Sensitivity dipole and the sensitivity points method

SENSITIVITY DIPOLE AND THE SENSITIVITY POINTS IETHOD I. S.dl.r Inetitut. for Auto •• tion .nd T.l.co.aunic.tion. -I. Pupin-. B.lgr.d •• Yugo.l.vi.

Introduction In .ddition to the gener.l a.thod. of coaputing •• n.itivity function. /1-3/ • • • v.ral .p.ci.l a.thod. h.v. b •• n dev.lop.d /4-8/ for c.rt.in cl ••••• of .y.tea.. Aaong th •• e the aethod of •• nsitivity point. /6.7.9/ i. p.rticularly us.ful for lin.ar .y.t •••• off.ring two iaportant advantag ••• (i)

All •• n.itivity functions can be obtained simultan.ou.y froa

a .ingl. aod.l which i. id.ntical to the mod.l of the ba.ic .y.t.a, (ii) To obtain .ensitivity functions it is .uffici.nt to apply at the input of the sensitivity aodel the output .ignal of the ba.ic sy.t •• or it. aodel. Th. a.thod of •• n.itivity points ie based on a structural interpr.tation of logarithmic s.n.itivity function •• the •• neitivity point. ar. defin.d a. tho •• point. in the .y.t •• up to which • • tarting fro. the input of the .ystem. the tran.aittanc. is .qu.l to the logarithaic •• nsitivity function S~.

Since the qu.stion how to d~ter.in. s.nsitivity

points in a structur. of arbitrary coaplexity has not y.t b •• n answ.rsd, this pap.r is intend.d to provide a mora general structur.l interp.r.tion of logarithaic s.nsitivity function..

IntrodUCing the notion of s.nsi-

tivity dipol •• it is possible not only to d.t.rain. the log.rithaic s.n.itivity functions and the corresponding sensitivity points, but al.o to det.r.in. the ainimum nuab.r of aodel. requir.d to obtain siaultan.o.ly .11 sensitivity functions.

The sen.itivity dipol., furth.r, perait. a

dir.ct rel.tionship to b. established b.tween s.n.itivity and the .tructur. of the ba.ic .yst...

The t.chniqu. used is that of .ignal

flow graphs and the results are .xpreseed in t.ras of the well known topological formulae for logarithllic .ensitivity functions /11-13/.

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State.ent of the proble. Logarith.ic aeneitivity function is defined ae iHn T

(1)

51n •

where T is the trans.ittance between two node. A and 8 of ths graph C and. ie the branch of the graph with respect to which the sensitivity is being sought. I

Let us formulate the following proble •• find a graph C. and the nodes m end N such thet the transfsr function T' (s) between these two nodes is equal to the aensitivity function S. T •of the graph C. The problem can be solved by expressing the sensitivity function sT in the topological form for the transmittance

•

T~(s)

•

L

pi (s) A'i (s)

(2)

A'(s)

i

Using this expression we can construct the required graph C~ which we ehall call the sensitivity graph of the basic graph C with respect to the branch m. Since s! can be expanded in Pi, 4'i and A in several waye, there will be a set of graphs C~ corresponding to a single function s!. As a solution to our problem we shall eccept euch graph C~ which most closely resembles the bdSic graph C. We shall restrict our further treatment to the class of graphs C containing only one direct path. Logerithmic sensitivity functions The topological formula for the transfer function of graphs with one direct path reads Pl (s).d 1 (s)

(3)

T(s) .4 (s)

Introducing this expression in equation (1) we obtain

+

31n.4 1

81ni3

cHn m

31n m

- '''''-

(4)

Since il

is linear in m, .. e have

a In il

1

aln m

m - -.:IA

la

is obtained from .1 ..:I containt the factor m.

.,here

(5)

by eliminating all terms .. hich do not

In an analogous .,ay

a l.n

1I 1

1 _

cHn m

1I m 1

( 6)

~ for

(7 ) for Introducing these expressions in equation (4)

we obtain

( 8)

+

I t is useful to note that the cofactor

ill is obtained as the determinant

of tha graph Gl .,hich results from the graph G after eliminating the direct path PI. Graph Gl may consist of several separate subgraphs Gll , G12 , ••• , GIn one of .,hich, a.g. Gll , contains the branch m. In this case the determinant of graph Gl is obtained as the product of determinants 4 11' Ll 12 , ••• , Ll ln of subgraphs Gll , G12 , ••• , GIn and it can be sho .. n that the following relation holds

Lt m 1

(9)

71 Sensitivity dipole Taking into account equation (8) let us first determine graph and nodes

m and

G

N such that the transmittance bet.,een these t.,o nodes is

T' (s) On account of squation (2)

the graph G'

accordancs with the following expression

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(10) should be constructed ,in

pi .1' _ i_ _ i i

(11 )

.1'

The simplest solution of this equation is for i

11

,6' (s)

,6 (s)

(12)

pi (s)

1

(13)

£11, (8)

(14)

Relation (12) determines identity between graph G'

and the basic graph

G with respect to the arrangement of loops and their characteristics. rrom relation (13) it follows that between nodes m and N of graph G' there is only one direct path and that its trans~ittance is equal to unity. Relation (14) states that this unity transmittance path should have such a position that it does not touch anyone of the loops which ars not touching the branch m. Depending on the structure, there may be several branchee which fulfill this condition. Branch m doee so by defini tion. On the basis of the above considerations one can conclude that the required graph G' ie obtained by "inserting" path MN with unity tranemittance in the branch m of the baeic graph G. We shall call this unity transmittance path the "eensitivity dipole". rigs. 1 and 2 ehow the basic graph G and the graph G' obtained by ineering the sensitivity dipole in the corresponding branch of the basiC graph. We shall now uee this result for structural interpretation of the transmittance (15 ) I f a new node N'

ie introduced in graph G'

(rig. 3) one can ehow that

the trans.ittance between nodes M and N' is determined by equation (15). The unity tranemittance path dipole-.

.N '

will be called the -reverse sensitivity

rig. 4 shows the graph obtained by inserting the reverse sensi·

tivity dipole in the corresponding branch of the basiC graph C.

Sensitivity functions

We shell uee the sensitivity dipole for a structural interpretation of the logarithmic sensitivity function in each of the three possible pOSitions of branch m in the basic graph.

- 11.6-

Position

I.

Branch m is in the direct path.

Then

(16) and the sensitivity graph

G~

is obtained by inserting the sensitivity

dipole in the branch m of the baeic graph G (rig. 5). Position

11.

Branch m belongs neither to the direct

path nor the loops which have no touching points with the direct path. Then (17 )

1

and the seneitivity graph G~ i8 obtained by inserting the reverse sensitivity dipole in the branch _ of the basic graph G (rig. 6). p

0

8 i

t ion

Ill.

Branch m belong8 to at least one of the

loops which do not touch the direct path.

Then

(lB) and the 8ensitivity graph

G~

coneists of two parts.

One is obtained by

inserting the seneitivity dipole in the branch m of the basic graph G and the other by inserting the sensitivity dipole in the branch m of the corresponding subgraph Gll (rig. 7).

Sensitivity points We shall now analyee the etructural conditions which several sensitivity functions to be obtained single sensitivity graph Grouping

ei~ultaneously

per~it

frOM a

G~.

o f

i

n put s.

Let us first consider

the case when the sensitivity functions with respect to branches ending in a common node are obtained by inserting the reverse eeneitivity dipole of ths correeponding branch (rig. B).

Input. of the reverse eeneitivity

dipolas may then be joined together into a single node and the sensitivity functions with re.pect to individual branches of interest are obtained simultaneously at nodee sensitivity points.

N~.

2•.•••

N

N~

which represent the

If thsrs is a branch leaving the co.mon node and

if this branch contains the sensitivity dipole. the input of the latter can also be added to the co.mon input (rig. 9).

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T ran

8

1 a t

ion

0

f

dip

0

1 a.

Consider the graph

containing ora subgraph with concentrically arranged loops shown in Fig. 10.

If thase loops close the branches which contain sensitivity dipolas,

a aimpla procedure will permit the inputs of the dipoles to be grouped together.

By inserting branches with unity transmittance,

can be split up into nodes

C', C"

the node C

One thus obtains branches which

are contained in all loops that contain the corresponding branches band c.

From the definition of the sensitivity dipole it follows

that the

dipoles from branches band c can be moved to corresponding unity transmittance branches since relation (14)

remains satisfied (Fig. 11).

Tile

advantage gained from dipole translation consists in the following: inputs of all dipoles contained in the branches with unity transmittance which are separated by summing nodes can be joined together in the first node C' .

In this case again the outputs of sensitivity dipoles become

sensitivity points

(Fig. 12) at which the corresponding sensitivity

functions Can be obtained Simultaneously. The minimum numbar of nodes into which the inputs are grouped r.epresents the minimum number of sensitivity models required for simultaneous generation of sensitivity functions.

Acknowledgment The author wishes to express his grattude to Mr. P. Kokotovic who initiated this work and contributed actively to its completion in the form as presented in the paper.

The au thor has also had valuable

eXChange of experiences with Mr. R. Rutman of the Institute of Automatic and Remote Control, Moscow.

REFERENCES

1. K.S. Miller, F.J. murray, " A ma thematical basis for an error analysis of differential analyssrs", m.I. To J. math. and Phys., Nos. 2-3, 1953. 2. H.F. meissinger, "Ths use of parameter influence coefficients in computer analysis of dynamic systems", western Joint Computer Conf., San Francisco, may, 1960. 3. R. Tomovic, "Sensitivity Analysis of Dynamic Systems", New York, 1964.

McGraw-Hill,

4. M.L. Bykhovski, "Fundamentals of Dynamic Accuracy of Electrical and meChanical Networks", Akademizdat, 1958.

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5. m.L. Bykhovski, "Sensitivity and Dynamic Accuracy of Control Systsms", Tekhnicheskaya kibarnetika, No. 5, 1964.

6. P. Kokotovi~, "Structural method for the analysis of parameter influences in linear feedback syetems", VIII National Conference on Electronics and Automation, Zagreb, 1963.

7. P.

Kokotovi~, "Ssnsitivity points msthod in th~ analysis and optimisation of linear control syetems", Avtomatika i telemekhanika, No. 12, 1964.

B. m. Vuikovi~, V. Ciri~, "Structural rules for the determination of sensitivity functions of nonlinaar nonstationary systems", Symposiu~ on Sensitivity Analysis, Dubrovnik, 1964. 9. P. Kokotovi~, "Simultaneous computation of sensitivity cosfficints" in the book "Sensitivity Analysis of Dynamic Systems", by R. Tomovi~, mcGralll-Hill, Na", York, 1963. 10. H.W. Bode, "Network Analysis and Feedback Amplifier De8ign", Van Noatrand, Princeton, 1945. 11. S.J. mason, H.J. Zimmermann, "Electronic Circuits, Signals and Systems", John Wiley, Nsw York-London, 1960. 12. J.G. Truxal, "Automatic Feedback Control Systeme", McGraw-Hill, Ne", York, 1955. 13. A. Lynch, "Linear Control Systems" in the book "Adaptive Control Systems", by E. mishkin and L. Braun, mcGraw-Hill, New York, 1961.

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m

A

I

I

B

r---o

~L ____ ____ ___ ,.I

I

I

L____ ____ ___ J

I I

IL. ____ ____ ___ .JI

L____ ____ ___ .JI

- 150-

I I

I IL ___________ ..JI

I I

I

L___________ .J

r-----------~-----------.~----+~

m

m N

r

II

, IL ________ .....JI

II

IL

L ___________ J

151

________ .....JI

I

IL ____ ____ ___ JI nl·1

I I ____ ____ ___ .JI L

152

I

I I L ____ ____ ___ J Hg.10

I

I I L ____ ____ ___ J flg.11

I I I L ____ ____ ___ J

153