Let the sliding variable and the sign of its total time
derivative _ be available for feedback.
This stabilization problem has been addressed and
solved in many ways (Emelyanov 1986, Levant 1993,
Bartolini et al. 1997, 1998 a, 1999, 2001 a, b). The restric-
tive condition (5) was assumed to hold locally in
(Emelyanov et al. 1986, Levant 1993) where specific con-
trol actions were devoted to keep the system within the
boundedness region. Recently, the global boundedness
assumption (5) has been explicitly relaxed by the
authors. In Bartolini et al. (2001 a) the ‘sub-optimal’ 2-
SMC algorithm was considered, and it was shown that if
the modulus of the drift term ’ðÞ can be upper bounded
by a known function affine in j _ðtÞj, then a set of con-
stant controller parameters guaranteeing the solution of
the problem exists. On the contrary, if j’ðÞj can be
upper bounded only by a known function of the phase
variables ðtÞ and _ðtÞ (without any assumption regard-
ing the growth rate w.r.t. j_ðtÞjÞ, then the convergence to
the 2-SM can be assured by suitable adaptation of the
controller parameters (Bartolini et al. 2001 b).
In order to make the treatment clearer and to present
the very idea of the controller design, in the following we
shall consider that conditions (5) hold globally.
A large number of 2-SMC algorithms studied in the
cited literature can be practically implemented by setting
the parameters of the following control law properly
uðtÞ¼ðtÞU signð
M
Þ
ðtÞ¼
1ifð
M
Þ
M
0
if ð
M
Þ
M
< 0
(
2½0; 1Þ
9
>
>
>
>
>
=
>
>
>
>
>
;
ð6Þ
where U > 0 is the minimum control magnitude,
> 1
is called the modulation factor, is the anticipation
factor, and
M
is the last extremal value of the sliding
variable (i.e. the value of at the last time instant at
which a local maximum, minimum or horizontal flex
point of ðtÞ has occurred).
M
can be initialized to
ðt
i
Þ (t
i
being the initial time instant) and then updated
either by checking the sign of _ or by inspection of the
past values of ðtÞ; in the latter case no information
about _ is needed. U,
and are the controller par-
ameters that must be tuned according to the inequalities
U >
F
G
m
2½1; þ1Þ \
2F þð1 ÞG
M
U
ð1 þ ÞG
m
U
; þ1
9
>
>
>
=
>
>
>
;
ð7Þ
The first part of (7) can be referred to as the dominance
condition, ensuring that the control has sufficient author-
ity to affect the sign of €. The second part of (7) repre-
sents the convergence condition, sufficient to guarantee
the stability of the SM and determining the rate of con-
vergence as well. Generally, these conditions lead to
transient trajectories twisting around the origin of the
N plane. The additional requirement of monotonic
convergence to zero of the sliding variable may be ful-
filled by imposing a stricter inequality than (7), i.e.
U >
F
G
m
2½1; þ1Þ \
F þð1 ÞG
M
U
G
m
U
; þ1
9
>
>
>
=
>
>
>
;
ð8Þ
Controller (6) satisfying either condition (7) or (8)
assures the establishment of the 2-SM behaviour in a
finite time T
c
s.t.
T
c
t
M
1
þ maxfT
c1
; T
c2
gð9Þ
T
c1
¼ U
G
m
þ G
M
G
m
U F
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 Þj
M
1
j
G
M
U þ F
s
1
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jF þ½ð1 ÞG
M
G
m
Uj
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G
m
U F
p
T
c2
¼ U
G
m
þ G
M
G
m
U þ F
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 Þj
M
1
j
G
m
U F
s
1
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jF ½ð1 ÞG
m
G
M
Uj
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G
M
U þ F
p
where t
M
1
is the time instant at which the first extremal
value of the sliding variable,
M
1
, occurs. The absolute
value of
M
1
defines also the maximum magnitude of the
time derivative of the sliding variable, _, after the time
instant t
M
1
, i.e.
j_ðtÞj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 ÞðG
M
U þ FÞj
M
1
j
q
8t 2½t
M
1
; þ1Þ
ð10Þ
In order to speed-up the convergence, the following con-
trol law should be implemented from the initial time
instant t
i
until the first extremal value is reached at t
M
1
uðtÞ¼U
i
signð ðt
i
ÞÞ
U
i
>
F
G
m
8t 2½t
i
; t
M
1
9
>
=
>
;
ð11Þ
The initializing control (11) assures that
t
M
1
t
i
þ
j_ðt
i
Þj
G
m
U
i
F
ð12Þ
From equations (9), (10) and (12), it is apparent that the
larger the control magnitude U, the shorter the conver-
gence time and the larger the maximal values of the time
derivative of the sliding variable. Nevertheless, once the
Applications of sliding mode control to mechanical systems 877