higher order sliding mode control
DESCRIPTION
Higher Order Sliding Mode Control. Department of Engineering. M. Khalid Khan Control & Instrumentation group. References. Levant, A.: ‘Sliding order and sliding accuracy in sliding mode control’, Int. J. Control , 1993,58(6) pp.1247-1263. - PowerPoint PPT PresentationTRANSCRIPT
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Higher Order Sliding Mode Control
M. Khalid KhanControl & Instrumentation group
Department of Engineering
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References
1. Levant, A.: ‘Sliding order and sliding accuracy in sliding mode control’, Int. J. Control, 1993,58(6) pp.1247-1263.2. Bartolini et al.: ‘Output tracking control of uncertain nonlinear second order systems’, Automatica, 1997, 33(12) pp.2203-2212.3. H. Sira-ranirez, ‘On the sliding mode control of nonlinear systems’, Syst.Contr.Lett.1992(19) pp.303-3124. M.K. Khan et al.: ‘Robust speed control of an automotive engine using second order sliding modes’, In proc. of ECC’2001.
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Review: Sliding Mode Control
Design consists of two steps
Selection of sliding surface
Making sliding surface attractive
Consider a NL system uxtgxtfx ),(),(
0),( xtss
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Robustness Chattering
High frequency
switching of control
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Pros and cons
Order reduction Full state availability
Robust to matched uncertainties
Simple to implement
Chattering at actuator
Sliding error = O(τ)
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Isn’t it restrictive?
Sliding variable must have relative degree one w.r.t.
control.
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Higher Order Sliding Modes
rth-order sliding mode:- motion in rth-order sliding set. Sliding variable (s) has relative degree r
rth-order sliding set: -
0)2()1( sssss rr
Consider a NL system ),,( uxtfx
Sliding surface 0),( xtss
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ButWhat about reachability condition?
So traditional sliding mode control is now
1st order sliding mode control!
There is no generalised higher order reachability condition available
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1-sliding vs 2-sliding
s
ds
2-sliding
τ
τ2s
ds
1-sliding
τ
Sliding error = O(τ) Sliding error = O(τ2)
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Sliding variable dynamics
Selected sliding variable, s, will have
relative degree, p= 1 relative degree, p 2
1-sliding design is possible.
2-sliding design is done to avoid chattering.
r-sliding (r p) is the suitable choice.
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2-sliding algorithms: examples
Consider system represented in sliding variable as,,|| ;),,(),,( Mmusstssts
Finite time converging 2-sliding twisting algorithm
0Ss
Sliding set: 0ss
0)(
0)()(
ssssignV
ssssignVtu
M
M
< 1
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PendulumThe model:
uyy )sin(25.0
Sliding variable: yys Sliding variable dynamics:
uyys )sin(25.0
uuyyys )sin(25.0)cos(25.0
Twisting Controller coefficients: α = 0.1, VM = 7
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Simulation
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Examples continue … Consider a system of the type
0 , ,|| ;),(),( Ssuststs Mm
Finite time 2-sliding super-twisting algorithm
0
01
1
||)(sign
||
)(sign||)(
uusW
uukuu
usstu
0ssSliding set:
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Review: 2-sliding algorithms
Twisting algorithm forces sliding variable (s) of relative degree 2 in to the 2-sliding set but uses
s Super Twisting algorithm do not uses but sliding variable (s) has relative degree only one.
s
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Is it possible to stabilise sliding surface with relative degree 2 in to 2-sliding set using only s, not its derivative?
Answer: yes!
1. by designing observer
2. using modified super-twisting algorithm.
Question:
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Modified super-twisting algorithm
0
01
1
||)(sign
||
)(sign)(
uusW
uukuu
ustu
0,,|| ;),,(),,( Ssusstssts Mm
System type:
Where λ, u0 , k and W are positive design constants
1. Sinusoidal oscillations for = u0
2. Unstable for < u0
3. Stable for > u0
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Phase plot
Sufficient conditionsfor stability
0 ,0
/0
Wk
u m
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Application: Anti-lock Brake System (ABS)
ABS model:
ukx
x
xJJ
NRx
RM
NNx
M
Rx
b
ww
vw
wv
wv
v
w
33
32
21
1)(
)(595.01
2
31514
43
212
11
11
25.0
1
xkxk
xkxk
xkkb
22)(
p
pp
),max( 21
12
xx
xx
ugf )()( Can be written as:
12.0desired
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Simulation ResultsController coefficients: 15 W,35 ,75 0 ku
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Results continued …
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Conclusions The restriction over choice of sliding variable can be relaxed by HOSM. HOSM can be used to avoid chattering
A new 2-sliding algorithm which uses only sliding variable s (not its derivative) has been presented together with sufficient conditions for stability.
The algorithm has been applied to ABS system and simulation results presented
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Future Work
The algo can be extended for MIMO systems.
Possibility of selecting control dependent sliding surfaces is to be investigated.
Stability results are local, need to find global results.
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