tn+ 1
∂v
n
∂ T
+ q δ [
+
2 ]+
∂η tn] δηt o[ δv
o[ δη2 ] ,
(21)
n
where εv
and ε
are the model local errors and Δ δv
= δv
−δv and Δ δη
=
n+ 1
η
t
t
t
n+ 1
n+ 1
n+ 1
n
tn+ 1
δη
−δη
t
. The functions o are truncating error vectors of the Taylor series expansions, all of
n+ 1
tn
T
T
T
T
2
∂
∂
∂
∂
them belonging to O[ h ]. Moreover, p δ
p δ
q δ
q δ
∂ ,
and
v
∂η , ∂v
∂η are Jacobian matrices of the system
which act as variable gains that strengthen the sampled-data disturbances along the path.
T
T
It is worth noticing that the Jacobian matrices ∂τn
n
∂
and ∂τ
v
∂η in (20) will be obtained from the
−
feedback law τn[ −
η
]
t , v t
of the adaptive control loop.
n
n
4. Sampled-data adaptive controller
The next step is devoted to the stability and performance study of a general class of adaptive
control systems whose state feedback law is constructed from noisy measures and model
errors.
A design of a general completely adaptive digital controller based on speed-gradient control
laws is presented in (Jordán & Bustamante, 2011). To this end let us suppose the control
goal lies on the path tracking of both geometric and kinematic reference as ηr and v r ,
tn
tn
respectively.
4.1 Control action
Accordingly to the digital model translation, we try out the following definitions for the exact
path errors
η = η + δη −η
t
(22)
n
tn
tn
rtn
− 1
− 1
v t = v + δv −J
˙ η + J
K
.
(23)
n
tn
tn
δ
p η
t
δ
n
rtn
tn
tn
A General Approach to Discrete-Time Adaptive Control Systems with Perturbed Measures for
Complex Dynamics - Case Study: Unmanned Underwater Vehicles
261
T
− 1
where Kp = Kp ≥ 0 is a design gain matrix affecting the geometric path error and Jδ means
tn
− 1
J [ η + δη ]
≡
+ δ −
≡
t
. Clearly, if η
0, then by (23) and (2), it yields v t
v t
v r
0.
n
tn
tn
n
n
tn
Then, replacing (18) and (19) in (22) for tn+1 one gets
η
=
− 1
+ η −η
+ δη
−δη
t
I − hJt J Kp η
(24)
n+ 1
n δtn
tn
rt
r
t
t
n
tn+ 1
n+ 1
n
+ ε
− 1
η
+ h Jt v t + Jt δv t + Jt J ˙ η
.
n+ 1
n
n
n
n
n δtn rtn
Similarly, with (18) and (19) in (23) for tn+ 1 one obtains
− 1
− 1
− 1
v t
= v + J ˙ η −J
˙ η
−J K
+
(25)
n+ 1
tn
δ
p η
t
δ
δ
n
rtn
t
r
t
n+1
tn+ 1
tn
n
+ − 1
− 1
Jδ
Kp η
+ εv + δv t −δv t + hM
p δ + τn .
t
t
n+ 1
n+ 1
n
t
n+1
n+ 1
n
We now define a cost functional of the path error energy as
T
T
Qt = η η +v v ,
(26)
n
t
t
n
tn
tn
n
which is a positive definite and radially unbounded function in the error vector space. Then
we state
Δ Qt = Q
− Q =
(27)
n
tn+ 1
tn
=
− 1
− 1
I − hJt J K
+ h J v + J δv + J J ˙ η
+
n δ
p
η
t
t
t
t
t
t
δ
n
tn
n
n
n
n
n
tn
rtn
2
2
+ η −η
+ ε
+ δη
−δη
−η +
r
η
tn
rt
t
t
t
n+ 1
n+ 1
n+ 1
n
n
+
− 1
− 1
− 1
− 1
v t + J
˙ η −J
˙ η
−J K
+ J
K
n
δ
p η
p η
t
δ
δ
δ
n
rtn
t
r
t
t
n+1
tn+ 1
tn
n
tn+1
n+ 1
2
2
+
− 1
hM
p δ + τ
− δv + ε
−v .
t
n + δv t
t
v
n
n+ 1
n
n+ 1
tn
The ideal path tracking demands that
lim Δ Qt = lim ( Qt
− Qt ) = 0.
(28)
t
n
n+ 1
n
n →∞
tn→∞
Bearing in mind the presence of disturbances and model uncertainties, the practical goal
would be at least achieved that {Δ Qt } remains bounded for t
n
n → ∞.
In (Jordán & Bustamante, 2011) a flexible design of a completely adaptive digital controller
was proposed. Therein all unknown system matrices ( Ci, Dq , D
i
l , B 1 and B 2) that influence the
stability of the control loop are adapted in the feedback control law with the unique exception
of the inertia matrix M from which only a lower bound M is demanded. In that work a
guideline to obtained an adequate value of that bound is indicated.
Here we will transcribe those results and continue afterwards the analysis to the aimed goal.
First we can conveniently split the control thrust τn into two terms as
τn = τ1 + τ ,
(29)
n
2n
262
Discrete Time Systems
where the first one is
τ
− 1
− 1
1 = −K
− 1 M J ˙ η + J K
+
(30)
n
v v tn
h
δ
p η
t
δ
n
rtn
tn
tn
+ − 1
− 1
Jδ
˙ η
−J
Kp η
−r δ ,
t
r
δ
t
t
n+1
tn+ 1
tn+1
n+ 1
n
T
with Kv = Kv ≥ 0 being another design matrix like Kp, but affecting the kinematic errors
instead. The vector r δ is
tn
6
r δ = ∑ U
v
+ U
+
(31)
t
δ
n
i. × Cvi
7v δ
tn
tn
tn
i=1
6
+ ∑ U 7+ i|vi |v + U
+ U
,
t
δ
n
tn
14g 1δ
15 g 2
t
δ
n
tn
i=1
where the matrices Ui in r δ will account for every unknown system matrix in p
in order
t
δ
n
tn
to build up the partial control action τ1 . Moreover, the U
n
iś represent the matrices of the
adaptive sampled-data controller which will be designed later. Besides, it is noticing that r δtn
and p δ contain noisy measures.
tn
The definition of the second component τ2 of τ
n
n is more cumbersome than the first
component τ1 .
n
Basically we attempt to modify Δ Qt farther to confer the quadratic form particular properties
n
of sign definiteness. To this end let us first put (30) into (27). Thus
Δ Qt = Q
− Q =
(32)
n
tn+ 1
tn
=
− 1
− 1
I − hJt J K
+ h J v + J δv + J J ˙ η
n δ
p
η
t
t
t
t
t
t
δ
n
tn
n
n
n
n
n
tn
rtn
2
2
+ η −η
+ ε
+ δη
−δη
−η +
r
η
tn
rt
t
t
t
n+ 1
n+ 1
n+ 1