Describing Function Basics

Describing Function Determination

Stability of Discontinuous Non-linear Systems

Example Problems

Consider idealized representation of dead zone as shown.

For sinusoidal input, output is a truncated sine wave.

Dead Zone Example

1( ) sin ; sin ;2

( ) sin for 2 2

d

dx t X t

X

d dy t K X t x

ω α

ω

−= =

= − >

( ) 0 for ( )2 2

( ) ( ) for ( ), ( )2 2

d dy t x t

d dy t Kx t x t x t

= − ≤ ≤ +

= − ≥ ≥ +

The frequency response can be evaluated as follows.

Dead Zone Example

2

1

1 1

2 sin 24sin sin ( ) 1

2

2 sin 21

d

d d

d d

dY K X t k td t KX

Y YNN

X K KX

π

α

α αω ω ω

π π π

α α

π π

= − = − −

= → = = − −

∫

The describing function

value depends on the extent

of the normalized dead zone

and it can be seen that if the

dead zone magnitude is

zero, Y1 = 1, recovering the

linear limit.

Consider idealized representation of saturation as shown.

Saturation Example

1

1

2 sin 2;

2 sin 2;

sin2

s s

s s

s

Y KX

N

K

S

X

α α

π π

α α

π π

α −

= −

= −

=

Effect of Describing Function

Describing function, which models the nonlinear element,

has an impact on overall close loop system behaviour.

Consider the generic unity feedback control system

containing the describing function, as shown below.

In general, the close loop characteristic equation gets

modified as follows, resulting in an impact on the close

loop system stability.

Stability Analysis

The corresponding closed loop system frequency response

can be obtained as follows.

Closed loop system, containing a nonlinearity, are prone to

exhibit limit cycles, as shown below.

Closed loop system stability can be investigated using

Nyquist/Nichols/root locus, by replacing -1 with -1/N.

( ) ( )1 ( ) 0

( ) 1 ( )

1( ) ; ( ) Describing Function Value

( )

C j NG jNG j

R j NG j

G j N jN j

ω ωω

ω ω

ω ωω

= → + =+

= − →

Stability Example

Consider the following open loop system.

Given below are Nyquist/Nichols plots, along with locus

of point ‘-1/N’, for the dead zone nonlinearity.

1( )

(1 )(1 0.5 )G j

j j jω

ω ω ω=

+ +

Obtain the describing function for an ideal relay, as given

below.

Further, if the above relay is in cascade with an open loop

system whose Nyquist plot is as given below, obtain the

periodic solutions and examine their stability.

Interactive Problem Solving

There are many situations where two or more nonlinear

effects are present simultaneously. The dead zone and

saturation can be present simultaneously, as shown below.

The fundamental component is as given below.

Compound Nonlinearities

2

1

0

sin 2 sin 24 2( )sin ( );

2 2

s dk s d

KXY y t k td t Y

π

α αω ω α α

π π

= = − + −

∫

Given below is the plot of the describing function for

variation in both ‘S’ and ‘d’.

As can be seen, the value of ‘N/K’ depends on both ‘S’ and

‘d’ and if d = S, N =0 for all inputs.

Dead Zone + Saturation Example

There are many situations, and particularly in sliding

mechanical components (e.g. gears, slider-crank etc.)

when friction is present along with backlash / dead zone,

as described below.

In this case, output member remains in contact with the

input member until velocity becomes zero. After that,

output member remains stand-still until the backlash is

taken up. At this point, output member instantaneously

acquires the velocity of the input member.

Friction Controlled Backlash

In such a case, we assume that collision that takes place

between input and output member is without rebound.

Thus, for a sinusoidal input, we obtain output as follows.

The describing function in this case can be obtained as,

Friction Controlled Backlash

( )2 1 2

1

1( ) 2 (1 ) cos (1 ) 2

cos ;b

N j

X b b

X X

φ φ φ φ π φ φ φπ

α φ

−

−

= − − + − − + −

−= =

Table below gives the values of the describing function

for various values of the backlash parameter ‘φφφφ’.

Friction Controlled Backlash

Summary

Describing function approach is a practical method to

arrive at an approximate transfer function of a

nonlinear system.

It is possible to create a more complex describing

function by including higher harmonics of the Fourier

series components.