T
+ η −η
η −η
+
rtn
rt
r
r
n+ 1
tn
tn+ 1
T
+ 2 I − hKp η
+
)+ η −η
+
t
h( Jt v t
˙ η
n
n
n
rtn
rtn
rtn+ 1
2
2
+
− 1
2
2
− 1
2
I − M M
s t + h M
p
−r
+ δv
− δv + ε
n
δt
δ
t
t
v
n
tn
n+ 1
n
n+ 1
T
+
− 1
− 1
2
I − M M s t + hM
p
−r
δv
− δv + ε
n
δt
δ
t
t
v
n
tn
n+ 1
n
n+ 1
264
Discrete Time Systems
and fΔ Q is a sign-undefined energy function of the model errors and measure disturbances
1 n
defined as
fΔ Q [ ε
, ε
, δη
, δv
] =
(40)
1
η
v
t
t
n
n+ 1
n+ 1
n+ 1
n+ 1
2
ε
− 1
− 1
η
+ δη
−δη −hJt Δ J Kpη + Jt δv t + Jt Δ J ˙ η
n+ 1
tn+ 1
tn
n
tn
tn
n
n
n
tn rtn
T
+2
I − hKp η +
+
+ η −η
×
t
h Jt v t
˙ η
n
n
n
rtn
rtn
rtn+ 1
ε
− 1
− 1
η
+ δη
−δη −hJt Δ J Kpη + Jt δv t + Jt Δ J ˙ η
.
n+ 1
tn+ 1
tn
n
tn
tn
n
n
n
tn rtn
Clearly, there are many variables involved like the system matrices, model errors and measure
disturbances which are not known beforehand.
T
−1
The idea now is to construct τ2 so that the sum a( M−1τ ) 2+b M τ + c in (36) be null. As n
2n
2n
there are many variables in the sum which are unknown, we can construct an approximation
of it with measurable variables. So, it results
− 1
2
T
− 1
¯ a M τ2
+b
τ + ¯ c
n
nM
2n
n=0.
(41)
Now, the polynomial coefficients ¯ a b n and ¯ cn are explained below. Here, there appear three
error functions, namely fΔ Q , and the new functions f
and f
, all containing noisy and
1
Δ
n
Q 2 n
Uin
unknown variables which are described in the sequel.
The polynomial coefficients result
2
¯ a = a= h
(42)
∗
b n = 2h( I−hKv)v t + 2 hM− 1(p −r )
(43)
n
δ
δ
tn
tn
2
2
T
2
cn = h
Jt v +˙ η
+ Δ J v
+ 2 J v +˙ η
Δ J v
+
(44)
n
tn
r
δ
t
t
t
δ
t
t
t
t
n
n
n
n
n
rtn
n
n
T
T
+2 h Jt v +˙ η
η −η
+2 h Δ J v
η −η
+
n
tn
r
δ
t
t
t
n
rtn
rt
n
n
r
r
n+ 1
tn
tn+ 1
T
+ η −η
η −η
+
rtn
rt
r
r
n+ 1
tn
tn+ 1
T
T
+2 I − hKp η
+
+ η −η
+
+
t
h Jt v t
˙ η
2
I − hKp η
hΔ Jδ v t
n
n
n
rt
t
n
rtn
rt
t
n
n
n+ 1
n
2
+ 2
h M− 1 p δ −r
+ 2 hM− 1(p −r δ ) T ( I−hK∗
,
t
δ
v )v t
n
δtn
tn
tn
n
with p δ being an estimation of p δ in (14) given by
tn
tn
v t −v t
p
n
n−1
δ = M
−τn.
(45)
tn
h
A General Approach to Discrete-Time Adaptive Control Systems with Perturbed Measures for
Complex Dynamics - Case Study: Unmanned Underwater Vehicles
265
The second component τ2 of τ
n
n was contained in the condition (41) like a root pair that
enables Δ Qt be the expression (47). It is
n
⎛
⎞
T
τ
⎝ −b
b b − 4 ¯ ac ⎠
n = M
± 1
1
,
(46)
2
2 ¯ a
2 ¯ a
6
with 1 being a vector with ones.
With the choice of (41) and (46) in Δ Qt one gets finally
n
T
Δ Qt = η hK
+
(47)
n
t
p hKp − 2 I η
n
tn
T
+v t hK∗
+ f
[ εη , ε
, δη
, δv
]+
n
v ( hK∗
v − 2 I) v tn
Δ Q 1
v
t
t
n
n+ 1
n+ 1
n+ 1
n+ 1
+ fΔ Q [ ε
, ε
, δη
, δv
]+ f [( U∗−U
2
η
v
t
t
n
n+ 1
n+ 1
n+ 1
n+ 1
Uin
i
i ) , M− 1 M].
The matrices U∗ that appear in
take particular constant values of the adaptive controller
i
fUin
matrices Uiś. They take the values equal to the system matrices in (1)-(2) (Jordán and
Bustamante, 2008), namely
∗
U =
i
Ci, with i = 1, ..., 6
(48)
∗
U =
7
Dl
(49)
∗
U =
i
Dq , with i = 8, ..., 13
(50)
i
∗
U
=
14
B 1
(51)
∗
U
=
15
B 2 .
(52)
Moreover, the error functions fΔ Q and f
in (47) are respectively
2 n
Uin
fΔ Q [ ε
, ε
, δη
, δv
] =2 h δv
− δv + ε
×
(53)
2
η
v
t
t
t
t
v
n
n+ 1
n+ 1
n+ 1
n+ 1
n+ 1
n
n+ 1
⎛
⎞
T
2
× − 1
b b − 4 ¯ a ¯ c
2
M M ⎝ − b ± 1
1⎠ −h
Δ J v
−
2 ¯ a
2 ¯ a
6
δtn tn
T
− 2
2
2 h
Δ Jδt v
η −η
− 2h J v +˙ η
Δ J v −
n
tn
r
t
t
δ
t
t
t
n
rt
n
n
r
n
n
n+ 1
tn
2
T
− 2 h I − hKp η Δ
+ δ
− δ + ε
+
t
Jδ v t
v t
v t
v
n
tn
n
n+ 1
n
n+ 1
T
+
− 1
−1
2
I − M M s t + hM
p
−r
δv