Smc intro

Sliding Mode Control –Sliding Mode Control –
An IntroductionAn Introduction
S. JanardhananS. Janardhanan
IIT DelhiIIT Delhi

Sliding Mode Control 2
OutlineOutline
►What is this ‘Sliding mode’ andWhat is this ‘Sliding mode’ and
how did its study start?how did its study start?
►How to design controller using thisHow to design controller using this
concept?concept?

Primitive Examples - ElectricalPrimitive Examples - Electrical

Primitive Examples-MechanicalPrimitive Examples-Mechanical

First ‘Formal’ StepsFirst ‘Formal’ Steps
►The first steps of sliding modeThe first steps of sliding mode
control ‘theory’ originated incontrol ‘theory’ originated in
the early 1950’s initiated bythe early 1950’s initiated by
S. V. Emel’yanov.S. V. Emel’yanov.
►Started as VSC – VariableStarted as VSC – Variable
Structure ControlStructure Control
 Varying system structure forVarying system structure for
stabilization.stabilization.

Variable Structure Control –Variable Structure Control –
Constituent SystemsConstituent Systems
Mode 1
Mode 2
1x a x= − 2
2 10
x a x
a a
= −
< <


Piecing together …Piecing together …

Properties of VSCProperties of VSC
►Both constituent systems were oscillatoryBoth constituent systems were oscillatory
and were not asymptotically stable.and were not asymptotically stable.
►‘‘Combined’ system is asymptotically stable.Combined’ system is asymptotically stable.
►Property not present in any of theProperty not present in any of the
constituent system is obtained by VSCconstituent system is obtained by VSC

Another Example – UnstableAnother Example – Unstable
Constituent SystemsConstituent Systems
0x x xξ α− + = 0x x xξ α− − = 

Analysis …Analysis …
►Both systems are unstableBoth systems are unstable
►Only stable mode is one mode of systemOnly stable mode is one mode of system
►IF the following VSC is employedIF the following VSC is employed
2
0,
2 4
x x x
ξ ξ
ξ α λ α− − = = − + 
0
, * , *
0
I xs
Mode s c x x c
II xs
λ
≤
= = + = −
>


Combined ..Combined ..

In this case,…In this case,…
►Again, property not present in constituentAgain, property not present in constituent
systems is found in the combined system.systems is found in the combined system.
►A stable structure can be obtain by varyingA stable structure can be obtain by varying
between two unstable structures.between two unstable structures.
►However, a more interesting behaviour canHowever, a more interesting behaviour can
be observed if we use a different ‘switching’be observed if we use a different ‘switching’
logic.logic.
2 2
0
, ,0 *
0
I xs
Mode s c x x c c
II xs
≤
= = + < <
>


The regionsThe regions

Sliding ModeSliding Mode
New trajectory that was not present in any of the two
original systems

Sliding Mode ?Sliding Mode ?
►Defined :Defined : Motion of the system trajectoryMotion of the system trajectory
along a ‘chosen’ line/plane/surface of thealong a ‘chosen’ line/plane/surface of the
state spacestate space..
►Sliding Mode Control :Sliding Mode Control : Control designedControl designed
with the aim to achieve sliding mode.with the aim to achieve sliding mode.
 Is usually of VSC typeIs usually of VSC type
 Eg : Previous problem can be perceived asEg : Previous problem can be perceived as
0
sgn( )
x x u
u xs x
ξ
α
− + =
= −
 

What is the advantage?What is the advantage?
►Consider a n-th order system represented inConsider a n-th order system represented in
the phase variable formthe phase variable form
►Also consider the sliding surface defined asAlso consider the sliding surface defined as
1
1 1
, 1,2, , 1i i
n n n
x x i n
x a x a x bu
+= = −
= − + + +
 
 
1 1 2 2 1 1
1 1 2 2 1 1
1
1 2 2 3 2 1 1
1
0T
n n n
n n n
n
n n n n i i
i
s c x c x c x c x x
x c x c x c x
x c x c x c x c c x
− −
− −
−
− − −
=
= = + + + + =
⇒ = − − − −
⇒ = − − − − + ∑


 

Advantage …Advantage …
► Thus entire dynamics of the system is governedThus entire dynamics of the system is governed
by the sliding line/surface parameters onlyby the sliding line/surface parameters only
► In sliding mode, dynamics independent of systemIn sliding mode, dynamics independent of system
parameters (aparameters (a11,a,a22,…).,…).
ROBUSTROBUST
1
1
1 2 2 3 2 1 1
1
, 1,2, , 1i i
n
n n n n i i
i
x x i n
x c x c x c x c c x
+
−
− − −
=
= = −
= − − − − + ∑
 
 

Required PropertiesRequired Properties
►For sliding mode to be of any use, it shouldFor sliding mode to be of any use, it should
have the following propertieshave the following properties
 System stability confined to sliding surfaceSystem stability confined to sliding surface
(unstable sliding mode is NOT sliding mode at all)(unstable sliding mode is NOT sliding mode at all)
 Sliding mode should not take ‘forever’ to startSliding mode should not take ‘forever’ to start

Stable SurfaceStable Surface
►Consider the systemConsider the system
►If the sliding function is designed asIf the sliding function is designed as
then confined to this surface ( ), thethen confined to this surface ( ), the
dynamics of can be written asdynamics of can be written as
1 11 1 12 2
2 21 1 22 2
x A x A x
x A x A x Bu
= +
= + +


[ ] 1
2
1 T
x
s K c x
x
 
= = 
 
1x
( )1 11 1 12 2 11 12 1x A x A x A A K x= + = −
1 2 0s Kx x= + =

The Surface …The Surface …
► If K is so designed that hasIf K is so designed that has
eigenvalues on LHP only , then the dynamicseigenvalues on LHP only , then the dynamics
of is stable.of is stable.
► Since , the dynamics of is alsoSince , the dynamics of is also
stable.stable.
► Hence, if the sliding surface is ‘designed’ asHence, if the sliding surface is ‘designed’ as
, the system dynamics confined to, the system dynamics confined to
s=0 is stable. (s=0 is stable. (Requirement 1Requirement 1))
Note : Strictly speaking, it is not necessary for sNote : Strictly speaking, it is not necessary for s
to be a linear function of xto be a linear function of x
( )11 12A A K−
1x
1 2 0Kx x+ = 2x
1 2
T
s Kx x c x= + =

Convergence to s=0Convergence to s=0
►The second requirement is that slidingThe second requirement is that sliding
mode should start at a finite time.mode should start at a finite time.
►Split the requirement into further bitsSplit the requirement into further bits
 Sliding mode SHOULD start.Sliding mode SHOULD start.
 It should do so in finite time.It should do so in finite time.

Run towards the surfaceRun towards the surface
► To be sure that sliding mode starts at some timeTo be sure that sliding mode starts at some time
t>0, irrespective of the initial statet>0, irrespective of the initial state x(0),x(0), we shouldwe should
be sure that the state trajectory is always movingbe sure that the state trajectory is always moving
towards s=0, whenever s is not zero.towards s=0, whenever s is not zero.
 Mathematics …Mathematics …
 This is called the ‘This is called the ‘reachability conditionreachability condition’’
2
0
0
d
s
dt
ss
<
⇒ <

A figure to help out …A figure to help out …
s=0
s<0
s>0
0s >
0s <

InsufficientInsufficient
►Consider the case,Consider the case,
►This gives the solution ofThis gives the solution of
► is not enoughis not enough (Violates(Violates
Requirement 2)Requirement 2)
2
,
0, 0
s s
ss s s
= −
= − < ∀ ≠


( )
( )
( ) 0
0,
t
s t e s
s t t
−
=
⇒ = → ∞
0ss <

ηη-reachability-reachability
► With only , s slows down too much whenWith only , s slows down too much when
close to zero to have finite time convergenceclose to zero to have finite time convergence
► Stronger condition is needed for finite timeStronger condition is needed for finite time
convergence.convergence.
► Defined asDefined as ηη-reachability-reachability conditioncondition
► s has a minimum rate of convergences has a minimum rate of convergence
0ss <
,
0
ss sη
η
≤ −
>


DiscontinuityDiscontinuity
► ObserveObserve
► So, at ,So, at , is discontinuousis discontinuous..
0s
ss s s s
s
s
η
η
η
<
= − < −
⇒ − < −
⇒ >
 


0s
ss s s s
s
η
η
>
= < −
⇒ < −
 

s0s =
s
s
η
-η

Discontinuous DynamicsDiscontinuous Dynamics
►Thus, for s>0, the system dynamics areThus, for s>0, the system dynamics are
and for s<0and for s<0
►Thus, at s=0, the dynamics is not wellThus, at s=0, the dynamics is not well
defined.defined.
►The dynamics along the sliding surface isThe dynamics along the sliding surface is
determined usingdetermined using continuation methodcontinuation method
( )x f x−
=
( )x f x+
=

Continuation MethodContinuation Method
►Using continuation method as proposed byUsing continuation method as proposed by
Filippov*, it is said that when s=0, the stateFilippov*, it is said that when s=0, the state
trajectory moves in a direction in betweentrajectory moves in a direction in between
andand
f +
f −
( )
( ) ( )
0
0
0
0
0
1 ,0 1
0
s
s
s
x f f f
s
s f
x
s s
f f
x xx f f
s s
f f f f
x x
µ µ µ+ −
=
=
− +
+ −
=
− + − +
= = + − ≤ ≤
∂
= =
∂
∂ ∂
∂ ∂⇒ = −
∂ ∂
− −
∂ ∂



*A. F. Filppov, “Differential Equations with discontinuous righthand sides”Kluwer
Academic Publishers,The Netherlands, 1988

Diagrammatically Speaking …Diagrammatically Speaking …

The reaching law approachThe reaching law approach
►InIn reaching law approachreaching law approach, the dynamics of, the dynamics of
the sliding function is directly expressed. Itthe sliding function is directly expressed. It
can have the general structurecan have the general structure
( ) ( )
( )
sgn
, 0
0, 0
s
s
s qf s k s
q k
sf s s
= − −
≥
> ∀ ≠


Few ExamplesFew Examples
►Constant rate reaching lawConstant rate reaching law
►Constant+Proportional rateConstant+Proportional rate
►Power-rate reaching lawPower-rate reaching law
( )sgns k s= −
( )sgns qs k s= − −
( ) ,0 1s k s
α
α= − < <

The Control SignalThe Control Signal
► Now, consider the conditionNow, consider the condition
► Thus,Thus,
► Or, control isOr, control is
► And the system dynamics is governed byAnd the system dynamics is governed by
( )sgns qs k s= − −
( ) ( )sgnT T T T
c x c Ax bu qc x k c x= + = − −
( ) ( ) ( )( )
1
sgnT T T T
u c b c A qc x k c x
−
= − + +
( ) ( )( ) ( ) ( )
1 1
sgnT T T T T
x A b c b c A qc x c b k c x
− −
= − + −

The Chattering ProblemThe Chattering Problem
► When, s is very close to zero, the control signalWhen, s is very close to zero, the control signal
switches between two structures.switches between two structures.
► Theoretically, the switching causes zeroTheoretically, the switching causes zero
magnitude oscillations with infinite frequency in x.magnitude oscillations with infinite frequency in x.
► Practically, actuators cannot switch at infinitePractically, actuators cannot switch at infinite
frequency. So we have high frequency oscillationsfrequency. So we have high frequency oscillations
of non-zero magnitude.of non-zero magnitude.
► ThisThis undesirableundesirable phenomenon is calledphenomenon is called chatteringchattering..

The pictureThe picture
Ideal Sliding Mode
Practical – With
Chattering

Why is chattering undesirable?Why is chattering undesirable?
►The ‘high frequency’ of chattering actuatesThe ‘high frequency’ of chattering actuates
unmodeled high frequency dynamics of theunmodeled high frequency dynamics of the
system. Controller performancesystem. Controller performance
deteriorates.deteriorates.
►More seriously, high frequency oscillationsMore seriously, high frequency oscillations
can cause mechanical wear in the system.can cause mechanical wear in the system.

Chattering avoidance/reductionChattering avoidance/reduction
►The chattering problem is because signumThe chattering problem is because signum
function is used in control.function is used in control.
 Control changes very abruptly near s=0.Control changes very abruptly near s=0.
 Actuator tries to cope up leading to ‘maximum-Actuator tries to cope up leading to ‘maximum-
possible-frequency’ oscillations.possible-frequency’ oscillations.
►Solution :Solution :
Replace signum term in control byReplace signum term in control by
‘smoother’ choices’‘smoother’ choices’

Chattering Avoidance…Chattering Avoidance…
►Some choices of smooth functionsSome choices of smooth functions
Saturation functionSaturation function Hyperbolic tangentHyperbolic tangent
( )sgn
,
0
s s
s
sat s
s
φ
φφ
φ
φ
 >
  
=   ≤  

>
( )1tanh k s

Disadvantage of ‘smoothing’Disadvantage of ‘smoothing’
►If saturation or tanh is used, then we canIf saturation or tanh is used, then we can
observe that near s=0observe that near s=0
►
►Where represents the saturation orWhere represents the saturation or
tanh function.tanh function.
►The limit in both cases is zero.The limit in both cases is zero.
►So, technically theSo, technically the sliding mode is lostsliding mode is lost
s( )f s
0
( )lim
s
s
kf s
s→
= −


What are the actual conditions forWhat are the actual conditions for
achieving Sliding Modeachieving Sliding Mode
►System is stable confined toSystem is stable confined to
► Control moves states towards this stableControl moves states towards this stable
sliding surfacesliding surface
►And does it in finite time.And does it in finite time.
0.s =
0, 0ss s< ∀ ≠
0s
ds
ds =
→ ∞


Some aspects of ContinuousSome aspects of Continuous
Sliding Mode ControlSliding Mode Control
►RobustnessRobustness
►Multivariable Sliding ModeMultivariable Sliding Mode
►‘‘Almost’ Sliding ModeAlmost’ Sliding Mode

Robustness of CSMCRobustness of CSMC
►When in sliding mode, entire system
dynamics is governed by sliding surface
parameters and not original system
parameters.
►Hence, sliding mode is robust.

DisturbanceDisturbance
►Consider the system with disturbanceConsider the system with disturbance
►Disturbance comes through input channelDisturbance comes through input channel
►How does sliding mode behave in such aHow does sliding mode behave in such a
situation.situation.
( )
1 11 1 12 2
2 21 1 22 2 1
x A x A x
x A x A x Bu B d t
= +
= + + +



Disturbance RejectionDisturbance Rejection
► The control law is designed so as to bring theThe control law is designed so as to bring the
system to the sliding surface.system to the sliding surface.
► Let us see dynamics confined to the slidingLet us see dynamics confined to the sliding
surfacesurface
► Thus,Thus,
► Therefore,Therefore,
► AndAnd
Again, dynamics independent of disturbance.Again, dynamics independent of disturbance.
Hence disturbance rejection.Hence disturbance rejection.
2 1x Kx=
( )1 11 12 1x A A K x= −
( )2 11 12 1x K A A K x= −

What if more than one input ?What if more than one input ?
►If system has more than one input, then theIf system has more than one input, then the
system can be transformed to the formsystem can be transformed to the form
►With having more than one elements.With having more than one elements.
►Thus, will also have multipleThus, will also have multiple
rows. Hence, the system can have morerows. Hence, the system can have more
than one sliding surfacethan one sliding surface
1 11 1 12 2
2 21 1 22 2
x A x A x
x A x A x BU
= +
= + +


2x
1 2 0s Kx x= − =

Approach to sliding surfaceApproach to sliding surface
►Sliding mode will start when all slidingSliding mode will start when all sliding
functions are zero. I.e, intersection of allfunctions are zero. I.e, intersection of all
sliding surfaces.sliding surfaces.
►Approach to the intersectionApproach to the intersection
 Direct to intersection (Eventual)Direct to intersection (Eventual)
 Surface by surfaceSurface by surface
►In particular order (Fixed Order)In particular order (Fixed Order)
►First approach (Free order)First approach (Free order)

Eventual Sliding ModeEventual Sliding Mode
►In this type of sliding mode, the stateIn this type of sliding mode, the state
trajectory moves to the intersection of thetrajectory moves to the intersection of the
sliding surfaces through a connected subsetsliding surfaces through a connected subset
in the state space.in the state space.
►It does not necessary stay on any one of theIt does not necessary stay on any one of the
sliding surfaces on approaching it.sliding surfaces on approaching it.

Eventual Sliding ModeEventual Sliding Mode

Fixed order Sliding ModeFixed order Sliding Mode
►In fixed order sliding mode, the stateIn fixed order sliding mode, the state
trajectory moves to one pre-specified slidingtrajectory moves to one pre-specified sliding
surface and staying on it moves to thesurface and staying on it moves to the
intersection of the first surface with the nextintersection of the first surface with the next
pre-specified sliding surfacepre-specified sliding surface
( ) ( ) ( ) ( )1 1 2
1
0 0 0 0
m
n
i
i
s s s s
=
→ = → = = → → =  

Free order sliding modeFree order sliding mode
►In free order sliding mode, the stateIn free order sliding mode, the state
trajectory remains on a sliding surface oncetrajectory remains on a sliding surface once
the state approaches it. However, there isthe state approaches it. However, there is
no particular order in which the surfaces areno particular order in which the surfaces are
reachedreached
( ) ( ) ( ) ( )1 1 2
1
1 2
0 0 0 0
, are not fixed apriori
m
n
i i i i
i
s s s s
i i
=
→ = → = = → → =  

Ordered Sliding ModeOrdered Sliding Mode

Chattering RefreshedChattering Refreshed
► A conventional sliding mode behaviour would haveA conventional sliding mode behaviour would have
a sliding surface dynamics of the forma sliding surface dynamics of the form
► However, due to finite bandwidth of the actuator,However, due to finite bandwidth of the actuator,
the input cannot switch fast enough near thethe input cannot switch fast enough near the
sliding surfacesliding surface
 Chattering – Finite frequency, finite amplitudeChattering – Finite frequency, finite amplitude
oscillations about the sliding surfaceoscillations about the sliding surface
( )sgns Qs K s= − −

Almost Sliding ModeAlmost Sliding Mode
►To remedy chattering, the strict requirementTo remedy chattering, the strict requirement
of “movement on sliding surface” is relaxed.of “movement on sliding surface” is relaxed.
►We try to get ‘Almost’ – sliding mode (QuasiWe try to get ‘Almost’ – sliding mode (Quasi
sliding mode)sliding mode)

Saturation function basedSaturation function based
Sliding Mode ControlSliding Mode Control
►Instead ofInstead of
Inside the band |s|<Inside the band |s|<φφ, the reaching law is, the reaching law is
linear aslinear as
This is also called ‘boundary layerThis is also called ‘boundary layer
techniquetechnique
( )sgns Qs K s= − −
s
s Qs K
φ
= − −

The motionThe motion
S=0
S=-φ
S=φ

DisadvantageDisadvantage
►‘‘Almost’ is NOT exactAlmost’ is NOT exact

‘‘Newer’ AvenuesNewer’ Avenues
►Two phases in sliding motion : ReachingTwo phases in sliding motion : Reaching
Phase and Sliding PhasePhase and Sliding Phase
►ImprovementsImprovements
►Reaching Phase - Higher Order SlidingReaching Phase - Higher Order Sliding
Mode ControlMode Control
►Sliding Phase - Terminal Sliding ModeSliding Phase - Terminal Sliding Mode
ControlControl

Higher order Sliding ModeHigher order Sliding Mode
►Basic Definition of Sliding Mode : s(x)=0 inBasic Definition of Sliding Mode : s(x)=0 in
finite time. Sliding surface reached in finitefinite time. Sliding surface reached in finite
time and stays on it.time and stays on it.
►Problem : Chattering resultsProblem : Chattering results
►Solution : Try to get ds/dt = 0, additionally inSolution : Try to get ds/dt = 0, additionally in
finite time.finite time.  Second order sliding mode.Second order sliding mode.
►Get the first n-1 derivatives of s(x) to zero inGet the first n-1 derivatives of s(x) to zero in
finite time.finite time.  n-th order sliding mode.n-th order sliding mode.
( ) 0s x ≠

AdvantageAdvantage
►Smooth control results. No Chattering.Smooth control results. No Chattering.
►Disadvantage : Not very straight forward.Disadvantage : Not very straight forward.

HOSM : Twisting AlgorithmHOSM : Twisting Algorithm
►Applicable to systems of relative degree 2.Applicable to systems of relative degree 2.
►Input appears in 2Input appears in 2ndnd
derivative of slidingderivative of sliding
function.function.
►Input is still discontinous. However, there isInput is still discontinous. However, there is
no chattering in states.no chattering in states.

Typical Twisting TrajectoryTypical Twisting Trajectory

Super Twisting AlgorithmSuper Twisting Algorithm
►For systems with relative degree 1.For systems with relative degree 1.
►Switching shifted to derivative of input.Switching shifted to derivative of input.
►Input continuous and so is derivative ofInput continuous and so is derivative of
sliding function.sliding function.
►No chattering here too.No chattering here too.

Typical Super-twisting TrajectoryTypical Super-twisting Trajectory

New IdeaNew Idea
►If we are concerned with getting an outputIf we are concerned with getting an output
to zero, why not set s=y!!to zero, why not set s=y!!
►Are there any extra conditions?Are there any extra conditions?
►Zero DynamicsZero Dynamics

Terminal Sliding ModeTerminal Sliding Mode
►Higher order sliding mode is about reachingHigher order sliding mode is about reaching
the sliding surface smoothly.the sliding surface smoothly.
►Terminal sliding mode deals with design ofTerminal sliding mode deals with design of
the sliding function such that the systemthe sliding function such that the system
reaches origin in FINITE TIME one thereaches origin in FINITE TIME one the
sliding surface is reached.sliding surface is reached.

The sliding surfaceThe sliding surface
►Terminal Sliding ModeTerminal Sliding Mode
Fast close to origin. Finite time convergence.Fast close to origin. Finite time convergence.
►Fast-Terminal Sliding ModeFast-Terminal Sliding Mode
1 2 ,0 1, 0kx x kα
σ α= + < < >
1 1 2 1 2 1 2,0 1, , 0,1 2k x k x x k kα β
σ α β= + + < < > < <

Terminal Sliding Surface …Terminal Sliding Surface …
In case of systems with more than 2 states,In case of systems with more than 2 states,
►For the system in phase variable form,For the system in phase variable form,
1
0 1
1 1 1
, 1,2, , 1
, 1,2, , 1
, 0
0 1 2
i i
i i
i i i i i i
i i
i i
x x i n
s x
s s s s i nγ δ
α β
α β
γ δ
+
− − −
= = −
=
= + + = −
>
< < < <
 
 

ReferencesReferences
► V. Utkin, “Variable Structure Systems with Sliding Mode”,V. Utkin, “Variable Structure Systems with Sliding Mode”,
IEEE Trans. Automat. ContrIEEE Trans. Automat. Contr., AC-12, No. 2, pp. 212-222,., AC-12, No. 2, pp. 212-222,
19771977
 An introductory paper on VSC and sliding mode control.An introductory paper on VSC and sliding mode control.
► J.Y.Hung, W.Gao, J.C.Hung, “Variable Structure Control –J.Y.Hung, W.Gao, J.C.Hung, “Variable Structure Control –
A Survey”,A Survey”, IEEE Trans. Ind. ElectronIEEE Trans. Ind. Electron., Vol. 20, No. 1, pp. 2-., Vol. 20, No. 1, pp. 2-
22,Feb. 199322,Feb. 1993
 A survey paper on VSC and sliding mode control concepts.A survey paper on VSC and sliding mode control concepts.
► B. Draženović , "The invariance conditions in variableB. Draženović , "The invariance conditions in variable
structure systems",structure systems", AutomaticaAutomatica, vol. 5, pp. 287, 1969, vol. 5, pp. 287, 1969
 The paper proving that in case of matched disturbance, one canThe paper proving that in case of matched disturbance, one can
eliminate disturbance effect using appropriate control. Cited moreeliminate disturbance effect using appropriate control. Cited more
than 250 times ‘officially’. Work done in one night.than 250 times ‘officially’. Work done in one night.

► C. Edwards and S. Spurgeon,C. Edwards and S. Spurgeon, Sliding Mode control: TheorySliding Mode control: Theory
and Applications,and Applications, Taylor and Francis, London, 1998Taylor and Francis, London, 1998
 A good book on the subject.A good book on the subject.
► L. Fridman and A. Levant, "L. Fridman and A. Levant, "Higher order sliding modesHigher order sliding modes," in," in
Sliding Mode Control in EngineeringSliding Mode Control in Engineering, Eds. W. Perruquetti, Eds. W. Perruquetti
and J. P. Barbot, Marcel Dekker Inc., 2002, pp. 53-101.and J. P. Barbot, Marcel Dekker Inc., 2002, pp. 53-101.
 An initial paper on HOSMAn initial paper on HOSM
► X. Yu and Z. Man, On finite time convergence: TerminalX. Yu and Z. Man, On finite time convergence: Terminal
sliding modes,” in Proc. 1996 Int. Workshop on Variablesliding modes,” in Proc. 1996 Int. Workshop on Variable
Structure Systems, Kobe, Japan, 1996. pp. 164–168Structure Systems, Kobe, Japan, 1996. pp. 164–168
 Initial paper on TSMInitial paper on TSM

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