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CHAPTER

FOUR

STEADY-STATE OPERATION

By steady-state operation is meant the equilibrium state attained such that there so no

change with respect to time of any of the system variables. The system remains at this

equilibrium state of operation until it is excited by a change in the resired input or in the

external disturbance. A transient condition is said to exist chapter, it is shown that

considerable information about the basic character of a system may be obtained from an

analysis of its steady-state operation.

4.1 STEADY-STATE ANALYSIS

The general block-diagram representation for a feedback control system is shown Fig

4.1a. for steady-state operation. c, v, and u will have constant values, and therefore terms

resulting from powers of D operating on these constant quantities will be zero. The equation

describing the steady-state operation of a control system is obtained by letting D = 0 in the

general block-diagram representation for the system. The block diagram that describes the

steady-state operation of the system of Fig. 4.1b, in which

KG1 = [G1 (D1)]D=0 KG2 = [G2 (D)]D=0 KH = [H (D1)]D=0

Where KG1 is obtained by letting D = 0 in the differential operator G1(D), etc. From Fig 4.1b,

the equation for steady-state operation is found to be

[(AvKHc)KG1 + Bu]KG1 = c

or c =

(4.1)

The constant A which appears in Eq. (4.1) is, in effect, the scale factor for the input dil. To

have the coefficient of the v term equal to unity, a must be slected such that


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When A inces chosen in accordance eith Eq. (4.2), the coefficient of the v term is unity , in

which case Eq. (4.1) becomes

c = v +

(4.3)

to have the controlled variable c equal to the command signal v (that is, c = v), it is necessary

that the coefficient of the u term be zero. This coefficient is zero if either or isinfinite. From Eq. (4.1) it follows that if is infinite, then c must be zero regadless of the

value of v or u. In effect, no control is posibble when is infinite. also not from Eq. (4.2)

that an infinite value of would necessitate A being infinite, which is physically

impossible. Thus only can be made infinite. This is accomplished by having and

integrator in the control elements to yield a 1/D term, which gives the effect of an infinite

constant during steady-state operation. This type of system is called an integral control

system.

The left portion of the control system enclosed by the dotted lines in 4.1b is the

controller. For the controller it follows that

(Av - c) = m

or c =

(4.4)

in fig 4.2a is shown a plot of the steady-state operating characteristics for a typical controller.

Lines of constant values for the command signal are plotted with the cotroller variable C as

the abscissa and the manipulated variable M as the ordinate. For v = 0, Eq. (4.4) show that

- (4.5)


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Where v = is the change in V from the reference value, m = is the change in M from

the reference value, and c = is the change in C from the reference

Value. For v = 0 for v = 0, then V must be constant. The term || is the slope of the

controller curves shown in fig. 4.2a.

For the case in which c = 0, Eq. (4.4) becomes

For c = 0, then C must be constant. A line of constant C is a vertical line in Fig, 4.2a. The

term || determines the vertical spacing between the lines of constant V.

Finally, for the case in which m = 0, Eq. (4.4) show that

For m = 0, then M must be constant. A line of constant M is a horizontal line. The term

|| determines the horizontal spacing between lines of constant V.

The right portion of Fig. 4.1b enclosed by the dotted line represents the system to be

controlled. The equation for the steady-state operation of the system to be controlled is


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Typical steady-state operating curves for the system to be controlled are shown in Fig. 4.2b.

The first partial || is the slope, the second partial || determines the vertical

spacing, and the last partial || determines the horizontal spacing. In summary, the

individual constants in the block diagram of Fig. 4.1b may be obtained from the steady-state

operating curves, and vice versa.

Illustration example 4.1 The block diagram for a feedback control system is shown in

Fig 4.3. The reference operating point is Vi = Ci = 100, Mi = 50, and Ui =10.

Determine the steady-state constants for this system nd then sketch the steady-state

operating curves. Select A in accordance with Eq. (4.2).

Solution: The steady-State constants re B = -5,

=0.5, and


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The value of A is

A =

= ()()

The slope of the operating curves for the controller is

The line of Vi = 100 may now be drawn through the reference point (M i = 50, Ci=100) with a

slope of (-0.5), as shown in Fig. 4.4a. The horizontal spacing is

For C=20, then V = 0.5 C=(0.5)(0.2)=10. Note in Fig. 4.4a that when M is constant (a

horizontal line), the horizontal spacing is such that as V changes by 10 units, then C changes

by 20 units. The vertical spacing is

For V=10, then M=1.0 V=(1.0)(10)=10.0. Note in Fig. 4.4a that when C is

constant (a vertical line), the vertical spacing is such that as V change by 10 units, then M

changes by 10 units.

If any two of the three quantities (slope, horizontal spacing, of vertical spacing) are

known, the third quantity is determined. Thus, only two of these three quantities are

independent. This fact is proved by noting that for the controller enclosed by the dashed box

Fig.4.1b, the output is the manipulated variable M and the inputs are the command signal V

and the controlled variable C. For a given controller the manipulated variable M is a function

of V and C. That is

M = M(V, C)


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The implicit form for this relationship is

G(M, V, C)=0

For an implicit function of n variables, the product of the n partial derivatives is (-1)n. For the

implicit function G of the three variables M, V, and C, it follows that

This illustrates the statement that if the vertical spacing || = A and the horizontal

spacing || = /A are known, then the slope || = - is automatically

determined.

The slope of the load lines for the system to be controlled is


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As shown in Fig. 4.4b, the load line for U i = 10 is drawn through the reference point (Mi =

50, Ci=100) with a slope of (0.5). The horizontal spacing is

For U=-2, then M=5 U=5(-2)=-10. Thus, when C is constant (a vertical line), the vertical

spacing between load lines is such that as M changes by -10 units, then U changes by -2

units. T5he vertical spacing is

As was the case for the lines of operation for the controller, the load lines can be completely

determined if any two of the three quantities (slope , horizontal spacing or vertical spacing)

are known. As is shown in Fig. 4.1b for the system to be controlled, the output is the

controlled variable C and the inputs are the manipulated variable M and the disturbance U.

Thus, the output C is a function of M and U. That is

C = C (M,U)

The implicit form for this relationship is

G(C, M, U)=0

The product of the partial derivatives is


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Because Figs 4.4a and b have the same coordinates (C, M), the two diagrams may be

superimposed upon each other as shown in fig 4.4c. substitution of the values of the constants

into Eq. (4.1) yields the overall equation for steady-state operation

The coefficient of the v term and the coefficient of the u term may be obtained directly from

Fig. 4.4c without the need to evaluate all the steady-state constant. For the overall system

shown in Fig.4.1b. The output is the controlled variable C and the inputs are the command

signal V and the disturbance U. Thus, for any given system, C is function of V and U. That is.

C=C(V, U)

Linearization gives


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r = v 5u=10. Note that in going from point A to point B, the controlled variable changes

from 100 to 110, so that c = 10. In going from point A to point F. The command signal

changes from 100 to 110, so that v = 10. The load U change from 10 to 8, so that u=-2.

Application of Eq. (4.12) gives c=v-5u=10-5(-2)=20. In going from A to F the controlled

variable changes from 100 to 120, so that c = 20

Because the input scale factor has been selected in accordance with Eq. (4.2), at the

reference load Ui=10 (u=0) the controlled variable is equal to the command signal (c=v).

Note in Fig.4.4c that at point A, V=C=100; at point B. V = C = 110; and at point C, V=C=90.

4.2 EQUILIBRIUM

In fig. 4.5a is shown typical operating line for a controller in which V is the value of the

command signal. When the value of the controlled variable is C1. The value of the

manipulated variable being supplied by the controlled is M1. In Fig 4.5b is shown a typical

load line for a system to be controlled in which U is the load. When the value of the

manipulated variable being supplied to the system is M1. The output from the system is C1.

As indicated in Fig. 4.5c by point A, the intersection of the line of operation of the command

signal V for the controller and the load line U for the system determines the equilibrium point

of operation for the system. That is at point A the amount of the manipulated variable M1

being supplied by the controller is the same as that required to maintain the system output at

C1. If the system output were C2, the point B would be the point of operation for the

controller and point C would be the point of operation for the system to be controlled. The

amount of the manipulated variable required to maintain the system at point C is MC. Because

the controller is only supplying the amount MB, the system output C decreases until

equilibrium is attained at point A.


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The characteristic shown in Fig.4.6a illustrate the effect of changing the command

signal from V1 to V2 with the load U maintained constant. Initially the system is at

equilibrium at point A, the intersection of the V1 controller curve and the U load curve. When

the command signal is changed from V1 to V2, the new operating point for the controller is at

point B, which is the intersection of the system to be controlled remain at point A, which is

the intersection of the output C1 and the U load curve. Because the controller is supplying

more of the manipulated variable than is required to maintain the system to be controlled at

point A (MB > M1), the output C increases. The new equilibrium point of operation will be at

point C, where the value of the manipulated variable supplied by the controller M2 is the

value required to maintain the system output at C2. If the input dial is calibrated in such a way

that the value of the command signal V1 is equal to the output C2, etc, then the system is

calibrated so that the output will always equal the input for a given loading condition. When

this is so Eq. (4.2) is automatically satisfied.

The characteristic shown in Fig. 4.6b illustrate the effect of changing the load from U1

to U2 with the command signal V maintained fixed. Initially the system is at equilibrium at

point A. The intersection for the controller curve V and the load line U1. When the load is

changed to U2. The new point of operation for the system to be controlled is at point B, which

is the intersection of the value of the output C1 and the load line U2 . The point of operation of


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the controller remains at point A, which is intersection of the value of the output C1 and the

controller line V. Because the amount of manipulated variable M1 being supplied by the

controller is less than the amount MB required to maintain the system to be controlled at point

B, the output C decreases. The new equilibrium point of operation will be at point C, which is

intersection of the new load line U2 and the controller line V. Note that with the command

signal V maintained fixed, the output has changed from C1 to C2 due to the change in load

The slope of the controller lines V is KGKH. For an integral type controller KG1 is

infinite. The resulting controller lines are vertical. For this case there is no change in the

output due to change in the load. Replacing the control element G1(D) = 1/(1+D) by the

element G1(D)=1/D would yield such a system. The resulting steady-state characteristic are

shown in Fig. 4.7a. Note that for steady-state operation, the value of the controlled variable C

is always equal to the command signal V regadless of the value of the disturbance U. Using

the points B and C in Fig. 4.7a to evaluate the coefficient || and using the point D

and E to evaluate the coefficient || gives


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Substitution of these results into Eq. (4.13) yields the equation for steady-state operation.

c = v

Thus, for an integral-type control system, the controlled variables c is equal to the command

signal v regadless of the disturbance.

For an open-loop system, there is no feedback path. When KH=0, the feedback path of

Fig. 4.1b is disconnected so that an open-loop system results. For this case the slope of the

controller lines (-KG1KH=0)is zero. As is shown in Fig 4.7b, for an open-loop system the lines

of operation for the controller are

Horizontal. Using the points B and C Fig. 4.7b to evaluate the coefficient || and the

points D and E to evaluate the coefficient || gives

Substitution of these results into Eq. (4.13) yields the equation for the steady-state operation

c = v10u

The slope the controller lines || KG1KH varies from zero for an open-

loop system (KH=0) to infinity for an integral control system (KG1=). A proportional control

system is one for which the controller lines have a finite slope, as shown in Fig. 4.4c. Usually

the slope of the control lines for a proportional controller is very steep, so that they are almost


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vertical. The steeper the slope, the less the variation in the controlled variable C due to a

change in the disturbance U. An open-loop system has the greatest change in the controlled

variable due to a disturbance, whereas an integral controller has no change in the controlled

vatiable due to a disturbance.

Proportional control systems usually exhibit better transient characteristics than

integral control system. In addition, proportional control systems provide the operator with a

feel as to what is going on. For example, a power-steering system feeds back some the

torque applied to the steering wheel so that the driver has a measure or feel for the turning

efforts being applied to the wheels. Satisfactory performance may usually be achieved by

making the control lines sufficiently steep; this makes the coefficient of the u term

sufficiently small so that variations in the external disturbance cause only slight errors. For

integral control systems, the coefficient of the u term is zero. For proportional control

systems, the coefficient of the u term is finite. The coefficient attains its maximum value for

an open-loop system.

4.3 PROPORTIONAL CONTROL SYSTEM

The differential eqution relating the output n0 to the input n10 and external disturbance

tL for the speed control system represented by the block diagram of Fig 3.30 is given by Eq.

(3.67). Letting D = 0 in Eq. (3.67) yields for the equation describing the steady-state

operation of the speed control system

n0=

(4.16)

Letting D=0 in the overall block diagram of Fig. 3.30 yields the block diagram for steady-

state operation shown in Fig. 4.8a. Comparison with the general block diagram of Fig. 4.8b

show that c=n0, v=nin, u=tL, A=C2K5, B=-C8, KG1=K1, KG2=C6, and KH=C4. Subtitution of

these results into Eq. (4.1) verifies Eq. (4.16)


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The term C2= is the scale factor for the speed-setting dial.

Because some of the terms in Eq. (4.17) are partial derivatives evaluated at the reference

operating condition, the value of the scale factor C2 is seen to vary for different reference

points. This would result in a nonlinear scale for the input speed dial. The use of a nonlier

scale may be avoided by connecting the throttle lever to a cam which in turn sets the desired

position of the top of the spring, as is shown in Fig. 4.9. It is then a relatively easy matter to

set up the speed control system so that Eq. (4.17) is satisfied for any reference condition.

When this is so, Eq (4.16) becomes

n0=nin

(4.18)

Equation (4.18) is the typical form of the steady-state relationship that exists between the

input, output, and external disturbance for a proportional control system.


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When the load torque TL is not equal to the reference value (that is tL0), then n0 is not equal

to nin. For example, suppose that this is the speed control system for the gas turbine of a jet

airplane and that TLi is the torque required for the airplane in level flight. When the airplane

is inclined to gain altitude, a greater load torque TL is required than that for level flight (that

is tL >0). Thus, Eq (4.18) shows that the output speed is slightly less than the desired value

for this flight condition.

The physical reason for this can be seen by looking at the schematic diagram of

Fig.3.25 for the speed control system. For level flight, the system is set up so that N0=Nin.

When the airplane is gaining altitude, the load torque is increased. This increased torque

results in a decreased speed, which in turn causes a lower position for x. Because of the lower

position for x. There is a greater flow of fuel. To have the airplane continue to gain altitude.

More flow is required than for level flight.

The steady-state operating curves for this speed control system are shown in Fig.4.10.

For an airplane in level flight, the curve of fuel flow Q required to maintain various speeds N0

is indicated by TLi = T2. The curve marked T3 would correspond to operation of the airplane

losing altitude at a certain angle of declination.

The operating line AB for the controller is obtained by fixing the speed setting at some value

Nin and the plotting corresponding values of fuel flow Q


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Coming from the controller, for various speeds of rotation N0. Note from Fig. 3.25 that an

increased speed N0 of the flyweights increases the centrifugal force, which causes x to move

up. This in turn causes y to move down, and this decreases the flow Q supplied by the

controller.

Steady-state operation exists at the intersection of the line of operation of the

controller and the torque line for the given flight condition, because t this intersection just

enough flow is being supplied to maintain the flight condition. For example, if the airplane is

in level flight and the operating line for the desired speed setting is AB, then the intersection

at point A of the line T2 and the operating line AB is the steady-state operating point for the

system. The speed-setting dial is calibrated by setting the value speed Nin on the dial equal to

the steady-state value of the output speed at the reference load T 2. For a given speed setting

such as yhat indicated by the line AB, if the load is uncreased to T3 while the desired speed is

unchanged, then the new operating point must be on the line of T3 at point B. Because AB is

not a vertical line, variations in the load are seen to cause variations in the output speed. A

proportional controller is sometimes called a droop controller and the line AB is referred to as

the droop line.

Illustration Example 4.2 A typical family of steady-state operating curves for a unity

freedback (KH=1) speed control system is shown in Fig. 4.11. At the reference

operating condition (point A), Nin=N0=4000, Qi=1000, and Ti= 200. Determine the

steady-state constant and the equation for steady-state operation. With Nin held fixed at

its reference value, what is the change in speed N0 when the load T change from the

reference value TI=200 to T=300?By what factor should the slope of the controller lines

be changed so as to reduce this change by a factor of 50?


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Solving for KG1KH gives

KG1KH=

To decrease the speed error by a factor of 50, the slope of the controller lines must be

increased by a factor of 99 (i.e, from - KG1KH=-1/4 to -99/4)

4.4 INTEGRAL CONTROL SYSTEMS

By eliminating the linkage between x and y of Fig. 3.25 and using the hydraulic

integrator shown in Fig. 4.12, the proportional control system is converted to an integral

control system. The block-diagram representation for the integrator is also shown in Fig.

4.12. The substitution of this diagram for that of the servomotor which it replaces in Fig. 3.28

yield the block-diagram representation shown in Fig. 4.13a.

The value of KG1 is computed as follows:


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Because KG1 is infinite, e must be zero for steady-state operation. Thus subtracting the

feedback signal from the reference input in Fig 4.13b gives

The preceding expression shows that the speed is independent of the load torque for an

integral control system. It is an easy matter to adjust the scale factor C2 for the input speed

dial so that C2K2/C4 =1, in which case

n0=nin (4.21)

The operation of an integral control system may be visualized as follows. From

Fig.4.12 it can be seen that if x momentarily changes and then returns to its line-on-line

position, the position of y has been changed permanently and so has the amount of flow

going to the engine. Therefore, changing the amount of flow to account for a new operating

torque does not change the steady-state position of x, which must be line on line. Because

neither x nor the spring compression changes, the output speed must always be equal to the

desired value in order that the flyweight force balances the spring force. (Note that, for the


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proportional control system, changing the fuel flow requires a permanent change in the

position x)

An integral control is easily recognized because there must be an integrating

component yielding a 1/D term in the block diagram between the comparator and the point

where external disturbance enters the system. The line operator of an integral controller is a

vertical line. The operating characteristic of an integral control system are shown in Fig. 4.14.

An integral controller is also called a floating controller because of the floating action

of the position y of the flow-setting valve. Two other terms used for an integral controller are

reset controller and isochronous controller.

4.5 PROPORTIONAL PLUS INTEGRAL CONTROL SYSTEMS

From a consideration of steady-state operation only, integral control systems seem

preferable to proportional systems. However, it is generally easier to achieve good transient

behavior with a proportional system than with an integral system (techniques for determining

the transient behavior of systems are presented in Chap. 5 through 12). It is possible to

combine the basic features of a proportional controller and an integral controller to from a

proportional plus integral controller.


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The action of a proportional plus integral controller in response to a change in the

input or external disturbance is initially similar to that of a proportional controller, but as the

new equilibrium point is reached, the control action becomes the same as that of an integral

controller. (In effect, the slope of the controller line continually increases).

A proportional plus integral controller combines the desirable transient characteristic

of a proportional controller and the feature of no steady-state error of the integral controller.

A proportional plus integral controller is shown in Fig. 4.15. The proportional controller

shown in Fig 3.25. The equation for the proportional action is


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Causes the position x to move down, as does e.The time constant 1 of the proportional unit

is small, so that y1 changes rapidly to increase the flow setting. The resulting motion of y1

returns e to its line-on-line position.

For the integrating unit, the quantity C/A is small, so that y2 continues to move at a

slower rate to provide corrective action. As the speed increases, the position x moves up. The

integration unit continues to provide corrective action until x is returned to its line-on-line

position (that is, x=0). In summary, for the proportional plus integral control, the initial effect

is provided primarily by the proportional action and the final effect is provided by the

integrator.

4.6 MODES OF CONTROL

In addition to proportional, integral, and proportional plus integral control, another

mode of control is derivative, or rate, action. For a derivative controller, the steady-state

expression for the control elements is

KG1= (KD)D=0=0

The output of a derivative controller is proportional to the rate of change of error. For

any constant value of the actuating signal e, the output of the control elements is zero. Thus, a

steady-state may exist in a derivative control system with any constant value of error signal.

Because a derivative controller operates on the rate of change of error and not the error itself,

the derivative mode of control is never used alone, but rather in combination with a

proportional, or integral, or proportional plus integral controller. The advantage of using

derivative action is that the derivative is a measure of how fast the signal is changing and thus

tends to give the effect of anticipation. The addition of derivative action is limited primarily

to systems which respond very slowly, such as large industrial processes.

The selection of the control elements G1(D) is seen to have a predominant effect upon

the steady-state operation of a system. For more complex control system, it becomes


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increasingly difficult, if not impossible, to distinguish the individual modes of control.

However, regardless of the various modes that may be present, it is a relatively simple matter

to determine whether KG1 is finite or infinite. For an infinite value, the integral action

predominates and there is no steady-state error due to variations in the external disturbance.

For a finite value, the system behaves as a proportional control system.

A major problem in the design of control system is the determination of the system

parameters to obtain satisfactory transient performance. The transient behavior of a system is

prescribed by the differential equation of operation for the system. In the next chapter, it is

shown how such differential equations may be solved algebraically by the use of Laplace

transforms. In Chap. 6, it is shown that the transient behavior is governed primarily by the

roots of the characteristic equation for the system. Thus, the transient characteristics of a

system may be ascertained directly from a knowledge of the roots of the characteristic

equation.

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