APPENDIX E

TABLE OF RANDOM-INPUT

DESCRIBING FUNCTIONS (RIDFs)

In this table we employ the probability function (cf. Sec. 7.2) denoted by

and its integral, the probability integral, denoted by

PZ(X)= = 1 1"exp (- g) dv

4 2 ~-a

These functions are plotted in Fig. E.2-1.

This table is given in three sections:

E.1 Gaussian-input RIDFs E.2 Gaussian-plus-bias-input RIDFs E.3 Gaussian-plus-bias-plus-sinusoid-input RIDFs

E.1 GAUSSIAN-INPUT RlDFs

x(t) = r(t) an unbiased Gaussian process

--

TABLE O F RANDOM-INPUT DESCRIBING FUNCTIONS (RIDFs) (Continued)

Nonlinearity Comments

16. Linear gain

--

TABLE OF RAN DOM-INPUT DESCRIBING FUNCTIONS (RIDFs) (Continued)

Nonlinearity Comments

See Sec. 7.2

n = 2 , 4 , 6 , ...

See Sec. 7.2

-y = 4 x (x 2 0)

= -4-x ( x < 0)

26. Odd square root See Fig. E.l-3

y = ~ 1 1 %

27. Cube root characteristic

r ( x ) is gamma function

574 RANDOM- INPUT DESCRIBING F U N C T I O N S

Figure E.1-2 RIDFs for litniter and tlrr.esholrlcharacteristics.

R A N D O M - I N P U T DESCRIBING F U N C T I O N S 575

FigureE.1-3 RZDFfor the simple polynonlial nonlinearity y = c,xn (n odd)

ory = c,xn-' 1x1 (n even).

RANDOM- INPUT DESCRIBING FUNCTIONS

Figure E.1-4 Harmonic nonlinearity RIDF.

E.2 GAUSSIAN-PLUS-BIAS-INPUT RlDFs

x( t ) = r ( t )+ B

The gain to the gaussian input component is given by:

and the corresponding gain to the bias input component is:

1 N,(a,B) = -jmy@+ B)exp (- $)dr~ Z O B

This section uses the additional function G(x) = xPI(x) + PF(x).

The functions PF(x), PI(x), and G(x) are plotted in Fig. E.2-1.

TABLE OF RANDOM-INPUT DESCRIBING FUNCTIONS (RIDFs) (Continued)

Nonlinearity Comments

7. Sharp saturation or limiter

8. Dead zone or threshold

9. Gain-changing nonlinearity

TABLE OF RANDOM- INPUT DESCRIBING FUNCTIONS (RIDFs) (Continued)

Nonlinearity Comments

y = x

16. Linear gain

TABLE O F RANDOM- INPUT DESCRIBING FUNCTIONS (RIDFs) (Continued)

Nonlinearity Comments

56. Biased ideal relay

R A N D O M - I N P U T DESCRIBING F U N C T I O N S 587

Figure E.2-I Graphs of PF(x), PZ(x), and G(x).

E.3 GAUSSIAN-PLUS-BIAS-PLUS-SIN USOID-INPUT RlDFs

x( t ) = r ( t )+ B + A sin ( o t + 8)

The gain to the gaussian input component is given by

the gain to the bias input component is

dry(r + B + A sin 0)exp (- 2) and the corresponding gain to the sinusoid input component is

TABLE OF RANDOM-INPUT DESCRIBING FUNCTIONS (RIDFs) (Continued)

Nonlinearity Comments

y = xS

18. Cubic characteristic

y = Msinmx J, and J, are the Bessel NR= Mm cos mB exp (- ?)J,,(mA) functions of orders 0 and 1 , respectively. M

NB =-sin mB exp (--B m y ) ~ o ( m ~ )

2M mSuzNd = A con mB exp (- i)l,(mA)

29. Harmonic nonlinearity

590 RANDOM- INPUT DESCRIBING FUNCTIONS

Figure E.3-I Three-input RZDFs for the ideal-relay nonlinearity.

In 3 parts

Figure E.3-Ia Gain to the gaussian input component. (ideal relay)

1.o 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.09 0.08

Figure E.3-lb

1.o 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

R A N D O M - I N P U T

I 2 3 4

DESCRIBING F U N C T I O N S 591

5 6 7 4 0

Gain to the bias input component. (ideal relay)

I I I I I0.1 0 1 2 3 4 5 6 7 A

u Figure E.3-lc Gain to the sinusoid input cotnponent. (ideal relay)

592 RANDOM- INPUT DESCRIBING FUNCTIONS

Figure E.3-2 Three-input RIDFs for the limiter nonlinearity.

In 1 1 parts

Figure E.3-2a Gain to the gaussian and bias input components. (limiter, IB1/6 = 0)

0 1 2 3 4 5 6

Figure E.3-26 Gain to the sinusoid input component. (limiter, IB)/6 = 0)

Figure E.3-2c Gain to thegaussian input component. (limiter, IBI/S = 0.5) 6

Figure E.3-2d Gain to the bias input component. (limiter, IBI/S = 0.5) 6

596 RANDOM-INPUT DESCRIBING FUNCTIONS

Figure E.3-2h Gain to the sinusoid input component. (limiter, 1B1/6 = I )

R A N D O M - I N P U T D E S C R I B I N G F U N C T I O N S 597

Figure E.3-2i Gain to theguussiun input component. (limiter, IBI/S = 2 ) 6

598 RAN DOM- INPUT DESCRIBING FUNCTIONS

Figure E.3-2j Gain to the bias input component. (limiter, lB1/6 = 2 ) 6

RANDOM-INPUT DESCRIBING FUNCTIONS 599

Figure E.3-2k Gain to the siriusoid input conlponent. (limiter, 1Bj/6 = 2 )

600 RANDOM- INPUT DESCRIBING F U N C T I O N S

Figure E.3-3 Three-input RZDFs for the relay with dead zone nonlinearity.

In 11 parts

60NB

1 0.8

0.6

0.5

0.4

0.3

0.2

0.1

0.08

0.06 0.05

0.04

0.03

0.02

0.01

0.008

0.006 0.005 0.004

0.003

0.002

o.OO11 1 0 1 2 3 4 5 6 7 4

6

Figure E.3-3a Gain to the gaussian and bias input compotlents. (relay with dead zofie,

I W = 0)

R A N D O M - I N P U T DESCRIBING F U N C T I O N S 601

Figure E.3-36 Gain to the sinusoid input component. (relay with dead zone, [BI/S = 0)

601 RANDOM- INPUT DESCRIBING FUNCTIONS

Figure E.3-3c Gain to the gaussian input component. (relay with dead zone, IB)/6 = 0.5)

6

RANDOM- INPUT DESCRIBING F U N C T I O N S 603

Figure E.3-3d Gain to the bias input component. (relay with dead zone, IB1/6 = 0.5)

604 RANDOM-INPUT DESCRIBING FUNCTIONS

Figure E.3-3e Gain to the sinusoid input component. (relay with dead zone, (B( /6= 0.5)

R A N D O M - I N P U T DESCRIBING F U N C T I O N S 605

10 9 8 7 6

5

4

3

2

1 0.9 0.8 0.7

0.6

0.5

0.4

0.3

0.2

0.1 0.09 0.08 0.07

0 I 2 3 4 5 6 ' A 6

Figure E.3-3f Gain to thegaussian input component. (relay with dead zone, lBl/6 = I )

606 RANDOM- INPUT DESCRIBING FUNCTIONS

Figure E.3-3g Gain fo the bias input component. (relay with dead zone, IB1/6 = I )

RANDOM- INPUT DESCRIBING F U N C T I O N S 607

Figure E.3-3h Gain to the sinusoid input component. (relay with dead zone, IB1/6 = I )

608 RANDOM- INPUT DESCRIBING FUNCTIONS

Figure E.3-3i Gaitr to the gaussian input component. (relay with dead zone, 1B1/6 = 2)

RANDOM- INPUT DESCRIBING F U N C T I O N S 609

Figure E.3-3j Gain to the bias input component. (relay with dead zone, IB1/6 = 2 )

610 R A N D O M - I N P U T DESCRIBING F U N C T I O N S

Figure E.3-3k Gain to the sinusoid input component. (relay with dead zone, IB1/6 = 2)