578 I E E E T R A N S A C T I O N S ON A U T O X A T I C CO-VTROL October

Fig. 1-.I nonlinear characteristic with a threshold, a linear. and a saturating region. represented by a product of two Gaussian-type functions e - B s and (1 -CCDSZJ and a linear function Ax.

Fig. ?-.A nonlinear characteristic x i r h a linear and a saturating region represented by a product of a Gaussian iunction and a linear iunction A x .

Fig. 3--A nonlinear characteristic with a threshold and a linear region represented by a product oi a

function Ax. Gaussian-type function ( 1 -CcDS2j and a linear

approximation to the nonlinear function y = f ( r ) leads t o a simple solution with a sinusoidal input signal. I n the following it will be shown that the same approximation leads also to a simple answer for stochastic noise input.

A nonlinear characteristic with a thresh- old, linear and saturation region can be ap- proximated by the product of a linear and two Gaussian type functions (see Fig. 1).

y = . ~ l ~ e - B “ ( 1 - Ce-Dz’). (6)

For C=O, this characteristic approximates a linear and saturation region (see Fig. 2 ) and for B = O it approximates a threshold and a linear region (see Fig. 3) .

\\-hen ( 6 j is inserted in (S), we find that

For the case of a linenr and saturating re- gion (C=O),

For the case of a threshold and linear region (B=O),

The function coefficients A , B, C, and D can be evaluated graphically from a given characteristic y=,f(x), by trial and error, by successive approximation or with a com- puter. Simple analytical methods of calcu- lating these coefficients have been outlined in the previous communication on the sub- ject.l

D. h,I. h ‘ k ~ o w Applied Physics Division

R. A. Hum Radio and Electric Eng. Div.

ra t ional Research Council Ottawa, Ontario, Canada

Experimental Study of a Random Linear System

general theorem on the stability of linear systems with Gaussian random parameters has been published recently by N. J. Bershad’ (see -1ppendix). This theorem establishes sufficient and necessary condi- tions to insure that the impulse response of a system converges to zero, in the mean-square sense, after a sufficiently long time. From a practical point of view, however, one is more interested in determining whether the im- pulse response tends to converge in a rela- tively short time. This paper presents the re- sults of an analog simulation to determine the range of parameters for m-hich this “short-time stabilit>-” exists and compares these regions with the results obtained from the theorem.

A second-order linear system of the form

:i: + 2 r w t + w?[1 + a ( t ) ] x = 0, rw > 0 (1) was simulated on a P-ACE TR-10 analog computer, as shown in Fig. 1. The noise source was Automation Laboratories’ ivIodel lOOB Gaussian Noise Generator, which has a “flat’ spectrum from 0 to 500 cps. The out- put of the low-pass filter has the exponential autocorrelation function

The computer was run in repetitive opera- tion mode; M and 5 were fed into an oscillo- scope to display the phase plane trajectory of the system. -1 typical display is shown in Fig. 2. The system was deemed unstable if an>- impulse responses exceeded the linear range of the operational amplifiers of the computer.

The results of the simulation are plotted in Fig. 3. T h e t x o more or less straight lines which divide the plane into three regions are

Manuscript received Ma? 8. 1964; revised July 6. 1964. Based on a thesis submitted by W. G. Tuel. Jr. to the iacultp oi R e n s d a e r Polytechnic Institute in partial fulfillment oi the requirements for th; was sponsored jointly by Grant KO. AF-AFOSR M.E.E. degree. The research reported in this paper

2i9-63 irom the Air Force Office oi Scientific Re- search. and by Grant S o . AF 19(628)-3859 from the .Air Force Cambridge Research Laboratory. Office of .Aerosoace Research. USAF.

I964 Correspondence 579

obtained from Theorems 1 and L 2 The region labeled “stable” is found from Theorem 1; the one labeled “unstable” is found from Theorem 2. The intermediate region is of undetermined character. I t is seen that the system is stable for a larger range of parameters than predicted by the theorem, especially for gain variations that are nar- row band with respect to the system. This is discussed below.

Fig. I-Simulation of random second-order system.

I

Fig. 2-Typical phase plane trajectories. * = I , r = O . S , G( s ) =l / ( s+l ) .

If the gain variation is narrow band with respect to the deterministic part of the system, ie., if the parameter variation is slow, then the random process may be ap- proximated by a random variable a with the same probability distribution. The system (1) then becomes a time-invariant system which is stable if a> -1. For a true Gaussian distribution, a < - 1 with some nonzero probability; however, all physical noise sources are amplitude limited and non-Gaussian to some extent. Hence, in a practical case, one can say that (1) is stable if prob [a < - 1 ] <E, \\-here e is some small number.

For a particular noise source used, the value ~ = 0 . 0 1 represents the amplitude limi- tation well. Then (1) is stable if prob [a< -11 <0.01. In terms of the Gaussian distribution

2 3 9 4 5

(a) (C)

Fig. 3-(a) Experimental results: x-unstab!e. &-stable; 0=1000; G(s) =lOO/(s+100). (b) Experimental results: x-unstable, &stable; = loo; G(s) =lO/(s+lO). ( c ) Esperimental results: x-unstable, 0-stable; w=lOOO; G(s) =lO/(s+lO).

16 SO0

12

8

4

0 I 2 3 4 3 . 5

I6 H w SO0

12

!stable I ,

I 2 3 4 3 . 5

(a) b)

Fig. 4-(a) Experimental results, clipped noise: x-unstable. &stable: w =1000; G(s) =lOO/(s+lOO). (b) Experimental results. clipped noise: s-unstable. 0-stable; 0=1000; G(s) =lOO/(s+lOO).

this implies that Z<O.lM. Since u* is the autocorrelation value b(O), if

oso = 2uZ (:) < 0.37 (t), the system should be stable. This condition is plotted as the line labeled “1 per cent probability” in Fig. 3. I t is seen that this provides a good explanation for the larger stability regions.

To further justify this conclusion, a di- ode limiter was inserted between the lox- pass filter and the system. The results of experiments with this configuration are

from the input So and the degree of limit- i r ~ g . ~ T h e results confirm the conclusion that the larger stability regions are due to amplitude limiting of the random process.

In summary, the experimentally deter- mined parameter ranges for which the im- pulse response of a second-order system con- verges in a short time generally agree with the values obtained by application of Rershad’s theorem. The difference between the experiment and the theorem, which is a larger stable region than predicted for nar- row-band parameter variations, is due to the non-Gaussian nature of any physical noise source. This explanation is confirmed by experiments conducted with artificially limited noise.

580 IEEE T R A X S A C T I O X S O N A U T O M A T I C C O X T I Z O L October

APPENDIX

Theorems on the stability of random linear systems:'

Theorem I : Given the differential equation

L,[v(91 + ao(t!u(l) = W ) , E'"'(0) = 0, i = 0, 1, . . ' , q - 1

where L , is a constant parameter gth-order differential operator, and ao( t ) is a sta- tionary, zero-mean. continuous in the mean- Gaussian process, then

lim (u?(t), < cze-?(-c280)r - 1-

where LqF4t)] = 6( t ) , and ce+ =f(t) 2 I ~ ( t ) ' , for c, I, t >O.

S O = J-1 I {a(l)a(t + r ) ) I dr

with the autocorrelation (a( t )a( t+~)\ hav- ing a rational power spectrum.

Theorem 2:

1 where m(t )2O, and For the differential equation in Theorem

with b;, SG>O, and P

so = so;, i=l

then

where 6, C-' denote Laplace and inverse Laplace transforms, respectively.

\V. G. TUEL, JR. P. li. DERUSSO

Dept. of Elec. Eng. Rensselaer Polytechnic Institute

Troy, X. Y .

Analysis of Approximation of Linear and Time-Invariant Systems Pulse Response by Means of Laguerre Finite Term Expansion

A simple calculation method for correla- tion functions through orthonormal series expansions was suggested by Lampard.' hIishkin and Braun refer to orthonormal series expansions for the determination of a mathematical model of nonlinear systems.? Also, Schetzen has studied the problem of optimum Laguerre finite term expansion of

1 D. G . Lampard, aA new method of determi:ing Manuscript received April 27. 1961.

correlation functions oi stationary time series, J. I E E (London); September. 1954.

Systems." McGraw-Hill Book Co.. Inc.. S e w York. 2 E. Mishkin and L. Braun, '.Adaptive Control

X. Y . ; 1961.

a generical function with the condition of minimizing mean-square error.3

The problem of approximation obtain- able, stopping a Laguerre series expansion to a fixed term, is therefore a n object of in- terest. This problem was examined by the author4 under the hypothesis that the func- tion to be expanded is known and has the form of a linear, time-invariant. and lumped parameters-)-stem pulse response

with z; = x ; + j ! ; and x , > 0. (1)

The generical Laguerre function is

and its Laplace transform is

For the function (1) we can consider the expansion con\-ergent in the mean,

f ( t ) = 2 kAdk&it) 0

being its generical coefficient

At = J 0 * j ( l W k ( l ) d l . (3)

If we decompose this expansion in the partial series in regard to each exponential term of the function ( 1 ), we obtain

where, clearly, Axi is the Kth coefficient of the expansion of i th exponential of f(t). Therefore, it is

oc

R,e-z;l = Ek .4i<;Sk(l) . 0

.kcording to (3) . we have

Keeping ( 2 ) and (5) in mind, we can con- sider the Fourier transform of (4)

P - 7nRi

i X i - ? j1 /2 ) ( jw - 111/2) t ] . (6)

It can now he seen that F ( j u ) is ex- pressed through the sum of p complex geo-

(2; + 111/2!(jw + m / 2 )

3 M . Scherzen." Optimum Laguerre Finite Term

Quart. Prop. Rept. S o . 71: October. 1963. Expansion of Function% M.I.T.. Cambridge. Mass.

mediante lo sviluppo in serie di funzioni di Lawerr: C. Bruni. *Xnalisi dell'approseimazione ottenibile

della risposta impulsiva di sistemi lineari e normali. Ric. Sci. R e d . A,, Rome. Italy. pp. 377-408; IIav. 1964.

metric series, and the generic modulus of ratio is

(z; - 111/2)(jw - n1/2) I I zi - 111/2 ~

(zi + 1 ~ / 2 ) ( j u + m/2) I 1

= J (Ti - m/2)' j (Xi + ?12/2)' + y;'

I n the simple h>-pothesis that every ex- ponential term has the same importance in the constitution of f(t), it is reasonable to think that the \vhole series converges as rapidll- to f( t ) as each geometric series converges to its own limit. Therefore, for even- geometric series relative to each ex- ponential term, we should minimize the quantity X<. I t is possible to get the same result for ever\- partial series, minimizing the figure of merit

as being ~ ( t ) the error due to the arrest of expansion to the generic nth term

Ei(f) = R;e-*,t - A a ; S k ( t ) = EL. Aki&(t ) .

.According to Parseval's theorem, it is 0 n+l

Ei = J o * ~ ; 2 ( t ) d t = & I I?, ,l+l

and, keeping ( 5 ) in mind,

Hence, in order to reduce E; t o a mini- mum, for every n, it is necessary to mini- mize, as noted, the quantity

dl; = j z i - m / 2 (x; - 1n/2)2 + y i 2

2; + n1/2 I ( x i + t i ~ / 2 ) ~ + ?'i2 ___ = J

The value of ]?I which satisfies this con- dition is

m ; O = 2d/si2+yi* = 2 I zi , and with this selection, the result is

I t can, therefore, be seen that for every pole or pair of conjugate complex poles, there is for nz, an optimum value nz0 which provides the maximum convergence of rela- tive partial series. Thus, if the function f(t) (1) is reduced simply to one or two ex- ponential terms with conjugate complex ex- ponents, the problem of optimal selection of 172 would be solved, and the relati\-e Value 11; min would give an idea of the rapidity of the expansion convergence. Function (1) is actually given by the superposition of several exponential terms; but, if we sup- pose, as dread>- pointed out, that the im- portance of different terms is approximately equal, it might be reasonable to choose the value of nz which minimizes the greatest of M i , and consider it for estimating con- vergence rapidity of the whole series.

For this procedure it is possible to give simple and convenient graphical representa- tions deriving from the considered formulas