Sliding Mode Control: Basic Theory, Advances and Applications · Sliding Mode Control: Basic...

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

[email protected]


• 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


• 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


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


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


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.


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


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


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


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


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


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


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, =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


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


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”


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)



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



( ) ( ) ( )( ) 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



( ) ( ) ( )( )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



( ) 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



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+≤∞



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 ,,



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



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

( ) ( ) ( )( )( ) ( )( )

∈∈∈== 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



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



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=⋅

∂∂



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=



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


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


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


L1 – Classic SMC


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


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


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


L1 – Classic SMC


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


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


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


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=



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



( ) ( )

( ) ( ) ( ) ( )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



∑

∑−

=

−

=−

+

+−=

+−=

−==

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



( ) ( ) ( ) ( ) ( )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


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



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



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



( ) ( )[ ]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



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 ⋅

∂∂⋅

⋅

∂∂−=

−



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



( ) ( ) ( ) ( ) ( )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


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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