sliding mode control: basic theory, advances and applications · discrete-time sliding mode control...

Dobbiaco Summer School - Dobbiaco (BZ) 22-26 June 2009 1

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]


• Approximate Sliding Modes• Discrete Time implementation• Effect of Parasitic Dynamics• Effect of Measurement Noise• Chattering Attenuation

➢ Control magnitude adaptation➢ Parameter tuning in 2-SMC➢ System dynamics shaping

Lecture 3

Implementation Issues of

Sliding Mode Control Systems


L3 – Approximate SM

Ideal Sliding Modes in Control Systems can be established if infinite frequency switching in the closed loop dynamics appears

Real devices has low-pass characteristics and therefore cannot perform infinite frequency switching

Switching delay appears

The system state is no more constrained on the sliding surface

WHAT is the EFFECT of Switching Delays?



Assume that the sliding surface is an attractive set of the closed-loop dynamics

At certain time instant t0 the system state is within an vicinity of the sliding

surface and the system dynamics is represented by the input-output and internal dynamics

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

∈∈∈==

= − qqnqσtttt

tttttR,R,R

,,,,,

uwxywyw

uwyyψϕ

( ) ( )( ) ( ) ( ) ( )( )tttttttf ,,,,, uwyuxxσ ϕ=⋅

∂∂

( ) qnqn −∈∈∈Φ=

R,R,R wyxxwy



General treatment of the analysis of the system behaviour nearby the sliding surface is quite complex and could need a Poicaré analysis

qnqqnq −− ∈∈∈∈⋅=

⋅

=

R,R,RRˆ21

2

1

22

1211 wyxxxCxx

C0CC

wy

Complete results can be quite easily obtained in the linear case for the classic first order sliding mode control systems

( ) ( ) ( ) ( ) ( )( ) ( )tt

tttttxCy

uBxAx⋅=

⋅+⋅=nqqqn <∈∈∈ ,R,R,R yux

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

⋅+⋅=⋅⋅+⋅+⋅=

tttttttttttt

wΨyΨwuBCwΦyΦy

21

21



Assuming that the control is designed taking into account for the nominal dynamics

( ) ( )[ ] ( ) ( ) ( ) ( ) ( )( )( )tUtttttt ywΦyΦBCu sgn211 +⋅+⋅⋅−= −

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

⋅+⋅=−⋅+⋅=

ttttttUttttt

wΨyΨwywΦyΦy

21

21 sgn~~

If the system were in ideal sliding mode the system dynamics will be characterized by its zero dynamics

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( )∫

+

−→ ⋅ →

⋅=−⋅= ε

εεττ

ε

t

t

ttdUttt

tUtttwΦy

wΨwywΦy

202

2 ~sgn2

sgn~



The dynamics of the error between the ideal and real sliding behaviour

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

⋅+⋅=−⋅+⋅=

ttttttUttttt

wΨyΨwywΦyΦy

~~~sgn~~~~

21

21

( ) ( )( ) ( ) ( )ttt

ttwww

yy−=

=~~

Assume that T is the switching delay and that the sliding dynamics can be upper bounded by a constant D

( ) DTt +≤ 0~~ yy

Assume also that the matrices in the error dynamics can be upper bound by proper constants during the switching delay

( ) ( ) PtQt << 21 ΨΨ



( ) ( )

( )

+≤

+++≤ ∫+

DTt

dPQDTTQtTt

t

0

221

00

~~

~~~0

0

yy

wyww ττ

Taking into account the BIBS assumption for the internal dynamics and that T is the switching delay

( ) [ ]TtttTt +∈∀=∆∆≤ 00,~ υx

The system trajectory remains confined within a O(T) vicinity of the ideal sliding trajectory



TheoremConsider system

Assume that conditions for conver-gence and stability of a r-SM arefulfilled by a homogeneous r–ordersliding mode controller

Then, if the switching device has a switching delay T, the real r-SM has the following finite time accuracy

( ) ( ) RR: →+= nu σxgxfx

Γ≤≤Γ<−==

Φ≤

−M

rm

k

r

σLLrkσLL

σL

102,,1,0 ,0

fg

fg

f

( )( )( )1,1 ,,,sgn −

−−= rrru σσσα Φ

( ) 1,,1,0 −=≤ − rkT krk

k υσ



ProofUnder the considered assumption the system trajectories are infinitely extendible in time for any Lebesgue-measurable bounded feedback control and at the ideal switching time the sliding variable and its time derivatives are bounded

( )

[ ] [ ]( ) krpTK

dtd pk

Ttttk

Ttttp

kp

−−=∀≤ −

+∈+∈

1,,1,0sup,,*

σσ

Applying the Lagrange theorem

( ) rkkk ,,1,0 =Σ≤σ

Integrating σ (k) k times and taking into account above inequalities

( ) ( ) ∞−

+ ∈−=Σ≤ Kkkr

kkk rkT υυσ 1,,1,01



Real actuation devices cannot implement infinite switching and therefore the system trajectory cannot be constrained on the sliding surface

The real sliding is a motion confined into a vicinity of the sliding surface

The thickness of the real sliding vicinity depends on the the switching delay T and on the control magnitude (Σ

r)

The real sliding accuracy can be improved by means of HOSM, if the switching delay is T<1

The accuracy can be also marginally improved by avoiding unnecessary large magnitude controls



Example ( ) ( ) ( ) ( )cyy

tuyykkyybbytm+=

−=++++

σπsin2

3121

0 1 2 3 4 5 6- 8

- 6

- 4

- 2

0

2

4

6

8x 1 0

- 4

T i m e [ s ]

τ= 1 e - 5

0 1 2 3 4 5 6- 8

- 6

- 4

- 2

0

2

4

6

8x 1 0

- 3

T e m p o [ s ]

τ = 1 e - 4


L3 – Discrete Time Implementation

The most common cause of delay switching is the digital 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



[ ] ,2,1,01 ==+ kk 0σ

What is a discrete time sliding mode?

[ ] [ ] ,2,1,01 ==−+ kkk 0σσ

The second is not convincing and does not imply the first

?

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

The system behaviour within a sampling period is almost unpredictable, apart from the maximum deviation from the sliding surface

In some conditions chaotic behavior within the boundary layer has been recognized



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

This allows for implementing learning and adaptive methods that can improve the accuracy by one order, i.e., O (T) → O (T2)

( ) ( ) ( ) ( ) ( ) ( ) ( )xxxxxxx Mm

n

ii btbbFtfutbtfx

nixx≤≤<≤

+=−== + ,0,,

,,1,2,11

( ) ( )

( )( )( ) ( )

( )( ) ( )( ) ( ]TttttTtb

TtxcTtFtu

xcutbtf

xcx

kkkkm

n

ikiik

n

iii

n

iiin

+∈+

++++−=

++=

+=

∑

∑

∑

−

=+

−

=+

−

=

,sgn

,,

21

11

1

11

1

1

σκ

ηκκ

σ

σ

x

cx

xx



The switching delay due to sampling causes an approximate sliding motion in a O (T) boundary layer of the ideal sliding

σ

t

u

tk-1

t

tk

tk+1



( ) ( )τ

κκτκ TTtutu eqav 321 ++≤−

In the ideal case the equivalent control can be estimated by a low-pass filter

Since the sliding variable in constrained in a O (T) boundary layer of the ideal sliding and the equivalent control remains bounded

( ) ( ) ( )tututu avav =+τ

The estimation error can be minimized and the actual value of the average control computed exactly at each sampling time

( ) ( ) TtutuT eqav 43

1 κκκτ ≤−⇒=

[ ] [ ] [ ]kuekuekuT

av

T

av

−+=+ −− ττ 11



The control input can be implemented as a combination of two components

Adapting step by step the time constant of the filter and the magnitude of the discontinuous control

( ) ,2,122

215

21

3

1

=>∀≤⇒

=

= −

−

−

−

−

−

jTKtTO

TU

Tj

j

j j

j

j

j

j

j

σ

κ

κκ

τ

[ ] [ ]( )( ) ( )

( )( ) ( )( ) ( ]TttttTtb

TtxcTtFTkuku kkk

km

n

ikiik

av +∈+

++++−=

∑−

=+

,sgn

21

11

5 σκ

ηκκκ

x

cx

[ ]

→ →2

3

1TOk Kkσ



Example of discrete time sliding mode control with recursive estimation and adaptation of the control


L3 – Effect of the parasitic dynamics

( ) ( )( )

( )( )σ

σσµ

switchus

uz

n

m

n

=→=

→×=→+=

RR:RRR:,

RR:,m

n1

xhzhz

gfxgxfx

If the switching control is applied to the plant by means of a dynamic actuator the relative degree between the sliding variable and the switching control increases and the ideal sliding cannot be achieved

( ) ( ) ( )( )( )

( )( )σ

σσµ

σσσ

switchussss

usss

LLLzsssLLsssLs

rnrr

mm

rnrnrr

rnrrrrrrrr

=→×=

→×=→×=

→×⋅+=

−−

−−−

−−−−−

RRR:,,,,RRR:,

RRR:,,,,RRR:,,,,,,,,,

1

1

11

111

whzhzψwψw

ww fgffgf



If the parameter µ is sufficiently small the actuator dynamics is a singular perturbation of the nominal dynamics

a) Poincaré analysis of the fast dynamics, freezing the slow dynamics

b) Phase trajectory analysis considering differential inclusions with switching delays

c) Homogeneity of the differential inclusion

( ) [ ] [ ] 1,, zs Mmrrr ΓΓ+ΛΛ−∈

Method a) is very much involved and hard to implement for nonlinear uncertain systems;

Method b) is relative simple only for relative degree 2 sliding dynamics

Method c) is general but require the homogeneity of the controller



All methods confirmed that the accuracy of the sliding mode depends on the singular parameter µ only, no matter the relative degree m of the parasitic dynamics is

( ) ( ) 1,,1,0 −== − rkkrk µσ O

In general information about the system behavior within the boundary layer are not available apart for linear system

Approximate method Exact methods

Describing Function Tzipkin locus

LPRS



f’(u) G (jω)+∆r’(t)=0 ∆u(t) ∆m(t) ∆w(t)

_

This methods refer to linear systems with nonlinear static feedback

Give only necessary conditions for the stability of limit cycles because they consider the steady state behavior only, and are based on the harmonic balance of the feedback loop

∑∞

=

−=1k

tjkjk

jk

tj eeMeGUe kk ωϑϕω

If the linear system has low-pass characteristics the Describing Function method can be applied

( ) ( ) 0,1 =+ ωω UNjG ( ) ( )111, jabU

UN +=ω



Example

( ) ( ) ( ) etaa

t

kkBJjRLjjkjG

+++=

ωωωω

N y q u i s t a n d D F p l o t s

R e a l A x i s

Imag

inar

y A

xis

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1- 2 0

- 1 8

- 1 6

- 1 4

- 1 2

- 1 0

- 8

- 6

- 4

- 2

0

ω

U

( )( ) ( )( ) rad 0.17594M-Urad/s 056.360

=ℜ⋅==+===ℑ crp

a

etacrjW

jWJL

kkBRp

ωπ

ωωω

R=0.4; % rotor resistance

L=0.001; % rotor inductance

ke=0.3; % voltage feedback constant

kt=0.3; % torque constant

Jm=0.01; % motor inertia

Jl=0.09 % load inertia

Bm=0.05; % motor friction coefficient

Bl=0.05; %load friction coefficient

J=Jm+Jl;

B=Bm+Bl;



Example

( ) ( ) ( ) etaa

t

kkBJjRLjjkjG

+++=

ωωωω

( )( ) ( )( ) rad 0.17594M-Urad/s 056.360

=ℜ⋅==+===ℑ crp

a

etacrjW

jWJL

kkBRp

ωπ

ωωω

R=0.4; % rotor resistance

L=0.001; % rotor inductance

ke=0.3; % voltage feedback constant

kt=0.3; % torque constant

Jm=0.01; % motor inertia

Jl=0.09 % load inertia

Bm=0.05; % motor friction coefficient

Bl=0.05; %load friction coefficient

J=Jm+Jl;

B=Bm+Bl;

N y q u i s t a n d D F p l o t s

R e a l A x i sIm

agin

ary

Axi

s- 2 0 - 1 5 - 1 0 - 5 0

x 1 0- 3

- 5

- 4

- 3

- 2

- 1

0

1

2

3

4

5

x 1 0- 4

S y s t e m : W p R e a l : - 0 . 0 0 5 6 8

I m a g : - 8 . 7 3 e - 0 0 7 F r e q ( r a d / s e c ) : 3 6 . 8

ω

U



0 1 2 3 4 5 6 7 8 9 1 0-8 0

-6 0

-4 0

-2 0

0

2 0

4 0

6 0

8 0

T im e

a n g les p e e dc u r r e n t

The system presents a periodic steady-state oscillation


L3 – Effect of the Measurement noise

A measurement noise super-imposed on the ideal sliding variable

( ) ( ) ( ) ( ) δσσ ≤+= tntntt ,ˆ

1-SMC ( ) ( )δσ O=t

2-SMC( ) ( )( ) ( )δσ

δσOO

==

tt

Possibly not convergent

r-SMC Possibly not convergent( )

,2,1,0

,

=

=−

i

O riri δσ



Robust sliding mode differentiators can make the HOSM convergent even in the presence of noise

For the generalized sub-optimal a peculiar adaptation of the anticipation parameter β allows for implementing a noise robust 2-SMC

( ) ( )( ) ( )

( )[ )

( ) ( ) 0ˆ ˆlast theis ˆ1;0

0ˆˆˆ 10ˆˆˆ 1

,ˆˆsgn

*

=∋∈

<−>≥−

=

−−=

exexex

exex

exex

ex

tt

ifsif

t

sUtuσσσ

βσβσσασβσ

α

βσα

( )( )

+

−+∈

>

1,*2UGG

UGGFGFU

mM

mM

m

αβ



Is it possible to estimate the sequence of the extremal values

by inspection of the measured values of σ in a proper time window if the measurements are noisy?

- 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

S l i d i n g v a r i a b l e σ

Slid

ing

varia

ble

deriv

ativ

e d

σ /dt

0 0 . 5 1 1 . 5 2 2 . 5 3-0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

1 . 2

T im e

Mea

sure

d sl

idin

g va

riabl

e

?

!

e x t r e m a l v a lu e s



Since the main problem is the detection of the flex points of the sliding variable σ(t), the control switchings needed as local minima are reached could be postponed by a fixed ratio of the distance between two subsequent extremal values, i.e., a local maximum and a local minimum

1,ˆˆˆˆ >−≤− N

NmM

mσσσσ

This choice guarantees the reduction of the estimation error of the flex points up to δ as the approximate sliding mode is reached



N affects the magnitude of the loop, and therefore the ultimate accuracy



NmM

mσσσσˆˆˆˆ −≥−

Mσβσ ˆˆ ≤

σσ ˆ:ˆ =m

σσ ˆ:ˆ =MN

mMM

σσσσˆˆˆˆ −≥−

mσβσ ˆˆ ≥

σσ ˆ:ˆ =M

σσ ˆ:ˆ =m

0ˆ <σ

0ˆ >σ

( )MM

Uuσσσ ˆ,ˆmaxˆ =

−=1( )MM

Uuσσσ ˆ,ˆmaxˆ =

−= 4

( )mm

Uuσσσ ˆ,ˆminˆ =

+=2

( )mm

Uuσσσ ˆ,ˆminˆ =

+= 3

Implementation of the switching logic require an automaton and modified stability conditions

n

son

Nβ

ββ

−=

+=

11

,2

1



With exact measurements and ideal switching the 2-Sliding set is reached in a finite time T

∞

( ) ( )( ) ( ) 2

102

210

,,

,,

τδσ

τδσ

FkFkk

FkFk

FU

FU

Tt

FU

FU

Tt

MM

MM

′+′′→

′+′→

∞

∞

→

→

With noisy measurements and switching delays only a boundary layer of the 2-Sliding set can be reached in a finite time T

0,0∞∞ →→

→→TtTt

σσ


L3 – Effect of the Measurement noiseExample

( )( ) dd yyty

nnyyutyyy−==

≤+=+++=σ,5sin2

2.0,ˆ,3sin22

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4- 4

- 3

- 2

- 1

0

1

2

3

4S u b -O p t im a l 2 - S M c o n t ro l l e r , τ = 0 . 0 1

T im e [ s e c ]

Tra

ckin

g er

ror

e x t r e m a l v a lu e s


L3 – Chattering Attenuation

Chattering cannot be eliminated but only attenuated!

Many definitions can be found:

• Discontinuous control

• Not precise attainment of the sliding

• Oscillation of the system state due to unmodelled dynamics

Continuous control can produce large and unpredictable state oscillationsNot precise attainment of the sliding can be due to design errors of the sliding surfaceUnmodelled dynamics is always present

What is Chattering ?



νk is an increasing function of Σ

k+1 , and in particular, by a chain rule, of

Σr , that depends on the discontinuous control magnitude

Chattering depends on the ultimate accuracy of the sliding motion

( ) ( ) ∞−

+ ∈−=Σ≤ Kkkr

kkk rkT υυσ 1,,1,01

The shape of νk depends on the overall closed loop dynamics

T can be either:• the switching delay of a relay device• the sampling time in a discrete time implementation• the equivalent time constant of a dynamic actuator


L3 – Chattering AttenuationChattering can be attenuated by means of:

Smooth approximations of the discontinuous functionIt is effective only if the matching uncertainty vanishes on the sliding surface

Implementation of HOSMIt will require the knowledge of a number of time derivatives of the sliding variable, apart from the Super-Twisting and Sub-Optimal algorithms that require Single-Input systems

Using much higher sampling frequencyIt can be sensible to high frequency noise (Aliasing phenomenon)


L3 – Chattering AttenuationChattering can be attenuated by means of:

Adaptation of the switching control magnitudeIt will require more complicated control schemes

Tuning the parameter of a 2-Order Sliding Mode ControllerIt is quite easy but some drawback on the reaching phase will follow, and it is quite simple for Single-Input linear systems only

Shaping the dynamics of the systemIt requires an additional filter it is quite simple for Single-Input linear systems only


Adaptation of the switching control magnitude

Most of the adaptation algorithms resort to the estimation of the equivalent control, and define the system input as a proper combination of the averaged control and a switching control

SystemVSCref. + _

σ u y+ +

FF Equivalent control

LPF2

LPF1

x++

uav

usw

uc

k

L3 – Control Magnitude Adaptation




If the parameter are properly chosen the system input asymptotically approaches the ideal equivalent control and the accuracy is improved

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

( )( )( ) ( )

( ) ( ) ( )( ) ( ) ασκτ

τσ

γϕ

σ

=+=+

−=

−=

++=

tktktututu

Utut

tttu

tutktututu

ad

swavavav

sw

c

swavc

sgn

,,

xx




( ) ( ) ( )( )

( ) ( )1122

22112

21

2122112

21

12

1sinsin12

xzcxzzzzz

zzutaDtDxxxx

xx

−+−=

−+−==

++++−+−==

σµ

ωµ



Another approach for adapting the switching control magnitude is not based on the on the estimation of the continuous equivalent control but on avoiding too large control authority

In real-sliding mode the derivatives of the sliding variable must be zero in media and the discontinuous control switches at very high, but finite, frequency

A real r-order sliding mode (r-sliding) in which the rth derivative of the sliding variable σ is always separated from zero is characterised by

( )

( )21

111

10

0,,,

RRkkk

rr

rrr

≤≤<≤≤≤ −

−−

στστστσ

τ is the switching delay

∈

− ττ1,

2 1

1

rsw k

Rf



Lemma: Let σ be the sliding variable of a variable structure control system, and let the discontinuous control be always separated from zero. Assume that σ(i) (i=0,1,...,r-1) are continuous functions. Consider a time interval T* of length Nτ, N∈R+, and let Nsw be the number of zero crossings of σ during the time T*. If Nsw ≥ r then, over the time interval T*, the system trajectories are confined within the domain

( ) ( )

−=≤∈= − 1,...,1 ,0 ,

! :D i2 riN

iR iir τσ xRx n

The time interval between two subsequent zero-crossing of the sliding variable in the steady state varies over [Tm , TM] that depend on the specific VSC algorithm, its parameters and τ

A proper choice for N in order to detect the real sliding could be related to the maximum "cycle-time" TM

N τ = r TM



Adaptation of the switching control magnitudeFirst order sliding modes

( ) ( )utt ,, 11 xx γϕσ +=( )

( ) 211

1

,0,

Γ≤≤Γ<Φ≤

tt

xx

γϕ

u(t)=-UM sign (σ(ti)) ti ≤ t < ti+1 ( ti+1 - ti = τ) i=0,1,2, ….

σ τ

A

B

C

D E t

R2τ

TM

Tm

τ

τ

+=

+=

1

2

2

1

1

1

RRT

RRT

M

m

σ τ

A

B

C

D E t

R2τ

TM

Tm

τ

τ

+=

+=

1

2

2

1

1

1

RRT

RRT

M

m

Φ=

−Γ+Γ=

MURR

η

ηη

11

1

2

2

1





η represents the “dominance factor”.

The larger the η the larger the boundary layer size and the highest the switching frequency

As far as the actual value of η guarantees the stability of the sliding mode, the switching frequency is higher than a certain minimum value, as well as the number of changes of sign of σ with a certain time interval

UM can be adapted on-line to maintain a prescribed, desired, “dominance factor” η*

+<Λ++≥Λ−

=+ 1 ,,1 ),0,(max

sw

sw1 rNNU

rNNUU

Mi

MiMi τ

τ




ut ++= )5sin(23σ

1 2 3 4 5-0,005

0,000

0,005

1-SMC without adaptation. The Sliding Variable.

Time [sec]

1 2 3 4 5

-0,004

-0,002

0,000

0,002

0,004

1-SMC with adaptationThe sliding variable

Time [sec]

UM=20 20=ΛN=10

τ =10-4


L3 – Parameters Tuning in 2-SMC

A feedback Second-Order Sliding Mode System Scheme is simply represented

PLANT

2-VSC

σ - Sliding variable

( ) ( ) ( )( ) ( )

RRR

∈∈∈

=+=

σσ

utt

tuttnx

CxBAxx

The controller is the Generalized Sub-Optimal

( ) ( )( ) ( )

( )[ )1;0

0ˆˆˆ 10ˆˆˆ 1

,ˆˆsgn

*

∈

<−>≥−

=

−−=

βσβσσασβσσ

α

σβσα

exex

exex

ex

ifif

t

Utu

If the plant is stable linear system with low-pass properties



The steady state analysis of the system can be approximately carried out by means of the Describing Function approach

( ) ( ) ( ) ( )[ ]{ }1*1*11*2 2 ++−+−+= αβαβαπ

jyUyN p

M

MpM

( ) ( ) BAIC 1−−= ωω jjG

( ) ( ) ( ) ( )[ ]( ) ββα

αβαβαπω−++

++−+−+−=11*

1*1*11*4 2

2 jUyjG

M

pM

The Describing Function of the Generalized Sub-Optimal controller

The harmonic response of the system

The Harmonic Balance equation



The steady state analysis of the system can be approximately carried out by means of the Describing Function approach

Re

Im

( )pMyN

1−

( )ωjG

A

O

ψ

( ) ( )( ) 211*

1*1*arctanβα

αβαψ−+

++−=

( ) ββαπ

−++=

11*22 2pM

M

yUAO

Parameters α* and β can be tuned in

order to minimize ypM


L3 – System Dynamics Shaping

The Describing Function approach can be used to properly shaping the Nyquist plot of the linear system by introducing a proper linear filter

With reference to the Generalized Sub-Optimal controller only parameter β can

be considered since optimization is not much sensitive with respect to α*

PLANT

FILTER

σ - Sliding variable

2-VSC



The introduction of a low-pass filter “compensates for” the actuator dynamics in the range of frequency of the steady state oscillations

Nominal plant

Nominal plant plus parasitic dynamics

B o d e D i a g r a m

F r e q u e n c y ( r a d / s e c )

Pha

se (d

eg)

Mag

nitu

de (d

B)

1 00

1 01

1 02

1 03

1 04

- 3 6 0

- 2 7 0

- 1 8 0

- 9 0

0- 2 0 0

- 1 5 0

- 1 0 0

- 5 0

0

5 0



The introduction of a low-pass filter “compensates for” the actuator dynamics in the range of frequency of the steady state oscillations

Nominal plant

Nominal plant plus parasitic dynamics

Overall Harmonic Response with the shaping low-pass filter τ=0.1s

B o d e D i a g r a m

F r e q u e n c y ( r a d / s e c )

Pha

se (d

eg)

Mag

nitu

de (d

B)

1 00

1 01

1 02

1 03

1 04

- 4 5 0

- 3 6 0

- 2 7 0

- 1 8 0

- 9 0

0- 3 0 0

- 2 0 0

- 1 0 0

0

1 0 0


L3 – References

• V.I. Utkin, Sliding Modes In Control And Optimization, Springer Verlag, Berlin, 1992.

• K.D. Young, V.I. Utkin, U. Ozguner“, A control engineers guide to sliding mode control”, IEEE Trans. Control Sys. Technology, 7, pp. 328–342, 1999.

• A. Levant, “Sliding order and sliding accuracy in sliding mode control”, Int. J. Control, 58, pp. 1247-1263, 1993.

• A. Levant, “Homogeneity approach to high–order sliding mode design”, Automatica, 41, pp. 823–830, 2005.

• C. Milosavljevic, “General conditions for the existence of a quasisliding mode on the switching hyperplane in discrete variable structure systems”, Automation Remote Control, 46, pp. 307–314, 1985.

• Bartolini G., A. Pisano, E. Usai, “An improved Second-Order Sliding Mode Control Scheme Robust Against the Measurement Noise”, IEEE Trans. Automatic Control, 49, pp. 1731-1736, 2004

• K. Furuta, “Sliding mode control of a discrete system”, System and Control Letters, 14, pp. 145-152, 1990


L3 – References

• W.-C.Wu, S. V. Drakunov, and U. Ozguner“, An O(T 2) boundary layer in sliding mode for sampled-data systems”, IEEE Trans. Automatic Control, 45, pp. 482-484, 2000.

• G. Bartolini, A. Pisano, E. Usai, “Digital second-order sliding mode control for uncertain nonlinear systems”, Automatica, 37, pp. 1371–1377, 2001.

• H. Lee, V.I. Utkin, “Chattering suppression methods in sliding mode control systems”, Annuals Reviews in Control, 31, pp. 178-188, 2007.

• L. Fridman, “Chattering analysis in sliding mode systems with inertial sensors”, Int. J. Control, 76, pp. 906-912, 2003.

• I. Boiko,L. Fridman, A. Pisano, E. Usai, “Analysis of Chattering in Systems with Second-Order Sliding Modes”, IEEE Trans. Automatic Control, 52, pp. 2085–2102, 2007.

• J.-X. Xu, Y.-J. Pan, T.-H. Lee, “Sliding Mode Control With Closed-Loop Filtering Architecture for a Class of Nonlinear Systems”, IEEE Trans on Circuit and Systems—II: Express Brief, 51, 2004.


L3 – References

• Levant A., Bartolini G., Pisano A., Usai E., "A Real-Sliding Criterion for Control Adaptation", Advances in Variable Structure Systems – Theory and Applications. Proceedings of the 7th IEEE Int. Workshop on Variable Structure Sysstems (VSS 2002), Sarajevo, YUG, July 2002, A. Sabanovic ed., pp. 205-213, ETF Sarajevo, 2002

• Y.B. Shtessel, Y.-J. Lee,“New approach to chattering analysis in systems with sliding modes”, Proceedings of the 35th IEEE CDC, Kobe, Japan, 4014–4019.

• I. Boiko, Discontinuous Control Systems: Frequency-domain Analysis And Design, Birkhauser, Boston, 2008

• I. Boiko, L. Fridman, R. Iriarte, A. Pisano, E. Usai, “Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators”, Automatica, 42, pp. 833–839, 2006.

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