observa dor neuronal

7/31/2019 Observa Dor Neuronal

1/14

-

Scheme or Tra ector Trac inwith Constrained Inputs

1


2/14


3/14

Now let us consider the recurrent high order neural observer

( ) ( ) ( )

T

T

A Wz y bv t K y c

c

= + +

=

(4)

( )where is selected such that is Hurtwitz and ( ) is anTK A Kc v t.

Definin the state and out ut estimation error as

( ) ( ) ( )

whose dynamics is given by

t t t =

( ) * ( ) ( ) ( ) ( )

T

T

t A Kc W z y Wz y bv t = + +

=

(5)

3


4/14

Adding and substracting the term ( ), we obtainWz y

( ) ( ) ( ) ( )c zt A Wz y W y bv t = + + + (6)

*

T

c

W W WA A c K

= =

( ) , ( ) ( )

The term ( ) is bounded according to

y y z y z y

y

=

( )( ) - ( )

z y z y L y

y

The stabilizing signal ( ) is determined via the Lyapunov

methodology.

v t

4


5/14

Stability analysis

{ }

1 1

2 2

T TV P tr W W = + (7)where P 0 is a positive definite symmetric matrix whichsolves the Ricatti equation

>

T

C C

T

A P PA -Q

Pb c

+ =

=with positive definiteQ symmetric matrix.

The time derivative of is given byV

{ }1 1 1( ) ( )2 2

TT T

c cV P A f bv t A f bv t P tr W W

= + + + + + +

5

( , ) ( )zf W y y Wz y= +


6/14

which can be simplified as

1

( ) ( , ) ( )T T T

V PWz PW v t = + + +

{ }1 Ttr W W+

we now define the learning adaptation law

1 T T T

2

which can be written term b term as

r z y y z yb

= =

iw = 21

Ti j

b yz (y)b

Replacing the learning law in the Lyapunov time derivative,

we obtain

62

1 1

( , ) ( )2

T T

zV Q b W y y y yv t b = + +

(10

)


7/14

Applying the inequality

T T 11which holds for any vectors , , to the second termka b R

2 2 2 2

,

1 1 12

,2 4

1 1 1

y y y yv tb

+ + +

2

1 ( )2 4

y yv tb

+ + +

f

2 2 2

2

1 1 11

g

T

fV Q b W y

= + +

7gV y =


8/14

Then, the observer stabilizing signal ( ) which guaranteesv t

2 2

2

11v(t) b W y

= + (12)

( )1 , 1, gR Vy W >

( )2

ep ac ng , we o a

11 1

n

1T

v

QV

= + 2 2

b W y

(13)0

( )2 2

2

1

( ) 1 TA Wz y b b W K y cy

= + + + +

8 Ty c =


9/14

Once the observer stabilizing signal is obtained,

INVERSE OPTIMALITY ANALYSIS

we proceed to analyze its optimality with respect to

a cost functional defined by

( ) ( ) ( )( )0

lim 2 , ,t

T

tJ v V l W v R W v d

= + +

(14)The Lyapunov function solves the Hamilton-Jacobi-Bellman

equation.

( ) ( )12, 2 , 0

2 is bounded when

T

f g gl W V VR W V

V t

+ =

( )We require , positive defined and radial unbounded

with respect to ,

l W

9,l

( ) ( )

12

2 ,

T

f g t gW V VR e W V

= + (15)


10/14

Substituting (15) into (14), the learning adaptation law and then applying

( )( ) 2 2

2

21 1, 1

Tl W Q yb W

+ +

,

being positiv

,

e definite and radiallly unbounded.

Hence, (15) is a suitable cost functional which is evaluated replacing

T = , ,

into (13) to obtain

=

10This optimal value is achieved by the stab ( )ilizing signal .v t


11/14

Consider the Van der Pol nonlinear oscillator d namical s stem

Simulation Example for Neural Observer.

1 2

20.5 0.5cos 1.1

x x

x x x x t

=

= +

1

0 0 0.25

y x

x=

=

We use the neural observer given by (4) with

= 400 2K 600 470=

[ ] (0) 0.5 0.5 =

high order terms and another simulation considering them.

For the second case, we consider 10 hi h order

11terms ( )( ) tanh( ) 0.65i

iz x ky k= =


12/14

RHONO without high order terms

12Time evolution of the estimated states 2Estimation error forx


13/14

RHONO with ten high order terms

13Time evolution of the estimated states 2Estimation error forx


14/14

RHONO with ten high order terms

14

observa dor neuronal

Documents