Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 1/54
Dept of Electrical and Electronic Eng.Dept of Electrical and Electronic Eng.University of CagliariUniversity of Cagliari
Summer School onODEs with Discontinuous Right-Hand Side: Theory and Applications
Dobbiaco (BZ) – Italy
Sliding Mode Control: Basic Theory, Advances and Applications
Elio USAI
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 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• Zero finding of nonlinear algebraic systems via VSS
Summary of the Talks
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 3/54
• 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
Lecture 1
Introduction to Sliding Mode Control in
Variable Structure Systems
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 4/54
L1 - SM in VSS
Variable Structure Systems are dynamical systems such that their behavior is characterised by different dynamics in different domains
( ) ( ) ( )( )N
RRR⊆∈
∈⊆∈⊆∈= +
QitUXtttft q
in
ii ,,,,, uxuxx
fi is a smooth vector field fi: RnxRqxR+→Rn
The state dynamics is invariant until a switch occurs
The system dynamics is represented by a differential equation with discontinuous right-hand side
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 5/54
L1 - SM in VSS
Switching between different dynamics
( )( )
( ) ( ) ( )( ) ( ) ( ) ( )( )
NRRR
⊆∈∈⊆∈⊆∈
==⇓
=
+
++++−−−−
QjitUX
ff
g
mi
ni
ji
kksw
kkkkkkkk
i
,,,
,, ,,,
0,
ux
uxxuxx
x
ττττττττ
ττ
The reaching of the guard gisw cause the switching from the
dynamics fi to the dynamics fj, according to proper rules
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 6/54
L1 - SM in VSS
What does it happen on the guard? ( )( ) 0, =kkswi
g ττx
( ) ( )+− =kk
ττ xx Continuous state variables
( ) ( )+− =kk
ττ uu Continuous control variables
( ) ( )+− ≠kk
ττ xx Jumps in state variables
( ) ( )+− ≠kk
ττ uu Discontinuous control variables
In Variable Structure Systems there is no jumps in the state variablesbut there could be discontinuity in the control variables.
The most interesting point is what happens on the guard.
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 7/54
L1 - SM in VSS
Variable Structure Systems may behave very differently from each of the constituting ones.
( ) ( ) ( )( ) ( ) ( )
2010
201
101
02010
aa systemtyatyaty systemtyatyaty
<<=++=++
• a1>0 the systems are both asymptotically stable
• a1=0 the systems are both marginally stable
• a1<0 the systems are both unstable
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 8/54
L1 - SM in VSS
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
y ( t )
dy(t
)/dt
P h a s e p l a n e
a0
1
= 0 . 5
a0
2
= 9 . 0
( )2.07.01.0
0sgn
001
001
=∆==
=∆−++
aaa
yyayayay
1.01 =a
Switched unstable dynamics
Both dynamics are asymptotically stable
- 1 0 - 5 0 5 1 0- 5
0
5
1 0
1 5
2 0P h a s e p l a n e
y ( t )
dy(t
)/dt
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 9/54
L1 - SM in VSS
1.01 −=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
1P h a s e p l a n e
y ( t )
dy(t
)/dt
- 6 - 4 - 2 0 2 4 6 8- 6
- 4
- 2
0
2
4
6
y ( t )
dy(t
)/dt
P h a s e p l a n e
a0
2
= 9 . 0
a0
1
= 0 . 5
( )2.07.01.0
0sgn
001
001
=∆=−=
=∆+++
aaa
yyayayay
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 10/54
L1 - SM in VSSA Sliding Mode behavior appears when the switching frequency tends to infinity
( ) 01 ∞→+ →−iii ττ
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 χH
T = {τi}i∈N : set of switching/jump time instantsIn = {xi} i∈N xi ⊆ D : set of initial states sequenceEd = {ηi}i∈N ηi=(i,j) ⊆ Q x Q : set of edge sequence
{ }EdInT ,,=Hχ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 11/54
L1 - SM in VSS
In a Zeno/Sliding Mode condition the system evolves along a guard
( )( ) ( ) ∞≥∀=⋅∂
∂ τttttg 0, xx
x
( ) ∞<=−= ∞
∞
=+∞→ ∑ ττττ
01lim
iiiii
The Zeno phenomenon appears if the execution set is such that
the Sliding Mode behavior is achieved in a finite time
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 12/54
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, =ttg x( ) ( ) ( )( )tttft ,,1 uxx =
( ) ( ) ( )( )tttft ,,2 uxx =
( ) ( ) ( )( )tttft ss ,,uxx =
( )( )x
x∂
∂ ttg ,
The motion of the system on a discontinuity surface is called Sliding Mode
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 13/54
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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 14/54
L1 - SM in VSS
( )( ) 0, =ttxσ Represents the boundary between distinct regions Sk
of the state space, possibly time varying
S1
S2
S4
S3
σ1=0
σ2=0
The behavior of the system on/across the guard σ = 0, defining the regions of the state space, depends on how the dynamics fk are related to the switching surface
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 15/54
L1 - SM in VSS
0>σ
0<σ 0=σ( )( ) ( )( )( )( ) ( )( )
>⋅∂
∂
<⋅∂
∂
0,,
0,,
1
2
ttftt
ttftt
xx
x
xx
x
σ
σattractive
switching surface
0>σ
0<σ 0=σ( )( ) ( )( )( )( ) ( )( )
<⋅∂
∂
>⋅∂
∂
0,,
0,,
1
2
ttftt
ttftt
xx
x
xx
x
σ
σrepulsive
switching surface
0>σ
0<σ 0=σ( )( ) ( )( )( )( ) ( )( )
>⋅∂
∂
<⋅∂
∂
0,,
0,,
1
2
ttftt
ttftt
xx
x
xx
x
σ
σacross
switching surface
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 16/54
L1 - SM in VSS
When considering Variable Structure Systems, the system dynamics can be represented by a discontinuous right-hand side differential equation
( ) ( ) ( )( ) kk tttft S∈∀= xuxx ,,
A discontinuous right-hand side differential equation can also be represented by a differential inclusion
( ){ } ( ) nqn
km tfffft
RRRR →××⊂∈
+:,,,,,, 21 uxx
FF
In the Sliding Mode a specific solution of the differential inclusion satisfying σ = 0 is “selected”
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 17/54
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)
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 18/54
L1 – VSC, problem statement
α±=>=+−
01
01
00
aayayay
a0=α a0=-α
α>0
001 =⋅+− uyayay
( )cyy
ucaa
+==
===
σσsgn
,2.0,1,1 21
- 8 - 6 - 4 - 2 0 2 4 6 8- 1 0
- 8
- 6
- 4
- 2
0
2
4
6
8
1 0
y
dy/d
t reachingsliding
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 19/54
L1 – VSC, problem statement
( ) ( ) ( )( ) kk tttft S∈∀= xuxx ,,
( ) ( ) ( )( )( ) ( )( ) ( ) ( )( ) k
kk
k
tttftttftttft
S∈∀=
=x
uxuxuxx
,,,,,,
In Variable Structure Control the switching between dynamics is enforced by a proper control variable
The problem is to define a suitable switching control so that the Sliding Mode is established, possibly in a finite time, and the resulting controlled dynamics fulfill the requirements
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 20/54
L1 – VSC, problem statement
( ) ( ) ( )( )tttft ,,uxx =
( ) ( )( ) ( ) ktttf V∈∀−=
⋅
∂∂ xσux
xσ sgn,,sgn
Theorem.Consider the system dynamics
If it is possible to define the control variable
in any ε -vicinity of the switching surface σ(x)=0
such that
then the surface σ(x)=0 is an invariant set in the state space and a sliding mode occurs on it.
( ) ( ) kk tt S∈∀= xuu
( ){ }0,:1
><= εεε xσxV
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 21/54
L1 – VSC, problem statement
( ) σσσ ⋅= TV21
Proof.Consider the positive definite function
Its time derivative is
Therefore V(σ) is a Lyapunov function and the origin of the p-dimensional space of variables σ is an asymptotically stable equilibrium point.
( )( )( )( ) 0sgn
sgn<⋅−=−=
⋅=⋅=σσσσσ
σσσσσ
TT
TT
diagdiagV
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 22/54
L1 – VSC, problem statement
Condition
implies that in a vicinity of the sliding surface the vector field is always directed towards the surface itself
( ) ( )( ) ( ) ktttf V∈∀−=
⋅
∂∂ xσux
xσ 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
( )piσi ,,2,1 => η
VV T η−<⋅−= σσ
( )η
τ 00
tt
σ+≤∞
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 23/54
L1 – VSC, problem statement
The vector field is discontinuous on the sliding surface and the set of switching time instants T={τ
1, τ
2, … , τ
∞} is a zero-measure set
is Lebeasgue integrable on time and a continuous solution x(t) exists in the Filippov sense
( ) ( ) ( )( ) kk tttft S∈∀= xuxx ,,
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 24/54
L1 – VSC, problem statement
Usually the dimension of variable σ 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 σi, even if it affects all
components of σ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 25/54
L1 – VSC, problem statement
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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 26/54
L1 – Internal and input-output dynamics
Consider the sliding variable as the output of a dynamical system
( ) ( ) ( )( )( ) ( )( )
∈∈∈== qqn
tttttft
R,R,R,,
σuxxσy
uxx
In classical Sliding Mode Control the sliding surface is chosen so that
εV∈∀=
∂∂ xuσ qrank All the output variables
have well defined relative degree 1
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 27/54
L1 – Internal and input-output dynamics
The system state can be represented by a combination of output and internal variables
( ) qnqn −∈∈∈Φ=
R,R,R wyxxwy
Diffeomorphism preserving the origin( )00 Φ=
×→Φ − qnqn RRR:
w: internal variablesy: output variables
The output variables are the sliding variables that we want to nullify
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 28/54
L1 – Internal and input-output dynamics
ϕ(y,w,u,t) is the input-output dynamics
ψ(y,w,t) is the internal dynamics
Is equivalent to
( ) ( ) ( )( )( ) ( )( )
∈∈∈== qqn
tttttft
R,R,R,,
σuxxσy
uxx
( ) ( ) ( ) ( )( )( ) ( ) ( )( )
∈∈∈=
= − qqnq
ttttttttt
R,R,R,,
,,,uwy
wywuwyy
ψϕ
( ) ( )( ) ( ) ( ) ( )( )tttttttf ,,,,, uwyuxxσ ϕ=⋅
∂∂
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 29/54
L1 – Internal and input-output dynamics
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 Bounded-Input Bounded-State stable, 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( ) ( )( )ttt ,, w0w ψ=
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 30/54
L1 – Internal and input-output dynamics
12
2212
21
−=+−−=
=
xxuxxxx
xx
σ
( ) ( )
( ) ( )σ
σσσ
σ
sgn2010;20
1
0111
21
−===
=+−+++−=
⋅
−=
uxx
wwuwww
w
x
Example
0 1 2 3 4 5 6 7 8 9 1 00
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5P h a s e p l a n e p l o t
x1
x 2
x1
( 0 ) ; x2
( 0 )
0 0 . 5 1 1 . 5 2 2 . 5 3- 9
- 8
- 7
- 6
- 5
- 4
- 3
- 2
- 1
0
1
T i m e [ s ]
σ
The unstable zero-dynamics causes the loss of the sliding mode behaviour
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 31/54
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
( ) ( )( ) ( )( ) ( )
( ) ( )qb
tttttt
nji
nn
,,1j,n,1,iR,RR:RRR:
,,
,
==→×→×
⋅+=
+
+AuxBxAx
Vector A and matrix B can be uncertain but some knowledge will be required to design the control law and assure the system stability
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 32/54
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
uux
I0
uxxxux
0x
xux
∂∂
+
∂∂+⋅
∂∂
=
tft
tftfq
,,,,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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 33/54
L1 – Classic SMC
Theorem.Consider the system dynamics
Chose the sliding variable set σ(x) such that the corresponding internal dynamics is BIBS 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 σ(x)= 0
( ) ( )( ) ( )( ) ( )tttttt uxBxAx ⋅+= ,,
( )( ) ( )xxA Ftt ≤,
( )( )tt ,xBxσ ⋅
∂∂
( ) ( ) ( ) 0sgn,1
>⋅
⋅
∂∂
+
∂∂−=
−
ηη σxBxσ
xσxu tF
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 34/54
L1 – Classic SMC
Proof.
Sliding variable dynamics
Lyapunov function
( ) ( )( ) ( )( ) ( )
( )( ) ( ) ( )σxσxxA
xσ
uxBxσxA
xσσ
sgn,
,,
+
∂∂−⋅
∂∂=
⋅⋅∂∂+⋅
∂∂=
ηFtt
tttttt
( ) σσσ ⋅= TV21
( )( ) ( ) ( )
( )0
sgn
sgn,
21
<−<−≤−=⋅−≤
+
∂∂−⋅
∂∂⋅=
⋅=
V
Ftt
V
T
T
T
ηηηη
η
σσσσ
σxσxxA
xσσ
σσ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 35/54
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 σ(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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 36/54
L1 – Classic SMC
Theorem.Consider the system dynamics
Chose the sliding variable set σ(x) such that the corresponding internal dynamics is BIBS 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 σ(x)= 0
( ) ( )( ) ( )( ) ( )tttttt uxBxAx ⋅+= ,,
( )( ) ( )xxA Ftt ≤,
( )( )tt ,xBxσ ⋅
∂∂
( )0
2
>Λ
+∂∂
−= ηη
σσx
σxu
m
F
( )( )
⋅
∂∂=Λ tteigm ,min xB
xσ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 37/54
L1 – Classic SMC
Proof.
Sliding variable dynamics
Lyapunov function
( ) ( )( ) ( )( ) ( )
( )( )( )
2
,
,,
σσx
σxxA
xσ
uxBxσxA
xσσ
m
Ftt
tttttt
Λ
+∂∂
−⋅∂∂=
⋅⋅∂∂+⋅
∂∂=
η
( ) σσσ ⋅= TV21
( )( )( )
( )( )
( )( ) 0,
,,
22
2
<−<−≤⋅
⋅
∂∂⋅
Λ−≤
⋅⋅∂∂
Λ
+∂∂
−⋅∂∂⋅=
⋅=
Vtt
ttF
tt
V
T
m
m
T
T
ηηη
η
σσxBxσσ
σ
σσxB
xσx
σxxA
xσσ
σσ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 38/54
L1 – Classic SMC
If a local stability suffices, the magnitude of the switching control can be set to a sufficiently large value
( )( )
m
FUU
Λ
+∂∂
>⋅−=η
xσx
σu sgn
If it is necessary to limit the magnitude of the switching control, some knowledge about the system dynamics can be exploited
( ) ( )( ) ( )( ) ( )( ) ( )( )( ) Φ≤
⋅++=tt
tttttttt,~
,,~,x
uxΓxxσϕ
ϕϕ
( )( ) ( )( ) ( )( )σxxΓu sgn,,1 Φ+⋅−= − tttt ϕ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 39/54
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
( ) ( )( )ttt ,, w0w ψ=
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 40/54
L1 – Invariance in SMC
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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 41/54
L1 – Invariance in SMC
( ) ( )
( ) ( ) ( ) ( )tbbFtf
utbtfxnixx
m
n
ii
,0,,
,,1,2,11
xxxx
xx
≤<≤
+=−== +
Example: the Single-Input case of a system in the Brunowsky canonical form
( ) ( )
( )
( ) ( ) σσσσ
σ
σ
2
21
11
1
11
1
1
sgn
,,
kb
kxcFu
xcutbtf
xcx
m
n
iii
n
iii
n
iiin
−≤⇒++
−=
++=
+=
∑
∑
∑
−
=+
−
=+
−
=
x
x
xx
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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 42/54
L1 – Invariance in SMC
∑
∑−
=
−
=−
+
+−=
+−=
−==
1
1
1
11
1 2,2,1
n
iiin
n
iiin
ii
xcx
xcx
nixx
σ
σ
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 σ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 43/54
L1 – Invariance in SMC
( ) ( ) ( ) ( ) ( )cyy
tuyykkybyybbytm+=
−=+++++
σπsinsgn 2
31321
( )σsgnUu −=
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 i t i o nv e lo c i t yσ
reaching
sliding
1st order sliding behavior
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 44/54
L1 – Equivalent Control
Equivalent dynamics method
( ) ( )( ) ( ) ( )( )
( )( ) ( ) ( )( ) 0,,,:
,, ,,
=∂
∂
=
tttftt
tttftttf
eqseq
eqeqs
uxx
xu
uxux
σ
It is used when the discontinuity is due to switching of an independent variable, the “control”, u(t)
0>σ
0<σ 0=σ
f1
f2
fSeq
( )( ) ( )( ) ( )( )
( ) ( )( ) ( )( )( ) ( )( ) ( )( )
>
=
<
=
0,,,
0,,,
0,,,
2
1
tttttf
tttttf
tttttf
t eq
xux
xux
xux
x
σ
σ
σ
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 45/54
L1 – Equivalent Control
The equivalent control can be derived by looking for the continuous control that makes the time derivative of the sliding variable to zero
( ) ( )( ) ( )( ) ( ) 0,, =⋅⋅∂∂+⋅
∂∂= tttttt equxB
xσxA
xσσ
( ) ( )( ) ( )( )ttttteq ,,1
xAxσxB
xσu ⋅
∂∂⋅
⋅
∂∂−=
−
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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 46/54
L1 – Equivalent Control
SystemVSCref. +
_σ u y
SystemInternalModel
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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 47/54
L1 – Equivalent Control
( ) ( )[ ]UUtt +−+=∈ ,,, xBxAx F
The controlled system dynamics belongs to a differential inclusion
The sliding variable σ can be considered as a performance index to be nullified to find the “right” solution
( ) ( ) F∈+= eqtt uxBxAx ,, ***
The equivalent control is the continuous control corresponding to the “right” solution
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 48/54
L1 – Equivalent Control
The equivalent control is generated by infinite frequency switchings
( ) ( ) ( )∞=
+=ω
ωωω jUjUjU eq
The spectrum of the discontinuous control contains that of the equivalent control and can be recovered by low-pass filtering
uuu avav =+τ
The equivalent control contains information about uncertainties
( ) ( )( ) ( )( )ttttteq ,,1
xAxσxB
xσu ⋅
∂∂⋅
⋅
∂∂−=
−
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 49/54
L1 – Equivalent Control
uuu avav =+τIf ueq is bounded with its time derivative then
∆≤=→→
∆
στ
τ eqav uu00
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
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 50/54
L1 – Equivalent Control
( ) ( ) ( ) ( ) ( )cyy
tuyykkybyybbytm+=
−=+++++
σπsinsgn 2
31321
( ) ( ) ( ) ( ) ( ) ytcmtyykkybyybbueq +++++++= πsinsgn 231321
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 ]
ua v
ue q
Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 51/54
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.
• 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.
• 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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