed back as part of the control. The parameters corre-sponding to Fig. 4.4 are

G(s) =1

s(s + 1)

,

D(s) = kp,

H(s) = 1 + kts.

Determine the system type and relevant error constant with respect to reference inputs.

Solution. The system error is

E(s) = R(s) Y(s)

= R(s) T(s)R(s)

= R(s)

DG(s)

1 + HDG(s) R(s)

=1 + (H(s) 1)DG(s)

1 + HDG(s)R(s).





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The steady-state system error from Eq. (4.47) is

ess = lims0

sR(s)[1 T(s)].

For a polynomial reference input, R(s) = 1/sk+1 , and hence

ess = lims0

[1 T(s)]

sk = lim

s0

1

sks(s + 1) + (1 + kts 1)kp

s(s + 1) + (1 + kts)kp

= 0 , k = 0,

=1 + ktkp

kp, k = 1.

Therefore the system is type 1and the velocity constant is K v = kp

1 + ktkp. Notice that

if kt > 0, this velocity constant is smaller than the unity feedback value of kp . Theconclusion is that if tachometer feedback is used to improve dynamic response, the

steady-state error is increased.

4.2.3 System Type with Respect to Disturbance

Inputs

In most control systems, disturbances of one type or another exist. In practice,thesedisturbancescansometimesbeusefullyapproximatedbypolynomialtimefunctions such as steps or ramps. This would suggest that systems also be clas-sified with respect to the systems ability to reject disturbance inputs in a wayanalogous to the classification scheme based on reference inputs. System type

with regard to disturbance inputs specifies the degree of the polynomial ex-pressing those input disturbances that the system can reject in the steady state.Knowing the system type, we know the qualitative steady-state response of thesystem to polynomial disturbance inputs such as step or ramp signals. Becausetype depends on the transfer function from disturbance to error, the systemtype depends on exactly where the disturbance enters into the control system.

The transfer function from the disturbance inputW(s)to the error E(s)is

E(s)

W(s)=

Y(s)

W(s)= Tw(s), (4.48)

because, if the reference is equal to zero, the output is the error. In a similar wayas for reference inputs, the system is type 0if a step disturbance input results

in a nonzero constant steady-state error and is type 1if a ramp disturbanceinput results in a steady-state value of the error that is a nonzero constant. Ingeneral, following the same approach used in developing Eq. (4.41), we assumethat a constant n and a function To,w(s) can be defined with the properties





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that To,w(0) = 1Kn,w and that the disturbance-to-error transfer function can bewritten as

Tw(s) = snTo,w(s). (4.49)

Then thesteady-stateerror to a disturbance input that is a polynomial of degreekis

yss =lims0

sTw(s)

1sk+1

=lims0

To,w(s)

sn

sk

. (4.50)

From Eq. (4.50), if n > k , then the error is zero, and if n < k , the error isunbounded. Ifn = k , the system is type k and the error is given by 1

Kn,w.

EXAMPLE 4.4 Satellite Attitude Control

Consider the model of a satellite attitude control system shown in Fig. 4.9(a), where

J = moment of inertia,

W = disturbance torque,

Hy = sensor gain, and

Dc(s) = the compensator.

With equal input filter and sensor scale factors, the system with PD control can beredrawnwithunityfeedbackasinFig.4.9(b)andwithPIDcontroldrawnasinFig.4.9(c).Assume that the control results in a stable system and determine the system types anderror responses to disturbances of the control system for

(a) System Fig. (4.9)(b) PD control

(b) System Fig. (4.9)(c) PID control

Solution.

(a) We see from inspection of Fig. 4.9(b) that, with two poles at the origin in the plant,the system is type 2 with respect to reference inputs. The transfer function fromdisturbance to error is

Tw(s) =

1

J s2 + kD s + kp (4.51)

= To,w(s), (4.52)





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Figure 4.9

Model of a satellite attitude

control: (a) basic system;

(b) PD control; (c) PID

control

(a)

R

D (s)

W

K

K

U

Js

1s1

uYu

(b)

R

W

Js21

Y

1.0

kpkD s

(c)

R

W

Js21

Y

1.0

kp kDss

kI

for which n = 0 and K o,w = kp . The system is type 0and the error constant is kp ,so the error to a unit disturbance step is 1kp

.

(b) With PID control, the forward gain has three poles at the origin, so this system istype 3 for reference inputs, but the disturbance transfer function is

Tw(s) =s

J s3 + kD s2 + kps + kI

, (4.53)

n = 1, (4.54)

To,w(s) =1

J s3 + kD s2 + kps + kI

, (4.55)

from which it follows that the system istype 1and the error constant is k I, so the

error to a disturbance ramp of unit slope will be 1kI .





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EXAMPLE 4.5 System Type for a DC Motor Position Control

Consider the simplified model of a DC motor in unity feedback as shown in Fig. 4.10,where the disturbance torque is labeled W(s).

(a) Use the proportional controller

D(s) = kp, (4.56)

and determine the system type and steady-state error properties with respect todisturbance inputs.

(b) Let the control be PI, as given by

D(s) = kp +kI

s, (4.57)

anddeterminethe systemtype andthe steady-state error properties fordisturbanceinputs.

Solution.(a) The closed-loop transfer function from W to E (where R = 0) is

Tw (s) =B

s(s + 1) + Akp

= s0To,w,

n = 0,

Ko,w =Akp

B.

Applying Eq. (4.50), we see that the system istype 0and the steady-state error to

a unit step torque input is ess = B

Akp . From the earlier section, this system is seen tobe type 1 for reference inputs and illustrates that system type can be different fordifferent inputs to the same system.

Figure 4.10

DC motor with unity

feedback

W(s)

A

B

1.0

D (s)s(ts1)

AYR





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(b) If the controller is PI, the disturbance error transfer function is

Tw (s) =Bs

s2( s + 1) + (kps + kI)A, (4.58)

n = 1, (4.59)

Kn,w =AkI

B, (4.60)

and therefore the system is type 1and the error to a unit ramp disturbance inputwill be

ess =B

AkI. (4.61)

4.3 Control of Dynamic Error: PID ControlWe have seen in Section 4.1 basic properties of feedback control, and in Sec-tion 4.2 we examined the steady state response of systems to polynomial refer-ence and disturbance input. At the end of Section 4.1 we observed that propor-tional control changed the time constant of the simple speed-control system.In this section the impact of more sophisticated controls on system character-istic equations is examined in the context of a standard controller structure.The most basic feedback is a constant Proportional to error. As we saw in Sec-tion 4.2, addition of a term proportional to the Integral of error has a majorinfluence on the system type and steady-state error to polynomials. The finalterm in the classical structure term proportional to the Derivative of error.Combined, these three terms form the classicalPIDcontroller, which is widelyThe PID (proportional-integral-

derivative) controller

used in the process and robotics industries.

4.3.1 Proportional Control (P)

When the feedback control signal is linearly proportional to the system error,we call the result Proportional feedback. This was the case for the feedbackused in the controller of speed in Section 4.1, for which the controller transferfunction is

U(s)

E(s)= Dc(s) = kp. (4.62)

As we saw in Section 4.1.4, the time constant of the feedback system was re-duced by a factor 1 + Akpby proportional control. If the plant is second order,as, for example, is a DC motor with nonnegligible inductance, then the transfer

function can be written as

G(s) =A

s2 + a1s + a2. (4.63)





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Section 4.3 Control of Dynamic Error: PID Control 187

In this case, the characteristic equation with proportional control is

1 + kpG(s) = 0, (4.64)

s2 + a1

s + a2

+ kp

=

0.

(4.65)The designer can control the constant term and the natural frequency, but notthe damping of this equation. Ifkp is made large to get adequate steady-stateerror, the damping may be much too low for satisfactory transient response.

4.3.2 Proportional plus Integral Control (PI)

Adding an integral term to the controller results in theProportional plus Inte-gral(PI) control equationProportional plus Integral

control

u(t) = kpe + kI t

t0

e()d, (4.66)

for which the Dc(s) in Fig. 4.5 becomes

U(s)

E(s)= Dc(s) = kp +

kI

s. (4.67)

This feedback has the primary virtue that, in the steady-state, its control outputcan be anonzeroconstant value even when the error signal at its input is zero.This comes about because the integral term in the control signal is a summationof all past values ofe(t). In fact, the integral term will not stop changing untilits input is zero, and therefore if the system reaches a stable steady state, theinput signal to the integrator will of necessity be zero. This feature means that aconstant disturbancew(see Fig. 4.4) can be canceled by the integrators output

even while the system error is zero.If PI control is used in the speed example, the transform equation for thecontroller is

U = kp(ref m) + kIref m

s, (4.68)

and the system transform equation with this controller is

( s + 1)m = A(kp +kI

s)(ref m) + T W. (4.69)

If we now multiply by s and collect terms, we obtain

( s2 + (Akp + 1)s + AkI)m = A(kps + kI)ref + AsW. (4.70)

Because thePIcontroller includes dynamics, use of this controller will changethe dynamic response in more complicated ways than the simple speed-upwe saw with proportional control. We can understand this by considering the





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characteristicequationofthespeed control with PIcontrol,asseen inEq.(4.70).The characteristic equation is

s2 + (Akp

+

1)s + Ak

I =

0.

(4.71)

The two roots of this equation may be complex and, if so, the natural frequency

is n =

AkI

, and the damping ratio is = Akp +12 n . These parameters areboth determined by the controller gains. If the plant is second order, then thecharacteristic equation is

1 +kps + kI

s

A

s2 + a1s + a2= 0, (4.72)

s3 + a1s2 + a2s + Akps + AkI = 0. (4.73)

In this case, the controller parameters can be used to set two of the coefficients,but not the third. For this we need derivative control.

4.3.3 Proportional-Integral-Derivative Control (PID)

The final term in the classical controller is derivative control,D, and the com-plete three-term controller is described by the transform equation we will use,namely,

Dc(s) =U(s)

E(s)

= kp +kI

s

+ kDs, (4.74)

or, equivalently, by the equation often used in the process industries, or

Dc(s) = kp[1 +1

TIs+ TDs], (4.75)

where the reset rate TI in seconds, and the derivative rate, TD , also inseconds, can be given physical meaning to the operator who must select valuesfor them to tune the controller. For our purposes, Eq. (4.74) is simpler to use.The effect of the derivative control term depends on the rate of change of theerror. As a result, a controller with derivative control exhibits an anticipatoryresponse, as illustrated by the fact that the output of a PD controller having aramperrore(t) = t1(t) input would lead theoutputofaproportionalcontroller

having the same input by kDkp

= TDseconds, as shown in Fig. 4.11.





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Section 4.3 Control of Dynamic Error: PID Control 189

Figure 4.11

Anticipatory nature of

derivative control

TD

u(t)

0 1 2 3 4 5

Time (sec)

PD

Proportional

Because of the sharp effect of derivative control on suddenly changingsignals, the D term is sometimes introduced into the feedback path as shownin Fig. 4.12(a), which would describe, for example, a tachometer on the shaftof a motor. The closed-loop characteristic equation is the same as if the termwere in the forward path, as given by Eq. (4.74) and drawn in Fig. 4.12(b), ifthe derivative gain is kD = kpktbut the zeros from the reference to the outputare different in the two cases. With the derivative in the feedback path, thereference is not differentiated, which may be a desirable result if the referenceis subject to sudden changes. With the derivative in the forward path, a stepchange in the reference input will, in theory, cause an intense initial pulse inthe control signal, which may be very undesirable.

To illustrate the effect of a derivative term on PID control, consider speedcontrol, but with the second-order plant. In that case, the characteristic equa-tion is

s2 + a1s + a2 + A(kp +kI

s+ kDs) = 0,

s3 + a1s2 + a2s + A(kps + kI + kDs

2) = 0. (4.76)

Collecting terms results in

s3 + (a1 + AkD)s2 + (a2 + Akp)s + AkI = 0. (4.77)

Figure 4.12

Alternative ways of

configuring rate feedback

(a)

(b)

1 kt,Ds

kpR G(s) Y

kp kDsR G Y





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The point here is that this equation, whose three roots determine the natureof the dynamic response of the system, has three free parameters in kP, kI,and kD , and by selection of these parameters, the roots can be uniquely and,in theory, arbitrarily determined. Without the derivative term, there would be

only two free parameters, but with three roots, the choice of roots of the char-acteristic equation would be restricted. To illustrate the effect more concretely,a numerical example is useful.

EXAMPLE 4.6 PID Control of Motor Speed

Consider the DC motor speed control with parameters 5

Jm = 1.13 102 N-m-sec2 /rad, b = 0.028 N-m-sec/rad, La = 101 henry,

Ra = 0.45 ohms, Kt = 0.067 N-m/amp, Ke = 0.067 V-sec/rad. (4.78)

Use the controller parameters

kp = 3, kI = 15 sec1, kD = 0.3 sec. (4.79)

Discuss the effects of P, PI, and PID control on the responses of this system to stepsin thedisturbanceand steps in thereferenceinput.Let theunused controllerparametersbe zero.

Solution. Figure 4.13(a) illustrates the effects of P, PI, and PID feedback on the step

disturbance response of the system. Note that adding the integral term increases theoscillatory behavior but eliminates the steady-state error, and that adding the derivativeterm reduces the oscillation while maintaining zero steady-state error. Figure 4.13(b)illustrates the effects of P, PI, and PID feedback on the step reference response, withsimilar results. The step responses can be computed by forming the numerator anddenominator coefficient vectors (in descending powers ofs ) and using the step functionin MATLAB. For example,afterthe values forthe parameters areentered,the followingcommands produce a plot of the response of PID control to a disturbance step:

numG = [La Ra 0];

denG = [Jm*La Ra*b + Ke*Ke + Ke*kD Ra*Ke*Ke + Ke*kp Ke*ki];

sysG = tf(numG,denG);

y = step(sysG).

5 These values have been scaled to measure time in milliseconds by multiplying the true La andJmby 1000 each.





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Section 4.4 Extensions to the Basic Feedback Concepts 191

6

8

6

4

2

0

2

4

Amplitude

0 1 2 3 4 5 6

Time (msec)

(a)

0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Amplitude

0 1 2 3 4 5 6

Time (msec)

(b)

P

PI

PID

P

PI

PID

Figure 4.13 Responses of P, PI, and PID control to (a) step disturbance input and (b) step reference input

4.4 Extensions to the Basic Feedback Concepts

4.4.1 Digital Implementation of Controllers

As a resultof therevolution in the cost-effectivenessof digital computers, therehasbeenan increasing useof digital logic in embeddedapplications, such ascon-trollers in feedback systems. With the formula for calculating the control signalin software rather than hardware, a digital controller gives the designer muchmore flexibility in making modifications to the control law after the hardwaredesign is fixed. In many instances, this means that the hardware and softwaredesigns canproceed almost independently, savinga great deal of time.Also, it iseasy to include binary logic and nonlinear operations as part of the function of

a digital controller. Special processors designed for real-time signal processingand known as digital signal processors, or DSPs, are particularly well suited foruse as real-time controllers. While, in general, the design of systems to use adigital processor requires sophisticated use of new concepts to be introducedin Chapter 8, such as the z-transform, it is quite straightforward to translate alinear continuous analog design into a discrete equivalent. A digital controllerdiffers from an analog controller in that the signals must besampledandquan-tized.6 A signal to be used in digital logic needs to be sampled first, and thenthe samples need to be converted by an analog-to-digital converter, or A/Dconverter,7 into a quantized digital number. Once the digital computer has cal-culated the proper next control signal value, this value needs to be convertedback into a voltage and held constant or otherwise extrapolated by a digital-to-

6 A controller that operates on signals that aresampled but notquantized is called discrete, whileone that operates on signals that are both sampled and quantized is called digital.

7 Pronounced A to D.





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analog converter, or D/A, in order to be applied to the actuator of the process.The control signal is not changed until the next sampling period. As a resultof the sampling, there are more strict limits on the speed or bandwidth of adigital controller than on analog devices. Discrete design methods that tend

to minimize these limitations are described in Chapter 8 . A reasonable ruleof thumb for selecting the sampling period is that during the rise time of theresponse to a step, the input to the discrete controller should be sampled ap-proximately six times. By adjusting the controller for the effects of sampling,the sampling can be adjusted to 2 to 3 times per rise time. This corresponds to asampling frequency that is 10 to 20 times the systems closed-loop bandwidth.The quantization of the controller signals introduces an equivalent extra noiseinto the system, and to keep this interference at an acceptable level, the A/Dconverter usually has an accuracy of 10 to 12 bits. For a first analysis, the effectsof the quantization are usually ignored. A simplified block diagram of a systemwith a digital controller is shown in Figure 4.14.

For this introduction to digital control, we will describe a simplified tech-nique for finding a discrete (sampled, but not quantized) equivalent to a givencontinuous controller. The method depends on the sampling period Ts beingshort enough that the reconstructed control signal is close to the signal that theoriginal analog controller would have produced. We also assume that the num-bers used in the digital logic have enough accurate bits so that the quantizationimplied in the A/D and D/A processes can be ignored. While there are goodanalysis tools to determine how well these requirements are met, here we willtest our results by simulation, following the well known advice that The proofof the pudding is in the eating.

Finding a discrete equivalent to a given analog controller is equivalent tofinding a recurrence equation for the samples of the control which will approx-imate the differential equation of the controller. The assumption is that wehave the transfer function of an analog controller and wish to replace it with a

discrete controller that will accept samples of the controller input,e(kTs ), froma sampler and, using past values of the control signal, u(kTs ), and present andpast samples of the input, e(kTs ), will compute the next control signal to besent to the actuator. As an example, consider a PID controller with the transferfunction

U(s) = (kp +kI

s+ kDs)E(s), (4.80)

Figure 4.14

Block diagram of a digital

controller

Sensor

H

U

T

YR A/De(kT) u(kT)

D/ADigital controller

D(z)

Clock

Plant

G





28/64


which is equivalent to the three terms of the time-domain expression

u(t) = kpe(t) + kI

t0

e()d+ kD e(t) (4.81)

= uP + uI + uD. (4.82)

Usingthefactthatthesystemislinear,thenextcontrolsamplecanbecomputedterm by term. The proportional term is immediate:

uP(kTs + Ts ) = kpe(kTs + Ts ). (4.83)

The integral term can be computed by breaking the integral into two parts andapproximating the second part, which is the integral over one sample period,as follows:

uI(kTs + Ts ) = kI

kTs +Ts0

e()d (4.84)

= kI

kTs0

e()d+ kI

kTs +TskTs

e()d (4.85)

= uI(kTs ) + {area undere()over one period} (4.86)

= uI(kTs ) + kITs

2{e(kTs + Ts ) + e(kTs )}. (4.87)

InEq.(4.87)theareainquestionhasbeenapproximatedbythatofthetrapezoidformed by the base Ts and vertices e(kTs + Ts ) and e(kTs ), as shown by thedashed line in Fig. 4.15.

The area can also be approximated by the rectangle of amplitude e(kTs )and width Ts , shown by the solid blue in Fig. 4.15, to give uI(kTs + Ts ) =

uI(kTs ) + kITs e(kTs ). These and other possibilities are considered in Chapter 8.In the derivative term, the roles ofu and e are reversed from integration,and the consistent approximation can be written down at once from Eq. (4.87)and Eq. (4.81) as

Ts

2{uD(kTs + Ts ) + uD(kTs )} = kD{e(kTs + Ts ) e(kTs )}. (4.88)

Figure 4.15

Graphical interpretation of

numerical integration

t

x

xf(x, u)

x dt

0 ti ti1

x(ti)

0

t





29/64


As with linear analog transfer functions, these relations are greatly simplifiedand generalized by the use of transform ideas. At this time, the discrete trans-form will be introduced simply as a prediction operator z, much as if we de-scribed the Laplace transform variable s as a differential operator. Here we

define the operator zas the forward shift operator in the sense that if U(z)isthe transform ofu(kTs ), then zU(z)will be the transform ofu(kTs + Ts ). Withthis definition, the integral term can be written as

zUI(z) = UI(z) + kITs

2 [zE(z) + E(z)] , (4.89)

UI(z) = kITs

2z + 1z 1

E(z), (4.90)

and from Eq. (4.88) the derivative term becomes the inverse

UD(z) = kD2

Ts

z 1

z + 1

E(z). (4.91)

The complete discrete PID controller is thus described by

U(z) =

kp + kI

Ts

2z + 1z 1

+ kD2

Ts

z 1z + 1

E(z). (4.92)

Comparing the two discrete equivalents of integration and differentiation withthe corresponding analog terms, it is seen that the effect of the discrete ap-proximation in the z -domain is as if everywhere in the analog transfer functionthe operator s has been replaced by the composite operator 2

Ts

z1z+1 . This is the

trapezoid rule8 of discrete equivalents:

Trapezoid RuleThe discrete equivalent toDa (s)isDd(z) = Da

2Ts

z 1z + 1

. (4.93)

EXAMPLE 4.7 Discrete Equivalent

Find the discrete equivalent of the analog controller with transfer function

D(s) =U(s)

E(s)=

11s + 13s + 1

, (4.94)

using the sample period Ts = 1.

8 The formula is also called Tustins Method after the English engineer who used the techniqueto study the responses of nonlinear circuits.





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Solution. The discrete operator is 2(z 1)z + 1 , and thus the discrete transfer function is

Dd(z) =U(z)

E(z)= D(s)s= 2

Ts

z1

z+1

(4.95)

=

11

2(z 1)z + 1

+ 1

3

2(z 1)z + 1

+ 1

. (4.96)

Clearing fractions, we get the discrete transfer function

Dd(z) =U(z)

E(z)=

23z 217z 5

. (4.97)

Converting the discrete transfer function to a discrete difference equation by using thedefinition of zas the forward shift operator is done as follows: First we cross-multiply

in Eq. (4.97) to obtain(7z 5)U(z) = (23z 21)E(z), (4.98)

and interpreting zas a shift operator, we find that this is equivalent to the differenceequation9

7u(k + 1) 5u(k) = 23e(k + 1) 21e(k), (4.99)

where we have replaced kTs + Ts with k + 1 to simplify the notation. To compute thenext control at time k Ts + Ts , therefore, we solve the difference equation

u(k + 1) =57

u(k) +237

e(k + 1) 217

e(k). (4.100)

Now lets apply these results to a control problem. Fortunately, MATLABprovides us with the Simulink capability to simulate both continuous and dis-crete systems, allowing us to compare the responses of the systems with con-tinuous and discrete controllers.

EXAMPLE 4.8 Equivalent Discrete Controller for Speed Control

A motor speed control is found to have the plant transfer function

Y

U=

45(s + 9)(s + 5)

. (4.101)

9 The process is similar to that used in Chapter 3 to find the ordinary differential equation towhich a rational Laplace transform corresponds.





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A PI controller designed for this system has the transfer function

D(s) =U

E= 1.4

s + 6s

. (4.102)

The closed-loop system has a rise time of about 0.2 sec and an overshoot of about20%. Design a discrete equivalent of this controller, and compare the step responsesand control signals of the two systems. (a) Compare the responses if the sample periodis 0.07, which is about three samples per rise time. (b) Compare the responses with asample period ofTs = 0.035, which corresponds to about six samples per rise time.

Solution.

(a) Using the substitution given by Eq. (4.93), the discrete equivalent for Ts = 0.07 is

given by replacing s by s 20.07z 1z + 1 in D(s)as follows:

Dd

(z) =

1.

4

2.07

z 1z + 1

+ 6

2.07

z 1z + 1

,

(4.103)

= 1.42(z 1) + 6 0.07(z + 1)

2(z 1), (4.104)

= 1.41.21z 0.79

(z 1). (4.105)

On the basis of this expression, the equation for the control is (the sample periodis suppressed)

u(k + 1) = u(k) + 1.4 [1.21e(k + 1) 0.79e(k)]. (4.106)

(b) For Ts = 0.035, the discrete transfer function is

Dd = 1.41.105z 0.895

z 1 , (4.107)

for which the difference equation is

u(k + 1) = u(k) + 1.4[1.105 e(k + 1) 0.895 e(k)].

A Simulink block diagram for simulating the two systems is given in Fig. 4.16, andplots of the step responses are given in Fig. 4.17(a). The respective control signals areplotted in Fig. 4.17(b). Notice that the discrete controller for Ts = 0.07 results in asubstantial increase in the overshoot in the step response, while with Ts =0.035, the

digital controller matches the performance of the analog controller fairly well.For controllers with many poles and zeros, making the continuous-to-discrete sub-

stitution called for in Eq. (4.93) can be very tedious. Fortunately, MATLAB providesa command that does all the work. If one has a continuous transfer function given by





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Step

MuxControl

ss6

PI Control1.4

Slider Kc

s99

Tau 1

Mux1Output

s55

Tau 2

z1

1.21z0.79Discrete

PI control 1.4

Slider Kd

s9

9Tau 1

s5

5Tau 2

Figure 4.16 Simulink block diagram to compare continuous and discrete controllers

Dc(s) = numD

denD represented in MATLAB as sysDa=tf(numD,denD), then the discreteequivalent with sampling period Ts is given by

sysDd = c2d (sysDa, Ts , 't'). (4.108)

In this expression, of course, the polynomials are represented in MATLAB form. Thelast parameter in thec2dfunction given by't'calls for the conversion to be done usingthe trapezoid method. The alternatives can be found by asking MATLAB forhelp c2d.For example, to compute the polynomials for Ts = 0.07 for the preceding example, thecommands would be

numDa = [1 6];

denDa = [1 1];

sysDa = tf(numD,denD)

sysDd = c2d( sysDa,0.07,'t')

0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (sec)

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (sec)

(b)

0

0.5

1.0

1.5

2.0

2.5

Continuous controller

Digital controller (T0.07 sec)

Discrete controller (T0.035 sec)Continuous controller

Digital controller (T0.07 sec)

Discrete controller (T0.035 sec)

Figure 4.17 Comparison plots of a speed-control system with continuous and discrete controllers: (a) outputresponses; (b) control signals





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4.4.2 ZieglerNichols Tuning of PID Regulators

As we will see in later chapters, sophisticated methods are available to developa controller that will meet steady-state and transient specifications for both

tracking input references and rejecting disturbances. These methods requirethat the designer have either a dynamic model of the process in the form ofequations of motion or a detailed frequency response over a substantial rangeof frequencies. Either of these data can be quite difficult to obtain, and the dif-ficulty has led to the development of sophisticated techniques of system modelidentification. Engineers early on explored ways to avoid these requirements.

Callenderetal.(1936)proposedadesignforthewidelyusedPIDcontrollerby specifying satisfactory values for the controller settings based on estimatesof the plant parameters that an operating engineer could make from experi-ments on the process itself. The approach was extended by J. G. Ziegler andN. B. Nichols (1942, 1943) who recognized that the step responses of a largenumber of process control systems exhibit a process reaction curvelike thatshown in Fig. 4.18, which can be generated from experimental step response

data. The S-shape of the curve is characteristic of many systems and can beapproximated by the step response ofTransfer function for a

high-order system with a

characteristic process

reaction curve Y(s)

U(s)=

Aestd

s + 1, (4.109)

which is a first-order system with a time delay of tdseconds. The constants inEq. (4.109) can be determined from the unit step response of the process. If atangent is drawn at the inflection point of the reaction curve, then the slope ofthe line is R = A/and the intersection of the tangent line with the time axisidentifies the time delay L = td.

Ziegler and Nichols gave two methods for tuning the PID controller forsuchamodel.InthefirstmethodthechoiceofcontrollerparametersisdesignedTuning by decay ratio of 0.25to result in a closed-loop step response transient with a decay ratio of approxi-

Figure 4.18

Process reaction curve

t

y(t)

LtdLag

t

A

tA

SlopeRAt reaction rate





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Figure 4.19

Quarter decay ratio

0.25

1 Period

t

y(t)

mately 0.25. This means that the transient decays to a quarter of its value afterone period of oscillation, as shown in Fig. 4.19. A quarter decay correspondsto = 0.21 and is a reasonable compromise between quick response and ade-

quate stability margins. The authors simulated the equations for the system onan analog computer and adjusted the controller parameters until the transientsshowed the decay of 25% in one period. The regulator parameters suggestedby Ziegler and Nichols for the controller terms, defined by

Dc(s) = kp(1 +1

TIs+ TDs), (4.110)

are given in Table 4.2.

TABLE 4.2 ZieglerNichols Tuning for the RegulatorD(s) = kp(1 + 1/TIs + TD s) , for a decay ratio of 0.25

Type of Controller Optimum Gain

Proportional kp = 1/RL

PI

kp = 0.9/RL,TI = L/0.3

PID

kp = 1.2/RL,TI = 2L,TD = 0.5L

Inthe ultimate sensitivity method, the criteria for adjusting the parametersTuning by evaluation at limit ofstability (ultimate sensitivity

method)are based on evaluating the amplitude and frequency of the oscillations of

the system at the limit of stability rather than on taking a step response. Touse the method, the proportional gain is increased until the system becomesmarginally stable and continuous oscillations just begin, with amplitude limitedby the saturation of the actuator. The corresponding gain is defined as Ku





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Figure 4.20

Determination of the

ultimate gain and period

Processry

Kue

(called the ultimate gain) and the period of oscillation isPu(called the ultimateperiod). These are determined as shown in Figs. 4.20 and 4.21. Pu should bemeasured when the amplitude of oscillation is as small as possible. Then thetuning parameters are selected as shown in Table 4.3.

TABLE 4.3 ZieglerNichols Tuning for the RegulatorDc(s) = kp(1 + 1/TIs + TDs) , Based on the UltimateSensitivity Method

Type of Controller Optimum Gain

Proportional kp

= 0.5Ku

PI

kp = 0.45Ku,

TI = Pu1.2

PID

kp = 0.6Ku,

TI =12 Pu,

TD =18 Pu

Experience has shown that the controller settings according to ZieglerNichols rules provide acceptable closed-loop response for many systems. The

process operator will often do final tuning of the controller iteratively on theactual process to yield satisfactory control.10

Figure 4.21

Neutrally stable system

Pu

t

y(t)

10 Tuning of PID controllers has been the subject of continuing study since 1936. A modernpublication on the topic is H. Panagopoulous, K. J. strm, and T. Hagglund, Proceedings ofthe American Control Conference, San Diego, CA, June 1999.





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Figure 4.22

A measured process

reaction curve

0.0 100.0 200.0 300.0 400.0

Time (sec)

1.2

1.0

0.8

0.6

0.4

0.2

0

y

EXAMPLE 4.9 Tuning of a Heat Exchanger: Quarter Decay Ratio

Consider the heat exchanger of Example 2.13. The process reaction curve of this systemis shown in Fig. 4.22. Determine proportional andPI regulatorgains forthe system usingthe ZeiglerNichols rules to achieve a quarter decay ratio. Plot the corresponding stepresponses.

Solution. From the process reaction curve, we measure the maximum slope to beR = 190 and the time delay to be L

= 13 sec. According to the ZeiglerNichols rules ofTable 4.2 the gains are as follows:

y

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

y

400.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0300.0200.0100.00.0

Time (sec)

(b)

400.0300.0200.0100.00.0

Time (sec)

(a)

PI PI

Proportional

Proportional

Figure 4.23 Closed-loop step responses





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Proportional :kp =1

RL=

9013

= 6.92,

PI : kp =0.9RL

= 6.22 and TI =L

0.3 =

130.3

= 43.3

Figure 4.23(a) shows the step responses of the closed-loop system to these two regula-tors. Note that the proportional regulator results in a steady-state offset, while the PIregulator tracks the step exactly in the steady-state. Both regulators are rather oscilla-tory and have considerable overshoot. If we arbitrarily reduce the gain kp by a factorof 2 in each case, the overshoot and oscillatory behaviors are substantially reduced, asshown in Fig. 4.23(b).

EXAMPLE 4.10 Tuning of a Heat Exchanger: Oscillatory Behavior

Proportional feedback was applied to the heat exchanger in the previous example until

the system showednondecaying oscillations in response to a short pulse (impulse) input,as shown in Fig. 4.24. The ultimate gain was Ku =15.3, and the period was measuredat Pu = 42 sec. Determine the proportional and PI regulators according to the ZeiglerNichols rules based on the ultimate sensitivity method. Plot the corresponding stepresponses.

Solution. The regulators from Table 4.3 are

Proportional :kp = 0.5Ku = 7.65,

PI : kp = 0.45Ku = 6.885 and TI =1

1.2Pu = 35.

The step responses of the closed-loop system are shown in Fig. 4.25(a). Note that theresponsesaresimilartothoseinExample4.9.Ifwereducekpby 50%, then theovershoot

is substantially reduced, as shown in Fig. 4.25(b).

Figure 4.24

Ultimate period of heat

exchanger

Impulseresponse

0 20 40 60 80 100 120

0.010

0.008

0.006

0.004

0.002

0.00

0.002

0.006

0.004

0.008

0.010

Time (sec)





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y

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

y

400.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0300.0200.0100.00.0

Time (sec)

(b)

400.0300.0200.0100.00.0

Time (sec)

(a)

PI

PI

Proportional Proportional

Figure 4.25 Closed-loop step response

4.4.3 Truxals Formula for the Error Constants

In this chapter we have derived formulas for the error constants in terms of thesystem transfer function. The most common case is the type 1 system, whoseerrorconstantis Kv ,thevelocityerrorconstant.Truxal(1955)derivedaformulafor the velocity constant in terms of the closed-loop poles and zeros, a formulathat connects the steady-state error to the dynamic response. Since control

design often requires a trade-off between these two characteristics, Truxalsformula canbe useful to know. Itsderivation is quite direct. Suppose theclosed-loop transfer function T(s) of a type 1 system is

T(s) = K(s z1)(s z2) (s zm)

(s p1)(s p2) (s pn). (4.111)

Since the steady-state error in response to a step input in a type 1 system iszero, the DC gain is unity. Thus,

T(0) = 1. (4.112)

The system error is given by

E(s) = R(s) Y(s) = R(s)

1

Y(s)

R(s)

= R(s)[1 T(s)]. (4.113)





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The system error due to a unit ramp input is given by

E(s) =1 T(s)

s2 . (4.114)

Using the Final Value Theorem, we get

ess =lims0

1 T(s)s

. (4.115)

Using LHpitals rule, we rewrite Eq. (4.115) as

ess = lims0

dT

ds(4.116)

or

ess = lims0dT

ds =1

Kv. (4.117)

Equation (4.117) implies that 1/Kvis related to the slope of the transfer func-tion at the origin, a result that will also be shown in Section 6.1.2. UsingEq. (4.112), we can rewrite Eq. (4.117) as

ess = lims0

dT

ds

1T

(4.118)

or

ess = lims0

d

ds[lnT(s)]. (4.119)

Substituting Eq. (4.111) into Eq. (4.119), we get

ess = lims0

d

ds

ln

K

mi=1(s zi )ni=1(s pi )

(4.120)

= lims0

d

ds

K+

mi=1

ln(s zi ) m

i=1

ln(s pi )

, (4.121)

or

1Kv

=

dln Tds

s=0=

ni=1

1

pi+

mi=1

1zi

. (4.122)

We observe from Eq. (4.122) that Kv increases as the closed-loop polesTruxals formulamove away from the origin. Similar relationships exist for other error coeffi-cients, and these are explored in the problems.





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EXAMPLE 4.11 Truxals Formula

control systems

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