164 IEEE TRAA'SACTIONS OAT AC'TOAfATIC CONTROL April

APPEKDIS V Proof of Lemma 2

1) pjiyi is the jth component of x o ; then the condition 1) is necessary.

2) If X O E R X , a straight line passing through x0 and parallel to rl intersects C, and C-, then yl+2yl-. Sow, in the remaining part of the proof we assume that the conditions 1) and 2) hold.

3) Suppose the condition 3) holds. Since Fs is con- vex, i t follows that xoEF.\-. The condition 1) implies xoESo. Then, xoEF~A.50. Since F.,-ASoCR.,-, we have x0ER.v.

4) The point yl+rl+ym+ . . . +y,lrn lies on C+. Then, the point xl,=ylrl+yyrz+ . . . +yr.rn such that

i y1 -yl+ I 5 1 belongs to R.Y.

5) This can be proved similarly.

REFERENCES [l] Pontvagin, L. S., 17. G. Boltyanskii, R. X?. Gamkrelidze, and

E. F. hlishchenko, The 4fathematical Theory of Optimal Processes, Ye\+- York: IViley, 1962, chap 6.

[2] Chang, S. S. L. , Optimal control in bounded phase space, Auto- matica, vol 1, Jan-hlar 1963, pp 55-67.

[3] Ho, Y . C., A computational technique for optimal control prob- lems with state variable constraint, J . X d h . AItaZ. a d Appl. ,

[4] HP, E'. C., and P. B. Brentani, On computing optimal control w t h inequality constraints, Minneapolis Honeywell Regulator Co. Rept. prepared for Symp. on hlultivariable Control Systems,

IS] Kalman, R. E., Optimal nonlinear control of saturating systems 1962 Joint .Automatic Control Conference. 1962.

by intermittent action, 1957 IRE Ti'escon Conr. Rec., pt 4, pp. 130-135.

[6] Desoer, C. A , , and J. \\-ing, The minimal time discrete system,

11 1-125. I R E Trans. on Automatic Control, 1-01 .AC-6, May 1961, pp

[f] Desoer, C. A. and J. If-ing, The minimal time discrete system

[8] Kodama, S. , Controllabilitl- of linear discrete-time systems with (General Theory). J Franklin Imf . , vol 252, 1961, pp 208-228.

input and state variable constraints. (Japanese) J. Inst. Elec. Comm. Engrs. of Japan (to appear).

191 Sagata, .A,, Optimal control of sampled-data control systems. (Japanesej D.E. dissertation, Osaka Cniversity, Osaka, Japan, 1964, p 100.

VOI 5 , 1962, pp 216-224.

Quasi-Linear Analysis of Oscillating Servo-Systems with Random Inputs

YEhT S. LIM, MEMBER, IEEE

Abstract-This paper presents an approximate analysis of oscil- lating servo systems with random inputs. The power spectrum rela- tions for these systems are established by extending the dual-input describing function method. The analysis shows that the system ex- hibits a quasi-linear mode of operation similar to that previously shown to exist for a sinusoidal input. Distortion components resulting from the closed loop system are examined, and conditions for quasi- linear operation (small distortion) are given in terms of signal level and bandwidth. Some analog computer results are presented to demonstrate the applicability of the quasi-linear theory to the random input case.

I. INTRODUCTION

T H E OSCILLATING servo is interesting in that under certain conditions it has high gain, almost linear behavior, and adaptive characteristics. This

t>Ipe of sl-stem has been analyzed by Lozier, AIacColl, Schuck, Gelb and Vender \,'elde, Bonenn and many others [1]-[6]. However, these anall-ses were con- cerned with sinusoidal inputs.

This paper presents an approximate statistical anal- ysis for oscillating servo systems, and provides a basis for the use of the quasi-linear dual-input describing

\lanuscript received September 8, 1964; rex-ised January 5, 1965. The author is with Bell Telephone Labs., Inc., IYhippany, X. J.

function (DIDF) method when the input is a random signal. The basic configuration of an oscillating servo is shown in Fig. 1. I t consists of a constant parameter linear plant G(w) whose phase shift characteristics passes through - 180 degrees, and a bang-bang controller ( * r-) with a sign-of-error control law. The input to the sl-stem is assumed to be stationary Gaussian with zero mean.

Fig. 1. Oscillating servo.

The computed DIDF of a sinusoid and a random com- ponent passing through the relay-type nonlinearity are given in Section 11, along with the description of various components at the output of the relay. In Section 111, the important relations of the input-error power spectra are derived. Also a graphical method for easy solution of the quasi-linear mode is described. Section IV dis- cusses the effect of distortion components of the closed-

1965 Lim: Oscillating Seroo-Systems with Random I n p u t s

loop system. Analog computer results are presented in Section V.

11. THE DUAL-INPL‘T DESCRIBING FUNCTIONS Consider the input z ( t ) to an ideal relay (see Fig. 2)

as the sum of a sinusoidal carrier component c ( t ) and a Gaussian random signal component e( t ) with zero means and autocorrelation Qe(r) ,

z ( t ) = c ( t ) + e ( t ) , c ( t ) = d sin wet. (1)

Fig. 2. Ideal relay function.

Using the correlation method of Rice [g] with the aid of integral representation of the relay function [ l l 1 , the autocorrelation function of the output is

for IZ + k = odd,

h n k 2 = 0 for 11 + k = even, ( 3 )

and where lFl(a, y , x) is a confluent hypergeometric function. The first two terms of (2) represent the signal and the carrier component, respectively, i.e.,

[@p(~)]s + [@u(r)]c = - ho12@e(T) - 2J1.10’ COS ~ 4 0 ~ . (4)

Since the correlation function of the input to the non- linear element is

A

2 @*(T) = ae(T) + - COS W O T ( 4 4

the describing functions may now be defined as

s , = [ - h o 1 2 ] l!2

and

where

The describing functions defined above are indeed iden- tical to the linearized equivalent gains in the sense of Booton [ 7 ] . The necessary and sufficient condition for minimum rms error between the output of the non- linear element and the output of the linear equivalents with gain K , for the signal component and K c for the sinusoidal component is that [8]

where cPcu(r) and cPcu(r) are the cross-correlations of c ( t ) and u(t) , e ( t ) , and z ~ ( t ) , respectively. For the particular case when the signal component is Gaussian, K c and K , can be shown equal to N, and respectively. The plot of (A / L’) . N , and ( A / L:) . A i c vs. 17 is given in Fig. 3. Xote that for small signal (7 small), the signal gain is almost half the carrier gain,

which is the same result for a sinusoidal signal. When A =0, then

which is the usual equivalent gain for Gaussian input.

1.2

. B

.4

0 .4 .8 1.2 1.6 2

Fig. 3. Dual input describing function.

166 IEEE TRANSACTIOXS ON A U T O M A T I C CONTROL April

For constant A , M, is nearly constant up to 7 = 0.4 and has only 17 per cent increase a t maximum when 7 = 0.8. For 77 > 0.8, saturation effect is evident. A l l other terms of (2) become important for large 9. These terms are useful in estimating the distortion in the closed-loop system. The mean square values (T = 0) of the first few terms of (2) are plotted as functions of 7 in Fig. 4 for comparison. Kote that these curves only give a partial indication of the distortion of the closed-loop system. The relative importance of these terms also depends on their power spectrum distribution and the frequency characteristics of the plant as will be discussed i n Section 11‘.

... 0 0.4 0.8 I . 2

Fig. 4. hIean square values for first six terms of ( a ) .

I f the power spectrum &(w) is defined as the Fourier Transform (FT) of @=(TI, i.e.,

1 “ +<(w) = - J e-j*r%e(7)d7

$ P ( w ) , the FT of @.,“(T), can be evaluated by the con- volution integral,

2n .-m

&(w) = J-:+.‘(. - w‘)+e(w’)dw’, k = 2, 3 , . . (9)

with $1(w) = & ( w ) . Then the spectrum of the distortion terms is

OC hnkZ

n=l 1;=1 k . + 7 [&(w + 1zwo) + &(w - n w o ) ] . (10)

The general characteristics of the distortion spectrum may be described as follows. The first summation terms of ( lo) , independent of the distribution of Q e ( w ) , repre- sent the higher harmonic components of W O . These terms have little significance if a low-pass linear element fol- lows. The second summation terms, independent of w O , also represent the distortion of partial interaction.

Assume that &(a) has a narrow band. As a result of the convolutions, the major peak of #l;(w), for k = 3 , 5, . . , always coincides with that of the signal spectrum &(a). This in effect changes the equivalent gain AT,. The dis- tortion from the second summation terms will be small onl?: if 77 is small. The third summation terms give the distortion due to complete interaction between the sig- nal and carrier. Since &(w f nu,) is simply a translation of #J~(w) by + m o o , those components with large n are insignificant i n view of the low-pass element which is assumed to follow. For x= 1, all the major peaks for k = 2 , 4, . . . always coincide with wo, thereby affecting the carrier frequency. For n=2, the average powers of the terms for k = 1 and 3 are relatively large (see Fig. 4), but their major peak locations are not fixed, and depend on the peak location of & ( w ) . The interference of these terms with the signal or the carrier component can be avoided if &(w) has a narrow band a t frequencies much lower than wo. This is one of the considerations given later in the discussion of frequency limitation for the closed-loop system.

111. APPROXIMATE STATISTICAL AN-ALYSIS OF THE CLOSED-LOOP SYSTEM

\Then the input to the system of Fig. 1 is zero, the oscillating frequency w o can be exactly determined by the Zypkin loci or Hamel loci method. The power spec- trum of z(f) consists of 6 functions a t frequencies w O ,

3w0. S U O , . . . The magnitude at w o is by far the largest. Suppose now that the input is a slowly varying signal, and the system is operating normally as an oscillating servo. The error spectrum will conceivably have a sharp spike at frequency near wo and at the low frequencies, a continuous spectrum which is almost identical to that obtained from an equivalent linear system (Fig. 5). I t is this idea that leads to the following assumption.

Fig. 5. Error power spectrum.

1) The spectrum equation. From Fig. 1,

x ( f ) = e ( / ) + y(t). (11)

Then the spectrum equation is given by

+&) = 4 d w ) + + Z , b ) + +v,Z(w> + +?Am). (12)

The key assumption is that z(t) is a sinusoid plus a random Gaussian component,

E(/) = d sin wot + e( t ) (13)

with A and c$~(w) unknown. The justification of this assumption will be discussed when the distortion is con- sidered. The spectrum of the input to the nonlinear ele- ment is the FT of (4a);

1965 Lim: Oscillating Servo-Systems witla R m d o m Inputs 167

A 2

2 &(w) = A&) + - S(w - wo). (14)

The cross-spectrum, from ( 8 ) , can be written as

A *

2 + z , ( ~ ) = G(w) [.\..&(w) + ;I7, -6 (w - w O ) ] (15)

A2 2

q!+,,.(w) = G ( - w ) LI~~&(u) + :Yc -6(u - OO)] (16)

and the power spectrum of y( t ) is

&(w) = G(w)G(-w) A 2

where &(a) is given in (10). Substituting (14) through (17) in ( l l ) ,

. [ S S 2 & ( ~ ) + X , ? - S(W - w 0 ) + +&)I (17)

&(u) = [l + ;Y:,G(wo) + iVcG( -wo) A2

2 4- :l~czG(~o)G( - W O ) ] - S(W - W O )

+ [l + LYIG(o) + LTSG(-w) + A'I?G(~)G(-w)]q5e(w)

+ G(w)G(-w)&(w). (1 8)

Since &(a) is the power spectrum of the input to the system, i t is necessary that the coefficient of the term G(w-wo) be zero, i.e.,

1 + S,G(wo) + :l',G(-~g) + S,'G(wo)G(-wo) = 0

which implies that

1 + :YT,G(~O) = 0. (19)

This is the well-known equation to determine the oscil- lating frequency wo and X c . Rearranging (18) gives

For a given &(a), (20) is too difficult to solve. A quasi- linear solution for &(w) can be obtained if the second term on the right-hand side of (20) is small and can be ignored. This means that all the distortion components are sufficiently attenuated around the loop. &(w), and -4 can be solved by the graphical method described in 2). Using (lo), the second term of (20) is then integrated to yield the ms value of the closed-loop distortion which can be compared with ae(0).

2) Graphical method for the quasi-linear solution. In order to facilitate the graphical method, the four equa- tions necessary for the four remaining unknowns A , N , , 7, and Qe(0), are rewritten as follows. Let the curves of Fig. 3 be represented by

(21)

(22)

with

Finally using Parseval's integral,

1

'e(0) = S , l 1 + LYsG(w) ! i 24z(w)dw.

Since N, is known from (19), (21) through (24) can be expressed as functions of 7,

(24)

-4 = L'I G(uo) [ fl(71) (25)

I t is now a simple matter of plotting of these four expressions against 7. The intersection of the last two in Fig. 6 gives the quasi-linear solution. Then A and ;V8 are obtained from Fig. 7, and the approximate # I ~ ( w ) from (20).

Fig. 6. Sketch of @*(O) vs. 7.

168 IEEE TRANSACTIONS O N A C T T O M A T I C C O N T R O L April

IV. DISTORTION ASD APPLICABILITY OF

QUASI-LIXEAR ANALYSIS The approach to the approximate solution described

above is based on the assumption that the input to the relay is the sum of a sinusoid plus a random Gaussian component. This assumption is valid only if the distor- tion is small. Otherwise, if the distortion is found to be large, the Gaussian assumption is incorrect and, physi- cally, the oscillating component may frequently disap- pear. Then the equivalent linear system can not be a good representative of the nonlinear system. To s tudy the distortion effect, i t is necessary to examine the inte- grand of the following:

ms value of distortion = # d ( W ) d W . (29)

I t is intended that the discussion be kept general and not tied to any particular system. Since the factor

G(u) I 1 + S s G ( w ) 1 , for a normally behaved G(w), falls off rapidly beyond wo, only a few terms of +d(w) are needed. -Also this factor will have a high peak i n the range [ two , wo] if the gain N, is large. Hence, the distribution of +d(w) in this range will strongly affect the distortion. Note that 9 , and the coefficients h , , 2 in various terms of + d ( w ) will increase if 7 is increased.

From the foregoing, the following results can be stated.

If +,(w) is such that 1) & ( w ) has narrow bandwidth with peak location well below wo, and 2) 7 is small, possibly not exceeding 0.8, then &(w) will have a small number of terms with large part of their power in the pass band and not concentrated in the range [wo.:2, w O ] . Consequently, the integral (29) will be small.

T o show this, suppose that &(w) has the form shown in Fig. 8(a) with its peak location at +wo. The forms of the first few terms of &(w) are sketched in Fig. 8(b)-(d) according to the convolutions of &(w). The general description of various components has been given in Section 11. In this particular case, the major peaks for (n , k ) = ( 2 , 1) and (2, 3) are just outside the pass band, only a minor peak for (n, K ) = (2 , 3) is located a t wo [Fig. 8(d)]. If the peak of &(w) is moved out toward

W O , these major peaks will be brought in near the pass band, “contaminating” the carrier component. This will result in greater distortion, since these components have relatively large powers (Fig. 4). The interference situation of various components with the carrier and the signal for different peak locations up of &(o) are sum- marized in Table I.

I t is seen from Table I that when w, is in the range [$coo, w o ] , all terms listed above have their power con- centrated in the accentuated range of the pass band. Hence, to keep the distortion small, the spectrum ~ & ( w )

A A W

- 2 - I O L I 2 w o 3

-

J z - 2 - I 0 I 2 2 w o

3

Fig. 8. Power spectra.

‘FAABLE I

Term 1 1 . k

1 2 1 1

with op

Interference with wg

X X = XIajor peak of &.A. (0).

X =hIinor peak of &.r;(w).

must be narrow. and its peak location must be well below coo. Unlike the linear servo, this is a severe restric- tion for oscillating servos.

Suppose now that condition 1) is satisfied. For 7 < 0.4, :VS is almost constant, and the components (0 , 3) and (1, 2) are very small compared to the signal and carrier (see Table I and Fig. 4). The operation is almost linear. Beyond, but not far from that point, as the input is in- creased, the distortion gradually increases but still is small. The system remains essentially linear, as the amount of distortion allows a test for the applicability of the quasi-linear analysis. Note that for large iVs, the factor

1965 Lim: Oscillating Servo-Systems w’th Random Inpu t s 169

a t frequencies near wo becomes not only very large but also extremely sensitive to a small change in -!J~,. There- fore, if the input is further increased, the distortion becomes not only large but increases very rapidl>T. I t is possible that saturation effect quickly shows up.‘ From Fig. 3, saturation effect is evidenced for r]>0.8 even for for constant -4. Of course there is no clear cut boundary lvhen the quasi-linear anal>-sis completely fails. In the saturation region, :he behavior of the system is too corn- plicated for analysis. Complete saturation is reached when the input is so large that the oscillating com-

The above expression in fact is a simplified solution from (27) and (28), and is of interest in that i t relates almost all the parameters of the system. \J7ith this, the effect of variation in each parameter upon 17 can be easily assessed. Relation (30) is shown in Fig. 9. The in- tersection point Pgives the solution forv. .Although small variation of @,(O) or K, changes r ] ? the system still re- mains in the quasi-linear range. This demonstrates the gain-adaptive characteristics of the oscillating servo. However, adaptability is limited by the distortion if K, becomes too small. On the other hand, since the amplitude of the oscillating component is proportional to Ke, the upper bound of K , is determined by the amditude of th i s comDonent tolerable to the slstem.

ponent disappears ( A = O ) . Thus, the sgstem is over- loaded, resulting in very large error. In any specific case, the practical range of operation is determined by the error requirement of the slrstem.

Since &(w) is determined by & ( w ) of the input to the system, conditions 1) and 2) can easily be transformed into corresponding conditions on @ , ( w ) . The distortion will be small if la) &(a) has a narrow bandwidth at fre- W C u p quencies much lower than wo, or sharply cut off a t w < Q w o , and 2b) &(O) is small, possibll: not exceeding that corresponding to r] = 0.8. 7

In general, the existence of self-oscillation is sufficient to insure the quasi-linear mode of operation. The rnain- tenance of oscillation requires that the signal at the input to the relay is small relative to the amplitude of V. COMPUTER STUDY

/

Fig. 9. Sketch of (30j.

oscillation. For random input, this requirement is reflected in the smallness of r].

The relationship between various parameters of the system operated in the quasi-linear range may be esli- mated as follows. Assume that &(w) is approximated by

4 z ( w ) = L for I w I 5 wc

4 ~ w ) = o for I w I > wc

and that the factor

1 + LTsG(w) I 2 behaves like u ~ , : ’ N , ~ K , ~ at low frequencies, where K , is the velocity constant of the plant. Then

Recalling that v2 = 2ae(0)/A, @,(O) = 2Lw,, and -4LVs = Ef2(r]), it follows that

until q is about unity where q is the ratio of the signal amplitude to For sinusoidal input at low frequencies, linearity maintains

carrier amplitude (61.

Analog simulation was made for a system with

c = 1.1

where K is adjustable to give different input levels t o the system. The responses are shown in Fig. 10 for a low input level and in Fig. 11 for a high input level. y e ( t ) is the output response of the system replacing the relay by a saturated linear element having the same equivalent gain A T s and the same saturation limit U. The rms error vs. rms input is shown in Fig. 12. The operat- ing conditions for Figs. 10 and 11 correspond to points A and B in Fig. 12 , respectively. The “calculated” curve is obtained by the graphical method which has been described. Note that there is good agreement between the “calculated” and the “experimental” curve until r ]

is about 0.8. Beyond that, corresponding to high input levels, the quasi-linear theory becomes invalid as is indicated in the theory.

For various input spectra satisfying condition la), similar results were obtained from the simulation.

1 i o IEEE TRANSACTIOATS O K A U T O M A T I C C O N T R O L April

Fig. 10. Responses for lo\v-input level. Fig. 11. Responses for high input le\-el.

1965 Lim: Oscillating Sewo-Systems with Random Inputs 171

.4

% .3

.2

. I

.08

.2 .5 2 3

Fig. 11. Rms error vs. rms input.

' 9

VI. COKCLCSIOS *A method extending the DIDF approach is presented

to obtain the power spectrum relations of a typical oscil- lating servo system when the input is a Gaussian ran- dom signal. The analysis in this paper shows that the s>-stem exhibits a quasi-linear mode of operation similar to that previously shown in the literature for a sinus- oidal input. The quasi-linear mode of operation re- quires that most of the input power be concentrated in the frequency range \\;here the loop gain is high, and that the input level be not so high as to suppress the self-oscillation. These limitations are interesting since they also represent the range of input signals that the equivalent linear system with the same constraint

1 Z L ! < G can follow. I . -

L\CKNOIVLEDGbfENT

The author is grateful to J. C. Lozier for many helpful discussions, to Miss N. Gripp, L. Gingerich, and to G. Colom for their assistance in obtaining the digital and analog results.

REFERENCES Lozier, J. C., Carrier-control relay servos, Elect. Engrg., vol 69, Dec 1950, pp 1052-1056. MacColl, L. A,, F~mdawzental Theory of Ser::o?~eclmnisnls. Princeton, S. J.: \-an Nostrand, 1945. Schuck, 0. H.: Adaptix-e flight control, Rept, Proc. 1st IF.4C Congress, Rioscow, 1960. Amsler, B. E., and R. E. Gorozdos, On the analysis of bi-stable control system, IRE T r a m on rlzdonzutic Control, vol AC-4, Dec 1959, pp 46-58. Gelb, A., and 11'. E. Vander Velde, On limit cycling control sys- tems, IRE Trans. on -4zutontatic Contro!, uol AC-4, Apr 1963, pp

Proc. IEE, vol 108c, Sep 1961, pp 287-295. Bonnen, Z., Frequency response of feedback relay amplifiers, 142-157.

Booton, R. C., The analysis of nonlinear control systems with random inputs, Proc. Symp. Nonlinear Circuit Analysis, Poly- tech. Inst. Brooklyn, Brooklyn, X. Y., 1953.

[SI Sarawagi, Y., and Y. Sunahara, Statistical studies of response of control s)-stem with a nonlinear element of zero-memory type,

search Institute. Kyoto University. Japan, 1958-1960. Pts 1-5. Tech Repts 45, 50, 57. 61. and 65, Engineering Re-

[9] Rice, S. o., Mathematical analysis of random noise, Bell Sxstevz Tech.. J., vol 23, Jul 1911, pp 282-332; vol 21, Jan 1945, pp 1G156.

[lo] Davenport, 1V. B., Jr. and W . L. Root, A f t I?ztroductw?z to the Tlteoqf of Ra~zdowz Signal and LVoise. New York: McGraw-Hill, 19.58.

Ill] Hsu, J. C., Integral representation of zero-memory nonlinear functions, BSTJ, vol 11, Nov 1962, pp 1813-1830.

[12] Desoer, C. ri., Nonlinear distortion in feedback amplifiers, IRE Trans. on Ciicuit Tkeory, vol CT-9, Mar 1962, pp 2-6.

[13] Smith, H. \V., The applicability of quasi-linear methods to non-linear feedback system with random inputs, Proc. 2nd IF.4C Congress, Switzerland, 1963.

-. . _ ~