Sliding mode: Basic theory and new perspectivesdelbuono/usai_talk_1.pdf · • Regularization of...

SDS 2008 - Capitolo (BA) June 2008 1/54

Dept of Electrical and Electronic Eng.Dept of Electrical and Electronic Eng.University ofUniversity of CagliariCagliari

5th Workshop on

Structural Dynamical Systems: Computational Aspects

Capitolo (BA) – Italy

Sliding mode:

Basic theory and new perspectives

Elio USAI

[email protected]


Summary

• Switching systems

• Introductive examples

• Zeno behaviour

• Sliding modes in switching systems

• Regularization of sliding a mode trajectory

• Sliding modes in control systems

• Stability of a sliding mode control system

• Invariance of a sliding mode control system

• Recovering the uncertain dynamics

• Approximability

• Discrete time implementation

• Higher order sliding modes


Switching systems

Switching systems are dynamical systems such that their behaviour

is characterised by different dynamics in different domains

( ) ( ) ( )( )

N

RRR

⊆∈

∈⊆∈⊆∈= +

Qi

tUXtttftm

i

n

ii ,,,,, uxuxx&

fi is a smooth vector field fi: RnxRmxR+→Rn

The state dynamics is invariant until a switch occurs

The system dynamics is represented by a differential equation with discontinuous right-hand side


Switching systems

Switching between different dynamics

( ) ( ) ( )( )

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

N

RRR

⊆∈

∈∈⊆∈⊆∈

==

⇓

=

+

++++−−−−

Qji

tUX

ff

g

lm

i

n

i

ji

kkkk

sw

kkkkkkkk

i

,

1,0,,

,, ,,,

0,,,

vux

uxxuxx

vux

ττττττττ

ττττ

&&

The reaching of the guard gisw cause the switching from the

dynamics fi to the dynamics fj, according to proper rules


Switching systems

What does it happen on the guard? ( ) ( ) ( )( ) 0,,, =kkkk

sw

ig ττττ vux

( ) 1,0∈kτv Forced switching

( ) ( )+− =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

Most interesting……….. the time evolution of the guard function!

( ) φτ ∈kv Autonomous switching


Introductive examples

Switching systems may behave very differently from each

of the constituting ones.

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

2010

201

101

0

20

10

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



-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

y(t)

dy(t

)/dt

Phase plane

a0

1

=0.5

a0

2

=9.0

-10 -5 0 5 10-5

0

5

10

15

20Phase plane

y(t)

dy(t

)/dt( )

2.07.01.0

0sgn

001

001

=∆==

=∆−++

aaa

yyayayay &&&&

1.01 =a

Switched unstable dynamics

Both dynamics are asymptotically stable



1.01 −=a

Switched asymptoticallystable 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

1Phase plane

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

Phase plane

a0

2

=9.0

a0

1

=0.5

( )

2.07.01.0

0sgn

001

001

=∆=−=

=∆+++

aaa

yyayayay &&&&



Mass-spring and Coulomb friction

( ) ( )( ) ( ) 0sgn =++ tkytybtym &&&

Switching time instantsm

k

y( )

k

mky kk πττ =→= 0&

-6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5Phase space

y(t)

dy(t

)/dt

0 2 4 6 8 10-4

-3

-2

-1

0

1

2

3

4

5

Time [s]

velocityposition



Mass-spring and Coulomb friction

( ) ( )( ) ( ) 0sgn =++ tkytybtym &&&

The guard is m

k

y

( ) 0=ty&

In a finite time τ∞ the mass stops!( )

( ) [ ]∞

∞

≥+−∈=

=τt

yty

ty

kb

kb

,

0&

The state is continuous

The switching is autonomous

( ) ( )( ) ( ) ∞≥=+−= τttym

kty

m

bty 0sgn &&& ?



Singular optimal control

( ) ( )

( ) ( )[ ]

( ) ( ) T

T

Tu

dtubfJ

ubf

xxxx

xx

xxx

==≤

+=

+=

∫,0,1 0

0

00

&

( ) ( ) ( ) ( )[ ]

( ) ( )( )xpx

xp

xxpxx

bbu

H

ubfubfH

T

T

T

+−=

∂

∂−=

+++=

0

00

sgn

&

( ) ( ) [ ]210 ,0 tttbb T ∈=+ xpxu switches at infinite frequency

u is defined looking at the higher

derivatives of the guard



The bouncing ball( )( ) ( )

( ) ( ) [ ]1,0

,2,10:

0

∈−=

==

≥−=

−+ ατατ

ττ

iiyy

iy

ygty

ii

&&

K

&&y

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Ele

vation( )

α

ατ

ατ

αττ

−

+++=

+==

+=−

∞

+

+−+

1

12

2

2

0

210

1

0

2

0

221

0

0

0

gyyy

gyyyy

gyy

gg

k

kk

k

gkk

&&

&&&

&



The bouncing ball

( ) ( ) ( ) ( )

[ ]1,0

01

∈

≥⋅+−−=

α

δα yytygty &&&

ywith abuse of notation

δ(•) is the Dirac’s function

In a finite time τ∞ the mass stops!( )

( )∞≥

=

=τt

ty

ty

0

0

&

( ) ( ) ( ) ( ) ∞≥=⋅+−−= τδα tytygty 01 &&& ?


Zeno behaviour

T = τii∈N : set of switching/jump time instantsIn = xi i∈N xi ⊆ D : set of initial states sequenceEd = ηii∈N ηi=(i,j) ⊆ Q x Q : set of edge sequence

EdInT ,,=Hχ

The execution χH of the hybrid system H can be constituted by finite or infinite elements.

Executions with infinite number of elements may be due to the ZenoZenophenomenon

The switching behaviour of a hybrid system can be described by its execution set


Zeno behaviour

The Zeno phenomenon appears when the execution χH of the

hybrid system is such that

( ) ∞<=−= ∞

∞

=+

∞→∑ ττττ

0

1limi

iiii

τ∞ (Zeno time) is a right accumulation point for the time instants sequence

( ) 01∞→

+ →−i

ii ττ The switching frequency tends to be infinite

In a Zeno condition the system evolves along a guard

( )( ) ( ) ∞≥∀=⋅∂

∂τtt

ttg0

,x

x

x&


Sliding modes in switching systems

The Zeno phenomenon is mainly related to the switching frequencyon a guard, but previous relationship shows the relation between the guard and the system dynamics

( )( ) 0, =ttg x( ) ( )( )ttft ,1 xx =&

( ) ( )( )ttft ,2 xx =&

( ) ( )( )ttft s ,xx =&

( )( )x

x

∂

∂ ttg ,

The motion of the system on a discontinuity surface is called sliding mode



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 is invariant when constrained on the sliding surface

Any system on the sliding surface behaves the same way



α±=>

=+−

01

01

0

0

aa

yayay &&&

a0=α a0=-α

α>0

( ) 0sgn01 =+− σyayay &&&

,2.0,1,1 21 ===

+=

caa

cyy&σ

-8 -6 -4 -2 0 2 4 6 8-10

-8

-6

-4

-2

0

2

4

6

8

10

y

dy/d

t



When considering switched systems, the system dynamics can be represented by a discontinuous right-hand side differential equation

( )( ) 0, =ttxσ

( )( )( ) ( )( )

( )( ) ( )( )

>

<=

0,,

0,,

2

1

ttttf

ttttft

xx

xxx

σ

σ&

Represents the boundary between two distinct regions of the state space, possibly time varying

The behaviour of the system on/across the guard σ, defining the two distinct regions of the state space, depends on how the dynamics f1 and f2 are related to the switching surface



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



0>σ

0<σ 0=σ( )( ) ( )( )

( )( ) ( )( )

>⋅∂

∂

<⋅∂

∂

0,,

0,,

1

2

ttftt

ttftt

xx

x

xx

x

σ

σattractive

switching surface

A Sliding Mode appears in the segment [a, b] of the switching surface if

a

b

( )( ) ( )( )

( )( ) ( )( )[ ] ( )( ) 0,:b a,

0,,

0,,

1

0

2

0

lim

lim=⊆∈∀

>⋅∂

∂

<⋅∂

∂

−

+

→

→ tt

ttftt

ttftt

xxx

xx

x

xx

x

σσ

σ

σ

σ

The Sliding Mode is stable in the segment [a, b] of the switching surface


Regularization of a sliding mode trajectory

Considering the switched dynamics

( )

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

>

=

<

=

0,,

0,,

0,,

2

1

ttttf

ttttf

ttttf

t s

xx

xx

xx

x

σ

σ

σ

&

Regularization of the sliding mode implies to find the continuous vector

function fs such that the state trajectory remains on the switching surface

Filippov’s continuation method

Equivalent dynamics method (“equivalent control”)



Filippov’s continuation method

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

[ ] ( )( ) ( )( ) ∀=∂

∂∈

−+=

0,,

:1,0

,,1, , 12

ttftt

ttfttfttf

Fs

Fs

xx

x

xxx

σα

αα

The dynamics on the sliding surface is defined by a convex combination of the vector fields defined in the two separated regions of the state space

0>σ

0<σ 0=σ

f1

f2

fsF

( )( ) ( )

( )( ) ( )

,1

,

21

2

21

1

ffgrad

fgrad

ffgrad

fgrad

−⋅

⋅=−

−⋅

⋅=

σ

σα

σ

σα



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

&



Filippov’s continuation method and the equivalent control method can give different solutions in nonlinear systemsFilippov’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

In real systems such motions appear in representing constrained systems, or in control systems in which a variable is forced to be zero

Presence of more than a single switching surface

In most cases the uniqueness problem does not arise


Sliding modes in control systems

Usually sliding mode control systems consider dynamics with affine control

( ) ( )

( )mmn

RRR

t

tBtf

∈∈∈

=

+=

σux

xσ

uxxx

0,

,,&

The control vector is such that

( )( ) ( )

( ) ( )mi

tiftu

tiftutu

ii

ii

i ,,2,1 0,,

0,,, K=

<

>=

−

+

xx

xxx

σ

σ

the sliding manifold is constituted by the intersection of m switching

surfaces, and uniqueness problems can arise


Stability of a sliding mode control system

Local stability of a sliding mode on the domain

S(t)=σσσσ: σi=0; i=1,2,…,m

can be stated by the following Lyapunov-like theorem

The subspace S(t) is a sliding domain if a function V(σσσσ,x, t) exists in a

domain Ω of the space σ1, σ2, …, σm containing the origin such that:

• V(σσσσ,x, t) is positive definite with respect to σ,σ,σ,σ, for all x and any t

• On the sphere ||σσσσ||=R, for all x and any t, the function V(σσσσ,x, t) remains bounded

• The total time derivative of function V(σσσσ,x, t)

is negative everywhere except on the discontinuity surfaces where it is not defined

• On the sphere ||σσσσ||=R, for all x and any t, the total time derivative of

V(σσσσ,x, t) is upper bounded by a negative constant

( )t

VBf

VVV

∂

∂++

∂

∂+

∂

∂

∂

∂= u

xx

σ

σ

&



The sufficient conditions for stability of the sliding mode are not easy to verify in general but some specific results can be given

( ) ( )

( )mmn

RRRt

tBtf∈∈∈

=

+=σux

xσ

uxxx

0,

,,&

( ) ( ) ( ) ( ) ( ) ( ) ( )( )uxxxσσσσσx

σσσ tBtftCBCVBCBV

TTT,,,

2

1

2

1 11

1

+⋅⋅=⇒⋅=

⋅

∂

∂=

−−

−

&

( ) ( ) ( ) ( )σuu sgnsgnsgn;sgn −==+⇒∀+>−= Bfikfbu iiii αασα

( ) ( ) ( )( ) ( ) ( )σσσσσσ VkBfCCBVTT β−=−≤−⋅=

−sgnsgn

1&

Convergence to the sliding mode in finite time



SISO systems( )( )

( )RuRR

t

ttun ∈∈∈

=

=σ

σx

xh

xfx

,

,,&

( ) ( )( )

nriu

ytuy

ir <−==

∂

∂= 1,,2,10;,, Kxϕr is the relative-degree

( )[ ]( ) [ ]

( ) ( )

( )( )tu

t

wwwtuT

rn

Tr

,,,

,,,,

,,,;,,,

,,,,,

21

1

wy

wyxwyw

wwy

y

Θ=′

Θ==

=′=

=

−

−

ϕϕ

ψ

ϕσ

σσσσ

&

K

K&&&The control can stabilise the output vector y, but the internal dynamics wcan be unstable for y =0

(zero-dynamics)



2212

21

uxxxx

xx

+−−=

=

&

&

( )

( ) ( )

σ

σ

σ

sgn20

10;20

1

21

122

12

−=

==

+−+−=

−=

u

xx

uxxx

xx

&

Example

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Phase plane plot

x1

x 2

x1(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

Time [s]

σ

The unstable zero-dynamics causes the loss of the sliding mode behaviour


Invariance of a sliding mode control system

( ) ( )

( ) ( ) ( ) ( )tbbFtf

utbtfx

nixx

m

n

ii

,0,,

,,

1,2,11

xxxx

xx

≤<≤

+=

−== +

&

K&

The system is invariant when constrained on the sliding manifold σ

( ) ( )

( )

( )σσσ

σ

σ

2

21

1

1

1

1

1

1

1

sgn

,,

kb

kxcF

u

xcutbtf

xcx

m

n

i

ii

n

i

ii

n

i

iin

−≤⇒

++

−=

++=

+=

∑

∑

∑

−

=+

−

=+

−

=

&

&

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



The system is invariant when constrained on the sliding manifold σ

∑

∑−

=

−

=−

+

+−=

+−=

−==

1

1

1

1

1

1 2,2,1

n

i

iin

n

i

iin

ii

xcx

xcx

nixx

σ

σ&

K&

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)

Take care of unstable zero-dynamics



( ) ( ) ( ) ( ) ( )cyy

tuyykkybyybbytm

+=

−=+++++

&

&&&&&

σ

πsinsgn 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

Time [s]

positionvelocity

σ



SystemVSCref. +

_

σ u y

SystemInternal

Model

ueq yd

+ +d

+ +d

The invariance property during the sliding mode means that the “Internal Model Principle” is fulfilled



( ) ( )[ ]UUtbtf +−+=∈ ,,, xxx 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∈+= equtbtf ,,* xxx&


Recovering the uncertain dynamics

uuu avav =+τ

If ueq is bounded with its time derivative then

∆≤=

→

→∆

σ

τ

τ eqav uu

0

0lim

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



The equivalent control is the control signal that assures 0≡σ&

( ) ( )

( )

( )0

,

,

,,

1

1

1

1

1

1

1

1

≡⇒

+

−=

++=

+=

∑

∑

∑

−

=+

−

=+

−

=

σ

σ

σ

&

&

tb

xctf

u

xcutbtf

xcx

n

i

ii

eq

n

i

ii

n

i

iin

x

x

xx The equivalent control contains the information on the uncertain dynamics

( ) ( ) ( )∞=

+=ω

ωωω jUjUjU eq

The equivalent control can be estimated by a low-pass filter

uuu avav =+τ



( ) ( ) ( ) ( ) ( )cyy

tuyykkybyybbytm

+=

−=+++++

&

&&&&&

σ

πsinsgn 2

31321

( ) ( ) ( ) ( ) ( )ytcmtyykkybyybbueq&&&& +++++++= πsinsgn

2

31321

0 0.5 1 1.5 2 2.5 3 3.5-30

-25

-20

-15

-10

-5

0

5

10

15

Time [s]


Approximability

∑

∑−

=

−

=−

+

+−=

+−=

−==

1

1

1

1

1

1 2,2,1

n

i

iin

n

i

iin

ii

xcx

xcx

nixx

σ

σ&

K&

The dynamics of the original system can be reduced to

In the case of switching errors – the switching frequency is fs≤∞ –the real trajectory x(t) is near to the ideal one x*(t) and

( ) ( ) 0* →−→∞sf

tt xx

( ) ( )[ ]( ) ( ) ( ) xxxx

xxx

NMtutbtf

UUtbtf

+≤+

+−+=

,,,

,,,&Assuming that the dynamics of the original system is bounded


Approximability

( )∆≤

⋅=

0x

xxCσStarting from a vicinity of the sliding surface

By the Bellman-Gronwall lemma

( )

( ) ( ) ( ) ( ) ∆≤−+∆≤−

++≤

∫

∫

HdtttLStt

dtNMTMt

T

T

0

**

0

0

xxxx

xxx

τ

τσσ

σ

D=∆

Σ+≤

Σ≤

0

&

If τ is the equivalent delay of the switching


Approximability

The most common cause of delay switching is the digital implementation of the controller

0 1 2 3 4 5 6-8

-6

-4

-2

0

2

4

6

8x 10

-3

Tempo [s]

τ=1e-4

( ) ( ) ( ) ( )cyy

tuyykkyybbytm

+=

−=++++

&

&&&

σ

πsin2

3121

0 1 2 3 4 5 6-8

-6

-4

-2

0

2

4

6

8x 10

-4

Time [s]

τ=1e-5


Discrete time implementation

Software implementation of the controller

Discrete-time sliding mode control

[ ] [ ] [ ]

[ ] [ ] [ ]( ) [ ]

[ ] [ ]( )kk

kkkkk

kkk

dd

dd

σu

uΓσxΦσ

σΒxAx

sgn

,,1

1

α−=

+=+

+=+

Discrete-time sliding mode control can appear also in systems with continuous right-hand-side (e.g., deadbeat control)

Discrete time sliding mode control is sensible only in the presence of uncertainties or disturbances



[ ] K,2,1,01 ==+ kk 0σ

What is a discrete time sliding mode?

[ ] [ ] K,2,1,01 ==−+ kkk 0σσ

The second is not convincing and does not imply the first

The usual implementation of the control law has two parts* the nominal part* the discontinuous part to cope with uncertainties



Usual accuracy of discrete time sliding modes is O (Tc)

Learning and adaptive methods O (Tc2)

Multirate sampling allows for output-feedback implementation of the controller

Effective approach is constituted by continuous time design and subsequent discretization analysis

Can chaotic behaviour appear ?!


Higher order sliding modes

The sliding mode concept can be extended to more complete integral manifolds

( ) 01 ===== −rσσσσ L&&&

r constraints are introduced in the system dynamics

The control acts on the r-th derivative of the sliding

variable σ

Accuracy is improved

( ) 1,,2,1,0, −== −riH

ir

i

iKτσ

Hi does not depend on τ



The internal dynamics has a reduced dimension n-r

The robustness properties are preserved

The approximability property is preserved

Allows for chattering attenuation

No general Lyapunov-like methods for the stability analysis are available (only very recent results for r=2 were presented)

Phase plane of homogeneity approaches to state the closed-loop stability

Derivatives of the sliding variable σ are needed



Higher Order Sliding Modes can be used to regularise nonlinear variable structure systems with sliding modes

( )( )

( )RuRR

t

ttun ∈∈∈

=

=σ

σx

xh

xfx

,

,,&

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

( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )vttuttuuu

ttuttu

ttu

t

ttu

ttuttut

t

th

,,,,,,

,,,,,,

,,,,,,

xxx

xfx

xx

xxfx

xhx

γϕφφφ

σ

φσ

+=∂

∂+

∂

∂+

∂

∂=

=∂

∂+

∂

∂=

&&&

&

If a discontinuous v constrains the system onto the 2nd order manifold

Filippov’s solution and the “Equivalent Control” coincide, and the control

u is continuous (Chattering attenuation)

0== σσ &



The control algorithms for Higher-Order Sliding Modes require

derivatives of σ

Relativedegree Feedback signals VSS controller

1 Classical 1-SMC, Super-Twisting( )ssign

2

( ) ( )ssignssign &, Twisting 2-SMC

( ) ssigns &, Sub-optimal 2-SMC

3

ss &,

Any

rUniversal HOSM )(,,...,, )1()2( −− rr ssignsss &

( ) ( ) ( )ssignssignssign &&& ,, Hybrid 3-VSC

Dynamic / Terminal 2-SMC



Dynamical sliding modes

( ) ( )∑

∑

−−−

−

+=

+=

2

1

11

1

1

ri

i

r

n

iin

ss

xcxs

γσ

Starting from a classical sliding variable s, a new switching variable σis defined

Sliding mode is achieved asymptotically

Terminal sliding modes

odd , ,21

1

1

qpq

p

ss

xcxs

q

p

n

iin

<<

+=

+= ∑−

&γσ

A nonlinear switching surface is defined

Sliding mode is achieved in finite time



Super-Twisting controller

( ) ( )( )Φ−Γ

Φ+Γ

Γ

Φ≥

Γ

Φ>

−=

+−=

λ

λα

λ

σλ

σα

m

M

m

m

z

zu

2

2 4sgn&

Φ: upper bound of the drift term

Γm, ΓM: positive lower and upper bounds of the gain term

-4 -3 -2 -1 0 1 2 3 4 5 6-6

-4

-2

0

2

4

6

σ

dσ

/dt

The control is continuous

( ) ( )u•+•= γϕσ&



Suboptimal-controller ( ) ( )

( )( )

( )( )

[ )( )

( )

+∞

+Γ

−Γ+Φ+∞∈

Γ

Φ>

∈

<−

≥−=

−−=

;1

12;1

1;0

0

01

sgn

β

βα

βσβσσα

σβσσα

βσσα

U

U

U

t

Utu

m

M

m

MM

MM

M

I

Φ: upper bound of the drift term

Γm, ΓM: positive lower and upper bounds of the gain term

The control is discontinuous

( ) ( )u•+•= γϕσ&&

-6 -4 -2 0 2 4 6 8-15

-10

-5

0

5

10

σ

dσ

/dt



Suboptimal-controller

β: anticipation factor

α: modulation factor

U: control gain

-6 -4 -2 0 2 4 6 8-15

-10

-5

0

5

10

σ

dσ

/dt

β=0.5

β=07

( ) ( )

( )( )

( )( )1;0

0

01

sgn

∈

<−

≥−=

−−=

βσβσσα

σβσσα

βσσα

MM

MM

M

t

Utu



Universal quasi-continuous HOSM-controller

i=1, 2,.., n-1

Homogeneous nested implementations of discontinuous controllers

ey≡σ


Final remarks

Sliding Modes are a usual behaviour in switching systems

Sliding Modes are a useful tool for controlling uncertain dynamical systems

Switching control with Sliding Modes is a simple way of applying the Internal Model Principle

By resorting to the Equivalent Control definition it is possible to retrieve some information about an uncertain system by low-pass filters

Ideal Sliding Modes are not implemented in practice because theyrequire infinite frequency switching, and only an approximate sliding can be achieved

Higher Order Sliding Modes generalise the concept of sliding modes to integral manifolds

Real Higher Order Sliding Modes improve the accuracy and attenuate the chattering effect

Sliding mode: Basic theory and new perspectivesdelbuono/usai_talk_1.pdf · • Regularization of...

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