Transcript PRUNam
UNAM Dr. Leonid Fridman
NEW TRENDS IN SLIDING CONTROL MODE
L. Fridman Universidad Nacional Autónoma de México División de Posgrado, Facultad de en Ingeniería Edificio ‘A’, Ciudad Universitaria C.P. 70 256, México D. F.
14 MAYO DE 2004
1
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Given a system
u
1 2
x
2
u
x
2 ( 0 ) 1 sgn( )
f
(
x
,
t
) 0
f(x,t) u
x 2
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
1 2 0 sin(
t
) sgn( ) 3
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Motivations
Given a system
x
x
2 1
x
2
u
f
(
x
,
t
)
Problem formulation:
Design control function
u
stability
t
lim
x
1
t
lim
x
2 in presence of bounded uncertain term 0
f
(
x
,
t
) uncertainties and external disturbances.
0
f(x,t) u
x to provide asymptotic
L
, that contains model 4
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Basics of Sliding Mode Control x(0) Desired compensated error dynamics (sliding surface):
1
cx
1 0
x
1
x
1 ( 0 )
e
ct
,
x
2
cx
1 ( 0 )
e
ct
•The purpose of the
Sliding Mode Controller
(SMC) is to drive a system's trajectory to a user-chosen surface, named •
sliding surface
, and to maintain the plant's state trajectory on this surface thereafter. The motion of the system on the sliding surface is named •
sliding mode
. The equation of the sliding surface must be selected such that the system will exhibit the desired (given) behavior in the sliding mode that will not depend on unwanted parameters (plant uncertainties and external disturbances). 5
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
1. Sliding surface design x 2 x(0)
reaching phase
x 1
sliding phase
x
2
cx
1
0
2. SMC design Sliding mode existence condition Equivalent control
u eq
x
2
f
(
c x
1
x
,
t
)
u
cx
2
f
0 (
x
,
t
)
u
ˆ
eq cx
2
cx
0 2 6
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
WHY Sliding mode control?
More than Robustness (insensitivity!!!!) to disturbances and uncertainties WHEN Sliding mode control?
Control plants that operate in presence of unmodeled dynamics, parametric uncertainties and severe external disturbances and noise: aerospace vehicles, robots, etc.
7
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Numerical example:
x
1 ( 0 )
f
1 .
(
x
,
t
) 0 , 2 sin
x
2 ( 0 ) 10
t
, 0 .
5 ,
u
x
2
c
1 , 2 .
5
sign
Features:
1. Invariance to disturbance 2. High frequency switching 8
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Continuous and smooth sliding mode control 1. Continuous approximation via saturation function
sign sat( /e) 1 e -1
Numerical example:
x
1 ( 0 ) 1 .
0 ,
f
(
x
,
t
) 2 sin
x
2 ( 0 ) 10
t
, 0 .
5 ,
u
x
2
c
1 , 2 .
5
sat
( / 0 .
01 ) 9
UNAM Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Simulations Features:
1. Invariance to disturbance is lost to some extend 2. Continuous asymptotic control 10
UNAM Dr. Leonid Fridman
Second order Sliding mode control
1. Twisting Algorithm
sign
sign
Features:
2.Robustness INSENSITIVITY!!!!
3.Convergence
O
(
h
2 ) 11
UNAM Dr. Leonid Fridman
New trends in sliding mode control
Chattering avoidance whit Twisting Algorithm (continuous control)
y
1
y
2
y
2
) )
, 0
m
M
, 0
y
1
y
2 (
t
)
sign
sign
Features:
for
1 ,
for
1 ,
2.Robustness
3.Convergence
O
(
h
2 ) 12
UNAM Dr. Leonid Fridman
Continuous Second order Sliding mode control
2. Super Twisting Algorithm
u
x
2 15 0 .
5
sign
( ) 20
sign
( )
d
0
Features:
1. Invariance to disturbance 2. Continuous control 13
UNAM Dr. Leonid Fridman Sliding mode observers/differentiators 3. Second Order ROBUST TO NOISE Sliding Mode Observer
y
1 (
t
)
y
2 (
t
)
x
(
t
)
x
(
t
)
v
1 (
t
),
v
2 (
t
).
x
ˆ 10
sign
(
y
2 8
sign
(
y
1
x
ˆ )
x
ˆ ) 14
UNAM Dr. Leonid Fridman
Higher order Sliding mode control
4. High order slides modes controllers of arbitrary order Features: 1.Convergence in finite time for
, ,..., (
r
1 )
2.Robustness
3.Convergence
O
(
h r
)
4.r-Smooth control
15
UNAM Dr. Leonid Fridman
Higher order Sliding mode control
High order slides modes controllers of arbitrary order
16
UNAM Dr. Leonid Fridman CHATTERING ANALISYS
Frequency analysis 1. Frecuency Methods modifications . Boiko, Castellanos LF IEEE TAC2004 2. Universal Chattering Test.
Boiko, Iriarte, Pisano, Usai, LF 3. Chattering Shaping. Boiko, Iriarte, Pisano, Usac, LF 17
UNAM Dr. Leonid Fridman CHATTERING ANALISYS (s,x) PLANT Singularly Perturbed Approach S
g g
1
f
2 ( ( (
z z
,
z
, ,
s s
,
s
, ,
x x
)
x
) ,
u
)
ACTUA TOR S Integral Manifold Averaging LF IEEE TAC 2001 LF IEEE TAC 2002 Second Order Sliding Mode Controllers
18
UNAM Dr. Leonid Fridman UNDERACTUATED SYSTEMS
1 2
x
2
f
n
(
x
1 , (
t
,
x
)
x
2 ,
u
)
m
(
t
,
x
) Uncertaint y
m n matched unmatched
SMC + H_{∞} Fernando Castaños & LF SMC + Optimal multimodel Poznyak, Bejarano & LF 19
UNAM Dr. Leonid Fridman OBSERVATION & IDENTIFICATION VIA 2 -SMC
~ 1
x
2
e
1 sgn(
e
1 ) ~ 2
g
(
t
,
x
1 ,
x
2 ,
u
) sgn(
e
1 ) Uncertainty identification Parameter identification Identification of the time variant parameters J. Dávila & LF 20
UNAM Dr. Leonid Fridman RELAY DELAYED CONTROL
s
sgn
s
(
t
1 )
Countable set of periodic
Countables etofperiod icso
solutions=sliding modes
Shustin, E. Fridman LF 93 Set of Steady modes 21
UNAM Dr. Leonid Fridman CONTROL OF OSCILLATIONS AMPLITUDE
Only
sign s
(
t
1 ) Is accessible FFS 93----- s(t-1) is accessible Strygin, Polyakov, LF IJC 03, IJRNC 04 22
UNAM Dr. Leonid Fridman APPLICATIONS
Investigation and implementation of 2-SMC
Shaping of Chattering parameters
23