appendix e table of random-input describing functions (ridfs) · appendix e table of random-input...
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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)