sliding mode control – an introduction s. janardhanan iit delhi

Sliding Mode Control Sliding Mode Control – An Introduction– An Introduction

S. JanardhananS. Janardhanan

IIT DelhiIIT Delhi

Sliding Mode Control 2

OutlineOutline

►What is this ‘Sliding mode’ What is this ‘Sliding mode’ and how did its study start?and how did its study start?

►How to design controller using How to design controller using this concept?this concept?


Primitive Examples - Primitive Examples - ElectricalElectrical


Primitive Examples-Primitive Examples-MechanicalMechanical


First ‘Formal’ StepsFirst ‘Formal’ Steps

►The first steps of sliding The first steps of sliding mode control ‘theory’ mode control ‘theory’ originated in the early originated in the early 1950’s initiated by 1950’s initiated by

S. V. Emel’yanov.S. V. Emel’yanov.►Started as VSC – Variable Started as VSC – Variable

Structure Control Structure Control Varying system structure for Varying system structure for

stabilization.stabilization.


Variable Structure Control – Variable Structure Control – Constituent SystemsConstituent Systems

Mode 1Mode 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 Both constituent systems were oscillatory and were not oscillatory and were not asymptotically stable.asymptotically stable.

► ‘‘Combined’ system is asymptotically Combined’ system is asymptotically stable.stable.

►Property not present in any of the Property not present in any of the constituent system is obtained by VSCconstituent system is obtained by VSC


Another Example – Unstable Another Example – Unstable Constituent SystemsConstituent Systems

0x x x 0x x x


Analysis …Analysis …

►Both systems are unstableBoth systems are unstable►Only stable mode is one mode of Only stable mode is one mode of

system system

► IF the following VSC is employedIF the following VSC is employed

2

0,2 4

x x x

0, * , *

0

I xsMode s c x x c

II xs


Combined ..Combined ..


In this case,…In this case,…

►Again, property not present in Again, property not present in constituent systems is found in the constituent systems is found in the combined system.combined system.

►A stable structure can be obtain by A stable structure can be obtain by varying between two unstable varying between two unstable structures.structures.

►However, a more interesting behaviour However, a more interesting behaviour can be observed if we use a different can be observed if we use a different ‘switching’ logic. ‘switching’ logic.

2 2

0, ,0 *

0

I xsMode 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 Motion of the system trajectory along a ‘chosen’ trajectory along a ‘chosen’ line/plane/surface of the state spaceline/plane/surface of the state space..

►Sliding Mode Control : Sliding Mode Control : Control Control designed with the aim to achieve designed with the aim to achieve sliding mode.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


What is the advantage?What is the advantage?

►Consider a n-th order system Consider a n-th order system represented in the phase variable formrepresented in the phase variable form

►Also consider the sliding surface Also consider the sliding surface defined asdefined 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 11

0Tn n n

n n n

n

n n n n i ii

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 Thus entire dynamics of the system is governed by the sliding line/surface governed by the sliding line/surface parameters onlyparameters only

► In sliding mode, dynamics independent of In sliding mode, dynamics independent of system parameters (asystem parameters (a11,a,a22,…). ,…).

ROBUSTROBUST

1

1

1 2 2 3 2 1 11

, 1, 2, , 1i i

n

n n n n i ii

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 For sliding mode to be of any use, it should have the following propertiesshould have the following properties System stability confined to sliding System stability confined to sliding

surface surface (unstable sliding mode is NOT sliding (unstable sliding mode is NOT sliding mode at all)mode at all)

Sliding mode should not take ‘forever’ to Sliding mode should not take ‘forever’ to startstart


Stable SurfaceStable Surface

►Consider the systemConsider the system

► If the sliding function is designed as If the sliding function is designed as

then confined to this surface then confined to this surface ( ), the dynamics of can ( ), the dynamics of can be written asbe 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 Txs 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 has If K is so designed that has eigenvalues on LHP only , then the eigenvalues on LHP only , then the dynamics of is stable.dynamics of is stable.

► Since , the dynamics of is Since , the dynamics of is also stable.also stable.

► Hence, if the sliding surface is ‘designed’ asHence, if the sliding surface is ‘designed’ as

, the system dynamics confined , the system dynamics confined to to

s=0 is stable. (s=0 is stable. (Requirement 1Requirement 1) )

Note : Strictly speaking, it is not necessary for Note : Strictly speaking, it is not necessary for s to be a linear function of xs to be a linear function of x

11 12A A K

1x

1 2 0Kx x 2x

1 2Ts Kx x c x


Convergence to s=0Convergence to s=0

►The second requirement is that sliding The second requirement is that sliding mode should start at a finite time.mode should start at a finite time.

►Split the requirement into further bits Split 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 To be sure that sliding mode starts at some time t>0, irrespective of the initial state time t>0, irrespective of the initial state x(0),x(0), we should be sure that the state we should be sure that the state trajectory is always moving towards s=0, trajectory is always moving towards s=0, whenever s is not zero.whenever s is not zero. Mathematics …Mathematics …

This is called the ‘This is called the ‘reachability conditionreachability condition’’

2 0

0

ds

dtss


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 enough is not enough (Violates (Violates Requirement 2)Requirement 2)

2

,

0, 0

s s

ss s s

( ) 0

0,

ts t e s

s t t

0ss


-reachability-reachability

► With only , s slows down too much With only , s slows down too much when close to zero to have finite time when close to zero to have finite time convergence convergence

► Stronger condition is needed for finite time Stronger condition is needed for finite time convergence.convergence.

► Defined as Defined as -reachability-reachability condition condition

► s has a minimum rate of convergence s has a minimum rate of convergence

0ss

,

0

ss s


DiscontinuityDiscontinuity

► ObserveObserve

► So, at , So, at , is discontinuousis discontinuous..

0s

ss s s s

s

s

0s

ss s s s

s

s0s

s

s

-


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 well Thus, at s=0, the dynamics is not well defined.defined.

►The dynamics along the sliding surface is The dynamics along the sliding surface is determined using determined using continuation methodcontinuation method

x f x

x f x


Continuation MethodContinuation Method

►Using continuation method as Using continuation method as proposed by Filippov*, it is said that proposed by Filippov*, it is said that when s=0, the state trajectory moves when s=0, the state trajectory moves in a direction in between and in a direction in between and

f

f

0

0

0

0

0

1 ,0 1

0

s

s

s

x f f f

ss f

xs sf f

x xx f fs sf f f f

x x

*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

► In In reaching law approachreaching law approach, the , the dynamics of the sliding function is dynamics of the sliding function is directly expressed. It can have the directly expressed. It can have the general structuregeneral 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


The Control SignalThe Control Signal

► Now, consider the conditionNow, consider the condition

► Thus,Thus,

► Or, control is Or, control is ► And the system dynamics is governed byAnd the system dynamics is governed by

sgns qs k s

sgnT T T Tc x c Ax bu qc x k c x

1sgnT T T Tu c b c A qc x k c x

1 1sgnT T T T Tx 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 When, s is very close to zero, the control signal switches between two structures.signal switches between two structures.

► Theoretically, the switching causes zero Theoretically, the switching causes zero magnitude oscillations with infinite magnitude oscillations with infinite frequency in x.frequency in x.

► Practically, actuators cannot switch at Practically, actuators cannot switch at infinite frequency. So we have high infinite frequency. So we have high frequency oscillations of non-zero frequency oscillations of non-zero magnitude.magnitude.

► This This undesirableundesirable phenomenon is called phenomenon is called chatteringchattering..


The pictureThe picture

Ideal Sliding ModePractical – With Chattering


Why is chattering Why is chattering undesirable?undesirable?

►The ‘high frequency’ of chattering The ‘high frequency’ of chattering actuates unmodeled high frequency actuates unmodeled high frequency dynamics of the system. Controller dynamics of the system. Controller performance deteriorates.performance deteriorates.

►More seriously, high frequency More seriously, high frequency oscillations can cause mechanical oscillations can cause mechanical wear in the system.wear in the system.


Chattering Chattering avoidance/reductionavoidance/reduction

►The chattering problem is because The chattering problem is because signum function is used in control. signum 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 Actuator tries to cope up leading to

‘maximum-possible-frequency’ ‘maximum-possible-frequency’ oscillations.oscillations.

►Solution :Solution : Replace signum term in control by Replace signum term in control by

‘smoother’ choices’‘smoother’ choices’


Chattering Avoidance…Chattering Avoidance…

►Some choices of smooth functionsSome choices of smooth functions

Saturation Saturation functionfunction

Hyperbolic Hyperbolic tangenttangent sgn

,

0

s ss

sat ss

1tanh k s


Disadvantage of ‘smoothing’Disadvantage of ‘smoothing’

► If saturation or tanh is used, then we If saturation or tanh is used, then we can observe that near s=0can observe that near s=0

► ►Where represents the saturation Where represents the saturation

or tanh function. or tanh function. ►The limit in both cases is zero.The limit in both cases is zero.►So, technically the So, technically the sliding mode is lostsliding mode is lost

s( )f s0

( )lims

skf s

s


What are the actual conditions What are the actual conditions for achieving Sliding Modefor achieving Sliding Mode

►System is stable confined toSystem is stable confined to► Control moves states towards this Control moves states towards this

stable sliding surfacestable sliding surface►And does it in finite time.And does it in finite time.

0.s

0, 0ss s

0s

ds

ds


Some aspects of Continuous Some 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 Disturbance comes through input channelchannel

►How does sliding mode behave in such How does sliding mode behave in such a situation.a 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 The control law is designed so as to bring the system to the sliding surface.the system to the sliding surface.

► Let us see dynamics confined to the sliding Let us see dynamics confined to the sliding surfacesurface

► Thus, Thus, ► Therefore,Therefore,► And And Again, dynamics independent of Again, dynamics independent of

disturbance.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 What if more than one input ?input ?

► If system has more than one input, If system has more than one input, then the system can be transformed to then the system can be transformed to the formthe form

►With having more than one elements.With having more than one elements.►Thus, will also have multiple Thus, will also have multiple

rows. Hence, the system can have rows. Hence, the system can have more than one sliding surfacemore than 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 sliding Sliding mode will start when all sliding functions are zero. I.e, intersection of functions are zero. I.e, intersection of all sliding surfaces.all 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 state In this type of sliding mode, the state trajectory moves to the intersection of trajectory moves to the intersection of the sliding surfaces through a the sliding surfaces through a connected subset in the state space.connected subset in the state space.

► It does not necessary stay on any one It does not necessary stay on any one of the sliding surfaces on approaching of the sliding surfaces on approaching it.it.


Eventual Sliding ModeEventual Sliding Mode


Fixed order Sliding ModeFixed order Sliding Mode

► In fixed order sliding mode, the state In fixed order sliding mode, the state trajectory moves to one pre-specified trajectory moves to one pre-specified sliding surface and staying on it moves sliding surface and staying on it moves to the intersection of the first surface to the intersection of the first surface with the next pre-specified sliding with the next pre-specified sliding surfacesurface

1 1 21

0 0 0 0m

ni

i

s s s s

R


Free order sliding modeFree order sliding mode

► In free order sliding mode, the state In free order sliding mode, the state trajectory remains on a sliding surface trajectory remains on a sliding surface once the state approaches it. once the state approaches it. However, there is no particular order However, there is no particular order in which the surfaces are reachedin which the surfaces are reached

1 1 2

1

1 2

0 0 0 0

, are not fixed apriori

mn

i i i ii

s s s s

i i

R


Ordered Sliding ModeOrdered Sliding Mode


Chattering RefreshedChattering Refreshed

► A conventional sliding mode behaviour A conventional sliding mode behaviour would have a sliding surface dynamics of would have a sliding surface dynamics of the formthe form

► However, due to finite bandwidth of the However, due to finite bandwidth of the actuator, the input cannot switch fast actuator, the input cannot switch fast enough near the sliding surfaceenough near the sliding surface Chattering – Finite frequency, finite amplitude Chattering – 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 To remedy chattering, the strict requirement of “movement on sliding requirement of “movement on sliding surface” is relaxed.surface” is relaxed.

►We try to get ‘Almost’ – sliding mode We try to get ‘Almost’ – sliding mode (Quasi sliding mode)(Quasi sliding mode)


Saturation function based Saturation function based Sliding Mode ControlSliding Mode Control

► Instead of Instead of

Inside the band |s|<Inside the band |s|<, the reaching , the reaching law is linear aslaw is linear as

This is also called ‘boundary layer This is also called ‘boundary layer techniquetechnique

sgns Qs K s

ss Qs K


The motion The motion

S=0

S=-

S=


DisadvantageDisadvantage

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


‘‘Newer’ AvenuesNewer’ Avenues

►Two phases in sliding motion : Two phases in sliding motion : Reaching Phase and Sliding PhaseReaching Phase and Sliding Phase

► ImprovementsImprovements►Reaching Phase - Higher Order Sliding Reaching Phase - Higher Order Sliding

Mode ControlMode Control►Sliding Phase - Terminal Sliding Mode Sliding Phase - Terminal Sliding Mode

ControlControl


Higher order Sliding ModeHigher order Sliding Mode

►Basic Definition of Sliding Mode : s(x)=0 Basic Definition of Sliding Mode : s(x)=0 in finite time. Sliding surface reached in in finite time. Sliding surface reached in finite time and stays on it. finite time and stays on it.

►Problem : Chattering resultsProblem : Chattering results►Solution : Try to get ds/dt = 0, Solution : Try to get ds/dt = 0,

additionally in finite time. additionally in finite time. Second order Second order sliding mode.sliding mode.

►Get the first n-1 derivatives of s(x) to zero Get the first n-1 derivatives of s(x) to zero in finite time. in 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 Disadvantage : Not very straight

forward.forward.


HOSM : Twisting AlgorithmHOSM : Twisting Algorithm

►Applicable to systems of relative Applicable to systems of relative degree 2.degree 2.

► Input appears in 2Input appears in 2ndnd derivative of derivative of sliding function.sliding function.

► Input is still discontinous. However, Input is still discontinous. However, there is no chattering in states.there is 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 Switching shifted to derivative of

input.input.► Input continuous and so is derivative Input continuous and so is derivative

of sliding function.of sliding function.►No chattering here too.No chattering here too.


Typical Super-twisting Typical Super-twisting TrajectoryTrajectory


New IdeaNew Idea

► If we are concerned with getting an If we are concerned with getting an output to zero, why not set s=y!!output 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 Higher order sliding mode is about reaching the sliding surface smoothly.reaching the sliding surface smoothly.

►Terminal sliding mode deals with Terminal sliding mode deals with design of the sliding function such that design of the sliding function such that the system reaches origin in FINITE the system reaches origin in FINITE TIME one the sliding surface is TIME one the sliding surface is reached.reached.


The sliding surfaceThe sliding surface

►Terminal Sliding ModeTerminal Sliding Mode

Fast close to origin. Finite time Fast close to origin. Finite time convergence.convergence.

►Fast-Terminal Sliding ModeFast-Terminal Sliding Mode

1 2 ,0 1, 0kx x k

1 1 2 1 2 1 2,0 1, , 0,1 2k x k x x k k


Terminal Sliding Surface …Terminal Sliding Surface …

In case of systems with more than 2 In case of systems with more than 2 states,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 n


ReferencesReferences

► V. Utkin, “Variable Structure Systems with Sliding V. Utkin, “Variable Structure Systems with Sliding Mode”, Mode”, IEEE Trans. Automat. ContrIEEE Trans. Automat. Contr., AC-12, No. 2, pp. ., AC-12, No. 2, pp. 212-222, 1977 212-222, 1977 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 J.Y.Hung, W.Gao, J.C.Hung, “Variable Structure Control – A Survey”, Control – A Survey”, IEEE Trans. Ind. ElectronIEEE Trans. Ind. Electron., Vol. ., Vol. 20, No. 1, pp. 2-22,Feb. 199320, No. 1, pp. 2-22,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 variable B. 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 The paper proving that in case of matched disturbance, one

can eliminate disturbance effect using appropriate control. can eliminate disturbance effect using appropriate control. Cited more than 250 times ‘officially’. Work done in one Cited more than 250 times ‘officially’. Work done in one night.night.


► C. Edwards and S. Spurgeon, C. Edwards and S. Spurgeon, Sliding Mode control: Sliding Mode control: Theory and Applications,Theory and Applications, Taylor and Francis, London, Taylor and Francis, London, 19981998 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 Higher order sliding modesmodes," in ," in Sliding Mode Control in EngineeringSliding Mode Control in Engineering, Eds. , Eds. W. Perruquetti and J. P. Barbot, Marcel Dekker Inc., W. Perruquetti and J. P. Barbot, Marcel Dekker Inc., 2002, pp. 53-101.2002, pp. 53-101. An initial paper on HOSMAn initial paper on HOSM

► X. Yu and Z. Man, On finite time convergence: Terminal X. Yu and Z. Man, On finite time convergence: Terminal sliding modes,” in Proc. 1996 Int. Workshop on sliding modes,” in Proc. 1996 Int. Workshop on Variable Structure Systems, Kobe, Japan, 1996. pp. Variable Structure Systems, Kobe, Japan, 1996. pp. 164–168164–168 Initial paper on TSMInitial paper on TSM

sliding mode control – an introduction s. janardhanan iit delhi

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