Dazzo Sensitivity

8/3/2019 Dazzo Sensitivity

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14

Parameter Sensitivity and State-Space

Trajectories

14.1 INTRODUCTION

In Chap. 13 it is shown that an advantage of a state-feedback control system

operating with high forward gain (hfg) is the insensitivity of the system

output to gain variations in the forward path. The design method of Chap. 13

assumes that all states are accessible. This chapter investigates in depth the

insensitive property of hfg operation of a state-feedback control system.This

is followed by a treatment of inaccessible states. In order for the reader

to develop a better feel for the kinds of transient responses of a system, thischapter includes an introduction to state-space trajectories. This includes

the determination of the steady-state, or equilibrium, values of a system

response. While linear time-invariant (LTI) systems have only one equili-

brium value, a nonlinear system may have a number of equilibrium solutions.

These can be determined from the state equations. They lead to the develop-

ment of the Jacobian matrix,which is used to represent a nonlinear system by

approximate linear equations in the region close to the singular points.

14.2 SENSITIVITY

The environmental conditions to which a control system is subjected affect

the accuracy and stability of the system. The performance characteristics


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of most components are affected by their environment and by aging.Thus, any

change in the component characteristics causes a change in the transfer func-

tion and therefore in the controlled quantity. The effect of a parameter change

on system performance can be expressed in terms of asensitivity function.This

sensitivity function S

M

is a measure of the sensitivity of the systems responseto a system parameter variation and is given by

SM dln M

dln

dln M

d

d

dln 14:1

where lnlogarithm to base e, M systems output response (or its control

ratio), and system parameter that varies. Now

dln M

d

1

M

dM

d

anddln

d

1

14:2

Accordingly, Eq. (14.1) can be written

SMMMoo

dM=Mo

d=o

M

dM

d

MMo

o

fractional change in output

fractional change in system parameter

(14.3)

where Mo ando represent the nominal values ofMand.When Mis a function

of more than one parameter, say1, 2 , . . . , k, the corresponding formulas for

the sensitivity entail partial derivatives. For a small change in from o , M

changes from Mo , and the sensitivity can be written as

SM MMoo

%M=M

o=o

14:4

To illustrate the effect of changes in the transfer function, four cases are

considered for which the input signal r(t) and its transform R(s) are fixed.

Although the responseY(s) is used in these four cases, the results are the same

when M(s) is the control ratio.

Case 1: Open-Loop System of Fig. 14.1aThe effect of a change in G(s), for a fixedr(t) and thus a fixedR(s), can be deter-

mined by differentiating, with respect to G(s), the output expression

Yos RsGs 14:5


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giving

dYos Rs dGs 14:6

Combining these two equations gives

dYos dGs

GsYos S

YsGs s

dYos=Yos

dGs=Gs 1 14:7

A change in the transfer function G(s) therefore causes a proportional change

in the transform of the output Yo(s).This requires that the performance speci-

fications ofG(s) be such that any variation still results in the degree of accu-

racy within the prescribed limits. In Eq. (14.7) the varying function in the

system is the transfer function G(s).

Case 2: Closed-Loop Unity-Feedback System of Fig. 14.1b

Hs 1

Proceeding in the same manner as for case 1, forG(s) varying, leads to

Ycs RsGs

1 Gs14:8

dYcs RsdGs

1 Gs214:9

dYcs dGs

Gs1 GsYcs

1

1 Gs

dGs

GsYcs

SYsGs

dYcs=Ycs

dGs=Gs

1

1 Gs14:10

Comparing Eq. (14.10) with Eq. (14.7) readily reveals that the effect of changes

ofG(s) upon the transform of the output of the closed-loop control is reduced

by the factor 1/j1 G(s)j compared to the open-loop control. This is an

important reason why feedback systems are used.

FIGURE 14.1 Control systems: (a) open loop; (b) closed loop.


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Case 3: Closed-Loop Nonunity-Feedback System of Fig. 14.1b

[Feedback Function Hs Fixed and Gs Variable]

Proceeding in the same manner as for case1, forG(s) varying, leads to

Ycs Rs Gs1 GsHs

14:11

dYcs RsdGs

1 GsHs214:12

dYcs dGs

Gs 1 GsHs Ycs

1

1 GsHs

dGs

GsYcs

SYsGs

dYcs=Ycs

dGs=Gs

1

1 GsHs14:13

Comparing Eqs. (14.7) and (14.13) shows that the closed-loop variation is

reduced by the factor 1/j1 G(s)H(s)j. In comparing Eqs. (14.10) and (14.13), if

the term j1G(s)H(s)j is larger than the term j1 G(s)j, then there is an advan-

tage to using a nonunity-feedback system. Further, H(s) may be introduced

both to provide an improvement in system performance and to reduce the

effect of parameter variations within G(s).

Case 4: Closed-Loop Nonunity-Feedback System of Fig. 14.1b

[Feedback Function Hs Variable and Gs Fixed]

From Eq. (14.11),

dYcs RsGs2dHs

1 GsdHs214:14

Multiplying and dividing Eq. (14.14) by H(s) and also dividing by Eq. (14.11)

results in

dYcs GsHs

1 GsHs

dHs

Hs

!Yc s %

dHs

HsYcs

SYsHs

dYcs=Ycs

dHs=Hs% 1 14:15

The approximation applies for those cases where jG(s)H(s)j>>1. When

Eq. (14.15) is compared with Eq. (14.7), it is seen that a variation in the feed-

back function has approximately a direct effect upon the output, the same as

for the open-loop case. Thus, the components of H(s) must be selected


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as precision fixed elements in order to maintain the desired degree of accuracy

and stability in the transformY(s).

The two situations of cases 3 and 4 serve to point out the advantage of

feedback compensation from the standpoint of parameter changes. Because

the use of fixed feedback compensation minimizes the effect of variations inthe components of G(s), prime consideration can be given to obtaining the

necessary power requirements in the forward loop rather than to accuracy

and stability. H(s) can be designed as a precision device so that the transform

Y(s), or the output y(t), has the desired accuracy and stability. In other words,

by use of feedback compensation the performance of the system can be made to

depend more on the feedback term than on the forward term.

Applying the sensitivity equation (14.3) to each of the four cases, where

M(s) Y(s)/R(s),where G(s) (for cases 1, 2, and 3), and H(s) (for case 4),

yields the results shown inTable 14.1. This table reveals that SM never exceeds a

magnitude of1, and the smaller this value, the less sensitive the system is to a

variation in the transfer function. For an increase in the variable function, a

positive value of the sensitivity function means that the output increases

from its nominal response. Similarly, a negative value of the sensitivity

function means that the output decreases from its nominal response. The

results presented in this table are based upon a functional analysis; i.e., the

variationsconsidered are in G(s) andH(s). The results are easily interpreted

when G(s) and H(s) are real numbers. When they are not real numbers,and where represents the parameter that varies within G(s) or H(s), the

interpretation can be made as a function of frequency.

The analysis in this section so far has considered variations in the

transfer functions G(s) and H(s). Take next the case when r(t) is sinusoidal.

Then the input can be represented by the phasor R( jo) and the output by

the phasor Y( jo). The system is now represented by the frequency transfer

TABLE 14.1 Sensitivity Functions

Case System variable parameter SM

1 G(s) dYo=Yo

dG=G 1

2 G(s) dYc=Yc

dG=G

1

1 G

3 G(s) dYc=Yc

dG=G

1

1 GH

4 H(s) dYc=Yc

dH=H% 1


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functions G( jo) andH( jo). All the formulas developed earlier in this section

are the same in form, but the arguments arejo instead ofs. As parameters vary

within G( jo) and H( jo), the magnitude of the sensitivity function can be

plotted as a function of frequency. The magnitude SM jo does not have the

limits of 0 to 1 given in Table 14.1, but can vary from 0 to any large magnitude.An example of sensitivity to parameter variation is investigated in detail in the

following section. That analysis shows that the sensitivity function can be

considerably reduced with appropriate feedback included in the system

structure.

14.3 SENSITIVITY ANALYSIS [3,4]

An important aspect in the design of a control system is the insensitivity of thesystem outputs to items such as: sensor noise, parameter uncertainty, cross-

coupling effects, and external system disturbances.The analysis in this section

is based upon the control system shown in Fig. 14.2 where T(s)Y(s)/R(s) and

where F(s) represents a prefilter. The plant is described by P(s) and may

include some parameter uncertainties (see Sec.14.4). In this system G(s) repre-

sents a compensator.The prefilter and compensator are designed to minimize

the effect of the parameter uncertainties. The goal of the design is to satisfy

the desired figures of merit (FOM). In this text it is assumed that F(s) 1.The effect of these items on system performance can be expressed in terms of

the sensitivity functionwhich is defined by

ST

T

@T

@

!14:16

where represents the variable parameter in T. Figure 14.2 is used for the

purpose of analyzing the sensitivity of a system to three of these items.Using the linear superposition theorem, where

Y YR YC YN

FIGURE 14.2 An example of system sensitivity analysis.


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and

TR YR

R

FL

1 L14:17a

TN YNN

L1 L

14:17b

TC YC

C

P

1 L14:17c

andL GP(the loop transmission function), the following transfer functions

and sensitivity functions (where P) are obtained, respectively, as:

S

TR

P

FG

1 L 14:18a

STNP

G

1 L14:18b

STCP

1

1 L14:18c

Since sensitivity is a function of frequency, it is necessary to achieve a

slope for Lm Lo( jo) that minimizes the effect on the system due to sensornoise. This is the most important case, since the minimum BW of

Eq. (14.17b) tends to be greater than the BW of Eq. (14.17a), as illustrated in

Fig. 14.3. Based on the magnitude characteristic of Lo for low- and high-

frequency ranges, then:

For the low-frequency range,where jL( jo)j>>1, from Eq. (14.18b),

STNP %

1

Pjo 14:19

For the high-frequency range,where jL( jo)j


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FIGURE 14.4 Bandwidth characteristics of Fig. 14.2.

FIGURE 14.3 Frequency response characteristics for the system of Fig. 14.2.


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order to minimize the sensor noise effect on the systems output. For most

practical systems n ! w 2 andZ10

log STP

do 0 14:21

STPN < 1 log STPN < 0 14:22aS

TPN

> 1 log STPN > 0 14:22bThus, the designer must try to locate the condition of Eq. (14.22b) in the high-

frequency range where the system performance is not of concern; i.e., the

noise effect on the output is negligible.

The analysis for external disturbance effect (see Ref. 4) on the system

output is identical to that for cross-coupling effects. For either case, lowsensitivity is conducive to their rejection.

14.4 SENSITIVITY ANALYSIS [3,4] EXAMPLES

This section shows how the use of complete state-variable or output feedback

minimizes the sensitivity of the output response to a parameter variation.This

feature of state feedback is in contrast to conventional control-system design

and is illustrated in the example of Sec. 13.9 for the case of system gain varia-tion. In order to illustrate the advantage of state-variable feedback over the

conventional method of achieving the desired system performance, a system

sensitivity analysis is made for both designs. The basic plant Gx(s) ofFig.13.9a

is used for this comparison. The conventional control system is shown in

Fig. 14.5a. The control-system design can be implemented by the complete

state-feedback configuration of Fig.14.5b or by the output feedback represen-

tation of Fig.14.5c,which uses Heqas a fixed feedback compensator. The latter

configuration is shown to be best for minimizing the sensitivity of the output

response with a parameter variation that occurs in the forward path, between

the state xn and the output y. For both the state-feedback configuration and its

output feedback equivalent,the coefficients kiare determined for the nominal

values of the parameters and are assumed to be invariant.The analysis is based

upon determining the value of the passband frequency ob at jMo( job)j

Mo 0.707 for the nominal system values of the conventional and state-

variable-feedback control systems. The system sensitivity function of

Eq. (14.3), repeated here, is determined for each system of Fig. 14.5 and

evaluated for the frequency range 0 oob:

SM s MMoo

M

dM

d

MMoo

14:23


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Note that Mis the control ratio. The passband frequency is ob 0.642 for the

system of Fig. 14.5a with nominal values, andob 1.0 for the state-feedback

system of Figs.14.5b and 14.5cwith nominal values.

The control ratios for each of the three cases to be analyzed are

Figure 14.5a:

Ms 10A

s3 p1 p2s2 p1p2s 10A

14:24

Figure 14.5b:

Ms 10A

s3 2Ak3 p1 p2s2 p1p2 2A5k2 k3p2s 10Ak1

14:25

Figure 14.5c:

Ms 10A

s3 2Ak3 p1 p2s2 p1p2 10Ak3 k2s 10Ak1

14:26

FIGURE 14.5 Control: (a) unity feedback; (b) state feedback; (c) state feedback,

H-equivalent. The nominal values are p1 1, p2 5 and A 10.


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The system sensitivity function for any parameter variation is readily

obtained for each of the three configurations ofFig. 14.5 by using the respec-

tive control ratios of Eqs. (14.24) to (14.26). The plots of SMp1 jo , SMp2 jo ,

and SMA jo vs. o, shown in Fig. 14.6, are drawn for the nominal values

ofp1 1,p2 5, andA 10 for the state-variable-feedback system. For theconventional system, the value of loop sensitivity equal to 2.1 corresponds to

0.7076, as shown on the root locus in Fig. 13.10a.The same damping ratio is

used in the state-variable-feedback system developed in the example in

Sec. 13.9. Table 14.2 summarizes the values obtained.

FIGURE 14.6 Sensitivity due to pole and gain variation.


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TABLE

14.2

System

SensitivityAnalysis

Controlsyste

m

Figure

Mjob

%

Mjob

SMp

1job

SMp

2job

SMAjob

Conventionalsystem

design

1.

Nomin

alplant

(A0.

21,p1

1,p2

5):

Gs

2:

1

ss

1s

5

14.5a

0.7

07

(ob

0.6

42)

1.0

9

1.2

8

1.2

9

2.

P1

2:

Gs

2:

1

ss

2s

5

14.5a

0.3

38

52.4

3.

P2

10:

Gs

2:

1

ss

1s

10

14.5a

0.6

58

6.9

4.

A0.42:

Gs

4:

2

ss

1s

5

14.5a

1.2

30

74.0


13/16

State-variab

lefeedbacksystem

des

ign[MisgiveninEqs.

(13.1

11),(14.2

5),and(14

.26),respectively]

5.

Nomin

alplant:

GsHeq

s

95

:

4s2

1:

443s

1:

0481

ss

1s5

14.5

b,c

0.7

07

(ob

1.0

)

0.0

36

0.0

5forX3

inaccessible

3.4

0forX

3

accessible

0.0

51

6.p1

2:

GsHeq

s

95

:

4s2

1:

443s

1:

0481

ss

2s5

14.5

b,c

0.6

83

3.3

9

7a.p2

10,

X3

accessible:

GsHeq

s

95

:

4s2

6:

443s

1:

0481

ss

1s10

14.5

b

0.1

60

77.4

7b.p2

10,

X3

inaccessible:

GsHeq

s

95

:

4s2

1:

443s

1:

0481

ss

1s10

14.5c

0.6

82

3.5

4

8.

A20

:

GsHeq

s

190

:

8s2

1:

443s1

:

0481

ss

1s

5

14.5

b,c

0.7

17

1.4

1


14/16

The sensitivity function for each system, at oob, is determined for a

100 percent variation of the plant poles p1 1, p2 5, and of the forward

gain A, respectively. The term %jMj is defined as

% Mj j Mjob Mojob

Mojob 100 14:27

where jMo( job)j is the value of the control ratio with nominal values of theplant parameters, and jM( job)j is the value with one parameter changed to

the extreme value of its possible variation.The time-response data for systems

1 to 8 are given inTable 14.3 for a unit step input. Note that the response for the

state-variable-feedback system (5 to 8, except 7a) is essentially unaffected by

parameter variations.

The sensitivity function SM represents a measure of the change to be

expected in the system performance for a change in a system parameter. This

measure is borne out by comparing Tables 14.2 and 14.3. That is, in thefrequency-domain plots ofFig. 14.6, a value SMA jo

1.29 at oob 0.642is indicated for the conventional system, compared with 0.051 for the state-

variable-feedback system.Also, the percent change %jM( jo)j for a doubling

of the forward gain is 74 percent for the conventional system and only 1.41

percent for the state-variable-feedback system. These are consistent with the

changes in the peak overshoot in the time domain of %Mp(t) 13.45 and

0.192 percent, respectively, as shown in Table 14.3. Thus, the magnitude

of %M( jo) and %Mp(t) are consistent with the relative magnitudes of% S

MA job

. Similar results are obtained for variations of the polep1.An interesting result occurswhen the polep2 5 is subject to variation.

IfthestateX3 (see Fig.14.5b) is accessible andis fed back directly through k3,the

sensitivity SMp2 jo is much larger than for the conventional system, having a

TABLE 14.3 Time-Response Data

Control system Mp(t) tp, s ts, s%Mpt

Mp Mpo

Mpo100

1 1.048 7.2 9.8

2 1.000 16.4 4.8

3 1.000 15.0 4.8

4 1.189 4.1 10.15 13.45

5 1.044 4.45 6

6 1.035 4.5 6

0.86

7a 1.000 24 4.4

7b 1.046 4.6 6.5 0.181

8 1.046 4.4 5.6

0.192


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value of 3.40 at ob 1.0. However, if the equivalent feedback is obtained

from X1 through the fixed transfer function [Heq(s)]0p2, Fig. 14.5c, the

sensitivity is considerably reduced to SMp2 j1 0.050, compared with1.28 for

the conventional system. If the components of the feedback unit are selected

so that the transfer function [Heq(s)] is invariant, then the time-response char-acteristicforoutputfeedbackthrough[ Heq(s)]isessentiallyunchangedwhenp2doubles in value (seeTable14.3) compared with the conventional system.

In analyzing Fig. 14.6, the values of the sensitivities must be considered

over the entire passband (0 oob). In this passband the state-variable-

feedback system has a much lower value of sensitivity than the conventional

system.The performance of the state-variable-feedback systems is essentially

unaffected by any variation inA, p1, orp2 (with restrictions) when the feedback

coefficients remain fixed, provided that the forward gain satisfies the

condition K! Kmin (see Probs. 14.2 and 14.3). The low-sensitivity state-variable-

feedback systems considered in this section satisfy the following conditions: the

systems characteristic equation must have (1) b dominant roots (


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nondominant pole drastically reduces SM job

, which is indicative of the

reduction of the sensitivity magnitude for the frequency range 0 oob.

Example 3. The (Y/R)T of Design Example 2 is modified to improve the

value ofts; see Eq. (13.127). Using Eq. (13.127), with p 70 and KG, in

Eq. (14.3) yields SM job 0.087 at ob 1.41.

The data inTable14.4 summarize the results of these three examples and

show that the presence of at least one nondominant pole in (Y/R)T drastically

reduces the systems sensitivity to gain variation while achieving the desired

time-response characteristics.

14.6 INACCESSIBLE STATES [5]

In order to achieve the full benefit of a state-variable-feedback system, all

the states must be accessible. Although, in general, this requirement may not

be satisfied, the design procedures presented in this chapter are still valid.

That is, on the basis that all states are accessible, the required value of the

state-feedback gain vectork which achieves the desiredY(s)/R(s) is computed.Then, for states that are not accessible,the corresponding kiblocks are moved

by block-diagram manipulation techniques to states that are accessible. As a

result of these manipulations the full benefits of state-variable feedback may

not be achieved (low sensitivity SM and a completely stable system), as

described in Sec. 14.4. For the extreme case when the output yx1 is the only

accessible state, the block-diagram manipulation to the G-equivalent (see

Fig. 14.8c) reduces to the Guillemin-Truxal design. More sophisticated meth-

ods for reconstructing (estimating) the values of the missing states do exist(Luenberger observer theory). They permit the accessible and the recon-

structed (inaccessible) states all to be fed back through the feedback coeffi-

cients, thus eliminating the need for block-diagram manipulations and

maintaining the full benefits of the state-variable feedback-designed

TABLE 14.4 Time-Response and Gain-Sensitivity Data

Example (Y/R)D MP tp, s ts, s K1, s1

SMKG

j!b ob 1.41

12s 2

s2 2s 2s 2 1.043 3.1 4.2 1.0 1.84

2 100s 2

s2 2s 2s 2s 50

1.043 3.2 4.2 0.98 0.064

3 100s 1:4s 2

s2 2s 2s 1s 2s 70

1.008 3.92 2.94 0.98 0.087

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