On the sensitivity of a system compensated by the

inverse Nyquist array

T.G. Koussiouris, Dip.-lng., M.Sc, Ph.D.

Indexing terms: Matrix algebra, Transfer functions

Abstract: Bounds for the elements of the sensitivity matrix S(ju>) of a system compensated by the inverseNyquist array method will be found and a graphical technique will be presented for determining areas in thecomplex plane in which the diagonal elements S,-,-(/u>) of SQ'OJ) lie. A graphical method is also described fordetermining frequency intervals in which the norm IIS(/u>)IL (or Il5(/u;)ll1) of the sensitivity matrix is lessthan one. The application of the proposed techniques is illustrated in two examples.

1 Introduction

Let the m-input, m-output linear system f have transfer-function matrix G(s) and let fc be the system resultingfrom the compensation of f as in Fig. 1. The open-looptransfer-function matrix of the compensated system is

Q(s) = L(s)G(s)K(s) (1)

while its closed-loop transfer-function matrix is

His) = Um+Q(s)F(s)]-1Q(s) (2)

Let us define

Q(s) ± Q~l(s) (3)

H{s) * H~\s) = Q(s) + F(s) (4)

and let 6 Q(s) be a vanishingly small perturbation of Q(s)while F(s) is fixed. From eqn. 4 we deduce

8H(s) = SQ(s)

If we define

8H(s) = [H(s) + 8 His)] -1 -His)

(5)

(6)

(7)

the multivariable feedback sensitivity matrices S(s), S(s)have been defined4'5 from the relations

8H(s)-H(s) = S(s)'8Q(s)'Q(s)

H{s) • SH(s) = Qis) • SQ(s) • S(s)

(8)

(9)

Generally, Sis) is different from Sis). It has been proved5

that

(10)

(11)Sis) = [Im+F(s)Q(s)]

It has also been proved1'2 that Sis) relates the deviation ec

from the nominal response of the closed-loop system to thedeviation eo of the open-loop system which gives thesame nominal response by

ee(s) = Sis)eois) (12)

Paper 782D, first received 20th November 1979 and in revised form24th March 1980Dr. Koussiouris is with the National Technical University of Athens,42 October 28th Street, Athens 147, Greece

IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

The sensitivity of the compensated system when sinusoidalinputs are applied is one of the criteria for the selection ofthe compensator. From eqn. 12 we have

eciJoS) = Sijcj)e0i/co) (13)

If /?(•) represents a vector norm, and 11*11 the inducedmatrix norm,3

(14)\\SiJw)\\'hieoiju))

Then, a sufficient condition on the closed-loop system inorder to show an improved behaviour to changes of theforward transfer-function matrix is that II5(/CJ)II shouldbe less than one within the frequency range of interest.

In designing the compensator of a system by the inverseNyquist array (i.n.a.) method we choose matrices Kis),Lis)such that QO'co) is column (or row) diagonallydominant9'10 for every CJ. The dominance of QijoS)is checked by means of Gershgorin's bands.9'10 If &represents the system having inverse transfer-functionmatrix

Pis) - diag «}„(*)) (15)

the remainder of the design is completed on the basis of theset of the individual single loops which are determined bythe system &. The matrix F(s) has the form

F(s) = diag ifu) (16)

where fii} / = 1, 2, .. .m, are constant gains such thatQiJcS) + diag (/fl) is column (respectively, row) diagonallydominant for every co.

Let ^ c be the system having inverse transfer-functionmatrix:

Pcis) = (17)

Fig. 1 General multivariable closed-loop system

111

0143- 7054/80/0401 77+11 $01-50/0

The sensitivity matrices of the system c are

= [Jm + diag (QH(s)) diag (/„)] -1

= d i a g

^ T(s) (18)

Suppose that two families of circles depending on the para-meters c£[0 , °°], yG[—n, n] are drawn in the complexplane as in Fig. 2. A circle of the first family has its centreat the point fu[-c2/(c2 - 1), 0] withradius \fucl(c2 - 1)|,and a circle of the second family has its centre at the point[-/ if /2, fu/(2 tan 7)] with radius |/K/sin 71- Then, theelement THQ'CJ) of TQ'cS) can be found easily from thepolar plot of QuiJcS) (which is an element of the i.n.a. ofQ(jcS), as follows. If OM = Qa(jcok), the modulus ofr,-,(/co) is equal to the parameter of the circle of the firstfamily which passes through M, while the argument ofTn(Jcok) is equal to the parameter of the circle of thesecond family which passes through M.

It has been proved8-9 that the stability of $c can bededuced from the stability of the system ^ c , while itsperformance (rise time, overshoot etc. of the ith outputdue to a step function at the rth input) can be estimatedfrom the performance of .^c by means of Ostrowski'sbands. In this paper, we shall show that the sensitivity offc can also be estimated from the sensitivity of thesystem JPC. Especially,

(a) bounds for the modulus |5,-,(/co) — 7 ,-(/co)| are to befound and a graphical method is to be described for deter-mining areas in the complex plane in which S«(/co) lies.

(b) bounds for the matrix norm \\S(JCJ)\\ will be deter-mined and a graphical methodjs to be described for findingfrequency intervals O = (OL, Ov) such that for every co G O,\\S(ju)\\<l.

2 Development of theory

2.1 Preliminary results

Before we proceed, we collect some results which areneeded later. By definition, the matrix H(joS) is column(respectively, row) diagonally dominant if the followingrelations are satisfied:

fe =

£ Ot\Htt(jcS)\t i = l , 2 , . . . , m (19 )

fe =

/ = 1,2 m

(20)

Then, we have

Lemma 1: If HQ'cS) is column (respectively, row)diagonally dominant, then

(a)0<di<\ ( 0< | ,< l )(b) H(joS) is nonsingular, and therefore [//(/co)] - 1 =

H(joS) exists(c) If Hik(JcS) represents the element standing in the ith

row and kth column ofHQ'cS),

(21)

(22){\Hki(joS)\

This lemma is due to Ostrowski,7 and an elegant proof canbe found elsewhere.8 Expr. 21 indicates a relation betweenthe elements of the matrix H(jcS). The following lemmagives relations between the elements of the matrices HQ'cS)and H(j<S):

Lemma 2: Suppose that H(s), H(s) are as in eqns. 2 and 4,respectively, and H(jco) is column diagonally dominant. If

(23)

then

\Hu(jo>)\

(24)

Fig. 2 Graphical method for determining elements 7},- (JLJ) of T(s)from the inverse Nyquist array.

= 0-5 and arg {TH(jtJk)}= 30°

/Voof: We expand the determinant of the matrix H{jtS)along the elements of the fcth column. If [H(ju>)]ki

represents the minor formed from rows 1, 2, ..., k — 1,

178 IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

k + 1 , . . . , m and columns 1,2,then

|deti/(/co)| =

, i — 1, i + 1 , . . . , w, From eqn. 25 we take

(25)

From expr. 21, we have

\HikQcS)\ =det/7(/to)

= ek(cj)detHQcj)

and, consequently,

<«*(«) "I [#(/*>)] ill (27)

Since HQ'cS) is column diagonally dominant and expr. 27holds,

foOl • 0fe (co) •

m

> Z IGw(/«)| w|

From exprs. 25 and 28, we take

- Z

x l - Z

Therefore,

det

IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

(28)

(29)

(30)

+ Z IGw(7«)l I [H{joS)\ u\

/")] ill

(31)

Then, we have

&et HQ'cS)

(32)

and the lemma has been proved.

Corollary: Suppose that H(s), H(s) are as in eqns. 2 and 4,respectively, and HQ'cS) is row diagonally dominant. If

£ = ( Z*. fe=1, fe =?t I

then(33)

<

I G n C M + / « l - 0 -(34)

Proof: If HQ'cS) is row diagonally dominant, its transposewill be column diagonally dominant and the corollaryfollows at once from lemma 2.

Let us define

4>t = max{0fe}, k =? 1,2, . . . , i - l , i

(35)

The 0,<co), i = 1, 2 , . . . , m, are the shrinking factors whichare used to draw the Ostrowski bands. Then, we have

Lemma 3: If 6it dt, <p{ are as in eqns. 19, 23 and 35,respectively,

0t (36)

Proof: Since 0 < 9k < 1, 0 < 0fe < 1 for k = 1, 2, . . . , m ,by definition

Z

maxm

Q.E.D.

179

2.2 Bounds for elemen ts of sensitivity ma trix

Let S(s) be the sensitivity matrix of fc, as in eqn. 10, andlet Sik(fcS) represent the element of SQCJ) standing in therth row and in the kth column. If the compensator isdetermined by the iji.a. method, HQcS) and QQ'oS) arediagonally dominant and theorem 1 holds.

Theorem 1

ifHQ'cS), Q(joS) are column diagonally dominant,

(0

(37)

(ii) <

< 1

\fii\

(38)

(iii)

(39)

Proof: Since QQ'cS) is diagonally dominant, it is nonsingular,and, from eqn. 10, we take

= Im -H(jcS)F = Im -H(jto) diag (/„) (40)

(a) From eqn. 40, we deduce that

S-h(fcj) = Hu(](jS)fuu (41)

and, from exprs. 21 and 24, we take

\SikQco)\ = \HikQu)\-\fkk\

\fkk\'Ok(co)(42)

Since d^cJ) < 0,(co) and 6k < 0,-, from the last relation wededuce that expr. 37 holds true.

(b) From eqn. 40, we have

(43)

Then, because 0 f(cj) < 0,(co) and expr. 24 holds,

(44)

(c) We have

= \fu\

l/iilE QkiU<

det

Since 0,- < 0,-, combining exprs. 27 and 45 we take

(45)

fe * i

(46)

and the theorem has been proved.

Corollary: If Q(joS), H(joS) are row diagonally dominant and

i = max{£fe}, k= 1 ,2 , . . . , / — 1 , / + 1 , . . . ,m ,

(47)

(0

(ii)

(48)

(49)

180 IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

(iii)

(50)

The proof is omitted because it is similar to that oftheorem 1.

2.3 Determination of frequency intervals for which

Proof:(a) Since T(jco) is diagonal and

we have from eqn. 54 and exprs. 37 and 39

(59)

If Xi represents the zth component of the vector x, we candefine the vector norms3

1=1

/2oo(x) = _ max {\Xi\}

Then the induced matrix norms are3

\\A\\1 = max V \Aik\\

(51)

(52)

H^Hoo = max \ X \Aiki = l , 2 , . . . , m

(53)

(54)

and theorem 2 holds.

Theorem 2

If H{j(jS), Q,{ju>) are column diagonally dominant, then

max

where

,

« = 1 m[\Qu(jcS)+fu\

\fu\-e,

l/fefelflfe

(55)

k * I

max

(56)

(57)

where\fkk\h

l/fcfcltffc (58)

= max £ \Sik(jw)-TikUcS)

m a x

1/fefe I

(b) Similarly, from exprs. 37,39, 53 and 59, we have

HS(Ao)ll,

= max f \Sik-Tik+Tik\\

< max \\Tkk\+\Skk-Tkk\+ I

max• = i,...,A\Qkk(M+fkk\

\fkk\h

ifkkW,i = l

Q.E.D.

Corollary: Ifft(jco), Q(jcj) are row diagonally dominant,

(0

ll^^^lloo < max | . " — - — [ - ^ / ( C J ) ) (60)

where _

^ « M = T7TT "

(ii)fe # i

( lgfefe(/co)l

(61)

(62)

where

Ak(co) =

i* ky (63)

IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980 181

The proof is similar to that of theorem 2.For the frequency cov G (OL,Ou) let us define

(64)

) (65)

where -fi,(coy) is as in eqn. 56 . In the i.n.a. diagram, forany coU} we draw the circles ^ , i = 1, 2 , . . . , m, so thatJFi is centred on QaU^v) with radius

+ fan &,(/«„)

(66)

For 0 L < co < Oy every circle sweeps up a band in thecomplex plane, and let Ot L, Oitu be the frequencies atwhich the circles ^{0itL), -^1(0tiu) a r e tangential to thestraight line Re {s}= —fa/2 while their centres lie to theright of that line. Then, we have

Theorem 3

is column diagonally dominant and

Ou = min {O( v}i = 1 , . . . , m

OL = max {Oi>L}i = 1 , . . . , m

for every co G [Ov, OL]

(67)

(68)

(69)

Proof: The locus JS ,- of the points for which ct = MO (MAis constant is a circle centred on the point [—facf/icf — 1),0] with radius \fuCi/icf — 1)|. As-shown in Fig. 3, becauseexpr. 66 is satisfied, the circle JTt is tangential to thelocus S?i.

The straight line Re {s} = —full is the locus of the

Qii.(joi)

Fig. 3 Graphical method' fan determining frequency intervals forwhich WSfjtjjW <1'

a S^i Locus of the points M: \OM\j\MA\ = c

b JfiCircle centred on Qa(.fu>v) tangent to^",-

182

points for which 6t = 1. Since expr. 55 holds, E \Sik \ < c,-,fe = I

and, because exprs. 67 and 68 are satisfied, for every

w e [ 0 L , O t , ] , max {c,} < 1i = l , . . . , mand, consequently,

\\SO'co)\\o < max {i = l m

Q.E.D.

Instead_ of eqn. 6 5 , c,(coy) can be defined to equalc i ( c o u l " ' ' ^ l ( w u ) - The new circles ^ can be drawn andnew OL, Ov can be determined. For simplicity we retainthe symbols of theorem 3 in the corollaries.

Corollary 1: If # ( / c o ) is column diagonally dominant, andthe circles ^ a r e drawn with radii given by eqn. 66 , where

*>v) (70)

(71)

dt(uv) = Ci(cj

then for co G [OL,dv]

\\S{jui)\h < 1

then, for co G [OL,Ou]

Corollary 2: If (/co) is row diagonally dominant, and thecircles ^ a r e drawn with radii given by eqn. 66, where

>v)) (73)

(74)

(75)

The proofs of corollaries 1 and 2 are similar to that oftheorem 3.

3 Description of graphical methods

In the description of the methods, we consider that Q(jco),ft(joj) are column diagonally dominant. As we mentionedthroughout this paper, the same techniques can be appliedto row-dominant matrices.

Suppose that we are interested in the sensitivity of theclosed-loop system in the frequency range 0= {oo:0L <co < Ou). For a number of frequencies coU5 u = 1, 2 , . . . ,N, (usually coy = OL + (u - 0 ( 0 ^ - OL)I(N - 1)) withinthis range, the vectors Vj(jojv) = [Tn(jcjv), . . . ,rm m(/c0y)]T are calculated by the computer, and by inter-polation the array of the m polar plots r,-,(/co) = Q.u(joS)l(QHUU) + fn)> OJ^O, can be presented on a graphicdisplay, as in Fig. 5. For any coU5 the circles /,-, / = 1, 2 , . . . ,m can be drawn such that /,• is centred on r,-,(/coy) and hasradius

\fn\-lM- S i )

(76)

For v = 1, 2 , . . . , JV, the circle /,• determines a band whichis called the z'th sensitivity band. According to theorem 1,

IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

the element SuQ'cS), w 6 0 , of the sensitivity matrix5(/co) lies within this band.

If the capability for calculations is restricted, the circle/,- can be drawn with radius

(77)

and a set of new bands can be determined. From expr. 39we deduce that the first band lies within the second one.

Using the i.n.a. diagrams we can determine frequencyintervals for which ||5(/co)||oo (or ||5(co)Hi) is less than

-2

-2 -1

-1

Re|6n(j<ju)

-2

Fig. 4 Gershgorin bands of the gas turbine for 0 < u> < 100 rad/s

IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

one. For a number of frequencies cov, the vectors UT(jwv) =[0ii( /wu),--- ,0mm(M,)]T,u= \,2,...,N, are calculated,and the circles JTt are drawn, as in Fig. 7, such that^",- iscentred on QUO'CJV) with radius p,(cjy). For w € O , therth circle sweeps out a band. Let OiL, Ot v be the fre-quencies for which the corresponding circle JTi is tangentialto the straight line Re {s}— —full. If OL, Ou are as ineqns. 67 and 68, from theorem 3 we have that for every^ e [^L> Ou] the norm ||5"(/co)||oo is less than one. If theradii p,(cjy) are determined as in corollary 1 of theorem 3and the corresponding OL, Ov are found, then for co£[OL,Ou\ the norm HS /co)!!! is less than one.

Illustrative examples

4.1 Sensitivity of a gas turbine

We shall first study the sensitivity of the design of a gasturbine.6 The inputs of the system are the demanded jet-pipe nozzle area and the demanded fuel flow rate, and itsoutputs are the high-pressure-spool speed and the low-pressure-spool speed. The inverse transfer-function matrix

Re[T

a. b

Fig. 5 Sensitivity bands of the gas turbine

a Drawn with radii /2,-(o;)b Drawn with radii R^ (CJ)

183

of the system is

1 f- 124(s + 2-037)(s + 10) 95-15(s + l-898)(s + 10)

6250 |p-0852(s + 1 -44)(s + 100) -0-01496(s + 1 -7)(s + 100)

The compensator K(s) has been chosen6 as

IUUIO^S -r zu-o) 131-94(s + 11-6)|1 H

K(s) =

(78)

s+ 158-500698(s + 146-3) 0-0206 (s + 101-4) 1 + 1

0-178s

(79)

The resulting Q(s) is

0-08s3 + 8-88s2 + 103-125 + 120 -0-01s2 - l-91s

- 5s2 + 59-6s 0-08s3 + 13-82s2 + 127s + 204 IQ(s) =

9-7)(80)

The matrix Q (jcS) is row diagonally dominant. The polarplots of the diagonal elements of the matrix Q(jcS) and thecorresponding Gershgorin bands have been drawn in Fig. 4.

The matrix F(s) has been chosen6 to equal the identitymatrix / 2 . Then, the polar plots of Tn (/co), T22 (/co) are asin Fig. 5. In the same Figure, the sensitivity bands aredrawn with radii equal to /?,(u)y) and /?,(a>,,), as ineqns. 76 and 77, respectively. The polar plots of thediagonal elements Sn (/co) and 522 OV) of the sensitivitymatrix are also displayed in Fig. 5. From this Figure, wecan see that the sensitivity bands which correspond to theradii ^-(GJJ,) are so narrow that the polar plots of SH(j(jS)and Tit{joS) cannot be distinguished.Let us define

\Sti-Tu\T; = — "— =

(81)

(82)

Obviously, TJ(WV) and f/Cw,,) give a measure of theefficiency of the bounds of eqn. 50. In Fig. 6, the diagramsof ^ ( w ) , T2(CO), ^(co) and T2(oS) are given. It can be

0 5

50 100

Fig. 6 Efficiency of bounds T, —T2, T, and f2 as functions offrequency

184

Re

Fig. 7 Bands for determining frequency intervals for which\\S(fuj)l\at<lOL = 0,Ou= 38-32 rad/s

IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

proved easily that, for the case of a two-input, two-outputsystem,

T\(uv) = r2(cov) (83)

for every OJV.In order to determine frequency intervals in which

WSQ'o^Woo ^ 1, the polar plots of the diagonal elements ofQijco) with the corresponding J2T; circles are ^rawn inFig. 7. Using interpolation, we can find that Ov, as in,eqn. 67, is equal to 38-32 rad/s, while the value of co, saycoc, for which IIJS(/CJC)| |O O= 1 is equal to 38-409 rad/s.The ratio (coc —OU)I(JJC = 0-00231 gives a measure of theaccuracy of the proposed method.

4.2 Sensitivity of a gas compressor

The following example concerns the design of thecontroller for an industrial compressor*10 which hastransfer-function matrix

G(s) =

0-1134g"°-72s

l-78s2 +4-48s +

0-3378g~°-3s

0-36 Is2 + V09s-\

0-9242-07s + 1

-0-318e~ i : 2 9 s

2-39s + 1

(84)

05r

Fig. 8 Gershgorin bands of gas compressor for 0 < u> < 9-5 rad/s

IEEPROC, Vol. 127, Pt. D, No. 4, JULY 1980

05 10

Fig. 9 Sensitivity bands of gas compressor drawn with radii R iand diagonal elements of sensitivity matrix

185

The feedforward compensator K(s) and the feedbackcompensator F(s) have been chosen:9'10

K(s) =

0-379 2-431

1-038 -0-298

(\ +0-3s\1 + 005 s

1 + 0-35

1 + 005s

F(s) = diag(5,3-5)

(85)

(86)

The resulting Q(jo>) matrix is row diagonally dominant.The polar plots of the diagonal elements of @(/a>) and thecorresponding Gershgorin bands are drawn in Fig. 8.

The polar plots of Tn (joS) and T22 (/CJ) are shown inFig. 9. In the same Figure, the sensitivity bands are drawnwith radii Rt(cjv), as in eqn. 76, and the diagonal elementsSn(/co) and S 2 2 ( / G J )

o f t n e sensitivity matrix. Thediagrams of the ratios TX = r2 , TX and f2 are given inFig. 10.^ In Fig. 11, the polar plots of the diagonal elements of

Q{jiS) with the corresponding circles Jft_ are presented.Using interpolation we can find that Ov is equal to2-07 rad/s. The frequency coc k : which \\S(jco)\\00 = 1 isequal to 2-313 rad/s. Then, the relative error of theestimation of the frequency interval is (ooc — OU)I<JOC =0105.

5 Conclusions

Graphical methods have been established for studying thesensitivity of the compensated system when thecompensator has been designed by the i.n.a. method. Theshrinking factors 0,(co) which are used to construct theOstrowski bands can also be used to design the sensitivitybands. However, since the function >> = JC/(1 — x) is rapidlyincreasing in the interval (0, 1), jrnuch better results areobtained by the use of the factors 0,(co). This is justified inthe examples.

In the first example, the matrix Q{joS) is nearlytriangular, and the proposed technique was expected to beaccurate. In the second example, although Q (joS) has nospecial form, the sensitivity bands drawn with radii Rt(u>)are narrow and the error in determining coc is tolerable.

10

0 5

10

Fig. 10 Efficiency of bounds r, =T2, T, and f2 as functions offrequency

186

6 References

1 CRUZ, J.B., and PERKINS, W.R.: 'The role of sensitivity in thedesign of multivariable linear systems'. Proceedings of theNational Electronics Conference, 1964, 20, pp. 742-745

2 CRUZ, J.B., and PERKINS, W.R.: 'A new approach to thesensitivity problem in multivariable feedback system design',IEEE Trans., 1964, AC-9, pp. 216-223

3 LANCASTER, P.: 'Theory of matrices' (Academic Press, 1969)4 MacFARLANE, A.G.J.: 'A survey of some recent results in

linear multivariable feedback theory', Automatica, 1972, 8,pp. 455-492

5 McMORRAN, P.D.: 'Parameter sensitivity and the inverseNyquist method'. CSC report no. 119, University of ManchesterInstitute of Science and Technology, 1970

6 McMORRAN, P.D.: 'Design of gas-turbine controller usinginverse Nyquist method', Proc. IEE, 1970,117, (10), pp. 2050-2056

7 OSTROWSKI, A.M.: 'Note on bounds for determinants withdominant principal diagonal', Proc. Am. Math. Soc, 1952, 3,pp. 26-30

8 ROSENBROCK, H.H.: 'State space and multivariable theory'(Nelson, 1970)

9 ROSENBROCK, H.H.: 'Computer aided control system design'(Academic Press, 1974)

10 ROSENBROCK, H.H.: 'Computer aided design of multivariablesystems'. CSC report no. 285, University of Manchester Instituteof Science and Technology, 1975

-5

Fig. 11 Bands for determining frequency intervals for which||5/'/wyiloo<7

OL = 0 , 0 ^ = 2 0 7 rad/s

IEE PROC, Vol. 127, Pt. D, No. 4, JULY 1980

Trifon Koussiouris was born in Athens on 23rd September1948. He received the Diplom-Ingenieur degree in Electricaland Mechanical Engineering from the National TechnicalUniversity of Athens in 1971, and the M.Sc. and Ph.D.degrees from the University of Manchester Institute ofScience and Technology, England in 1975 and 1978, res-pectively. From 1971 to 1974 he served in the Greek Armyas a telecommunications officer. At present he is lecturingat the National Technical University of Athens. His researchinterests include multivariable systems theory, multi-variable controller design and optimal control.

ErrataSHIEH, L.S., and TAJVARI, A.: 'Analysis and synthesisof matrix transfer functions using the new block-stateequations in block-tridiagonal forms', IEE Proc. D, ControlTheor. &AppL, 1980,127, (1), pp. 19-31

On page 19, column 1, the line following eqn. \c shouldread:

where y(t) is an m x 1 output vector, r(f) is an m x 1input.. .

On page 20, column 2, eqn. 8c, the bottom row should read:

(H2Kl yl instead of (H2K3)"1

On page 23, column 2, paragraph 1, line 7, delete the twosentences from 'Because...' to end of paragraph

On page 26, column 2, the sentence following eqn. 25bshould read:

Since P is positive definite and Nt are assumed to beimaginary matrices, v is positive definite.

On page 27, column 1, line 8 should read:

It is noticed that Nt are imaginary matrices;Mt.. .

On page 27, column 1, the following should be insertedbefore the second paragraph from the bottom:

An additional sufficient condition is that, if both Nt andMi are real symmetric matrices and G2 in eqn. 19 is a realsymmetric negative-definite matrix, then the system ineqn. 2 is asymptotically stable. This sufficient conditioncan be verified from the fact that all eigenvalues of a real,symmetric, negative-definite, system matrix G, in eqn. 19,are negative real.

On page 27, column 1, last paragraph, the second lineshould read:

and thepairs (H2iK2i_1) are positive definite and the pairs(K2i+1H2iK2i-1)~

1/2 are imaginary matrices, then the.. .

On page 27, column 1, the tenth line from the bottomshould read:

all Kt and Ht are positive definite, and G2 in eqn. 19 is areal symmetric negative-definite matrix, then the system isnot . . .

On page 28, column 2, the fourth line from the bottomshould read:

Mi and M2 are real symmetric and G2 < 0 in eqn. 19,symmetric and positive definite, the. . .

On page 30, column 2, line 9 should read:

same sign Nx =N[, and .Mi and Af2 are real and symmetricand G2 < 0 in eqn. 19, positive definite.. .

ETC67D

SINGH, Y.P., and SUBRAMANIAN, S.: 'Frequency-response identification of structure of nonlinear systems',IEE Proc. D, Control Theor. & Appl., 1980, 127, (3),pp. 77-82:

On page 78, column 1, Fig. 3, second block from the topon right-hand side: angle at origin marked 0X should read 0X

On page 81, column 1, the second half of line 3 shouldread:

G2(s) =0168

s(s 20)

ETC 66D

IEE PROC, Vol. 127, Pt. D, No. 4, JULY 1980 187