chattering introduction

chattering INTRODUCTION

■ Causes of Chattering

1. Fast dynamics disregarded in an ideal model (unmodeled dynamics)

2. Switching frequency limitation

* Discrete time system (discretization chattering)

s

0 t

n

m

x

x

00 uxs ,

00 uxs ,

0 xcxs

3

0

( ) *( ) 0( ),

( ) *( ) 0( ) ( ), ,

( , ) and ( , ) are smooth functions.

lim0( ) 0.

lim ( ) 0.

x t x t

y t y t t

f x y g x y

( , )

( , ), , , 0< 1.n m

x f x y

y g x y x y

0 ( , ) 0 * ( *) and

* [ *, ( *)] - reduced order system.

g x y y h x

x f x h x

* ( *)y h x

( , ), y g x y x const

Formally:

Preposition:If the equilibrium point of the system

is asymptotically stable, then

<< 1

*

11

Example

)(sin 1212

21

tduxcbxaxx

xx

21

)(

xx

Msignu

where a and b are negative constant while c and d are positive constant values. Those equations govern a simple ‘inverted pendulum’ system when is angular displacement and the system is known to be unstable for

.

1x

ac

12

13

Unmodeled Dynamics

)()1(

1)(

2su

ssw

)0()(

112

00

22

usignuu

xx

uwww

wxxx

Plant dynamics

14

)0()(

0),,0,0,0,0(,1

),,,,,(

010

23

23

12

1

usignuu

FuwwxxF

xx

Stability Analysis

2231 2

1 V

0121

1),,,,,( uwwxxFV

15

16