Generalized Control Theory d350400x012

8/9/2019 Generalized Control Theory d350400x012

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Generalized Control Theory

Sheldon G. Lloyd


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Introduction

The purpose of this manual is to introduce the basicideas of automatic control theory. It is intended eitheras a primer for those embarking on a rigorous study ofcontrol theory or to provide a background for thoseseeking only a general knowledge of the analytical

approach to control dynamics.


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Section I Block Diagram Techniques

Block Diagram Construction

One of the very powerful tools used in system analysisis the block diagram. It provides an exceedingly simple

method of pinning down the exact responsibility ofeach element involved in a given system. For ourpurposes any combination of elements, however theymay be grouped, that fall within any prescribed set ofboundaries is defined as a system.

To illustrate the construction of a block diagram,assume that the behavior of any element in a systemcan be described by the ratio of its output to its input.For example, a diaphragm actuator has an input ofdiaphragm pressure, (P), and an output of stemmovement, (Y). The actuator would then be describedby the ratio (Y/P). For another illustration, the input toa valve is stem motion, (CY), and its output is flow,

(W). The valve can now be described by the ratio(W/Y). With the help of one more ratio, a simplesystem can be described. Consider a level process.The input is flow, (W), and the output is level or head,(h). The describing ratio for the process is (h/W).Combining the elements, a system such as shown inFigure 1-1 might result.

Figure 1-1

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The elements are arranged so that the output of onebecomes the input of the next. In Figure 1-2 eachelement has been replaced by a block. The function of

each block is represented by the describing ratio of theelement it replaces.

The block diagram should be thought of as aninformation system rather than a duplicate of thephysical system it describes. The inputs and outputs ofa block diagram consist of information about somevariable in the system and do not necessarilyrepresent the addition or removal of energy or mass.

Figure 1-2

P h

ACTUATOR VALVE PROCESS

P

Y

Y

W

W

h

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The diagram indicates that if the actuator block ismultiplied by (P), the output is (Y), and (Y)times thevalve block produces (W), and (W) times the processblock produces the final output (h). In other words, thesolution of each block is superimposed, in amathematical sense, on the next succeeding block.

No information about the final output is returned to theinput in this arrangement, i.e., there is no feedback inthis system. Any such arrangement is called an openloop control system. All open loop control systemshave the following characteristics in common:

1. Under constant load (outflow from the tank in theexample) the output can be controlled by the input onlyif each element in the system is carefully calibrated.

2. This system is completely incapable ofcompensating for changes in load or changes incalibration due to part wear.

3. This system is inherently stable; there is nopossibility whatsoever for this system to cycle.

Because the first two characteristics of the open-loopsystem seriously limit its control accuracy, it might bedesirable to add some form of feedback. That is, insome way allow information about the output variableto be communicated back to the input and exert a

corrective influence. A common method of introducingfeedback in a system such as the example would be toinclude a proportional controller. It would measure thelevel, (h), and compare it with a reference value orsetpoint, (r). The controller would produce adiaphragm pressure (P), proportional to the differencebetween (h) and (r).

To incorporate this into the block diagram, one morenotation is needed, the summing point. This isrepresented simply by:


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z

y

x

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which means that x + y = z. The quantities (x) and (y)are added algebraically to cover the usual case of adifference in sign. The new system is shown in Figure13. Note that, within the controller, the error, (e),equals (rh) although the describing ratio for thecontroller as a unit is (P/h).

Figure 1-3

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Figure 1-4

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Figure 1-5

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The only input shown is the set point or reference, (r).The output is the level, (h). The servo-mechanismengineer would be concerned with how the outputvariable, (h), varies following changes in the referencevalue, (r). In other words, he is concerned with afollowing problem.

In process control, the value (r) is usually heldconstant and the input variable is really the change inload. The control engineer is interested in how theoutput, (h), varies with a change in load or he isconcerned with a load problem. It is perfectlylegitimate to use information about the load as an inputto the information system or block diagram eventhough the load represents the removal of mass fromthe physical system.

To show how load changes effect the system, only onemore summing point need be included. Also, it shouldbe made clear that the level, (h), depends on the netflow, (Wn), which is the difference between the inflowthrough the valve (W1), and the load or outflow, (Wo).

Wn = W1Wo

Figure 14 shows the block diagram with a load input.The diagram becomes a little more clear as a loadproblem if it is rotated counter clock- wise slightly asshown in Figure 15.

The negative sign at the summing point merelyindicates that the output of the valve has the oppositeeffect of the load. This is an example of a negativefeedback control loop often termed a closed-loopcontrol system. Closed-loop control systems have thefollowing characteristics:

1. Compensation for load changes and element wearwill occur in accordance with the elements in thefeedback path.

2. Depending on the combination of elements in theloop, an unstable or oscillating system may exist.

These two statements really say that the introductionof feedback may improve the quality of control, but theprice paid for improvement is that the problem ofstability will have to be dealt with.


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Block Diagram Algebra

In the preceding examples each block represented theoutput-input ratio for some element in the system. Theoutput of each element was obtained by multiplyingthe describing ratio by an input. Using this principleand by superimposing successive solutions, a

feedback control system was described. The input tothe control system was the load, (Wo), and the systemoutput was level, (h). To see how the output varieswith the input, it is necessary to determine thedescribing ratio for the entire closed-loop system,(h/Wo). For simplicity, assume that the ratio in theforward path is (G)and that the product of the ratios inthe feedback path is (H). For the level control examplethen:

G = hwnand H = (P/h)(Y/P)(Wi/Y) = W1/h

The diagram for this example is shown in Figure 1-6.

Figure 1-6

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The output of the feedback path must be the product

of the input to the feedback, (h), and the feedbackblock, (H) or (hH). The output of the forward path mustbe the product of the input to (G)and (G)itself. Theinput to (G) is the algebraic sum of the negativefeedback and the load or:

input to G = WohH

The output from (G) must be:

G(Wo hH) = GWo GhH

and the output must equal (h), therefore:

GWo GhH = h

or:

GWo= h+ GhH= h(1 + GH)

hWo

+ G(1 ) GH)

The behavior of the closedloop system with negativefeedback can now be represented by one block, seeFigure 1-7.

Figure 1-7

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To illustrate how more complex systems can besimplified with this one relationship, two examples aregiven in Figures 18 and 1-9.

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Figure 1-8

Notice that in every case the most subordinate loop isclosed first and that series elements are multipliedwhen there are no branches either entering or leavingbetween them.

The analysis of any control system, regardless of howcomplex or elementary, should begin with a blockdiagram. All information concerning transient response(stability) and steadystate response can bedetermined by operations performed on the blockdiagram and on information gained from it.


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Figure 1-9

Mathematical Descriptions

Thus far it has been assumed that the behavior of anyelement can be described by the ratio of its output toits input. Within certain limitations, this is possible.When working with linear elements (a linear elementwill be defined later), the describing ratios that have

been used would be in the form of a transfer function.A transfer function is, by definition, the ratio of theLaplace transform of the output to the Laplacetransform of the input.

Obtaining a transfer function for an equation isnormally a very simple operation. In general, eachdifferential is replaced by the Laplace operator, (s), asecond differential, by (s2), and so on. An integral is

replaced by (1/s). A variable such a (x) is written L(x)to indicate the Laplace transform of (x). Table 1 showssome examples.

Table 1

The Laplace transform of this is this

x L(x)

dx

dts L(x)

d2x

dt2s2L(x)

d3x

dt3s3L(x)

xdt 1/s L(x)

dx

dt+ x = y sL(x) + L(x) = L(y)

Table 2

Element Typical describing equation(1) Laplace transformed equation Transfer function

controller P = Kh L(P) = KL(h) L(P)/L(h) = K

actuator T/a dy/dt + 1/a y = P T/a sL(y) + 1/a L(y) = L(P) L(y)/L(P) = a/(TS + 1)

valve W1 = NY L(W1) = NL(y) L(W1)/L(y) = N

process Wn = c dh/dt L(Wn) = CsL(h) L(h)/L(W) = 1/CS1. These equations are merely presented as typical and do not necessarily refer to previously discussed equations.

The transformed equation can be manipulated like anyalgebraic equation. Differentiation can be performedmerely by multiplying the transformed equation by (s),integration by (1/s). Notice that before transformationthe equation is a function of time, (t), but aftertransformation it is a function of the Laplace operator,(s). Returning the equation to a function of time iscalled finding the inverse function. If a number ofoperations have been performed on the transformedequation, finding the inverse function may be a difficulttask. To help with this, sets of tables are available formost of the common forms of transfer functions. Theentire procedure becomes somewhat analogous to theuse of logarithms. Logarithms are added andsubtracted to accomplish the operations of

multiplication and division. The real solution is then theanti-log, most easily obtained from a set of tables.Laplace transformed equations are multiplied anddivided by the Laplace operator, (s), to accomplish theoperations of differentiation and integration. The realsolution is then the inverse function, most easilyobtained from a set of tables.

The procedure presented here is entirely mechanicaland no mathematical proof is given. The nature of theLaplace operator itself will not be discussed other thanto say that it can be treated like any algebraic variable.It should be pointed out for the benefit of thoseintending further study of Laplace transforms that thetechnique presented here does not yield the complete


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Laplace transform equation. However, all quantitiesthat have been neglected are initial conditions, and it isalways possible in control systems to assume a datumplane that renders the initial conditions zero.

The utility of the Laplace transform lies in the fact thatthe output-input ratio or transfer function can be written

for an equation containing differential quantities as wellas for algebraic relationships. This permits the use ofblock diagrams and the superposition principleoutlined previously.

To illustrate the use of transfer functions, thedescribing ratios in the level control example will bedefined. Figure 110 shows how the block diagramwould be written in transfer function form. The value ofeach transfer function is given in the last column ofTable 2.

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Figure 1-10

As was inferred earlier, the Laplace transform existsonly for linear elements. A linear element is one whose

behavior can be described by a linear algebraic ordifferential equation with constant coefficients. That is,an equation containing no products or differentialproducts of the dependent variable and whosecoefficients do not vary.

In Table 3, equation No. 4 is nonlinear because (x2)is a product of (x) and itself. In equation No. 5, thesecond quantity is the product of the (x) variable andits differential. In equation No. 6, (y) appears to theminus one power.

Table 3

Linear Equations Nonlinear Equations

(1)ax ) b + y (4)ax2 ) b + y

(2)d2x

dt2) dxdt

) ax + y (5)d2x

dt2) xdx

dt) bx + y

(3)d2x

dt2+ a

dy

dt) by ) c (6)dx

dt+ a

dy

dt) by ) c

As serious as this limitation first appears, a great manyphysical elements that are of interest in control can bedescribed by linear equations. Those that can notusually can be considered linear for small excursionsabout some operating point. By investigating a numberof operating points throughout the range ofperformance of a system, a good overall picture canbe obtained. The use of linear theory in this way hasfound widespread and successful application in theservo-mechanisms and communications fields.


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Section II Oscillation and Stability

The topic under discussion in this section is systemstability. Working from the block diagram techniquesdeveloped in Section I, a method of explaining theprinciple of oscillation and determining stability is

presented. The problem of indicating relative stabilityis also discussed.

Frequency Response Method

In studying the dynamic behavior of a given controlelement, it is logical to supply the element with adynamic or changing input. A comparison of the outputwith the input would reveal the dynamic behavior ofthe element. A satisfactory analysis could be madewith any type of varying input. It has evolved, however,that the interpretation of the results becomes mucheasier and more effective if the input is restricted to a

sine-wave of appropriate amplitude and if theresponse of the element is observed as only thefrequency is varied. This technique is aptly calledfrequency response.

If a linear element is supplied with a sufficiently slowsine-wave input, the output will be a sine-wave of thesame frequency and will occur in phase(simultaneously) with the input(1). The outputamplitude may be greater or less than the input andmay differ dimensionally. The ratio of the outputamplitude to the input amplitude is called gain.During very low frequency inputs when there is nophase shift, the gain is commonly referred to as thezero frequency gain or static gain. As the frequencyof the input is increased, two things will occur: (1) thephase between the output and the input will shift withthe output usually lagging, (2) the gain of the elementwill change with the output usually being attenuated. Ifthe gain at any frequency is divided by the static gain,the resulting quantity is called the magnitude ratio.Knowledge of the static gain and of the phase shift andmagnitude ratio throughout an appropriate frequencyspectrum will completely describe the dynamicbehavior of the element. When a series of linearelements are connected output to input and thesystem is subjected to the frequency responsetechnique of analysis, the overall phase shift is thesum of the individual phase shifts, the overallmagnitude ratio is the product of the individualmagnitude ratios, and the overall static gain is theproduct of the individual static gains.

How the quantities of phase shift and gain vary foreach element in a feedback or closedloop system willdetermine the stability of that loop. For an illustration ofthe conditions of stability, refer to the level controlexample in Section I and the diagram in Figure 21.

Figure 2-1

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Suppose the chain of information is brokensomewhere in the loop, say between the actuator andthe valve stem as shown in Figure 2-1. Now study theopenloop system starting with the valve stem andending with the actuator stem and including all theelements of the closed-loop system. Assume that thevalve stem is provided with a sine-wave input andobserve the output at the actuator stem. The ratio ofthe amplitude of the actuator output to the amplitude ofthe valve input is called open-loop gain. It will be theproduct of the individual gains of each element in thesystem. The open-loop phase shift will be the sum ofall the individual phase shifts in the loop.

Generally, each element will contribute some phaselag or negative phase shift; and, since the feedback isnegative, an additional 180o lag will be acquired at thesumming point. Consider, now, changing thefrequency of the sine-wave input to the valve until theopen-loop phase shift shows that the output is laggingthe input by a full 360o.The output will appear to beback in phase with the input. At this frequency, if theopen-loop gain is one, the valve and actuator stemswill be moving with the same amplitude. It follows that,if the stems were connected at any given instant, theoutput of the actuator would become the input of thevalve. Since each output cycle is the result of oneprevious input cycle, the oscillations would beself-sustaining. This is the fundamental principle ofoscillation. It can be shown that any linear closed-loopsystem that has an open-loop gain of one at 360o

phase lag will cycle continuously. Furthermore, it willoscillate at the exact frequency that produces 360o

phase lag. If the gain at 360o phase lag is greater thanone, the system will oscillate with ever increasingamplitude. This is because the amplitude at any pointin the loop, at any instant, will equal the product of theopen-loop gain and the amplitude of the previous cycleat that point. If, however, the open-loop gain is lessthan one at 360ophase lag, the oscillations will dieout.

A very important fact concerning the stability of linearsystems is now evident: stability is determined by theopen-loop gain at 360o phase lag and is completelyindependent of load disturbances.

1. Technically, there are exceptions to this statement, but they may be safely ignoredfor the purposes of this discussion.


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Figure 2-2

The magnitude ratio is either plotted on a logarithmic scale or thelogarithm of the magnitude ratio is plotted on a linear scale. Acommon practice, and the one used here, is to convert the mag-nitude ratio, (M), into decibels or db.

db = 20 log10M

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Since the feedback is almost always negative, it isconventional not to speak of thel80o phase shift atthe summing point. Reference is normally made onlyto the180o phase shift accumulated by the elementsin the loop, and stability is then said to depend uponthe open-loop gain at 180o phase lag.

Frequency response data can be determinedexperimentally or by mathematical analysis. Theinformation is usually determined for individualelements and plotted in any of several ways. Twocommon methods of plotting frequency response dataare the Bode plot and the Nyquist diagram. They areillustrated in Figures 2-2 and 2-3 respectively. TheBode plot is on rectangular coordinates. The phaseshift and the log of the magnitude ratio are both plottedas functions of the log of the frequency. The Nyquistdiagram is a polar plot. The vector of the plot has alength equal to the magnitude ratio and an angle equalto the phase shift. While these plots are normallymade for individual elements, the values of all theelements in a control loop can be combined on oneplot to determine the stability of that loop.

Figure 2-3

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Relative Stablility

We have shown that the stability of a system dependsupon the openloop gain and the openloop phaseshift as functions of frequency. We have also statedthat the transfer function of an element completelydescribes its dynamic behavior. Therefore, there mustbe some relationship between frequency response andtransfer functions. This relationship will be shownwithout presenting the mathematical basis for it.

Assume that some element is described by the first

order differential equation:

dy

dt+ ay = bx

Think of (x) as the input and (y) as the output of theelement. This equation is linear and, if the coefficient(a) and (b) are constant, it can be Laplacetransformed.

dy

dt+ ay = bx

L(y) s + L(y)a = L(x)b

L(y) (s + a) = L(x)b

The ratio of the Laplace transforms of output to input isthe transfer function of the element.

Laplace transform of output

Laplace transform of input+L(y)

L(x)+ b

s) a

Divide numerator and denominator of the right handside by (a):


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L(y)

L(x)+

ba(1a)s ) 1

If we now define two new symbols so that:

Go= (b/a) and T= 1/a

We can write the transfer function as:

L(y)

L(x)+ Go 1Ts) 1

The quantity (Go) is a function of the two constants, (a)and (b), and so is itself a constant. It is, in fact, thestatic gain or zero frequency gain of the element. Thequantity, (T), depends only on the constant, (a), and sois also a constant. Because of its location next to theoperator, (s), it is called a time constant. The entirequantity [1/(Ts + 1)] involves a differential term, (s),and is therefore time dependent. The frequencyresponse of this element is shown in Figure 2-4. The

frequency response curves are really a plot of the timedependent part of the transfer functions, that is, [1/(Ts+ 1)] versus frequency. The time constant, (T), isrelated to the frequency at which the magnitude ratiofalls off, as shown in Figure 2-4. The phase is shownin the figure to lag as frequency increases. Because ofthis laging characteristic and because the definingequation is first order, the element is called a firstorder lag.

A second order differential equation of commonoccurrence is:

m

d2y

dt2 ) b

dy

dt ) ky + x

The transfer function for the equation is:

L(y) s2m + L(y)sb + L(y)k = L(x)

L(y) (s2m + sb + k = L(x)

L(y)

L(x)+ 1

ms2 ) bs) k+ (1k) 1

mk

s2 ) bks) 1

If we define the following new terms:

Go+

1k, T

+m

k

, and z

+(b

2) 1

km

We can then write the transfer function as:

L(y)

L(x)+ Go 1T2s2 ) 2Tzs) 1

Again, the term (Go) is the static gain and is aconstant. The quantity (T) is the time constant and isrelated to the magnitude ratio curve as shown inFigure 2-5. The term () is called the damping ratio.

Figure 2-4

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Its effect on the magnitude ratio is also shown inFigure 2-5. Notice that the magnitude ratio curveeventually becomes asymptotic to a line whose slopeis (1) for a first order lag and (2) for a second orderlag. This rule can be extended to any order system. Asecond rule is that the maximum phase shift for a firstorder element is 90_, for a second order element180_, and so on.

To summarize, we can say that both the transferfunction and the frequency response curves state:

Actual gain+Output

Input+

(static gain) (magnitude ration), at an angle

or in symbols:

G = (Go) (M) 6

where:

G = actual gain at any frequencyGo = static or zero frequency gainM = magnitude ratio = phase shift between output and input


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Figure 2-5

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Figure 2-6

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To illustrate the use of frequency response in

determining stability, consider the system in Figure2-6. The frequency response curves for this systemare both plotted and labeled in Figure 27. Theopenloop transfer function is the product of the twoindividual transfer functions:

openloop transfer function =

Go11

T12s2 ) 2T1zs ) 1

Go21

T2s ) 1

The total open-loop phase shift is the sum of theindividual phase shifts and is shown in Figure 27. Theopenloop gain is the product of both static gains andboth magnitude ratios.

open-loop gain = G = (Go1Go2)(M1M2)

For convenience the magnitude ratio is converted todecibels which is a logarithmic scale. Therefore, themagnitude ratios (M1) and (M2) in Figure 27 can bemultiplied simply by adding the decibel readings.

The stability criteria states that the gain must be lessthan unity at 180o phase lag. That is:

G = (Go1GQ2M1M2) < 1, when 1 + 2 =180o

The magnitude ratio curve shows an open-loop valueof15 db at the frequency which produces180o

phase shift.

M1M2= 15 db = 0.178

For a stable loop, then:

(Go1Go2) +1

M1M2+ 1

0.178+ 5.61

In other words, if the product of the static gains, (Go1)and (Go2), is numerically greater than 5.61, the systemis unstable.

We are now in a position to consider the concept ofrelative stability. Assume the example system with astatic gain of 6 and oscillating violently. Now imaginethat by turning a knob we can gradually decrease thegain. When the gain reaches a value slightly below5.61, the system must be stable, but we would not

expect it to suddenly snap into very solid, rock stablebehavior. In fact, with gains just slightly below 5.61 thesystem will respond to disturbances with a great dealof oscillation, but the oscillations will always diminishand die out in time. As the gain is decreased furtherbelow the value 5.61, the system will get less and lessoscillatory and eventually become extremely slow andsluggish. The question is now, Relatively how stableshould the system be for lively response with aminimum of oscillation? There is no general answer tothis question. If the open-loop transfer function isexactly known, an optimum value of gain can then becalculated, but the results will be different for eachindividual system. Some general rules can be shown

to be quite helpful in many cases, however. Twocommon measures of relative stability are gainmargin and phase margin. In a sense thesemeasures provide a factor of safety on stability. Gainmargin is the factor by which the openloop gain mustbe multiplied to just reach the unstable condition. Thusin our example if the static gain were adjusted to equal1.87, the gain margin would be calculated as follows:

if G01G02 = 1.87, Gain margin =5.611.87

= 3


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Figure 2-7

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Phase margin is defined as the difference between180o lag and the actual openloop phase lag at thefrequency where the openloop gain is unity. If we

want in our example a phase margin of 45o:

open-loop phase lag = 180o45o= 135o lagThe frequency that gives 135o lag is shown in Figure27 as 7.7 cps. The open-loop magnitude ratio is:

M1M2 + * 11.4db + 0.27

The necessary static gain to provide 45o phase marginthen is:

Go1Go2M1M2 + 1

Go1Go2 +1

M1M2+ 1

0.27+ 3.7

Therefore, if the static gain is adjusted to a value of3.7, the system will have a phase margin of 45o.In thisexample the reader can verify that a phase margin of45o will provide the same relative stability as a gainmargin of (1.5). For low order systems, acceptable


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relative stability can be obtained with phase margins ofabout 45o or gain margins around 2.

Other TechniquesThe discussion in this manual has been restricted tothe frequency response approach. Another significantapproach in the analysis of linear or linearized systemsis the root locus method. In this method, the roots ofthe system equation are plotted as the openloop gainis varied, thereby obtaining a locus of roots as afunction of gain. Interpretation of this plot then yieldsinformation about the system behavior.

A host of other techniques and concepts have beendeveloped for dealing with both linear and non-linearsystems. Techniques such as describing functions,transforms other than the Laplace transform, computersimulation, optomizing and adaptive control are allextremely important concepts in control theory.However, the author feels that the subjects introduced

in this manual are in no way belittled by these moreadvanced techniques. Linear theory and the frequencyresponse approach still has effective and widespreadapplication and is an essential first step toward themore complex topics.


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Section III System Response

Figure 3-1

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Section I presented the block diagram as a tool fordescribing a control system and for determining whererelationships exist among the individual elements.Working from this foundation, Section II discussed theproblem of oscillation which was shown to be theresult of the open-loop parameters in a feedback loop.

We shall now consider the relationship between inputand output for the overall system. In other words,closedloop performance.

Consider a single loop system with negative feedbackas shown in Figure 3-1. The elements in the forwardpath are described by (G) and in the feedback path by(H). The input to the entire system is (Wo) and theoutput is (h). In Section I it was shown that theclosed-loop transfer function for such a system is:

L(h)

L(Wo)+ G

1 ) GH

The output of the system is obtained merely bymultiplying the transfer function by the input.

L(h) + L(Wo)G

1 ) GH

A very important fundamental is now obvious; theoutput of a control system is a function of both theclosed-loop transfer function and the input. This is

contrasted to the stability which has been shown to beindependent of the input.

A second important concept in closedloop controlarises when a system with high openloop gain isconsidered. The reference here is to the static gain. Inthis case the product, (GH), will be large and thefollowing approximation is very good:

L(h)

L(Wo)+ G

1 ) GH+ G

GH+ 1H

,whereGH1

Thus, if the static open-loop gain is high, theclosed-loop function reduces to the inverse of the

feedback elements. The concept is obviously valuablewhere non-linear elements can be kept in the forwardpath. This also makes it possible to accurately vary thesystem response by manipulating only the feedbackelements.

Conclusion

It has been the purpose of this manual to present thebasic ideas of linear control theory in a generalizedfashion. Control systems are dynamic in behavior. Ifthe ever increasing requirements of control systemsare to be met, it will be done by facing the dynamics

problems involved. It is hoped that these few pageshave done a little toward creating a understanding ofthe basic utility of control theory.


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Fisher Controls International, Inc.205 South Center StreetMarshalltown, Iowa 50158 USAPhone: (641) 7543011Fax: (641) 7542830Email: [email protected]: www.fisher.com

The contents of this publication are presented forinformational purposes only, and while every effort has

been made to ensure their accuracy, they are not to be

construed as warranties or guarantees, express or implied,

regarding the products or services described herein or

their use or applicability. We reserve the right to modify or

improve the designs or specifications of such products at

any time without notice.

EFisher Controls International, Inc. 1974;

All Rights Reserved

Fisher and FisherRosemount are marks owned by

Fisher Controls International, Inc. or

FisherRosemount Systems, Inc.

All other marks are the property of their respective owners.

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Generalized Control Theory d350400x012