Levant 1993

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Sliding order and sliding accuracy in sliding modecontrolARIE LEVANT

a

a Mechanical Engineering Department, Ben-Gurion University of the Negev, Beer Sheva,

Israel.

Version of record first published: 15 Mar 2007.

To cite this article: ARIE LEVANT (1993): Sliding order and sliding accuracy in sliding mode control, International Journal of

Control, 58:6, 1247-1263

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INT J CONTROL 1993, VOL 58, No 6 1247-1263

Sliding

order and

sliding accuracy in sliding mode control

ARIE

LEVANTH

The synthesis of a control algorithm that stirs a nonlinear system to a given

manifold and keeps it within this constraint is considered. Usually, what is

called sliding mode is employed in such synthesis. This sliding mode is

characterized, in practice, by a high-frequency switching of the control. It turns

out that the deviation of the system from its prescribed constraints sliding

accuracy) is proportional to the switching time delay. A new class of sliding

modes and algorithms is presented and the concept of sliding mode order is

introduced. These algorithms feature a bounded control continuously depend

ing on time, with discontinuities only in the control derivative. It is also shown

that the sliding accuracy is proportional to the square of the switching time

delay.

1.

Introduction

Consider a smooth dynamic system described by

i

=

f t,

x,

u

1

where x is a state variable t hat t akes on values in a smooth manifold

X,

t - t ime, U

E

Rm-control. The

design objective is the synthesis of a con trol

U

such t ha t the constraint

a t, x =

0 holds.

Here a: x X

and both

f

and

a are smooth enough mappings . Some choices of the const ra ints are discussed

by Emelyanov et al. 1970),

Utkin

1977, 1981), Dor ling and Zinober 1986),

DeCarlo et al. 1988 .

This design approach enjoys the advantage of the reduct ion of the system

order

In add ition , the resulting system is independent of cer ta in var ia tions of

the right-hand side of 1).

The latt er

fact makes this

approach

effective in

control

under

conditions of uncertainty.

The quali ty of the contro l design is closely rel at ed to the sliding accuracy.

Exist ing approaches to this design

problem

usually do

not

maintain, in practice,

the prescribed constraint exactly. Therefore there is a need to int roduce some

new means in

order

to p rov ide a capability for the compari son of the different

approaches.

2. Real sliding

We call every motion that takes place strictly on the constra int manifold

a = 0 an ideal sliding. We also informally call every motion in a small

neighbourhood

of

the

manifold a

real sliding

Utkin 1977, 1981).

The

common

sliding mode exists due to an infinite frequency of the control switching.

However because of switching imperfections this frequency is finite. The sliding

Received

8

August

1991.

t Mechanical Engineering Department, Ben-Gurion

University

of the Negev Beer

Sheva

Israel.

Formerly,

L. V. Levantovsky.

0020-7179/93 10.00

©

1993 Taylor

Francis Lid

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1248 A.

Levant

mode notion should be understood as a limit of motions when switching

imperfections vanish and the switching frequency tends to infinity Fillippov

1960, 1985, Aizerman and Pyatnitskii 1974 a, b).

Definition 1: Let

t,

x t ,

s) be a family of trajectories indexed by

e

E

R/

with

common initial condition

to,

x to» and let i »

to

or

t

E

[to,

T] . Assume that

there exists

tl;; to or tl E [to, T]

such that on every segment

[t , t

U

where

r» or on

[tl

TJ) the f unction

a t, x t,

e» te nds uniformly to z ero with

e

tending to zero. In this case we call such a family

a real s lid in g family

on the

constraint

a =O

We call the motion on the interval

[to, tt l a transient process,

and the motion on the interval [r

00 or

[t

I, TJ

a steady state process.

We will use the following terminology: a rule for formi ng the con tr ol signal is

termed a

control

algorithm.

We call a control algorithm an ideal) sliding

al gor ithm on the con st rain t a

=

0 if it yields an ideal sliding for every initial

condition in a finite time. 0

Definition 2: A control algorithm de pe ndi ng on a parameter

e E R/

is called a

real sliding algorithm

on the cons tr aint

a

=

0 if, with

e

->

0, it forms a

real

sliding

family for every initial condition. 0

Definition 3: Let

y e

be a r eal -valued functi on such t hat

y e ->

0 as

e -> O

A

real sliding algo ri th m on the con st rain t

a =

0 is said to be of order

r r

> 0)

with respect to

y e

if, for any compact set of initial cond it ion s and for any time

interval

I

,

T

z

],

there exists a constant C, such t ha t the steady-state process

satisfies

la t, x t,

e»1

; Cly(e)l

for

t E

[TI> T

z

].

In the particular case when y e is the smallest time

smoothness the words with respect to

y

may be omitted.

i nt er val of con tr ol

o

Some examples

are

now in or de r. 1) High gain fee dba ck systems Yo ung

et

al.

1977, Saksena

et al.

1984): such systems constitute real sliding algorithms of

the first

order

with r es pect to «: , where

k

is a large gain. 2) R egu la r sliding

mode: this system constitutes an ideal sliding mode theoretically) and also

constitutes real sliding of the first

order,

provided switching time delay is

accounted for.

Let t,x t , s) be a real sliding family with e->O,

t

belongs to a bounded

interval. Let

a t, x)

be a smooth constraint function and

r

> 0 be the real

sliding order with respect to

, e),

where Ii )

>

0 is the smallest time interval of

smoothness of the piecewise smooth function

x t,

s).

Proposition 1: Let

1= [r] be the maximum integer

num ber not

exceeding r. If

the

lth

derivative a l) =

d/dt)la t,

x t,

s)

is uniformly

bounded

in e

fo r

the

steady-state part o f

x t , s ,

then there exist posit ive constants CI> Cz,

,

C

1_ I ,

such that

fo r

the steady-state process the

following

inequalities

hold

Ilall ,;

C1 1 1 Ilall ,; CZ I Z , Ild/-llll ,; C

_

1

Proposition 2: Let {j

>

0 an d

p >

0

be an integer,

an d

let

{s.}

be a

sequence

with e, -> 0 Assume that

fo r

every e, there exists a time interval on which the

steady-state process is

smooth, an d

s uc h th at

fo r everyone

o f these intervals the

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Order and accuracy in sliding

mode

control

1249

pth derivative

of

a satisfies Iial

p

II

on all

of

the interval. Then the real sliding

order r satisfies r

p.

I t

follows from these p rop os it ion s t hat in

or r

to get the rth

or r

of real

sliding with a discrete switching it is r equire d to satisfy the equalitie s

a

=

a

=

.

=

aU-I

=

°

n the ideal sliding.

It

is obvious that in a regular variable

s tr uctu re system only the first or r of the real sliding may be achieved with

discrete switching.

To prove these propositions we need the following simple lemma.

Lemma 1:

There is a constant

T:> 0,

such that for every real function

w r)

E C l

defined on an interval

of

length t: there exists an internal point t I on the interval

such that

Iw r1) tl)1 J sup wlr

r1

I

This l emm a is pr ove n by the successive application of La gr ange s t he or em .

Pr op os it io n 2 follows from the lemma. To prove Pr op os it io n 1 it is necessary to

use Lemma 1 for every integer r\> 1,;;; ,;;; I - 1, and then integrate all)1 1

times.

3. Ideal sliding

Consider the differential equation

y

= v y )

where

y E

R , v R R is a locally b ou nd ed measur ab le L ebesgu e) v ecto r

function. This e qua ti on is unde rst ood in the Filippov s sense Filippov 1960,

1963, 1985); that is, the equation is replaced by an equivalent differential

inclusion

Y

E

V y

In the particular case when the vector-field v is continuous almost every

w here, the s et- valued function V(Yo) is the convex closure of the set of all

possible limits of v yO ) as

YO ->

Yo, where {yO } are continuity points of v. The

s oluti on of the equ at io n is defi ned as the abs ol utely cont in uo us function y t),

satisfying the differential inclusion almost everywhere.

Definition 4: Let

r

be a smooth manifold. The set

T

itself is called the

first-order sliding point set. The second-order sliding point set is defined as the

set of the points y

E r,

where

V y

lies in the tangential space

Tyr

to the

manifold T at the point

y.

0

Definition 5: It is said t ha t t he re exists a first or second)

or r

sliding mode on

the manifold F in the vicinity of a first second)

or r

sliding point Yo if, in this

vicinity of the point Yo, the first second)

or r

sliding set is an integral set, i.e.

consists of the Filippov s sense solutions. 0

The above definitions are easily e xte nde d to include non- aut onom ous dif

ferential equ at io ns by i nt ro du ct io n of the fictitious equ at io n i = 1. A sliding

mode is considered to be stable if the corresponding integral sliding set is stable.

Regular sliding modes satisfy conditions that the set of possible velocities V

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equation

Order and accuracy in sliding mode control 1251

a = a; t, x a ~ t

x f t ,

x, u

eq

= 0

which is assumed to have a unique solution. Under certain obvious conditions,

4 will be satisfied approximately with switching imperfections or with a

second-order real sliding Proposition 1 . For the first-order real sliding this

result was proven by Utkin 1977, 1981 when the process 1 is dependent

linearly on the control, and was generalized by Bartolini and Zolezzi 1986 .

Here, there is no need for any conditions regarding the type of dependence on

the control.

4. Examples of high-order sliding

We now give several examples regarding the ideal and real sliding of the

second order. First, we formulate the conditions under which the problem is to

be solved. For simplicity we assume that a

E R

u

E Rand t, o r

a t

= aCt, x t , u t are available. The goal is to force the constraint a to

vanish.

Assume positive constants ao,

m

,

M

Co

are given. We now impose the

following conditions.

1 Concerning the constraint function

a

and equation 1

i= f t , x , u

we assume the following: lui,,;;

ce a

=constant> 1, f is a C

function,

aCt,

x is a C

2

function. Here,

x E

X where

X

is a smooth finite-dimen

sional manifold. Any solution of 1 is well defined for all t provided

u t

is continuous and satisfies lu t)1

;;

te for each t.

2 Assume there exists

Ul

E 0, 1 such that for any continuous function

u

with lu t)1 > U for all t, a t u t > 0 for some finite time t.

Remark: This condition implies that there is at least one t such that

aCt) = 0 provided u has a certain structure.

Consider a differential operator

L

u

) =

. · f t ,

x, u)

at

where L

u

is the total derivative with respect to 1 when u is considered

as a constant. Define a as

aCt, x, u) =

Lua t,

x = a; t, x

a ~ t

x f t , x, u) 0

3 There are

positive

constants ao,

m

,

M

uo,

< 1, such that if

la t,x 1

<

ao then

aa

« «

;;

K

M

au

for all u, and the inequality lui> implies au> O

The set {t,x, u: la t,x 1

<

ao} is called the linearity region.

4 Consider the boundedness of the second derivative of the constraint

function a with every fixed value of control. Within the linearity region

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1252 A. Levant

I

o]< ao for all t, x, u

the

inequality

ILuLua(t, x)1 <

holds.

The

variable structure system theory deals with the following class of systems

i

=

a t,

x bi x u

where x E R , Under conventional assumptions the task of keeping the con

straint

a =

0 is reduced to the task

stated

above. A new control

}.t and

a

constraint function

r

are to be defined in this case by the transformation

u = }.tk

1

-kawith Ikal,,;;; 1

forms a real sliding a lg or ith m of the first order with respect to k I when k

>

00.

t

is easy to show that a is also of

the

order of «: ,

The

algorithm

u = - sign a 5)

is the ordinary sliding algorithm on a, i.e. it is of

the

first order.

the

values of

a are m eas ur ed at discrete times to,

t >

t2, , t; - t;_1 =

t

> 0 we get the

first-order real sliding algorithm

u t

= - sign a(t;), where t; ;;; t

<

1;+1

The

AI -algorithm Emelyanov and Korovin 1981, Emelyanov 1984)

{

with

lui>

1 6)

it = _ o-sign

a

with lui,,;;; 1

constitutes, with

CY-->

00 a first-order real sliding algorithm with respect to cy -

I

.

Under the above assumptions there exists a unique function

ueq(t, x

satisfying

a

t, x, ueq(t,

x =

0 in the linear region.

The

second-order sliding set

may therefore be given by

a=O,

u

=

ueq t,x

and

is

not

empty.

The

next

t he or em is easy to p ro ve.

Theorem

Assume conditions

1), 3), 4)

are satisfied and let the control

algorithm be

it = 1/1 t,

x, u)

7)

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Order and accuracy in sliding mode control

1253

(9)

(10)

where jJ is a bounded measurable Lebesgue function. Assume also that for

every second-order sliding point M and for every neighbourhood

V M )

J.t V M

n 1jJ-l«_co, -CoIK

m]

>0

J.t V M n 1jJ-l [C

o

IK

m

, co))) > 0

where J t is a Lebesgue measure. Then there is a second-order sliding mode on the

constraint a =

0

and the motion in this mode is described by

(4)

x = f t, x, ueq t,

x

Proof: In view of § 3 the system (1), (7) is equivalent to a differential inclusion

x= f t , x , u }

u E e t,

x,

u

C R

It follows from the conditions of the theorem that for every second-order sliding

point t ,

x ,

u

the set e includes the interval

ColK

m,

ColKml

So every

trajectory

t,x t ,

u t

that

satisfies (1) with

il

E

[-CoIK

m, ColKml

is a solut ion

of (1) and (7) if it lies on the

second order

sliding set a =

a

= O

It

follows from conditions (3)

and

(4) and from the implicit function

theorem that lill

ColK

m

Therefore we can choose

il

=

il

eq

= LI/u

eq

(

t, x) 8

The theorem

follows now from the fact

that the second order

sliding

set a =

0,

u = u

eq

is invar iant with respect to (1) and (8). 0

Theorem

1 implies tha t the re is a second-order sliding mode in the system

(1) and (6) for any sufficiently large

a.

But in general, it is not stable.

However it may be shown that in certain cases this sliding mode is stable with

an exponential decay to zero of [o]

and lal When a-> co

the

properties

of the

AI algorithm (6) approximates the properties of the regular fi rs t-order sliding

algorithm (5).

Consider a so-called twist ing algorithm (Levantovsky 1985, Emelyanov et

al . 1986 a, b)

i

- u with lui> 1

il

-am sign a with oo 0, lui 1

-aM

sign a with

oo

> 0, lui 1

where aM> am >

Assume

am > 4KMlao, am > c.j«;

}

KmaM -

Co > KMa

m

Co

According to

Theorem

1

there

exists a

second order

sliding

mode.

Theorem 2:

Under assumptions

1 - 4

and conditions 10 the twisting algo

rithm (9) is a second-order sliding algorithm.

It may be shown that in systems (1), (6) and (1), (9) the state paths are

encircling the

second order

sliding manifold a a

Whereas

the

state

path

of the

AI

algorithm does not converge to the manifold a =

a

= 0 (Fig. 1), the

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1254

A Levant

Figure 1.

state path of the algorithm 9) converges to the manifold in a finite time and

makes an infinite number of rotations Fig. 2). h result is achieved by the

switching of the parameter Y in 9).

h

actual calculation of the derivative

a

may cause a serious obstacle in applicat ions. Let us use the first difference in 9)

instead of

a

itself. Suppose that the measurements of a are being made at times

10,

12, , I; - 1;-1

= l:> 0. Define

{

0,

tia I;

=

a lj, X I »

a l j -b X I;-I»,

u

i °

i 1

Figure 2.

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Order

and

accuracy in sliding

mode

control

Le t t

be the cur ren t time an d assume

t

E

t;, t;+I

and

1255

(12)

(11)

j

-u t;), lu t; 1

>

1

U

=

-IXM

s ign

a t; ,

a t ; ~ a t ;

>

0, lu t; I,;;; 1

-lXm

sign

a t; ,

a t ; ~ a t ; ;;; 0,

lu t; I,;;;

1

Theorem

3:

Under the conditions

o f

Theorem

2

with

,--- 0

the algorithm

(11)

is

a second-order real sliding algorithm. There are positive constants ao, l such that

in the steady-state

mode

inequalities

o ao?,

101 ;;; a f hold.

t

follows from Proposition 2 that this is the best possible accuracy which

may be achieved by means of a control with its derivative being switched.

Consider the so-called drift algorithm (Emelyanov

et at.

1986 c)

j

-u t;),

.

lu t; 1

> 1

li =

-IXM

sign

~ a t ; ,

a t ; ~ a t ; > 0,

lu t; l,;;;

1

- l X m s i g n ~ a t ; ,

a t ; ~ a t ;

;;;

0,

lu t; I,;;;

1

Here, IXM >

m

>

After

substitution of

for the first-order sliding

mode

on the const ra int 0 = 0 would be achieved. This implies a = constant. But, since

th e art if icial switching time delay is introduced by (12), we ensure a real sliding

on

0 with most of the time being spent in the set oo < Th e algorithm does

no t force th e tra jectory to reach the linearity region Io] <

ao but

forces this

trajectory to stay there once it reaches this set. Inequalities like the

ones

appearing in

Theorem

3 are ensured within a finite time by the algorithm. Th e

accuracy of th e rea l sliding on 0 = 0 increases with decreasing r; in fact, it is

proport ional to r. Hence, the durat ion of the t rans ient process is proport ional to

•

Such an algorithm

does

not satisfy the definit ion of a real sliding algorithm.

Let

and let

t; 1

- t;

=

; 1

=

, a t;»,

i

=

0,

1,2,

(13)

j

M

, S = vlsl

P

m

vISI

P

<

vlSI

P

<

v

P

;;;

m

(14)

where

0·5 ;;;

p

1,

M > m >

0,

v> O.

We call the region lal

ao -

< i where < i is any real number, such that

<

< i <

ao,

the reduced linearity region.

Theorem

With initial condi tions within the reduced linearity region, v an d

IXm/IXM

sufficiently small an d

m

sufficiently large, the algorithm

(12), (13), (14)

constitutes a second-order real sliding algorithm on the constraint a

=

0 with

respect to

m

O.

Th e

specific restrictions on m IXm IXM V may be found in the paper by

Emelyanov

et al.

(1986 c).

Th e

character is tic forms of the t rans ient process in

th e cases of the twisting and drift algorithms are shown in Figs 3 and 4,

respectively.

Th e

drift algorithm has no overshoot if p aram eters a re chosen

properly (Emelyanov et al. 1986 c).

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1256

A. Levant

Figure 3.

0·5506

-0.0013 L- - -::-=::-:-::-- _

Figure 4.

15

u lui>

1

u = 1 -cysign a - g a», lui 1

The algorithm with a prescribed law of variation of a Emelyanov et al.

1986a has the form

where rr

>

0, and the continuous function

g a)

is smooth everywhere except on

= o is assumed here that all the solutions of the equation

a

= g vanish

in a finite time, and the function g a)g a) is bounded. For example

g = -Asign

l l

Y

where A> 0,0·5 ;; Y< 1, may be used.

h or m 5: With initial conditions within the reduced linearity region and for

sufficiently large the algorithm

15

constitutes a second order sliding algorithm

on the constraint a = o

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Order and

accuracy in sliding

mode

control

1257

The substitution of

fla(t;) -

l:g a t;) for a -

g(a)

turns the algorithm 15

into a

real

sliding algorithm.

The orde r o f real

sliding

depends

on the funct ion g

but does not exceed 2. In Fig. 5 the characteristic form of the transient process

is shown with g

=

-AlaI

2

/

3

sign a. Near the transfer point t* with

t

e t*

a(t) = 1/27Alt -

t

3

•

All t he above examp le s of the sliding algorithms use the derivatives of a

calculated with respect to the system.

The

following is an example

t ha t does not

utilize this property (Emelyanov

et al.

1990 .

18

16

17

u

= Ul

Uz

{

u

lui> 1

ill =

- a s ig na , l ui

1

{

-Alaolpsigna

lal > ao

Uz

=

-

alP

sign

a

Ial ao

where

a ,

A> 0,

E 0, 1), and the initial values UI tO) are to be chosen from the

condition

lui =

IUt to)

uz(to, xo 1 re

The

following inequalities

are

to be sat isfied

a > Co/k

m,

a > 4K

M/ao

19)

I I

Z

p(AKmF>

KMa C

o)(2KMF-

20)

h or m 5: Assume

that condit ions

19 ,

20) are satisfied

and

0

P

1/2.

Then the algorithm 16 , 17 , 18 is a second-order sliding algorithm.

It may be shown

that

with = 1,

a

and

A/a

sufficiently large

there

is a st able

second-order

sliding

mode.

In this case lal lal tends to zero with exponential

upper and

lower bounds (Levantovsky 1986 .

Under

the

conditions of

the

0·4075

-0·0004::--- == -=-==- _

o

Figure 5.

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1258 Levant

21

theorem, discrete measurements turn these algorithms into real sliding al

gorithms. We remark here that, with

p

= 1, no real sliding algorithm will be

obtained, since the rate of the convergence is only exponential. Such an

algorithm may be efficiently implemented because of its simplicity. The bound

on the convergence decrement may be assigned any desirable value.

Remarks:

1 Utilizing 13 and 14 , all the above-listed algorithms with discrete

variable measurements become stable with respect to small model disturbances

and measurement errors. It may be shown that the accuracy of such modified

algorithms without measurement errors is the same as the accuracy of the initial

algorithms.

2 It may be shown that for all except for the last algorithms, smoothness of

f

in 1 and of a is not necessary. It is sufficient for

f

to be only a locally

Lipschitzian function and the same is true for the partial derivatives of

a

t x .

3 In the case where

X

is a Banach space the conclusions of the above

theorems still hold provided the statements are changed as follows: a and 0

vanish in a finite time or

lal

Ct

r :;;;

Cz

after a finite time.

5. An outline of the proof of Theorems 5

All the described algorithms either stir the system into the linearity region or

perform in this region. I t is easy to show that they are prevented from leaving

the region Levantovsky 1985, Emelyanov et al 1986b . We consider now the

motion of the system within the linearity region. Clearly

£

a =

Lua

= LuLua

~ u a

it

dt

Z

dt

u

Let C

=

LuLua K

= u

Li «

so that we have

a

= C

Kit

According to Assumptions 3 and 4

ICI o 0 < «;

:;;; K K

M

Hence, the following differential inclusion is valid

a E

[ Co o] [K

m

KM]it

22

23

The real trajectory starting at the axis 0 on the plane a 0 lies between the

extreme solutions of 23 . .

Figure 6 shows the set of the solutions of 23 and the solution of 21 which

are associated with the twisting algorithm 9 . An infinite number encircling the

origin occurs here within a finite time. The total time duration of the origin

encircling is proportional to the total variation of a which is estimated by a

geometric series.

The trajectories of the drift algorithm 12 are shown in Fig. 7. A real sliding

along the axis 0 = 0 may be seen here. Due to the switching of 0 the motion

with ao

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Figure 6.

Figure 7.

1259

M

oM

3

is proportional to

r.

So, the time that the drift takes to reach a = 0 is

proportional to

r-

1

with r

=

constant. With the variable r 14 the convergence

time turns out to be bounded Emelyanov et al 1986 c .

Fig. 8

the

trajectories of the algorithm 15 are shown. On the line

0= g a a common sliding mode

of

the first order arises. Considering now the

existence of the sliding mode on the manifold 0 - g a = 0 we have

~

g a» = g ~ a o = LuLua - g ~ g a ~ u a

dt

= c - g ~ a g a - Kasign o - g a»

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1260

A. Levant

Figure 8.

The last term must dominate the first two terms but it can do so only when

g;, a)g a) is bounded. That puts a restrict ion on g a).

Consider now the algorithm 16 , 17 and 18 . In this case

= C

KUl

- ; pK _a_

_

lal

1

-

p

The last term here involves

°

and a gain which is unbounded when

a-+

The

real trajectory lies inside the set of the solution points of the associated inclusion

equation with lui

<

1

E

-[Kmtl - Co, KMtl Co]signa - ; p[K

m

M] _o -

lal

1

-

p

Figure

9

depicts this case.

I follows from the accurate estimations that for

p

= 1/2 the sequence {O } of

the intersection points with the axis

a = 0 satisfies 10j I 0;1,,;;; q < 1 which

implies 10;1- 0 as i co. For 0 < P < 1/2 the convergence to the origin is even

faster. Also, the sum of the encircling time sequence is est imated by a geometric

series.

Mathematical modelling

We consider now the following example

Xl =

5xj 10X2 4X3

Xl

sin

t

u

2

- 1 Xl - X2

X2

= 6XI

-

3X2

- 2X3 3 xj X2

X3 COSt

X3

= Xl

3X3

x cos5t

4sin5t

10 1 0·5cos

IOt JL u

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Figure 9.

1261

Let

= X3/

0 which holds for the entire state space with

lui;; 1. Let

u = lalPsign a

f

-u

lui>

1

Ul =

-8:igna, lui ;;

1

Here Euler s method has been employed for the numerical solution with the

step of 10

4

.

Figures 10 and 11 depict a t and

u t

with

=

0·5. In Fig. 12 the

graphs J. t,

u t

and

J.eq t,

x t are shown, where J. = J eq is found from the

03208

Figure 10.

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1262 A Levant

0·3208

1 · 0 0 2 8 ~ L ::; == _ :

o

Figure 11.

Figure 12.

equation X = is seen from the graph that u converges to the equ iv al ent

control u

eq

. With the initial conditions

x O

= 2, 2 10) the accuracy

[o] : ;; 1·65 x

10

5

was achieved.

Figures 3, 4 and 5 show the graph of aCt . The figures correspondingly

describe the twisting algorithm 11) with am

=

8, aM

=

40 and the measu rement

interval m

=

5 x 10

4

; the drift algorithm 12), 13) and 14) with

am =

8,

a = 40,

= 0·05, y = 0·02, p = 0·5, m = 5 x 10

4

;

and the algorithm with a

prescribed law of variation of a 15) with a = 16, g = 1 1a1

2

/

3

sign a

=

5

X

10

4

.

This system is slightly di ff er ent from the p revio us examp le 24),

25) and 26) the last term in the first equation of 24) is absent). In all cases,

an accuracy of lal 10

3

was achieved. For the twisting and drift algori thms

with a reduction of m to

m/lOO

the accuracy changes from 6·6 x

10

4

and

1· 3 x

10

3

to 7·04

X

10-

8

and 7·12 x

10

8

,

respectively.

The

use of the regular sliding algorithm u = - sign aCt; gives, with the

measurement

interval =

5

X

10

4

,

the accuracy lal:o ;; 1·2 x 10

2

and after

reducing m to m/100 it gives an accuracy of Io]

: ;;

1·04 x 10

4

• N ot e, t ha t with

this alg or it hm the b eh av io ur of the system 24), 25) and 26) in the ideal sliding

mode cannot be described uniquely Utkin 1981, Bartolini and Zolezzi 1986).

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263

KNOWLEDGMENT

The author

wishes to thank Dr N.

Berman

for his invaluable help in

overcoming the language barrier and for his useful comments.

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