Sliding Mode Control: Basic Theory, Advances and Applications

Dept of Electrical and Electronic Eng.
Dept of Electrical and Electronic Eng.
University of Cagliari
University of Cagliari
Sliding Mode Control:
Basic Theory, Advances and Applications
Elio USAI
eusai@diee.unica.it
Institut für Regelungs- und Automatisierungstechnik
Graz University of Technology

Basics on SMC in VSS - Prof. Elio USAI - TU Graz - June 2016 2/54
• Introduction to Sliding Modes in Variable Structure Systems
• Sliding Mode Control of uncertain systems: basics
• Higher-Order sliding modes: basics
• Higher-Order sliding mode control design
• Implementation issues of sliding mode controllers with
engineering applications
• Simulation/solution of ODE with Algebraic constraints via VSS
• Real-time differentiation via higher-order sliding modes
• State variable estimation and input reconstruction in dynamical
systems via VSS
• Some applications
Summary of the Talks

• Sliding Modes in Variable Structure Systems
• Control problem statement
• Internal and input-output dynamics
• Convergence and stability conditions for systems with known gain function
• Convergence and stability conditions for systems with unknown gain function
• Invariance and reduced order dynamics
• Equivalent control
• Filippov's solution
Lecture 1
Basics on Sliding Mode Control
in
Variable Structure Systems

L1 - SM in VSS
Variable Structure Systems are dynamical systems such that their
behavior is characterised by different dynamics in different domains
     
 
N
R
R
R







 
Q
i
t
U
X
t
t
t
f
t q
i
n
i
i ,
,
,
,
, u
x
u
x
ẋ
fi is a smooth vector field fi: Rn
xRq
xR+
Rn
The state dynamics is invariant until a switch occurs
The system dynamics is represented by a differential
equation with discontinuous right-hand side

L1 - SM in VSS
Switching between different dynamics
 
 
     
       
 
N
R
R
R




















Q
j
i
t
U
X
f
f
g
m
i
n
i
j
i
k
k
sw
k
k
k
k
k
k
k
k
i
,
,
,
,
,
,
,
,
0
,
u
x
u
x
x
u
x
x
x










˙
˙
The reaching of the guard gi
sw
cause the switching from the
dynamics fi to the dynamics fj, according to proper rules

L1 - SM in VSS
What does it happen on the guard?  
  0
, 
k
k
sw
i
g 

x
   


 k
k

 x
x Continuous state variables
   


 k
k

 u
u Continuous control variables
   


 k
k

 x
x Jumps in state variables
   


 k
k

 u
u Discontinuous control variables
In Variable Structure Systems there is no jumps in the state variables
but there could be discontinuity in the control variables.
The most interesting point is what happens on the guard.

L1 - SM in VSS
Variable Structure Systems may behave very differently
from each of the constituting ones.
     
     
2
0
1
0
2
0
1
1
0
1
0
2
0
1
0
a
a
system
t
y
a
t
y
a
t
y
system
t
y
a
t
y
a
t
y








˙
˙
˙
˙
˙
˙
• a1>0 the systems are both asymptotically stable
• a1=0 the systems are both marginally stable
• a1<0 the systems are both unstable

L1 - SM in VSS
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
y ( t )
d
y
(
t
)
/
d
t
P h a s e p la n e
a 0
1
= 0 .5
a 0
2
= 9 .0
 
2
.
0
7
.
0
1
.
0
0
sgn
0
0
1
0
0
1









a
a
a
y
y
a
y
a
y
a
y ˙
˙
˙
˙
1
.
0
1 
a
Switched unstable dynamics
Both dynamics are
asymptotically stable
- 1 0 - 5 0 5 1 0
- 5
0
5
1 0
1 5
2 0
P h a s e p la n e
y ( t )
d
y
(
t
)
/
d
t

L1 - SM in VSS
1
.
0
1 

a
Switched asymptotically
stable dynamics
Both dynamics are unstable
- 0 .4 - 0 .2 0 0 .2 0 .4 0 .6 0 .8 1 1 .2
- 1
- 0 .8
- 0 .6
- 0 .4
- 0 .2
0
0 .2
0 .4
0 .6
0 .8
1
P h a s e p la n e
y ( t )
d
y
(
t
)
/
d
t
- 6 - 4 - 2 0 2 4 6 8
- 6
- 4
- 2
0
2
4
6
y ( t )
d
y
(
t
)
/
d
t
P h a s e p la n e
a 0
2
= 9 . 0
a 0
1
= 0 . 5
 
2
.
0
7
.
0
1
.
0
0
sgn
0
0
1
0
0
1










a
a
a
y
y
a
y
a
y
a
y ˙
˙
˙
˙

L1 - SM in VSS
A Sliding Mode behavior appears when the switching frequency
tends to infinity
  0
1


 

i
i
i 

If the switching frequency tends to infinity in a finite time, the sliding
mode can be considered as a Zeno phenomenon in Hybrid Systems
which are characterized by their execution set H
T = {i}iN : set of switching/jump time instants
In = {xi} iN xi  D : set of initial states sequence
Ed = {i}iN i=(i,j)  Q x Q : set of edge sequence
 
Ed
In
T ,
,

H


L1 - SM in VSS
In a Zeno/Sliding Mode condition the system evolves along a guard
 
    







t
t
t
t
g
0
,
x
x
x
˙
  



 





 



0
1
lim
i
i
i
i
i
The Zeno phenomenon appears if the execution set is such that
the Sliding Mode behavior is achieved in a finite time

L1 - SM in VSS
The Zeno phenomenon is mainly related to the switching frequency
on a guard, but previous relationship shows the relation between the
guard and the system dynamics
 
  0
, 
t
t
g x
     
 
t
t
t
f
t ,
,
1 u
x
x 
˙
     
 
t
t
t
f
t ,
,
2 u
x
x 
˙
     
 
t
t
t
f
t s
s ,
,u
x
x 
˙
 
 
x
x

 t
t
g ,
The motion of the system on a discontinuity surface is called Sliding Mode

L1 - SM in VSS
 Sliding Modes are Zeno behaviours in switching
systems
 The system is constrained onto a surface in the
state space, the sliding surface
 When the system is constrained on the sliding
surface, the system modes differ from those of the
original systems
 The system can be invariant when constrained on
the sliding surface
 Systems belonging to a specific class and
constrained onto the sliding surface behave the
same way

L1 - SM in VSS
 
  0
, 
t
t
x
σ Represents the boundary between distinct regions Sk
of the state space, possibly time varying
S1
S2
S4
S3
s1
=0
s2
=0
The behavior of the system on/across the guard  = 0 defining the
regions of the state space, depends on how the dynamics fk are
related to the switching surface

L1 - SM in VSS
0


0

 0


 
   
 
 
   
 













0
,
,
0
,
,
1
2
t
t
f
t
t
t
t
f
t
t
x
x
x
x
x
x


attractive
switching surface
0


0

 0


 
   
 
 
   
 













0
,
,
0
,
,
1
2
t
t
f
t
t
t
t
f
t
t
x
x
x
x
x
x


repulsive
switching surface
0


0

 0


across
switching surface
 
   
 
 
   
 













0
,
,
0
,
,
1
2
t
t
f
t
t
t
t
f
t
t
x
x
x
x
x
x


∂σ (x(t) ,t)
∂ x
f 2 (x(t),t )>0
∂σ (x(t ),t)
∂ x
f 1( x(t),t )>0
S1
S1
S1
S2
S2
S2

L1 - SM in VSS
When considering Variable Structure Systems, the system
dynamics can be represented by a discontinuous right-hand side
differential equation
     
  k
k t
t
t
f
t S


 x
u
x
x ,
,
˙
A discontinuous right-hand side differential equation can also be
represented by a differential inclusion
 
    n
q
n
k
m t
f
f
f
f
t
R
R
R
R 





:
,
,
,
,
,
, 2
1 u
x
x
F
F

˙
In the Sliding Mode a specific solution of the differential inclusion
satisfying s = 0 is “selected”

L1 – VSC, problem statement
Variable Structure Control of dynamical systems is a nonlinear
control technique in which the control variable is usually chosen so
that a sliding mode behavior on a proper surface of the state plane
is enforced
PRO CONS
✔ Robustness with
respect to matching
disturbances
✔ Robustness with
respect to system
uncertainty
✔ Simple implemen-
tation and tuning
● Theoretically, infinite
frequency switching is
required
● Unpredictable
oscillations can appear
in real implementations
(chattering)








0
1
0
1
0
0
a
a
y
a
y
a
y ˙
˙
˙
a0= a0=-

0
0
1 


 u
y
a
y
a
y ˙
˙
˙
 
cy
y
u
c
a
a






˙


sgn
,
2
.
0
,
1
,
1 2
1
- 8 - 6 - 4 - 2 0 2 4 6 8
- 1 0
- 8
- 6
- 4
- 2
0
2
4
6
8
1 0
y
d
y
/
d
t
reaching
sliding
0

     
  k
k t
t
t
f
t S


 x
u
x
x ,
,
˙
     
 
   
     
  k
k
k
k
t
t
t
f
t
t
t
f
t
t
t
f
t
S




x
u
x
u
x
u
x
x
,
,
,
,
,
,
˙
In Variable Structure Control the switching between dynamics is
enforced by a proper control variable
The problem is to define a suitable switching control such that the
Sliding Mode is established, possibly in a finite time, and the
resulting controlled dynamics fulfill the requirements

     
 
t
t
t
f
t ,
,u
x
x 
˙
   
    k
t
t
t
f V













x
σ
u
x
x
σ
sgn
,
,
sgn
Theorem.
Consider the system dynamics
If it is possible to define the control variable
in any e -vicinity of the switching surface s(x)=0
such that
then the surface s(x)=0 is an invariant set in the state space and a
sliding mode occurs on it.
    k
k t
t S


 x
u
u
 
 
0
,
: 1


 

 x
σ
x
V

  σ
σ
σ 
 T
V
2
1
Proof.
Consider the positive definite function
Its time derivative is
Therefore V(s) is a Lyapunov function and the origin of the p-
dimensional space of variables s is, at least, an asymptotically
stable equilibrium point.
 
 
 
  0
sgn
sgn










σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
˙
˙
˙
˙
˙
˙
T
T
T
T
diag
diag
V

Condition
implies that in a vicinity of the sliding surface the vector field is
always directed towards the surface itself
   
    k
t
t
t
f V













x
σ
u
x
x
σ
sgn
,
,
sgn
If the control vector u(t) is such that
the time derivative of the Lyapunov function is such that
the sliding mode behavior is reached in a finite time
 
p
i
σi ,
,
2
,
1 
˙ 

V
V T





 σ
σ ˙
˙
 

 0
0
t
t
σ




The vector field is discontinuous on the sliding surface and the set
of switching time instants T={t1
, t2
, … , t
} is a zero-measure set
is Lebeasgue integrable on time and a continuous solution x(t)
exists in the Filippov sense
     
  k
k t
t
t
f
t S


 x
u
x
x ,
,
˙

Usually the dimension of variable s defining the sliding surface is
the same of the control vector
p=q
Each control variable ui
is usually defined as a discontinuous
function of a single sliding variable si
, even if it affects all
components of s

The Sliding Mode Control problem cannot be reduced to find a control
law such that the previous Theorem is satisfied since control
specifications has to be fulfilled
Sliding Mode Control of dynamical systems is a two step design
approach:
➢ Define a proper sliding surface such that once the system is
constrained onto it the control specifications are fulfilled
➢ Define a feedback control logic such that the system state is
constrained onto the sliding surface
A switching control is compulsory when uncertainty in the system
dynamics is dealt with

L1 – Internal and input-output dynamics
Consider the sliding variable as the output of a dynamical system
     
 
   
 







 q
q
n
t
t
t
t
t
f
t
R
,
R
,
R
,
,
σ
u
x
x
σ
y
u
x
ẋ
In classical Sliding Mode Control the sliding surface is chosen so that

V











x
u
σ
q
rank
˙ All the output variables
have well defined relative
degree 1

The system state can be represented by a combination of output and
internal variables
  q
n
q
n 











R
,
R
,
R w
y
x
x
w
y
Diffeomorphism preserving the origin
 
0
0 



 q
n
q
n
R
R
R
:
w: internal variables
y: output variables
The output variables are the sliding variables that we want to nullify

(y,w,u,t) is the input-output dynamics
(y,w,t) is the internal dynamics
Is equivalent to
     
 
   
 







 q
q
n
t
t
t
t
t
f
t
R
,
R
,
R
,
,
σ
u
x
x
σ
y
u
x
ẋ
       
 
     
 







  q
q
n
q
t
t
t
t
t
t
t
t
t
R
,
R
,
R
,
,
,
,
,
u
w
y
w
y
w
u
w
y
y


˙
˙
   
       
 
t
t
t
t
t
t
t
f ,
,
,
,
, u
w
y
u
x
x
σ






The internal dynamics is strictly related on the choice of the sliding
variables as functions of the state
The internal dynamics strongly affects the performance of the system
and therefore the fulfillment of the control specifications depends on
how the sliding variables are defined
The internal dynamics is needed to be Input-to-State Stable (ISS), at
least
The system state is stabilizable if its input-output dynamics is
controllable (stability can be enforced by the control) and its zero
dynamics is stable
   
 
t
t
t ,
,w
0
w 

˙

1
2
2
2
1
2
2
1












x
x
u
x
x
x
x
x
x

˙
˙
  
   





sgn
20
1
0
;
2
0
1
0
1
1
1
2
1





























u
x
x
w
w
u
w
w
w
w
˙
˙
x
Example
0 1 2 3 4 5 6 7 8 9 1 0
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
P h a s e p la n e p lo t
x
1
x
2
x 1
( 0 ) ; x 2
(0 )
0 0 . 5 1 1 . 5 2 2 . 5 3
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
T im e [ s ]

The unstable zero-dynamics
causes the loss of the sliding
mode behaviour

L1 – Classic SMC
Finding a control law such that the stability condition for the sliding
mode can be fulfilled for a generic nonlinear system is not easy.
Nonlinear systems affine in the control law
   
   
   
   
q
b
t
t
t
t
t
t
n
j
i
n
n
,
,
1
j
,
n
,
1,
i
R,
R
R
:
R
R
R
:
,
,
, 

˙











A
u
x
B
x
A
x
Vector A and matrix B can be uncertain but some knowledge will be
required to design the control law and assure the system stability

L1 – Classic SMC
The control affine assumption can be considered not much strong
since any system can be reduced to such a form if a proper
augmented dynamics is considered
     
v
u
u
x
I
0
u
x
x
x
u
x
0
x
x
u
x














































t
f
t
t
f
t
f
q
,
,
,
,
ˆ
,
,
ˆ
ˆ
˙
˙
˙
This generalization is relatively simple for Single-Input systems but
give rise to very large and complicated systems in the multi-input case

L1 – Classic SMC
Theorem.
Chose the sliding variable set s(x) such that the corresponding
internal dynamics is ISS stable.
Assume that the uncertain matrix A is bounded by a known function
Assume that the known square matrix
is non singular
The following state feedback control law assures the finite time
stability of the sliding surface s(x)= 0
   
   
   
t
t
t
t
t
t u
x
B
x
A
x 

 ,
,
˙
 
   
x
x
A F
t
t 
,
 
 
t
t ,
x
B
x
σ



      0
sgn
,
1
























 σ
x
B
x
σ
x
σ
x
u t
F

L1 – Classic SMC
Proof.
Sliding variable dynamics
Lyapunov function
   
   
   
 
     
σ
x
σ
x
x
A
x
σ
u
x
B
x
σ
x
A
x
σ
σ
sgn
,
,
,
























F
t
t
t
t
t
t
t
t
˙
  σ
σ
σ 
 T
V
2
1
 
     
 
0
sgn
sgn
,
2
1

































V
F
t
t
V
T
T
T





σ
σ
σ
σ
σ
x
σ
x
x
A
x
σ
σ
σ
σ ˙
˙

L1 – Classic SMC
Perfect knowledge of the gain matrix B is problematic in practice
A switching control law which is able to enforce a sliding mode
behavior can be defined if the gain matrix B is uncertain but has some
properties
It must be possible to define the sliding variable vector s(x) such that
the gain matrix of the sliding variables dynamics:
• never vanishes
• it is positive definite
• a lower bound for its eigenvalues is known

L1 – Classic SMC
Theorem.
Chose the sliding variable set s(x) such that the corresponding
internal dynamics is ISS stable.
Assume that the uncertain matrix A is bounded by a known function
Assume that the square matrix is positive definite
Assume that
The state feedback control law
assures the finite time stability
of the sliding surface s(x)= 0
   
   
   
t
t
t
t
t
t u
x
B
x
A
x 

 ,
,
˙
 
   
x
x
A F
t
t 
,
 
 
t
t ,
x
B
x
σ



 
0
2






 

σ
σ
x
σ
x
u
m
F
 
 
















 t
t
eig
m ,
min x
B
x
σ

L1 – Classic SMC
Proof.
Sliding variable dynamics
Lyapunov function
   
   
   
 
 
 
2
,
,
,
σ
σ
x
σ
x
x
A
x
σ
u
x
B
x
σ
x
A
x
σ
σ
m
F
t
t
t
t
t
t
t
t



















˙
  σ
σ
σ 
 T
V
2
1
 
 
 
 
 
 
  0
,
,
,
2
2
2















































V
t
t
t
t
F
t
t
V
T
m
m
T
T




σ
σ
x
B
x
σ
σ
σ
σ
σ
x
B
x
σ
x
σ
x
x
A
x
σ
σ
σ
σ ˙
˙

L1 – Classic SMC
If a local stability suffices, the magnitude of the switching control can
be set to a sufficiently large value
 
 
m
F
U
U









x
σ
x
σ
u sgn
If it is necessary to limit the magnitude of the switching control, some
knowledge about the system dynamics can be exploited
   
   
   
   
 
  





t
t
t
t
t
t
t
t
t
t
,
~
,
,
~
,
x
u
x
Γ
x
x
σ



˙
 
   
   
 
σ
x
x
Γ
u sgn
,
,
1




 
t
t
t
t 

L1 – Invariance in SMC
A nth
order dynamical system constrained onto a q-dimensional
surface of the state space presents a reduced order dynamics when
in sliding mode
n state variables, q constraints n-q “free” motions
The zero dynamics
is the reduced order dynamics
   
 
t
t
t ,
,w
0
w 

˙

All the matching uncertainties (model mismatching and external
disturbances) are compensated for by the control and do not affect
the zero dynamics
The zero dynamics of a Variable Structure System with a Sliding
Mode is invariant
All different systems having the same zero dynamics behave the
same way

   
       
t
b
b
F
t
f
u
t
b
t
f
x
n
i
x
x
m
n
i
i
,
0
,
,
,
,
1
,
2
,
1
1
x
x
x
x
x
x







 
˙

˙
Example: the Single-Input case of a system in the Brunowsky
canonical form
   
 
 
  





2
2
1
1
1
1
1
1
1
1
sgn
,
,
k
b
k
x
c
F
u
x
c
u
t
b
t
f
x
c
x
m
n
i
i
i
n
i
i
i
n
i
i
i
n























˙
˙
x
x
x
x
The system is uncertain with
known bounds
ci are chosen such that the
corresponding polynomial is
Hurwitz
Finite time convergence to
the sliding manifold is
assured


















1
1
1
1
1
1 2
,
2
,
1
n
i
i
i
n
n
i
i
i
n
i
i
x
c
x
x
c
x
n
i
x
x


˙

˙
The system behaves as a
reduced order system with
prescribed eigenvalues
Matching uncertainties, included in the uncertain function f, are
completely rejected
In the sliding mode it is not possible to “recover” the original system
dynamics (semi-group property)
The system is invariant when constrained on the sliding manifold 

         
cy
y
t
u
y
y
k
k
y
b
y
y
b
b
y
t
m









˙
˙
˙
˙
˙
˙


sin
sgn 2
3
1
3
2
1
 

sgn
U
u 

0 0 . 5 1 1 .5 2 2 .5 3 3 .5
- 5
- 4
- 3
- 2
- 1
0
1
2
3
4
5
T im e [ s ]
p o s itio n
v e lo c ity

reaching
sliding
1st
order sliding behavior

L1 – Equivalent Control
Equivalent dynamics method
   
     
 
 
     
  0
,
,
,
:
,
,
,
,




t
t
t
f
t
t
t
t
t
f
t
t
t
f
eq
s
eq
eq
eq
s
u
x
x
x
u
u
x
u
x

It is used when the discontinuity is due to switching of an independent
variable, the “control”, u(t)
0


0

 0


f1
f2
fSeq
 
   
   
 
   
   
 
   
   
 











0
,
,
,
0
,
,
,
0
,
,
,
2
1
t
t
t
t
t
f
t
t
t
t
t
f
t
t
t
t
t
f
t eq
x
u
x
x
u
x
x
u
x
x



˙

The equivalent control can be derived by looking for the continuous
control that makes the time derivative of the sliding variable to zero
   
   
    0
,
, 








 t
t
t
t
t
t eq
u
x
B
x
σ
x
A
x
σ
σ̇
   
   
 
t
t
t
t
t
eq ,
,
1
x
A
x
σ
x
B
x
σ
u 















Filippov’s continuation method and the equivalent control method
can give different solutions in nonlinear systems
Not uniqueness problems in finding the solution of the
sliding mode dynamics

System
VSC
ref. +
_
 u y
System
Internal
Model
ueq yd
+ +
d
+ +
d
The invariance property during the sliding mode means that the
“Internal Model Principle” is fulfilled
The equivalent control compensates for uncertainties and generates
the right input to the system to achieve the desired performance

    
U
U
t
t 



 ,
,
, x
B
x
A
x F
˙
The controlled system dynamics belongs to a differential inclusion
The sliding variable  can be considered as a
performance index to be nullified to find the “right” solution
    F


 eq
t
t u
x
B
x
A
x ,
, *
*
*
˙
The equivalent control is the continuous control
corresponding to the “right” solution

The equivalent control is generated by infinite frequency switchings
      


 


 j
U
j
U
j
U eq
The spectrum of the discontinuous control contains that of the
equivalent control and can be recovered by low-pass filtering
u
u
u av
av 


The equivalent control contains information about uncertainties
   
   
 
t
t
t
t
t
eq ,
,
1
x
A
x
σ
x
B
x
σ
u 
















u
u
u av
av 


If ueq is bounded with its time derivative then








 eq
av u
u
0
0
lim
The cut-off frequency of the low-pass filter must be
• Greater than the bandwidth of the equivalent control
• Lower than the real switching frequency
In practice only an estimate of ueq can be evaluated

         
cy
y
t
u
y
y
k
k
y
b
y
y
b
b
y
t
m









˙
˙
˙
˙
˙
˙


sin
sgn 2
3
1
3
2
1
         y
t
cm
t
y
y
k
k
y
b
y
y
b
b
ueq
˙
˙
˙
˙ 






 
sin
sgn 2
3
1
3
2
1
0 0 .5 1 1 .5 2 2 . 5 3 3 .5
- 3 0
- 2 5
- 2 0
- 1 5
- 1 0
- 5
0
5
1 0
1 5
T im e [ s ]
u a v
u e q

L1 – Filippov solution
 
 
 










u
t
u
t
,
,
,
,
,
x
x
x






F
˙
The controlled system dynamics belongs to a differential inclusion
The solution x* such that state trajectory is tangent to the
sliding manifold  belong to a convex set
      0
,
1
,
0
,
1 




 

x
x ˙
˙ 



 grad
|

L1 – Filippov solution
The average velocity is defined by
If the system dynamics is affine in the control variable the
equivalent control and the Filippov solution agree
 
   
 
   ,
1
, 
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If the system dynamics is nonlinear the Filippov solution
may be not unique

L1 – References
●
V.I. Utkin, Sliding Modes In Control And Optimization, Springer
Verlag, Berlin, 1992.
●
C. Edwards, S.K. Spurgeon, Sliding mode control: theory and
applications, Taylor and Francis Ltd, London, 1998.
●
Y. Shtessel, C. Edwards, L. Fridman, A. Levant, Sliding Mode
Control and Observation, Birkhäuser, New York, 2014
●
A.F. Filippov, Differential Equations with Discontinuous Right–
Hand Side, Kluwer, Dordrecht, The Netherlands, 1988.
●
B. Draženović, “The invariance conditions in variable structure
systems”, Automatica, 5, pp. 287-295, 1969.

L1 – References
• Pisano A., Usai E., “Sliding Mode Control: a Survey with
Applications in Math”, Mathematics and Computers in
Simulation, vol.81, pp. 954-979, 2011
• R.A. De Carlo, S.H. Zak, G.P. Matthews, “Variable Structure
Control of Nonlinear Multivariable System: A Tutorial,
Proceedings of IEEE, vol. 76, no. 3, pp. 212-232, 1988
• G. Bartolini, T. Zolezzi, “Variable Structure Systems Nonlinear in
the Control Law”, IEEE Transactions on Automatic Control,
vol.30, pp. 681-684, 1985
• K.H. Johansson, J. Lygeiros, S. Sastry, M. Egerstedt, “Simulation
of Zeno Hybrid Automata, IEEE-CDC 1999, 3538-3543, Phenix,
Arizona, USA, December 1999

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