δ −r δ )
−
tn
tn
∂Ui
∂
T
−
−T
∗
r δ
2 hM ( I − hK
tn
v)v t
,
n
∂Ui
with the property
∂Δ Qtn = ∂Δ Qtn + Δ
∂
,
(58)
U
Uin
i
∂Ui
where
Δ U = δ
+ δ
,
(59)
in
M− 2 Ain
M− 1 Bin
−T
− 1
−T
− 1
− 1
− 1
and δ
−
≥
−
≥
M− 2 =
M M
M M
0 and δM− 1 = M
M
0. Here Ai and B are
n
in
sampled state functions obtained from (56) after extracting of the common factors δM− 2 and
δM− 1, respectively.
It is worth noticing that Δ Qt and Δ Q , satisfy convexity properties in the space of elements
n
tn
of the Ui’s.
Moreover, with (58) in mind we can conclude for any pair of values of Ui, say U of
, it is
i
Ui
valid
∂Δ Q ( U )
Δ
t
Q
n
i
t ( U ) −Δ Q ( U ) ≤
U −U
≤
(60)
n
i
tn
i
∂U
i
i
i
∂Δ
( )
≤
Qt U
n
i
−
∂
U
U
U
i
i
.
(61)
i
This feature will be useful in the next analysis.
268
Discrete Time Systems
In summary, the practical laws which conform the digital adaptive controller are
Δ
∂Δ Q
U
tn
i
= U − Γ
.
(62)
n+1
in
i
∂Ui
Finally, it is seen from (57) that also here the noisy measures ηδ and v δ will propagate into
tn
tn
∂Δ Q
the adaptive laws
tn
∂
.
Ui
5. Stability analysis
In this section we prove stability, boundness of all control variables and convergence of the
tracking errors in the case of path following for the case of 6 DOFś involving references
trajectories for position and kinematics.
5.1 Preliminaries
∗
Let first the controller matrices Ui’s to take the values U ’s in (48)-(52). So, using these constant
i
system matrices in (1),(4)-(6) and (14), a fixed controller can be designed.
∗
For this particular controller we consider the resulting Δ Qt from (47) accomplishing
n
T
Δ ∗
Qt = η hK
+
(63)
n
t
p hKp − 2 I η
n
tn
T
+
∗
∗
v t hK
+
n
v
hKv − 2 I v tn
+
− 1
f ∗
Δ
[ ε
Q
η
, εv
, δηt , δv t , M M],
n
n+ 1
n+ 1
n
n
where f ∗
Δ Q is the sum of all errors obtained from (47) with (53) and (54). It fulfills with
n
p δ =r
t
δ
n
tn
f ∗
Δ
=
+
[
=
]+
[
=
]
Q
fΔ
fΔ
p δ
r δ
f
p δ
r δ .
(64)
n
Q 1 n
Q 2 n
tn
tn
Uin
tn
tn
Later, a norm of f ∗
Δ Q will be indicated.
n
− 1
Since εη
+ δη
−δη
, δv t
− δv t + εv
∈ l∞ and M M ∈ l∞, then one concludes
n+ 1
tn+ 1
tn
n+ 1
n
n+ 1
f ∗
Δ
∈
Q
l∞ as well.
n
∗
So, it is noticing that Δ Q <
t
0, at least in an attraction domain equal to
n
B = η
∈ R6 ∩ B∗
t , v t
n
n
0
,
(65)
with B∗0 a residual set around zero
B∗= η
∈R6
∗ −
≤
0
t , v t
/Δ Q
f ∗
0
(66)
n
n
tn
Δ Qn
and with the design matrices satisfying the conditions
2 I > K
h
p ≥ 0
(67)
2 I > K∗
h
v ≥ 0,
(68)
A General Approach to Discrete-Time Adaptive Control Systems with Perturbed Measures for
Complex Dynamics - Case Study: Unmanned Underwater Vehicles
269
which is equivalent to
2 M ≥ 2 M > K
h
h
v ≥ 0.
(69)
The residual set B∗0 depends not only on εη
and εv
and the measure noises δηt and δv t ,
n+ 1
n+ 1
n
n
− 1
but also on M M. In consequence, B∗0 becomes the null point at the limit when h → 0, δηt ,
n
δv t → 0 and M = M.
n
5.2 Stability proof
The problem of stability of the adaptive control system is addressed in the sequel. Let a
Lyapunov function be
15
6
∼T
∼
Vt = Q + 1 ∑ ∑ u
Γ − 1 u
−
(70)
n
tn
2
j
i
j i
i=1 j=1
i
n+ 1
n+ 1
15
6
T
− 1 ∑ ∑ ∼
∼
u
Γ − 1 u
,
2
j
i
j i
i=1 j=1
i
n
n
∼
∗
∗
with u j
= u j−u
, where u j and u are vectors corresponding to the column j of the
i
j
j
n
in
∗
adaptive controller matrix Ui and its corresponding one U in the fixed controller, respectively.
i
Then the differences Δ Vt = V
− V can be bounded as follows
n
tn+1
tn
15
6
Δ
∼
Vt = Δ Q + 1 ∑ ∑ Δu T
Γ −1
u
+ ∼u
(71)
n
tn
2
j
j
j
i
i
i
i
i=1 j=1
n
n+1
n
15
6
15
6
= Δ
T
∼
T
Qt + ∑ ∑ Δu
Γ −1 u
− 1 ∑ ∑ Δu
Γ −1 Δu
n
j
j
j
i
i
i
2
j
i
i
i
i=1 j=1
n
n
i=1 j=1
n
n
T
15
6
≤
∂Δ
Δ
Q
∼
Q
tn
t − ∑ ∑
u
n
∂u
j i
i=1 j=1
j
n
T
15
6
∂Δ
≤
Q
Δ
∼
Q
tn
t − ∑ ∑
u
n
∂u
j i
i=1 j=1
j
n
≤ Δ Q∗t < 0 in B ∩ B∗
n
0 ,
with Δu j
a column vector of Ui
−Ui .
i
n+1
n
n
The column vector
Δu j
at the first inequality was replaced by the column vector
in
−Γ ∂Δ Q
∂Δ
t
Q
n
tn
i
∂
and then by −Γ
in the right member according to (58) and (60)-(61).
u
i
j
∂u j
So in the second and third inequality, the convexity property of Δ Qt in (60) was applied for
n
∗
any pair U = Ui , U = U .
n
i
This analysis has proved convergence of the error paths when real square roots exist from
T
b nb n− 4¯ a ¯ cn of (46).
270
Discrete Time Systems
T
If on the contrary 4 ¯ a ¯ cn > b nb n occurs at some time tn, one chooses the real part of the complex roots in (46). So a suboptimal control action is employed instead, In this case, it is valid
−
− − 1
τ
1
− 1
M
2 =
M b
( I − hK∗
.
(72)
n
2 a
n=
h
v )v tn
So it yields a new functional Δ Q∗∗
t
in
n
T
∗∗
Δ
∗∗
∗
Vt ≤ Δ Q = Δ Q + ¯ c
b
n
tn
tn
n −
1
4 h 2 nb n< 0 in B ∩ B0 ,
(73)
∗
where Δ Qt is (63) with a real root of (46) and B∗∗
n
0 is a new residual set. It is worth noticing
T
that the positive quantity
¯ cn − 1
4 h 2 b nb n
can be reduced by choosing h small. Nevertheless,
B∗∗0 results larger than B∗0 in (71), since its dimension depends not only on εη and εv but
n+ 1
n+ 1
T
also on the magnitude of
¯ cn − 1
4 h 2 b nb n
.
This closes the stability and convergence proof.