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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
To link to this article: http://dx.doi.org/10.1080/00207179308923053
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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
8/19/2019 Levant 1993
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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
8/19/2019 Levant 1993
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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
8/19/2019 Levant 1993
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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.
8/19/2019 Levant 1993
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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).
8/19/2019 Levant 1993
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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.
8/19/2019 Levant 1993
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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
8/19/2019 Levant 1993
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Order and accuracy in sliding mode control
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»
8/19/2019 Levant 1993
15/18
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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Order and accuracy in sliding mode control
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.
8/19/2019 Levant 1993
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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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Order and accuracy in sliding mode control
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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