Applied Nonlinear Control - Chapter 7: Sliding Control - Nguyễn Tân Tiến

ror x~ by further 
low-pass filtering, according to definition (7.3) 
s x~1storder filter
(7.29) 1)(
1
−+ np λ
)()( εOf d +∆− x
φofchoice sofdefinition
Fig. 7.9 Structure of the closed-loop error dynamics 
Control action is a function of x and dx . Since λ is break-
frequency of filter (7.3), it must be chosen to be “small” with 
respect to high-frequency un-modeled dynamics (such as un-
modeled structural modes or neglected time-delays). 
Furthermore, we can now turn the boundary layer thickness φ 
so that (7.29) also presents a first-order filter of bandwidth λ . 
It suffices to let 
λ=φ
x )( dk (7.30) 
which can be written from (7.27) as 
)( dk xφφ =+ λ& (7.31) 
(7.27) can be rewritten as 
φxxx λ+−= )()()( dkkk (7.32) 
⊗ Note that: 
- The s-trajectory is a compact descriptor of the closed-
loop behavior: control activity directly depends on s , 
while tracking error x~ is merely a filtered version of s 
- The s-trajectory represents a time-varying measure of the 
validity of the assumptions on model uncertainty. 
- The boundary layer thickness φ describes the evolution 
of dynamics model uncertainty with time. It is thus 
particularly informative to plot )(ts , )(tφ , and )(tφ− on 
a single diagram as illustrated in Fig. 7.11b. 
Example 7.3________________________________________ 
Consider again the system described by (7.10): 
uxxtax +−= 3cos)( 2&&& . Assume that λη /)0( =φ with 1.0=η , 
20=λ . From (7.31) and (7.32) 
 ( ) ( )
φ
φ
&&
&&
−+=
++−+=
η
ληη
xx
xxxxxk dd
3cos5.0
3cos5.03cos5.0)(
2
22
where, )3cos5.0( 2 ηλ ++−= xxd&& φφ . The control law is now 
Applied Nonlinear Control Nguyen Tan Tien - 2002.5 
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 
___________________________________________________________________________________________________________ 
Chapter 7 Sliding Control 35 
]/)~20~[()3cos5.0(
~3cos5.1
)/()(ˆ
2
2
φφ
φ
xxsatxx
xxxx
ssatxkuu
d
+−+−
−+=
−=
&&&
&&&&
η
λ 
⊗ Note that: 
- The arbitrary constant η (which formally, reflects the 
time to reach the boundary layer starting from the 
outside) is chosen to be small as compared to the average 
value of )( dk x , so as to fully exploit our knowledge of 
the structure of parametric uncertainty. 
- The value of λ is selected based on the frequency range 
of un-modeled dynamics. 
The control input, tracking error, and s -trajectories are 
plotted in Fig. 7.11. 
0 1.0 2.0 40.5 1.5 2.5 3.0 3.5
Co
nt
ro
l I
np
ut
Time (s)
-4
-3
-2
-1
0
1
2
3
4
6
5
0 1.0 2.0 40.5 1.5 2.5 3.0 3.5
-5
-4
-3
-2
-1
0
1
2
3
5
4
Time (s)
Tr
ac
ki
ng
 E
rr
or
 (x
10
-3
)
Fig. 7.11a Control input and resulting tracking performance 
-8
-6
-4
-2
0
2
4
6
8
S-
tr
aj
ec
to
rie
s (
x1
0-
2 )
0 1.0 2.0 40.5 1.5 2.5 3.0 3.5
Time (s)
φ
φ−
s
Fig. 7.11b s-trajectories with time-varying boundary layer 
We see that while the maximum value of the time-varying 
boundary layer thickness φ is the same as that originally 
chosen (purposefully) as the constant value of φ in Example 
7.2, the tracking error is consistently better (up to 4 times 
better) than that in Example 7.2, because varying the thickness 
of the boundary layer allow us to make better use of the 
available bandwidth. 
__________________________________________________________________________________________ 
In the case that 1≠β , one can easily show that (7.31) and 
(7.32) become (with )( dd xββ = ) 
d
dk β
λ φ
x ≥)( ⇒ )( dd k xφφ βλ =+& (7.33) 
d
dk β
λ φ
x ≤)( ⇒ 
d
d
d
k
ββ
λ )(
2
xφφ =+& (7.34) 
ddkkk βλ φ/xxx +−= )()()( (7.35) 
with initial condition )0(φ defined as: 
λβ /))0(()0( dd k xφ = (7.36) 
Example 7.4________________________________________ 
A simplified model of the motion of an under water vehicle 
can be written (7.21): uxxcxm =+ &&&& . The a priori bounds 
on m and c are: 51 ≤≤ m and 5.15.0 ≤≤ c . Their estimate 
values are 5ˆ =m and 1ˆ =c . 20=λ , 1.0=η . The smooth 
control input using time-varying boundary layer, as describe 
above is designed as follows: 
xxs ~~ λ+= & 
xxxs d &&&&&& ~λ+−= 
)~( xxmuxxcsm d &&&&&& λ−−+−= 
)~(ˆˆˆ xxmxxcuu d &&&&& λ−+=→ 
)sgn()~(ˆˆ
)sgn(ˆ
skxxmxxc
skuu
d −−+=
−=
&&&&& λ 
)sgn()~()ˆ()ˆ(
)~()sgn()~(ˆˆ
skxxmmxxcc
xxmskxxmxxcxxcsm
d
dd
−−−+−=
−−−−++−=
&&&&&
&&&&&&&&&&&
λ
λλ
Condition (7.5): sss η−≤& 
smssm η−≤& 
( ) smsskxxmmxxcc d ηλ −≤−−−+− )sgn()~()ˆ()ˆ( &&&&& ( ) smsxxmmxxcxxccssk d ηλ +−−++−≥ )~()ˆ(ˆ)ˆ()sgn( &&&&&&&( ) ηλ msxxmmxxcck d +−−+−≥ )sgn()~()ˆ()ˆ( &&&&& 
ηλ msxxmmxxcck d +−−+−≥ )sgn()~()ˆ()ˆ( &&&&& 
And the controller is 









−=
+=
−=
−=
+−−+−=
−+=
)/sat(ˆ
~~
)(
)(
)max(~ˆmaxˆmax)(
)~(ˆˆˆ
2
φ
φ
φφ
skuu
xxs
xkk
xk
mxxmmxccxk
xxmxxcu
d
d
d
λ
λ
ηλ
λ
&
&
&
&&&&
&&&&&
The results are given in Fig. 7.12 
0 1.0 2.0 40.5 1.5 2.5
-5
-4
-3
-2
-1
0
1
2
3
5
4
D
es
ire
d 
Tr
aj
ec
to
rie
s
Time (s)
acceleration (m/s2 )
velocity (m/s)
distance (m)
-10
-5
0
5
10
15
20
25
30
35
0 1.0 2.0 40.5 1.5 2.5 3.0 3.5
Time (s)
C
on
tr
ol
 In
pu
t
a. References b. Control input 
0 2 4 61 3 5
-25
-20
-15
-10
-5
0
5
10
15
Time (s)
Tr
ac
ki
ng
 E
rr
or
 (x
10
-2
)
0 2 4 61 3 5
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
S-
tra
je
ct
or
ie
s (
x1
0-
2 )
Time (s)
φ
s
φ−
c. Tracking error b. s- trajectories 
Fig.12 
__________________________________________________________________________________________ 
⊗ Remark: 
- The desired trajectory dx must itself be chosen smooth 
enough not to excite the high frequency un-modeled 
dynamics. 
- An argument similar to that of the above discussion 
shows that the choice of dynamics (7.3) used to define 
sliding surfaces is the “best-conditioned” among linear 
dynamics, in the sense that it guarantees the best tracking 
performance given the desired control bandwidth and the 
extent of parameter uncertainty. 
- If the model or its bounds are so imprecise that F can 
only be chosen as a large constant, then φ from (7.31) is 
Applied Nonlinear Control Nguyen Tan Tien - 2002.5 
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 
___________________________________________________________________________________________________________ 
Chapter 7 Sliding Control 36 
constant and large, so that the term )/sat( φsk simply 
equals βλ /s in the boundary layer. 
- A well-designed controller should be capable of 
gracefully handling exceptional disturbances, i.e., 
disturbances of intensity higher than the predicted 
bounds which are used in the derivation of the control 
law. 
- In the case that λ is time-varying, the term 
xu ~λ&−=′ should be added to the corresponding uˆ , while 
the augmenting gain )(xk according by the quantity 
)1( −′ βu . It will be discussed in next section. 
7.3 The Modeling/Performance Trade-Offs 
The balance conditions (7.33)-(7.36) have practical 
implications in term of design/modeling/performance trade-
offs. Neglecting time-constants of order λ/1 , condition (7.33) 
and (7.34) can be written 
dd
n kβελ ≈ (7.39) 
Consider the control law (7.19): )]sgn(ˆ[ˆ 1 skubu −= − , we see 
that the effects of parameter uncertainty on f have been 
“dumped” in gain k . Conversely, better knowledge of 
f reduces k by a comparable quantity. Thus (7.39) is 
particularly useful in an incremental mode, i.e., to evaluate the 
effects of model simplification on tracking performance: 
)/( ndd k λβε ∆≈∆ (7.40) 
In particular, margin gains in performance are critically 
dependent on control bandwidth λ : if large λ ’s are available, 
poor dynamic models may lead to respectable tracking 
performance, and conversely large modeling efforts produce 
only minor absolute improvements in tracking accuracy. And 
it is not overly surprising that system performance be very 
sensitive to control bandwidth. 
Thus, give system model (7.1), how large λ can be chosen ? 
In mechanical system, for instance, given clean measurements, 
λ typically limited by three factors: 
i. structural resonant modes: λ must be smaller than the 
frequency Rν of the lowest un-modeled structural resonant 
mode; a reasonable interpretation of this constrain is, 
classically 
RR νπλλ 3
2≈≤ (7.41) 
although in practice this bound may be modulated by 
engineering judgment, taking notably into account the 
natural damping of the structural modes. Furthermore, in 
certain case, it may account for the fact that Rλ may 
actually vary with the task. 
ii. neglected time delays: along the same lines, we have a 
condition for the form 
A
A T3
1≈≤ λλ (7.42) 
where AT is the largest un-modeled time-delay (for 
instance in the actuators). 
iii. sampling rate: with a full-period processing delay, one 
gets a condition of the form 
samplingS νλλ 5
1≈≤ (7.43) 
where, samplingν is the sampling rate. 
The desired control bandwidth λ is the minimum of three 
bounds (7.41-43). Ideally, the most effective design 
corresponds to matching these limitations, i.e., having 
λλλλ ≈≈≈ SAR (7.44) 
7.4 Multi-Input System 
Consider a nonlinear multi-input system of the form 
∑
=
+=
m
j
jiji
n
i ubfx i
1
)( )()( xx , mi ,,1L= , mj ,,1L= 
where 
T
muuu ][ 21 L=u : the control input vector [ ]Tnn xxx &L)2()1( −−=x : the state vector 
7.5 Summary