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

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.