chattering INTRODUCTION
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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 0 0 u x s , 0 0 u x s , 0 x cx s
description
m. n. chattering INTRODUCTION. ■ Causes of Chattering. 1. Fast dynamics disregarded in an ideal model (unmodeled dynamics) 2. Switching frequency limitation. * Discrete time system (discretization chattering). Singular Perturbation Theory.
Transcript of 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
*
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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
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)0()(
0),,0,0,0,0(,1
),,,,,(
010
23
23
12
1
usignuu
FuwwxxF
xx
Stability Analysis
2231 2
1 V
0121
1),,,,,( uwwxxFV
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