Higher Order Sliding Mode Control

M. Khalid KhanControl & Instrumentation group

Department of Engineering

References

1. Levant, A.: ‘Sliding order and sliding accuracy in sliding mode control’, Int. J. Control, 1993,58(6) pp.1247-1263.2. Bartolini et al.: ‘Output tracking control of uncertain nonlinear second order systems’, Automatica, 1997, 33(12) pp.2203-2212.3. H. Sira-ranirez, ‘On the sliding mode control of nonlinear systems’, Syst.Contr.Lett.1992(19) pp.303-3124. M.K. Khan et al.: ‘Robust speed control of an automotive engine using second order sliding modes’, In proc. of ECC’2001.

Review: Sliding Mode Control

Design consists of two steps

Selection of sliding surface

Making sliding surface attractive

Consider a NL system uxtgxtfx ),(),(

0),( xtss

Robustness Chattering

High frequency

switching of control

Pros and cons

Order reduction Full state availability

Robust to matched uncertainties

Simple to implement

Chattering at actuator

Sliding error = O(τ)

Isn’t it restrictive?

Sliding variable must have relative degree one w.r.t.

control.

Higher Order Sliding Modes

rth-order sliding mode:- motion in rth-order sliding set. Sliding variable (s) has relative degree r

rth-order sliding set: -

0)2()1( sssss rr

Consider a NL system ),,( uxtfx

Sliding surface 0),( xtss

ButWhat about reachability condition?

So traditional sliding mode control is now

1st order sliding mode control!

There is no generalised higher order reachability condition available

1-sliding vs 2-sliding

s

ds

2-sliding

τ

τ2s

ds

1-sliding

τ

Sliding error = O(τ) Sliding error = O(τ2)

Sliding variable dynamics

Selected sliding variable, s, will have

relative degree, p= 1 relative degree, p 2

1-sliding design is possible.

2-sliding design is done to avoid chattering.

r-sliding (r p) is the suitable choice.

2-sliding algorithms: examples

Consider system represented in sliding variable as,,|| ;),,(),,( Mmusstssts

Finite time converging 2-sliding twisting algorithm

0Ss

Sliding set: 0ss

0)(

0)()(

ssssignV

ssssignVtu

M

M

< 1

PendulumThe model:

uyy )sin(25.0

Sliding variable: yys Sliding variable dynamics:

uyys )sin(25.0

uuyyys )sin(25.0)cos(25.0

Twisting Controller coefficients: α = 0.1, VM = 7

Simulation

Examples continue … Consider a system of the type

0 , ,|| ;),(),( Ssuststs Mm

Finite time 2-sliding super-twisting algorithm

0

01

1

||)(sign

||

)(sign||)(

uusW

uukuu

usstu

0ssSliding set:

Review: 2-sliding algorithms

Twisting algorithm forces sliding variable (s) of relative degree 2 in to the 2-sliding set but uses

s Super Twisting algorithm do not uses but sliding variable (s) has relative degree only one.

s

Is it possible to stabilise sliding surface with relative degree 2 in to 2-sliding set using only s, not its derivative?

Answer: yes!

1. by designing observer

2. using modified super-twisting algorithm.

Question:

Modified super-twisting algorithm

0

01

1

||)(sign

||

)(sign)(

uusW

uukuu

ustu

0,,|| ;),,(),,( Ssusstssts Mm

System type:

Where λ, u0 , k and W are positive design constants

1. Sinusoidal oscillations for = u0

2. Unstable for < u0

3. Stable for > u0

Phase plot

Sufficient conditionsfor stability

0 ,0

/0

Wk

u m

Application: Anti-lock Brake System (ABS)

ABS model:

ukx

x

xJJ

NRx

RM

NNx

M

Rx

b

ww

vw

wv

wv

v

w

33

32

21

1)(

)(595.01

2

31514

43

212

11

11

25.0

1

xkxk

xkxk

xkkb

22)(

p

pp

),max( 21

12

xx

xx

ugf )()( Can be written as:

12.0desired

Simulation ResultsController coefficients: 15 W,35 ,75 0 ku

Results continued …

Conclusions The restriction over choice of sliding variable can be relaxed by HOSM. HOSM can be used to avoid chattering

A new 2-sliding algorithm which uses only sliding variable s (not its derivative) has been presented together with sufficient conditions for stability.

The algorithm has been applied to ABS system and simulation results presented

Future Work

The algo can be extended for MIMO systems.

Possibility of selecting control dependent sliding surfaces is to be investigated.

Stability results are local, need to find global results.