Discrete Time Systems by Mario A. Jordan and Jorge L. Bustamante - HTML preview

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