Contributions to the theory of sliding mode control
Vincent Brégeault
december, 3rd 2010
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Contributions to the theory of sliding mode control
1
Introduction to sliding mode control
2
Adaptive sliding mode control
3
Introduction to higher order sliding mode control
4
Introduction to time optimal control
5
Variable structure and time optimal control
6
Conclusion
Conclusion
2
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
1
Introduction to sliding mode control
Considered system
Classical SMC
Equivalent control
Chattering
2
Adaptive sliding mode control
3
Introduction to higher order sliding mode control
4
Introduction to time optimal control
5
Variable structure and time optimal control
6
Conclusion
VSS+TOC
Conclusion
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Considered system
SISO uncertain non linear system under canonical controllability form
ẋ1 = x2
ẋ2 = x3
..
.
ẋn ∈ ψ(x, t ) + [−C (x, t ), C (x, t )] + [Γm (x, t ), ΓM (x, t )]u
y = x1
[−C (x, t ), C (x, t )] is a matched perturbation,
[Γm (x, t ), ΓM (x, t )] an uncertainty on the gain.
0 < Γm (x, t ) 6 ΓM (x, t ) < ∞
C (x, t ) < Γm (x, t )uM
Conclusion
4
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Classical sliding mode control
[Utkin, 1992, Utkin et al., 1999]
Design a sliding hyperplane dened by
{x so that σ(x) = xn + an∗−1 xn−1 + . . . + a1∗ x1 = 0}
Conclusion
5
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Classical sliding mode control
[Utkin, 1992, Utkin et al., 1999]
Design a sliding hyperplane dened by
{x so that σ(x) = xn + an∗−1 xn−1 + . . . + a1∗ x1 = 0}
and a control law u = −uM sign(σ(x))
which forces the system to the sliding surface in nite time
Conclusion
5
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Classical sliding mode control
[Utkin, 1992, Utkin et al., 1999]
5
Design a sliding hyperplane dened by
{x so that σ(x) = xn + an∗−1 xn−1 + . . . + a1∗ x1 = 0}
and a control law u = −uM sign(σ(x))
which forces the system to the sliding surface in nite time
u
uM
0
−uM
Conclusion
t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Equivalent and nominal control
6
The actual control is discontinuous, but 2 useful continuous controls :
Nominal control : If perturbations are not taken into account, unom can be
computed in advance :
ẋn = ψ(x, t ) + Γunom = −
unom (x) =
n −1
X
i =1
ai xi +1
Pn−1
i =1 ai xi − ψ(x, t )
Γ(x, t )
Use : add nominal control to reduce amplitude of discontinous control :
u = unom − uM sign(σ)
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Equivalent and nominal control
6
The actual control is discontinuous, but 2 useful continuous controls :
Nominal control : If perturbations are not taken into account, unom can be
computed in advance :
ẋn = ψ(x, t ) + Γunom = −
unom (x) =
n −1
X
i =1
ai xi +1
Pn−1
i =1 ai xi − ψ(x, t )
Γ(x, t )
Use : add nominal control to reduce amplitude of discontinous control :
u = unom − uM sign(σ)
Equivalent control : If perturbations are taken into account, ueq can be
known afterwards, by ltering.
Use : observer, suppress discontinuity, dimension amplitude of control
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
Some ways to reduce it :
use saturation function instead of sign (boundary layer)
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
Some ways to reduce it :
use saturation function instead of sign (boundary layer)
use knowledge of the plant (nominal control)
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
Some ways to reduce it :
use saturation function instead of sign (boundary layer)
use knowledge of the plant (nominal control)
use asymptotic convergence observer
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
Some ways to reduce it :
use saturation function instead of sign (boundary layer)
use knowledge of the plant (nominal control)
use asymptotic convergence observer
add dynamics in the control
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
Some ways to reduce it :
use saturation function instead of sign (boundary layer)
use knowledge of the plant (nominal control)
use asymptotic convergence observer
add dynamics in the control
Adding a dynamic can be done, for example :
extending the sytem : u is a new state variable, u̇ is the new control
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
Some ways to reduce it :
use saturation function instead of sign (boundary layer)
use knowledge of the plant (nominal control)
use asymptotic convergence observer
add dynamics in the control
Adding a dynamic can be done, for example :
extending the sytem : u is a new state variable, u̇ is the new control
using the super twisting algorithm
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Chattering
Denition : Unwanted high frequency oscillation of the control and the output
Causes :
time delay, neglected (fast) dynamics
measurement/observation noise
Some ways to reduce it :
use saturation function instead of sign (boundary layer)
use knowledge of the plant (nominal control)
use asymptotic convergence observer
add dynamics in the control
Adding a dynamic can be done, for example :
extending the sytem : u is a new state variable, u̇ is the new control
using the super twisting algorithm
adapting the amplitude
7
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
2 design steps = 2 questions :
Sliding surface : It sets the dynamics of the system in sliding mode.
Which one to choose ?
Conclusion
8
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion
2 design steps = 2 questions :
Sliding surface : It sets the dynamics of the system in sliding mode.
Which one to choose ?
Reaching law : Discontinuous = robust, but chattering.
How to reduce the chattering while keeping (most of) the robustness ?
8
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
1
Introduction to sliding mode control
2
Adaptive sliding mode control
Control law
Worst case and enhancement
Electropneumatic benchmark
Test of the adaptive control
3
Introduction to higher order sliding mode control
4
Introduction to time optimal control
5
Variable structure and time optimal control
6
Conclusion
VSS+TOC
Conclusion
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
Other adaptive approches
Fuzzy logic : do not garantee precision
[Munoz and Sbarbaro, 2000, Tao et al., 2003]
or overestimate the amplitude [Huang et al., 2008]
VSS+TOC
Conclusion
10
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Other adaptive approches
Fuzzy logic : do not garantee precision
[Munoz and Sbarbaro, 2000, Tao et al., 2003]
or overestimate the amplitude [Huang et al., 2008]
increase amplitude, then use equivalent control : [Lee and Utkin, 2007]
10
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Other adaptive approches
Fuzzy logic : do not garantee precision
[Munoz and Sbarbaro, 2000, Tao et al., 2003]
or overestimate the amplitude [Huang et al., 2008]
increase amplitude, then use equivalent control : [Lee and Utkin, 2007]
This approach : boundary layer [Plestan et al., 2010, Plestan et al., ture]
10
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Adaptive control law [Plestan et al., 2010]
11
Theorem
σ
δ
t1
t2
t
control
K (t ) & if σ ∈ [−δ(t ); δ(t )] and
K (t ) % outside.
output
The control law u = −K (t ) sign(σ) is stable
when amplitude K (t ) varies as
(
K̄ |σ| sign(|σ| − δ(t )) if K > 0 or sign(|σ| − δ(t )) > 0
K̇ =
0
if K = 0 and sign(|σ| − δ(t )) 6 0
C
K
t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Adaptive control law [Plestan et al., 2010]
11
Theorem
K (t ) & if σ ∈ [−δ(t ); δ(t )] and
K (t ) % outside.
δ(t ) must be
output
The control law u = −K (t ) sign(σ) is stable
when amplitude K (t ) varies as
(
K̄ |σ| sign(|σ| − δ(t )) if K > 0 or sign(|σ| − δ(t )) > 0
K̇ =
0
if K = 0 and sign(|σ| − δ(t )) 6 0
σ
as small as possible, for precision
δ
t1
t2
t
control
bigger than amplitude of chattering
(depending on K (t ))
C
K
t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Adaptive control law [Plestan et al., 2010]
11
Theorem
K (t ) & if σ ∈ [−δ(t ); δ(t )] and
K (t ) % outside.
δ(t ) must be
output
The control law u = −K (t ) sign(σ) is stable
when amplitude K (t ) varies as
(
K̄ |σ| sign(|σ| − δ(t )) if K > 0 or sign(|σ| − δ(t )) > 0
K̇ =
0
if K = 0 and sign(|σ| − δ(t )) 6 0
σ
as small as possible, for precision
If θ majorant of delay
δ(t ) > 2ΓM θ K (t )
δ
t1
t2
t
control
bigger than amplitude of chattering
(depending on K (t ))
C
K
t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
control
Consider the system as LTI, because sign(σ) constant.
(
σ̇ = −Γm K + C
C
⇔ ẋ = Ax +
0
K̇ = K̄ σ
12
output
Worst case and enhancement
σ
δ
t
C
K
t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
C
Majorant : σM = p
K̄ Γm
control
Consider the system as LTI, because sign(σ) constant.
(
σ̇ = −Γm K + C
C
⇔ ẋ = Ax +
0
K̇ = K̄ σ
12
output
Worst case and enhancement
σ
δ
t
C
K
t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
12
control
Consider the system as LTI, because sign(σ) constant.
(
σ̇ = −Γm K + C
C
⇔ ẋ = Ax +
0
K̇ = K̄ σ
C
Majorant : σM = p
K̄ Γm
Add a linear term : u = −K (t ) sign(σ) − Kl σ
output
Worst case and enhancement
σ
δ
K
Kl π
−√
C
4K̄ Γm −Kl2
The majorant of σ become : σM = p
e
K̄ Γm
t
C
t
Intro SMC
Adaptive SMC
Intro HOSMC
Electropneumatic benchmark
Intro TOC
VSS+TOC
Conclusion
13
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
Electropneumatic benchmark
System :
krT
S
ṗP =
[φ (p ) + ψP (pP , sign(uP ))uP −
p v]
VP (y ) P P
rT P
krT
S
ṗN =
[φ (p ) + ψN (pN , sign(uN ))uN +
p v]
VN (y ) N N
rT N
v̇ =
1
M
ẏ = v
[S (pP − pN ) − Ff − F ]
Outputs :
position : y (relative degree 3)
pP + pN
mean of pressures :
2
(relative degree 1)
(control stiness or consumption of air)
VSS+TOC
Conclusion
13
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Test of the adaptive control [Brégeault et al., 2010]
14
0.05
10
0
0
−0.05
0
1
2
3
4
5
6
7
8
9
−10
0
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
10
6.5
0
6
5.5
−10
0
5
1000
4.5
0
4
0
1
2
3
4
5
6
7
8
9
10
Position (m) and mean of pressures
(bar)
−1000
−2000
0
uP (V), uN (V), perturbation (N)
4
x 10
10
0.1
0.05
5
0
−0.05
−0.1
0
0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
6
x 10
1.5
15
1
10
0.5
0
5
−0.5
−1
0
1
2
3
4
5
6
7
8
9
10
0
0
Errors of position (m) and pressures
amplitude K
(bar)
K̄y = 8000, 1 (t ) = 2.5 Ky (t )T ; K̄p = 8000 and p (t ) = 10 K2 (t )T
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
1
Introduction to sliding mode control
2
Adaptive sliding mode control
3
Introduction to higher order sliding mode control
Denitions
Errors
Homogeneity
Examples of second order algorithms
Proof of convergence of the super twisting
4
Introduction to time optimal control
5
Variable structure and time optimal control
6
Conclusion
VSS+TOC
Conclusion
Intro SMC
Adaptive SMC
Intro HOSMC
Denitions [Levant, 1993]
Intro TOC
VSS+TOC
Conclusion
16
Denition (Ideal n-order sliding mode)
The sliding variable σ and its n-1 successive derivatives are continuous and reach
0, in the absence of chattering.
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Denitions [Levant, 1993]
16
Denition (Ideal n-order sliding mode)
The sliding variable σ and its n-1 successive derivatives are continuous and reach
0, in the absence of chattering.
Denition (Real n-order sliding mode)
Precision in O(τ n ), in the presence of a source of chattering
√ of amplitude
majored by τ . Usually time delay, measurement error τ = n .
Theorem
If there is a real nth order sliding mode, then σ = O(τ n ), σ̇ = O(τ n−1 ), . . . ,
σ (n−1) = O(τ )
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Error due to time delay
Theorem ([Levant, 1993])
If
σ (n) is continuous and bounded on an interval τ
σ remains in a viciny of 0
then σ = O(τ n )
Example
A classical sliding mode is
rst order with respect to the sliding variable σ
nth order with respect to x1 for chattering due to pure time delay
Conclusion
17
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Error due to measurement and observation noise
18
If the state is known with the precision
1
1
1
= [O() O( 2 ) . . . O( n−1 ) O( n )]T
x2
for example, if it comes from a dierentiator
such as [Levant, 2003].
For a classical sliding mode, the error is
σ(∆x) = λ1 ∆x1 + . . . + λn−1 ∆xn−1 + ∆xn
1
= λ1 O() + λ2 O( 2 ) + . . .
1
1
+ λn−1 O( n−1 ) + O( n )
1
= O( n ) = O(τ )
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Homogeneity [Bacciotti and Rosier, 2001, Levant, 2005] 19
Example
Linear system : homogeneous with degree 0 :
f
(κx) = κf (x)
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Homogeneity [Bacciotti and Rosier, 2001, Levant, 2005] 19
Denition
R
A vector eld f ∈ n is homogeneous of degree q by the dilation
dκ (x1 , . . . , xn ) → (κm1 x1 , . . . , κmn xn ), with mi > 0 and κ > 0 if
f
(dκ x) = κq dκ f (x)
Example
Linear system : homogeneous with degree 0 :
f
(κx) = κf (x)
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Homogeneity [Bacciotti and Rosier, 2001, Levant, 2005] 19
Denition
R
A vector eld f ∈ n is homogeneous of degree q by the dilation
dκ (x1 , . . . , xn ) → (κm1 x1 , . . . , κmn xn ), with mi > 0 and κ > 0 if
f
(dκ x) = κq dκ f (x)
Denition (Equivalent denition)
The dierential equation
ẋ = f (x)
is invariant with respect to the transformation (t , x) → (κ−q t , dκ x).
Example
Linear system : homogeneous with degree 0 :
f
(κx) = κf (x)
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Homogeneity [Bacciotti and Rosier, 2001, Levant, 2005] 19
Denition
R
A vector eld f ∈ n is homogeneous of degree q by the dilation
dκ (x1 , . . . , xn ) → (κm1 x1 , . . . , κmn xn ), with mi > 0 and κ > 0 if
f
(dκ x) = κq dκ f (x)
Denition (Equivalent denition)
The dierential equation
ẋ = f (x)
is invariant with respect to the transformation (t , x) → (κ−q t , dκ x).
Example
Linear system : homogeneous with degree 0 :
f
(κx) = κf (x)
Pure chain of integrators : homogeneous with degree −1
and weights n,n − 1,. . . ,1.
Homogeneity kept with a suitable homogeneous control
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Properties of homogeneity
[Bacciotti and Rosier, 2001, Levant, 2005]
Denition (Contractivity)
[Levant, 2005] A dierential inclusion is contractive
i there exist 2 compacts D1 and D2 and a time
T > 0 so that
dκ D1 ∈ D1 for κ < 1,
D2 belong to the interior of D1 and contain the
origin,
all the trajectories starting in D1 reach D2 at
the time T .
20
x2
D2
D1
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Properties of homogeneity
[Bacciotti and Rosier, 2001, Levant, 2005]
Denition (Contractivity)
[Levant, 2005] A dierential inclusion is contractive
i there exist 2 compacts D1 and D2 and a time
T > 0 so that
dκ D1 ∈ D1 for κ < 1,
D2 belong to the interior of D1 and contain the
20
x2
D2
D1
x1
origin,
all the trajectories starting in D1 reach D2 at
the time T .
Theorem
[Levant, 2005] For a homogeneous system with a negative degree, the following
properties are equivalent :
Asymptotic stability ⇔ nite time stability ⇔ Contractivity.
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Examples of second order algorithms
Twisting algorithm
(
−αm sign(σ1 )
u=
−αM sign(σ1 )
σ2
si σ1 σ2 < 0
si σ1 σ2 > 0
with αm and αM so that
αM > 4
u = αm
Γm
σmax
u = −αM
σ1
u = αM
C
Γm
γm αM − C > ΓM αm + C
αm >
21
u = −αm
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Examples of second order algorithms
Suboptimal algorithm
(
−αm sign(σ1 )
u=
−αM sign(σ1 )
22
σ2
si σ1 σ2 < 0
si σ1 σ2 > 0
u (t ) = λ(t )uM sign(σ1 (t ) −
σ1 (tM )
σ1
σ1 (tM )
)
2
(
1
if |σ1 (t )| > σ1 (tM )
with λ(t ) =
∗
λ if |σ1 (t )| < σ1 (tM )
and tM , the last moment the state reaches the σ1 axis
(σ2 = 0).
λ∗ ∈]0; 1]∩]0,
uM > max(
σ2
σ1 (tM )
2
σ1 (tM )
σ1
3Γm
[
ΓM
C
4C
,
)
λ∗ Γm 3Γm − λ∗ ΓM
σ1 (tM )
2
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Examples of second order algorithms
Super twisting algorithm
√ p
u (t ) = uI (t ) + λ1 L |σ1 | sign(σ1 )
u̇I (t ) = λ2 L sign(σ1 )
with L =
C
and
Γm
λ2 > 1
q
p
λ1 > −2λ2 + 2 λ22 + 2λ2 + 2
Common values : λ2 = 1.1 et λ1 = 2.
(1)
23
σ̇1
σ1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
A new proof of convergence of the super twisting
Conclusion
24
Available proofs
Original proof [Levant, 1998] : numerical, gives the smallest coecients, but
tied to super twisting
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
A new proof of convergence of the super twisting
24
Available proofs
Original proof [Levant, 1998] : numerical, gives the smallest coecients, but
tied to super twisting
Majorant based proof [Davila et al., 2005] : quite small coecients, more
easily extendable to similar control laws
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
A new proof of convergence of the super twisting
24
Available proofs
Original proof [Levant, 1998] : numerical, gives the smallest coecients, but
tied to super twisting
Majorant based proof [Davila et al., 2005] : quite small coecients, more
easily extendable to similar control laws
Lyapunov based proofs [Moreno and Osorio, 2008] : more easily extendable
to other control laws, but large coecients
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
A new proof of convergence of the super twisting
24
Available proofs
Original proof [Levant, 1998] : numerical, gives the smallest coecients, but
tied to super twisting
Majorant based proof [Davila et al., 2005] : quite small coecients, more
easily extendable to similar control laws
Lyapunov based proofs [Moreno and Osorio, 2008] : more easily extendable
to other control laws, but large coecients
Steps of the proof
Homogeneity : only need to study the trajectory in one half plane : stable i
kσ f k < kσ 0 k
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
A new proof of convergence of the super twisting
24
Available proofs
Original proof [Levant, 1998] : numerical, gives the smallest coecients, but
tied to super twisting
Majorant based proof [Davila et al., 2005] : quite small coecients, more
easily extendable to similar control laws
Lyapunov based proofs [Moreno and Osorio, 2008] : more easily extendable
to other control laws, but large coecients
Steps of the proof
Homogeneity : only need to study the trajectory in one half plane : stable i
kσ f k < kσ 0 k
study only the worst case
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
A new proof of convergence of the super twisting
24
Available proofs
Original proof [Levant, 1998] : numerical, gives the smallest coecients, but
tied to super twisting
Majorant based proof [Davila et al., 2005] : quite small coecients, more
easily extendable to similar control laws
Lyapunov based proofs [Moreno and Osorio, 2008] : more easily extendable
to other control laws, but large coecients
Steps of the proof
Homogeneity : only need to study the trajectory in one half plane : stable i
kσ f k < kσ 0 k
study only the worst case
simplify the dierential equation to obtain analytical results
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
A new proof of convergence of the super twisting
σ̇
σ̇0
10
1
5
σc
0
−5
0
5
10
15
σM
20
σ̇f
−10
2a
σ P0
σ f2
σ P02
2b
σ̈ =
0
σM2
σM1
25
σ
Conclusion
24
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
1
Introduction to sliding mode control
2
Adaptive sliding mode control
3
Introduction to higher order sliding mode control
4
Introduction to time optimal control
Open loop control
Closed loop control
Compute the implicit equation
5
Variable structure and time optimal control
6
Conclusion
VSS+TOC
Conclusion
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Open loop time optimal control
26
Perfectly known observable and controllable SISO LTI system with input
v ∈ [−vM ; vM ]:
ẋ = Ax + bv
Theorem (Pontryagin's theorem for SISO LTI systems)
The time optimal control: [Athans and Falb, 1966, Boltjanski, 1969]
is a bang bang control with nite number of switchings
τ3
Conclusion
τ2
τ1
u
0 t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Open loop time optimal control
26
Perfectly known observable and controllable SISO LTI system with input
v ∈ [−vM ; vM ]:
ẋ = Ax + bv
Theorem (Pontryagin's theorem for SISO LTI systems)
The time optimal control: [Athans and Falb, 1966, Boltjanski, 1969]
is a bang bang control with nite number of switchings
this control sequence is unique
τ3
Conclusion
τ2
τ1
u
0 t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Open loop time optimal control
26
Perfectly known observable and controllable SISO LTI system with input
v ∈ [−vM ; vM ]:
ẋ = Ax + bv
Theorem (Pontryagin's theorem for SISO LTI systems)
The time optimal control: [Athans and Falb, 1966, Boltjanski, 1969]
is a bang bang control with nite number of switchings
this control sequence is unique
τ3
τ2
τ1
u
0 t
Theorem (Feldbaum's theorem)
[Athans and Falb, 1966, Boltyanski and Gorelikova, 1997]
Order n system with n real poles ⇒ at most n phases (n − 1 switchings)
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Closed loop time optimal control for SISO LTI systems
Conclusion
27
Theorem
The time optimal closed loop control has the form
v = −vM sign(fvM (x))
fvM (x) = 0 is the equation of a switching surface computed for an amplitude of
control vM .
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Closed loop time optimal control for SISO LTI systems
Conclusion
27
Theorem
The time optimal closed loop control has the form
v = −vM sign(fvM (x))
fvM (x) = 0 is the equation of a switching surface computed for an amplitude of
control vM .
In general, switching surface 6= sliding surface ⇔ set of trajectories of the system
x2
x2
1
ω
x1
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Closed loop time optimal control for SISO LTI systems
Conclusion
27
Theorem
The time optimal closed loop control has the form
v = −vM sign(fvM (x))
fvM (x) = 0 is the equation of a switching surface computed for an amplitude of
control vM .
In general, switching surface 6= sliding surface ⇔ set of trajectories of the system
x2
x2
1
ω
x1
x1
Lemma ([Brégeault and Plestan, 2009])
For systems with real poles only,
switching surface ⇔ trajectories of the system driven by a time optimal control
with at most n − 1 phases
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Computing the implicit equation for real poles systems
Conclusion
28
Function [x1 , . . . xk ]T = f k (s , τ1 , . . . , τk )
Theorem ([Brégeault and Plestan, 2009])
∀k 6 n, f k , is a bijection between {−1; +1} ×
R+k and Rk .
⇒ Equation of the switching surface : xn − xnS (x1 , . . . , xn−1 , vM ) = 0 for systems
in canonical controllability form
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Computing the implicit equation for real poles systems
Conclusion
28
Function [x1 , . . . xk ]T = f k (s , τ1 , . . . , τk )
Theorem ([Brégeault and Plestan, 2009])
∀k 6 n, f k , is a bijection between {−1; +1} ×
R+k and Rk .
⇒ Equation of the switching surface : xn − xnS (x1 , . . . , xn−1 , vM ) = 0 for systems
in canonical controllability form
⇒ Algorithm to compute the switching surface step by step, one dimension at a
time
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Computing the implicit equation for real poles systems
Conclusion
28
Function [x1 , . . . xk ]T = f k (s , τ1 , . . . , τk )
Theorem ([Brégeault and Plestan, 2009])
∀k 6 n, f k , is a bijection between {−1; +1} ×
R+k and Rk .
⇒ Equation of the switching surface : xn − xnS (x1 , . . . , xn−1 , vM ) = 0 for systems
in canonical controllability form
⇒ Algorithm to compute the switching surface step by step, one dimension at a
time
For pure chains of integrators :
The time optimal switching surface is homogeneous of degree −1.
x )
So, fvM (x) = kxkH fvM (
kxkH
⇒ reduces the dimension of the set of points by 1, and increases precision near
the origin.
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
1
Introduction to sliding mode control
2
Adaptive sliding mode control
3
Introduction to higher order sliding mode control
4
Introduction to time optimal control
5
Variable structure and time optimal control
Parametrization of the system
Control law
Proof of stability
Asymptotic precision
Examples
General case : VSS
Reduction of the chattering
Example
6
Conclusion
VSS+TOC
Conclusion
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
Parametrization of the system
Reference system : totally known LTI system. . .

ẋ1 = x2





ẋ

2 = x3



ẋ = Ax + bv ⇔  ...


n

X



ẋ
∈
a i xi + v

n

i =1
VSS+TOC
Conclusion
30
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Parametrization of the system
Reference system : totally known LTI system. . .

ẋ1 = x2





ẋ

2 = x3



ẋ = Ax + bv ⇔  ...


n

X



ẋ
∈
a i xi + v

n

i =1
. . . because all the uncertainties are in the new virtual control
v ∈ [−C 0 ; C 0 ] + [Γm ; ΓM ]u
so that −C 0 + Γm uM > 0.
Conclusion
30
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Parametrization of the system
Reference system : totally known LTI system. . .

ẋ1 = x2





ẋ

2 = x3



ẋ = Ax + bv ⇔  ...


n

X



ẋ
∈
a i xi + v

n

i =1
. . . because all the uncertainties are in the new virtual control
v ∈ [−C 0 ; C 0 ] + [Γm ; ΓM ]u
so that −C 0 + Γm uM > 0.
If u = ±uM , |v | > Γm uM − C 0 > 0
30
ΓM uM + C
Γm uM − C
0
−Γm uM + C
−ΓM uM − C
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Parametrization of the system
ΓM uM + C
Γm uM − C
0
−Γm uM + C
−ΓM uM − C
We can theoretically generate any control within [−(Γm uM − C 0 ); Γm uM − C 0 ]
thanks to high frequency switching (equivalent control).
30
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Control law
31
ΓM uM + C
Theorem
The control
Conclusion
u = −uM sign(fvN (x))
with vN the amplitude of the reference control chosen so
that:
0 < vN 6 Γm uM − C 0
is a nth order sliding mode control, provided the
reference system has only real poles
Γm uM − C
0
−Γm uM + C
−ΓM uM − C
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Ideal sliding mode
Prove the stability of the switching surface :
Time optimal switching surface ⇔ τn (x) = 0
32
τ3
τ2
τ1
u
0 t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Ideal sliding mode
Prove the stability of the switching surface :
Time optimal switching surface ⇔ τn (x) = 0
Lyapunov function: τn (x)
32
τ3
τ2
τ1
u
0 t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Ideal sliding mode
Prove the stability of the switching surface :
Time optimal switching surface ⇔ τn (x) = 0
Lyapunov function: τn (x)
Nominal case (pure time optimal) :
τn (x(t )) = τn (t = 0) − t
⇒ τ̇n = −1
32
τ3
τ2
τ1
u
0 t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Ideal sliding mode
Prove the stability of the switching surface :
Time optimal switching surface ⇔ τn (x) = 0
Lyapunov function: τn (x)
32
τ3
τ2
τ1
u
0 t
Nominal case (pure time optimal) :
τn (x(t )) = τn (t = 0) − t
⇒ τ̇n = −1
Real case (with uncertainties) :
The direction of the control bvN sign(fvN (x)) is so that τn decreases as fast as
possible.
As Γm uM − C 0 > vN , τn decreases at least as fast : τ̇n 6 −1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Ideal sliding mode
Prove the stability of the switching surface :
Time optimal switching surface ⇔ τn (x) = 0
Lyapunov function: τn (x)
32
τ3
τ2
τ1
u
0 t
Nominal case (pure time optimal) :
τn (x(t )) = τn (t = 0) − t
⇒ τ̇n = −1
Real case (with uncertainties) :
The direction of the control bvN sign(fvN (x)) is so that τn decreases as fast as
possible.
As Γm uM − C 0 > vN , τn decreases at least as fast : τ̇n 6 −1
⇒ attractive surface ⇒ sliding mode
⇒ the system behaves like a LTI system subject to a time optimal control
⇒ stable nth order ideal sliding mode.
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Real sliding mode
33
Delays : nth order sliding
x2
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Real sliding mode
33
Delays : nth order sliding
Measurement/observation error : state reaches S + E ,
1
1
1
with E = [O() O( 2 ) . . . O( n−1 ) O( n )]T
x2
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Real sliding mode
33
Delays : nth order sliding
Measurement/observation error : state reaches S + E ,
1
1
1
with E = [O() O( 2 ) . . . O( n−1 ) O( n )]T
Shape of the surface :
parametric equation of the surface :
(s , k1 τ, k2 τ, . . . , kn−1 τ, 0) with small τ > 0
Integrating the system with this control yields
xi = αi (s , vN , k2 , . . . , kn−1 )τ n+1−i + O(τ n+2−i )
n
X
ẋn =
ai xi + |v | sign(fvN (x)) = O(τ ) + |v | sign(fvN (x))
i =1
x2
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Real sliding mode
33
Delays : nth order sliding
Measurement/observation error : state reaches S + E ,
1
1
1
with E = [O() O( 2 ) . . . O( n−1 ) O( n )]T
Shape of the surface :
parametric equation of the surface :
(s , k1 τ, k2 τ, . . . , kn−1 τ, 0) with small τ > 0
Integrating the system with this control yields
xi = αi (s , vN , k2 , . . . , kn−1 )τ n+1−i + O(τ n+2−i )
n
X
ẋn =
ai xi + |v | sign(fvN (x)) = O(τ ) + |v | sign(fvN (x))
i =1
⇒
xS = [O(τ n )O(τ n−1 ) . . . O(τ )]T
x2
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Example : double integrator
34
x2
System:
x1
σ̇1 = σ2
σ̇2 ∈ [−C ; C ] + [Γm ; ΓM ]u
Control:
√
u = −uM sign σ2 + 2vN
with vN 6 Γm uM − C
p
|σ1 | sign(σ1 )
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Example : double integrator
34
x2
System:
x1
σ̇1 = σ2
σ̇2 ∈ [−C ; C ] + [Γm ; ΓM ]u
Control:
√
u = −uM sign σ2 + 2vN
p
|σ1 | sign(σ1 )
with vN 6 Γm uM − C
2nd order sliding mode control with prescribed convergence law:
p
u = −uM sign σ2 + β |σ1 | sign(σ1 )
with
β2
6 Γm uM − C
2
[Levant, 2007]
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Example : triple integrator
35
ẋ1 = x2
ẋ2 = x3
ẋ3 ∈ [−ψM ; ψM ] + u
Equation
from
[Pao and Franklin, 1993]
6
20
4
15
2
10
0
5
−2
0
−4
−5
−6
0
1
2
3
4
5
6
7
8
Figure: Control u and equivalent
control of v : veq
9
10
−10
0
1
2
3
4
5
6
7
8
9
Figure: x (), x (r), x (. . . )
1
2
3
10
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
General LTI systems and VSS control
Conclusion
36
Theorem
The control
u = −uM sign(fvN (x))
with vN the amplitude of the reference control chosen so that:
0 < vN 6 Γm uM − C 0
is a variable structure control which stabilizes the system in nite time.
The convergence time is no greater than the corresponding time optimal control
law with amplitude vN .
x2
1
ω
x1
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
General LTI systems and VSS control
Conclusion
36
Theorem
The control
u = −uM sign(fvN (x))
with vN the amplitude of the reference control chosen so that:
0 < vN 6 Γm uM − C 0
is a variable structure control which stabilizes the system in nite time.
The convergence time is no greater than the corresponding time optimal control
law with amplitude vN .
x2
1
ω
|v|
vN
x1
0
V(t)
δt
t
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Reduction of the chattering
Problem : The nominal control is neither continuous in time nor in space.
⇒ use of saturation or nominal control do not work
Conclusion
37
Intro SMC
Adaptive SMC
Intro HOSMC
Reduction of the chattering
Intro TOC
VSS+TOC
Conclusion
37
Problem : The nominal control is neither continuous in time nor in space.
⇒ use of saturation or nominal control do not work
Solution : Add a dynamic : compute the time optimal switching surface for an
LTI system of order n + 1, and the corresponding nominal dynamics of u and xn .
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Reduction of the chattering
Conclusion
37
Problem : The nominal control is neither continuous in time nor in space.
⇒ use of saturation or nominal control do not work
Solution : Add a dynamic : compute the time optimal switching surface for an
LTI system of order n + 1, and the corresponding nominal dynamics of u and xn .
Theorem
The control law using
the nominal value of u (triangular)
a super twisting of sliding variable xn − xnnom (x1 , . . . , xn−1 ),
C + Lγ vN
coecients L =
, λ1 = 1, λ2 = 1.1
Γm
stabilizes the system in nite time
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Example : double integrator
38
Neglected rst order dynamic (time constant: 10ms), and sinusoidal matched
perturbation.
2.5
1.5
2
1
1.5
0.5
0
1
−0.5
0.5
−1
0
−1.5
−0.5
−2.5
y
dy
−1.5
−2
u
ueq
−2
−1
0
5
10
veq
−3
15
−3.5
−perturbation
0
5
10
15
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
1
Introduction to sliding mode control
2
Adaptive sliding mode control
3
Introduction to higher order sliding mode control
4
Introduction to time optimal control
5
Variable structure and time optimal control
6
Conclusion
VSS+TOC
Conclusion
Intro SMC
Adaptive SMC
Conclusion
Conclusion:
Adaptive sliding mode
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
40
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion:
Adaptive sliding mode
Intermediate proof of convergence of the super twisting algorithm
Conclusion
40
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion:
Adaptive sliding mode
Intermediate proof of convergence of the super twisting algorithm
Algorithm to compute the time optimal control switching surface
Conclusion
40
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion
Conclusion:
Adaptive sliding mode
Intermediate proof of convergence of the super twisting algorithm
Algorithm to compute the time optimal control switching surface
New control law : VSS+TOC (HOSMC for real poles, VSS for complex
poles, reduction of chattering for real poles)
40
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion
Conclusion:
Adaptive sliding mode
Intermediate proof of convergence of the super twisting algorithm
Algorithm to compute the time optimal control switching surface
New control law : VSS+TOC (HOSMC for real poles, VSS for complex
poles, reduction of chattering for real poles)
Perspectives:
Adaptive control for higher order sliding mode
40
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion
Conclusion:
Adaptive sliding mode
Intermediate proof of convergence of the super twisting algorithm
Algorithm to compute the time optimal control switching surface
New control law : VSS+TOC (HOSMC for real poles, VSS for complex
poles, reduction of chattering for real poles)
Perspectives:
Adaptive control for higher order sliding mode
Improve algorithms to compute the switching surfaces
40
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion
Conclusion:
Adaptive sliding mode
Intermediate proof of convergence of the super twisting algorithm
Algorithm to compute the time optimal control switching surface
New control law : VSS+TOC (HOSMC for real poles, VSS for complex
poles, reduction of chattering for real poles)
Perspectives:
Adaptive control for higher order sliding mode
Improve algorithms to compute the switching surfaces
Take saturation into account in VSS+TOC smooth control laws
40
Intro SMC
Adaptive SMC
Intro HOSMC
Intro TOC
VSS+TOC
Conclusion
Conclusion
Conclusion:
Adaptive sliding mode
Intermediate proof of convergence of the super twisting algorithm
Algorithm to compute the time optimal control switching surface
New control law : VSS+TOC (HOSMC for real poles, VSS for complex
poles, reduction of chattering for real poles)
Perspectives:
Adaptive control for higher order sliding mode
Improve algorithms to compute the switching surfaces
Take saturation into account in VSS+TOC smooth control laws
Extend VSS+TOC algorithms to MIMO or nonlinear cases
40
Bibliography
Athans, M. and Falb, P. L. (1966).
Optimal Control. An introduction to the theory and its applications.
McGraw Hill. Lincoln library publications.
Bacciotti, A. and Rosier, L. (2001).
Liapunov functions and stability in control theory.
Lecture notes in control and information sciences. Springer.
Boltjanski, V. G. (1969).
Mathematische Methoden der optimalen Steuerung.
Carl Hanser, 2. edition.
Boltyanski, V. and Gorelikova, S. (1997).
Optimal synthesis for non oscillatory controlled objects.
Journal of Applied Analysis, 3(1):121.
Brégeault, V. and Plestan, F. (2009).
High order sliding mode control based on a time optimal control scheme.
In ROCOND'09.
Brégeault, V., Plestan, F., Shtessel, Y., and Poznyak, A. (2010).
41