Identificaçao Linear a Parametros Variantes No Tempo de Sistemas Nao-lineares

7/23/2019 Identificaao Linear a Parametros Variantes No Tempo de Sistemas Nao-lineares

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a

a


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2nd

4th


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m

k= sen(x)

m k= sen(x)

m

k= cos(x)

m

k= cos(x)

m

k= cos(x)

m

k= sen(x)

m1

6


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m1

4

m2

m2

10


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a

m

k= cos(x)

m

k=cos(x)

a

k= cos(x)

m

k= cos(x)

m

k= cos(x)

m

k= sen(x)

m

k=sen(x)

a

k= sen(x)

m

k= sen(x)

m k= sen(x)

a

m1 F1

m2 F2

a

m1 x1

m2 x2


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m1 v1

m2 v2

BZ

m

k= cos(x)

m

k= sen(x)


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a

JId JV al

JId JV al

a

JId JV al

m

k= cos(x)

JId JV al

a

JId JV al

JId JV al

JId

JId

JId JV al

a

JId JV al

JId JV al

JId JV al

m

k =

sen(x)

m

k= sen(x)

m

k =


14/106

cos(x)

m

k= cos(x)

m

k= cos(x)

m

k= sen(x)

m1 x1

m1

v1

m2 x2

m2

v2


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y(t)

u(t)

yk k

uk k

J

JId

JV al

G(j) G(s)

N(, s)

D(, s)

L

JS() JS()

< A, B >

a >> b

a

b


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yp

ye

J=||yp ye||

Gp(j)

Ge(j)

J=||Gp(j) Ge(j)||

Gs(j)

= [w1w2 wm]

Gne (, s) =

nk=0

kPk(s) =N(, s)

D(, s) =0 s

n +1 sn1 + +n1 s+n

sn +1 sn1 + +n1 s+n

Pk(s) = snk

0 sn +1 sn1 + +n1 s+n

= [0 1 n]T

= [1 2 n]T

= [T T]T


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Gne (, s)

= [T T]T

JS() =||D(, j) G(j) N(, j)||22

D(, j) G(j) =

D(, j1)G(j1)

D(, j2)G(j2)

D(, jm)G(jm)

Gne (s) D(s) N(s)

JS() =m

i=1

|D(, ji) G(ji) N(, ji)|2

JS() =m

i=1

(ji)(ji)

(ji) =D(, ji)G(ji) N(, ji)

(ji) = (ji)

= [TT]T

JS()

JS() =

JS0

JS1

JSn

JS1

JS2

JSn

T= 0(2n+1) 1

t k k = 0, , n

l

l= 1, , n

JSt

=m

i=1

(ji)

t(ji) + (ji)

(ji)

t

Rn(ji) (ji)

n

(ji)

k l

(ji)

k= [ji]

nk =Rnk(ji)

(ji)

l = [ji]nl

G(ji) =Rnl(ji)G(ji)


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JSk

=m

i=1

[Rnk(ji)(ji) (ji)R

nk(ji)]

=

mi=1

2Re[Rnk(ji)

(ji)]

=m

i=1

2Re{Rnk(ji)[D(, ji)G

(ji) N(, ji)]}

=m

i=1

2Re[Rnk(ji)N(, ji) Rnk(ji)G

(ji)D(, ji)]

l

JS

l

=m

i=1 [Rnl(ji)G(ji)(ji) + (ji)R

nl(ji)G(ji)]

=m

i=1

2Re[Rnl(ji)G(ji)(ji)]

=m

i=1

2Re{Rnl(ji)G(ji)[D(, ji)G

(ji) N(, ji)]}

=m

i=1

2Re[Rnl(ji)(ji)D(, ji) Rnl(ji)G(ji)N

(, ji)]

(ji) =G(ji)G

(ji) Re[]

JSk

= 0 JSl =

0

n

k

l

Q = Y

Q1 Q2Q3 Q4

= Y1

Y2

Rn = Rn(ji) G = G(ji)

= (ji) =G(ji)G(ji) Q1, Q2, Q3, Q4 Y1 Y2

Q1=m

i=1

Re

RnR

n RnR

n1 RnR

0

Rn1R

n Rn1R

n1 Rn1R

0

R0Rn R0Rn1 R0R0

R(n+1) (n+1)


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Q2 =m

i=1Re

RnR

n1G RnR

n2G RnR

0G

Rn1R

n1G Rn1R

n2G Rn1R

0G

R0R

n1G R0R

n2G R0R

0G

R(n+1) n

Q3=m

i=1

Re

Rn1R

nG Rn1R

n1G Rn1R

0G

Rn2R

nG Rn2R

n1G Rn2R

0G

R0R

nG R0R

n1G R0R

0G

R

n (n+1)

Q4=m

i=1

Re

Rn1Rn1 Rn1Rn2 Rn1R0

Rn2R

n1 Rn2R

n2 Rn2R

0

R0R

n1 R0R

n2 R0R

0

Rn n

Y1=m

i=1Re

RnR

nG

Rn1R

nG

R0R

nG

Rn+1

Y2=m

i=1

Re

Rn1R

n

Rn2R

n

R0R

n

R

n


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x(t) = A((t)) x(t) +B((t)) u(t)

y(t) = C((t)) x(t) +D((t)) u(t)

A

B

C

D

(t)

(t)

(t) x(t)

(t) = [Tx (x(t))

Tp(t)]

Tx (x(t))

Tp(t)


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Gi(s) i {0, 1, , v} v

Gi(s)

M

Gi(s)

Gi(s) =bi0s

n +bi1sn1 + +bi(n1)s+bin

sn +ai1sn1 + +ai(n1)s+ain

aij bi R j {1, 2 , n} {0, 1 , n}

Gi(s)

i

yi(t) = yi(t) +yN i

ui(t) = ui(t) +uN i

yN i uN i R i

Gi(s) y(t) u(t)

y(t)

u(t)

Gi(s) Gi(s)

Yi(s) = L[yi(t)]

Yi(s) = L[yi(t) +yN i]

Yi(s) = Yi(s) +yN i

s

Ui(s) =Ui(s) +

uNis

Gi(s) = Yi(s)

Ui(s)=

Yi(s) + yNi

s

Ui(s) + uNis=

s Yi(s) +yN i

s Ui(s) +uN i


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yN i = bi(n+1) uN i = ai(n+1)

Gi(s) =bi0s

n+1 +bi1sn + +bins+bi(n+1)

sn+1 +ai1sn + +ains+ai(n+1)

j {0, 1, , (n+ 1)}

aj = [a1j a2j avj ]T

bj = [b0j b1j bvj ]T

j = [1j 2j vj ]T

ai bi

G(s, ) =b0()s

n+1 +b1()sn + +bn()s+bn+1()

sn+1

+a1()sn

+ +an()s+an+1()

G(s, )

G

G(s, )

A() B()

C() D()

C()[sI A()]1B() +D()

x(t)y(t)

= A() B()C() D()

x(t)u(t)

ai() bi()

f() =0

p + 1p1 + + p p N

p

aj =

p1 p11 1

p2 p12 1

1

pk p1k 1

0

1

p

=Xj

bj

bj =

p1 p11 1

p2 p12 1

1

pk p

1k 1

0

1

p

=Xj


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ai bj

M(q, ) =B(q, )

A(q, )=

b0() +b1() q1 + +bnb() q

nb

1 +a1() q1 + +ana() qna

q1

n= na +nb+ 1

ai()

bj() i [0, 1, , na] j [0, 1, , nb] ai

bj

ai() = a1

i

+a2i

+ +aNi

N1

bj() = b1j +b

2j + +b

Nj

N1

M(q, )

aki b

lj

k, l [1, 2, , N]

a11 a21 a

N1

a12 a22 a

N2

a1na a2na a

Nna

b10 b20 b

N0

b1nb b2nb b

Nnb


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k k k

yk1

ykna

uk

uknb

[1 k 2k

N1k ]

k =(k)

A

B

< A, B >=tra o(A B) =tra

o(B A)

A

A

yk =< , k >

J=J() = 1

T

Tk=0

E{(k, )2}

(k, ) =yk< , k >

k k

= k k1=1

2 g(k1)

g(k1)

g() =dJ()

d

J() =(k, )2


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k+1 = k+1 1

2

d

d

(k, )2

=k

(k, ) =yk< , k >=yk (Tk) d

dX

(XTB) =B

1

2

d

d(2) = (

d

d(yk (

Tk)))

= (k)

=k

k = k k

k = yk< k, k >=yk

k k

k+1 = k+ k k

0

k

k uk yk

ai() bj (


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T

1/T

y(kT)

k

y(k)

yk T

x(t) = limt0

x(t+ t) x(t)t

T 0 x(t)

T

x(k)=x(k+ 1) x(k)

T

A(q) y(k) = B(q)

F(q)u(k) +

C(q)

D(q)v(k)

y(k) = B(q)

F(q)A(q)u(k) +

C(q)

D(q)A(q)v(k)

y(k) = H(q) u(k) +G(q) v(k)

q1

y(k)q1 =y(k 1) v(k)

A(q)

B(q)

C(q)

D(q)

F(q)

A(q) = 1 a1 q1 any q

ny

B(q) = 1 b1 q1 bnu q

nu

C(q) = 1 c1 q1 cnv q

nv

D(q) = 1 d1 q1 dnd q

nd

F(q) = 1 f1 q1 fnf q

nf

H(q)

G(q)

H(q) q = z


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Z

h(k)

H(z)

H(q)

q1

z1

C(q) =D(q) =F(q) = 1

A(q) y(k) = B(q) u(k) +v(k)y(k) =

B(q)

A(q)u(k) +

1

A(q)v(k)

H(q) = B(q)A(q)

G(q) = 1

A(q)

p

q

y(k) = A1y(k 1) +A2y(k 2) + +Anyy(k ny) +

B1u(k 1) +B2u(k 2) + +Bnuu(k nu) + e(k)


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Ai R

p p

Bi Rp r

y(k) = [y1(k) y2(k) yp(k)]T

u(k) = [u1(k) u2(k) ur(k)]T

e(k) = [e1(k) e2(k) ep(k)]T

A(q) = I A1 q1 Any q

ny

B(q) = B1 q1 + +Bnu q

nu

y1(k)

y2(k)

=

0, 4 1, 2

0, 3 0, 7

y1(k 1)

y2(k 1)

+

0, 35 0, 3

0, 4 0, 5

y1(k 2)

y2(k 2)

+

1 0

0 1

u1(k 1)

u2(k 1)

y(k) = T (k 1) + e(k)

= [A1 A2 Any B1 B2 Bnu]T

(k 1) = [yT(k 1) yT(k 2) yT(k ny)

uT(k 1) uT(k 2) uT(k nu)]T

k

xk+1 = Ak xk+Bk uk

yk = Ck xk+Dk uk

a

yk =a1k yk1 a2k yk2 + b0k uk + b1k uk1 + b2k uk2


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Ak1 Bk1

xk = a1k 1

a2(k+1) 0 Ak1

x1(k1)

a2k yk

2+b2k uk

2 xk1

+ b1k a1k b0(k1)

b2(k+1) a2(k+1) b0(k

1) Bk1

uk1

a

Ak =

a1(k+1) 1

a2(k+2) 0

Bk = b1(k+1) a1(k+1) b0(k)

b2(k+2) a2(k+2) b0(k)

Ck = [1 0]

Dk = b0k

n

Ak =

a1(k+1) 1 0 0

a2(k+2) 0 1 0

an(k+n1) 0 0 1

an(k+n) 0 0 0

Bk =

b1(k+1) a1(k+1) b0(k)

b2(k+2) a2(k+2) b0(k)

bn(k+n) a2(k+n) b0(k)

Ck = [1 0 0 0]

Dk = b0k

a

a

yk =a1k yk1 a2k yk2+b0k uk+b1k uk1+b2k uk2


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k


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k= 3

y3 = [1 0]

a13[b12 u1 a12 b01 u1] +b23 u1 a23 b01 u1+b13 u2 a13 b02 u2

a24[b12 u1 a12 b01 u1] +b24 u2 a24 b02 u2

+b03 u3y3 = a13 [b12 u1 a12 b01 u1] +b23 u1 a23 b01 u1+b13 u2 a13 b02 u2+ b03u3

y3 = a13 [a12 b01 u1+b02 u2+b12 u1] y2

a23 b01 u1 y1

+b03u3+b13 u2+b23 u1

y3 = a13 y2 a23 y1+b03u3+b13 u2+b23 u1

Ak Bk Ck Dk k+1 k+2 k+n

n

y= f(x)

N

x

y1 = f(x1)

y2 = f(x2)

=

yN = f(xN)


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y=X

X Rn n i i X i

n i R

n

xi R

y

x

1 x

2

xn

X

X

=X1y

n= N N >> n

X RN n

N > n

X RN n y RN 1 Rn 1 X

XT

XTy=XTX

XTy Rn 1

XTX Rn n

XTX

= [XTX]1XTy

[XTX]1XT

y

y=

T +

N > n

y=X +


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J=N

i=1 (i)2 =T= || ||2

J

y

J

J = (y X)T(y X)

= yTy yTX T

XTy +T

XTX

J

J

= 0

J

= (yTX)T XTy + (XTX+XTX)

= (XTy) XTy + 2XTX

= 2(XTy) + 2XTX

2(XTy) + 2XTX = 0

= [XTX]1 XTy

J 2J

2 = 2XTX >0

2XTX

u(t)

y(t)

e(t)

yk =a1 yk1 a2 yk2 ana ykna+ b0 uk+ b1 uk 1 + + bnb uknb


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ai bj i {1, 2, , na} j {0, 1, 2, , nb}

uk yk

Tk = [yk1 yk2 ykna uk uk1 uknb]

T = [a1 a2 ana b0 b1 bnb]

k R

(na+nb+1) 1

R(na+nb+1) 1

yk =Tk +ek

N

ai bj

y0

y1

yN1

=

T0

T1

TN1

+

e0

e1

eN1

Y = + e

J=N

k=1

(yk Tk )

2

X

(XTX)

X


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X Rn n

X= B

B Rn 1

I Rn n

B

B

B

A= X|B Rn (n+1)

i

n

i

k

aki

i

aii

k= 1

n

k=i akj

akj aij

A = I|

X

B


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G

u

y

uk yk k {1, 2, , m} m

yk

yk =a1 yk1 a2 yk2 an ykn + b0 uk + b1 uk1 + + bn ukn

ai bj i {1, , n} j {0, 1, , n}

G

G

ai

bj

k

yk =a1kyk1 a2k yk2 ank ykn + b0k uk + b1kuk1 + + bnk ukn

k

k

k+ 1

G

[min, max]

ai bj


45/106

aik bjk

aik =i0+i1 k1+ +iN Nk1

bjk =j0+j1 k1+ +jN Nk1

i j [0, , N] N

G

i

j

uk

k


46/106

y

G

y G

y

y

J= 1

m

m

k=1

(yk yk)2

aik bjk

yk =

ni=1

yki aik+

nj=0

ukj bjk

yk = n

i=1

yki (i0+i1 k1+ +iN Nk1) +

nj=0

ukj (j0+j1 k1+ +jN Nk1)

i i

yk =

ni=1

yki [ 1 k1 Nk1 ] [i0 i1 iN]T +

nj=0

ukj [ 1 k1 Nk1 ] [j0 j1 jN]

T

i [1, 2, , n]

Tik yki [ 1 k1 Nk1 ]

i [i0 i1 iN]T

j [0, 1, , n]

T(j+n+1)k ukj [ 1 k1 Nk1 ]

(j+n+1) [j0 j1 jN]T

Tk = [ T1k

T2k

T(2n+1)k ]


47/106

=

1

2

2n+1

yk = Tk

k k yk1 yk2

ykn uk ukn k1

i j G

J= 1

m

mk=1

(yk Tk )

2

m

Y

=

T1

T2

Tm

Y =

y1

y2

ym

Y =

Y m 1

m (2n+1)N

T

T Y P

= T Q

Q (2n+1)N 1 P (2n+1)N (2n+1)N


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G

p

yk = a1k yk1 a2k yk2 ank ykn

+b10k u1k+b

11k u

1k1+ +b

1nk u

1kn

+b21k u2k1+b

21k u

2k1+ +b

2nk u

2kn

+ +

+bp0k upk+b

p1k u

pk1+ +b

pnk u

pkn

yk = n

i=1

yki aik+

pr=1

nj=0

urkj brjk

aik i [1, , n] y

k

brjk j [0, , n] r [1, , p]

y

ur

ai brj

aik = i0+i1 k1+ +iN Nk1

brjk = r

j0+r

j1 k1+ +r

jN Nk1

aik b

rjk

yk =

ni=1

yki (i0+i1 k1+ +iN Nk1) +

pr=1

nj=0

urkj (r

j0+r

j1 k1+ +r

jN Nk1)

i

ri

yk = n

i=1

yki [ 1 k1 Nk1 ] [i0 i1 iN]

T +

p

r=1n

j=0 urkj [ 1 k1 Nk1 ] [rj0 rj1 rjN]T


49/106

i [1, 2, , n]

Tik yki [ 1 k1 Nk1 ]

i [i0 i1 iN]T

j [0, 1, , n] r [1, 2, , p]

rTjk urkj [ 1 k1

Nk1 ]

rj [r

j0 r

j1 r

jN]T

T1k [T1k

T2k

Tnk ]

T2k [1T0k

1T1k

1Tnk ]

T3k [2

T0k 2

T1k 2

Tnk ]

T(p+1)k [pT0k

pT1k

pTnk ]

1 [T1

T2

Tn ]

T

2 [1T

0 1T

1 1T

n ]T

3 [2T0

2T1

2Tn ]

T

p+1 [pT0

pT1

pTn ]

T

Tk [T1k

T2k

T(p+1)k]

1

2

p+1

yk = Tk

Tk

1 N[n(p+1)+p]

N[n(p+1)+p] 1

p= 1


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u

y

G i j

i

rj

vk

wk

(i)k+1=

(i)k +w

(i)k

yk+a1k yk1+ +ank ykn=b1k uk1+ +bmk ukm+vk

{w

(i)k } vk

{w

(i)k } {w

(j)k }

i=j

k+1 = IN(2n+1) N(2n+1)

F

k+wk

yk = Tk

HTk

k+vk

F =I

HTk

xk = Fk1 xk1

Pp(k) = Fk1 Pe(k1) FT

k1+Qk1

Kf(k) = Pp(k) HTk [Hk Pp(k) H

Tk +Rk]

1

xk = xk+Kf(k)[yk Hk xk]

Pe(k) = [I Kf(k) Hk] Pp(k)

x

x

Pp

Pe Q


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R

Kf


52/106

yk

n

n

na

nb

na nb

yk

uk

D= 0

JId JV al

a

na nb1

nb2

b0

a

JId

JV al

na nb1 nb2 N b0 JId

m1 13

13

m2 16

16

T

T

uk =cu yk =cy k =c k T


53/106

u

y

n

n

t1 t2 t3 tn y y1 y2

yn u u1 u2 un 1 2 n


54/106

t1 y1 u1 1

G1

G1 y1

G2 t2

y1

T y2T

T

y1 y2

Gn

y1 y2 yn1 yn

na

nb

N

k


55/106


56/106


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58/106

y=

1 0

0 1

x1

x2

+

0

0

F

k= cos(x)

=x(t)

m = 1Kg

b = 1Ns/m

k = sen(x)

3

10

m

k= cos(x)

JId

JV al

JId JV al

a

na nb N b0 JId JV al

k = cos(x)

8

5

8 6

k= sen(x)

10

3

9 6


59/106

JId JV al

m

k = cos(x)


5

7

3

8 5

m

k=cos(x)

m

k=cos(x)


60/106

a

k= cos(x)

m

k=cos(x)

m k= cos(x)


61/106

m

k= sen(x)

m

k= sen(x)

a

k = sen(x)


62/106

m

k= sen(x)

m

k= sen(x)

F1 F2

m1

m1 xm1=F1+Fk2+Fb2 Fk1 Fb1

Fk1 = k1 xm1


63/106

a

Fk2 = k2 (xm2 xm1)

Fb1 = b1 xm1

Fb2 = b2 (xm2 x1)

m1 xm1 = F1+k2(xm2 xm1) +b2(xm2 xm1) k1 xm1 b1 xm1

xm1 = 1

m1[F1+k2(xm2 xm1) +b2(xm2 xm1) k1 xm1 b1 xm1]

m2

m2 xm2 = F2 Fk2 Fb2

m2 xm2 = F2 k2(xm2 xm1) b2(xm2 xm1)

xm2 = 1

m2[F2 k2(xm2 xm1) b2(xm2 xm1)]

x1 = xm1 x1 = xm1=x2

x2 = xm1 x2 = xm1

x3 = xm2 x3 = xm2=x4

x4 = xm2 x4 = xm2

x1

x2

x3

x4

=

0 1 0 0

k1+k2m1

b1+b2m1

k2m1

b2m1

0 0 0 1

k2m2

b2m2

k2m2

b2m2

x1

x2

x3

x4

+

0 0

1m1

0

0 0

0 1m2

F1

F2


64/106

m1 m2

y=

1 0 0 0

0 1 0 0

0 0 1 00 0 0 1

x1

x2

x3

x4

+

0 0

0 0

0 00 0

F1

F2

k1 m1

k1 = cos(xm1) = cos(x1)

m1 =

m2= 3Kg

k1 = 1000 cos(x1)N/m

k2 = 1000N/m

b1=b2= 1Ns/m

na nb1 F1 nb2

F2 N

xm1 xm2 m1 m2

vm1 vm2

9

16

JId JV al

a

na nb1 nb2 N b0 JId JV alm1

13

5

9 2

m2 16

6

12

5


65/106


66/106

m1 x1

m2 x2

m1

v1


67/106

m2 v2

c(t) (t)

q(t)

(t)

q

=

Z 1 Z 0

M 0 M 0

0 0 0 1

0 0

2

a 2a

q

+

0

0

0

2

a

c


68/106

q

=

N 0 N 0

0 1 0 0

q

Z Z M M N N

Z = K M cos() [an 2 +bn || +cn(2 M/3)]

Z = K M dn cos()

M = Kq M2 [am

2 +bm || +cm (7 + 8M/3)]

M = Kq M2 dm

N = Kz M2 [an

2 +bn || +cn (2 M/3)]

Nz = Kz M2 dn

(t)

= 0

=||

K = 0, 7 P0180Smvs

m = 13, 98slugs

Kz = 0, 7 P0 Smg vs = 1036, 4 p s/s

Kq = 0, 7 P0180SdIy

Ax = 0, 7 P0SCam

an = 0, 000103 3

d = 0, 75 p s bn = 0, 00945 2

Iy = 182, 5 slug.p s2

cn = 0, 1696 1

Ca = 0, 3 dn = 0, 034 1

= 0, 7

am = 0, 000215 3

a = 150rad/s bm = 0, 0195 2

cm = 0, 051 1

g = 32, 2 p

s/s2

dm = 0, 206

1

P0 = 973, 3 lbs/p s2

S = 0, 44 p s2


69/106


70/106


71/106

Z Z M M N N

c

G(s, )|=0 = b2(0) s

2 +b3(0) s +b4(0)

s4 +a1(0) s3 +a2(0) s2 +a3(0) s +a4(0)

a1() = 2 a Z a2() = a M 2 a Z

a3() = Z a M 2 a a4() = M a

b2() = 0

b3() = Z 2a

b4() = M 2a

b2() = N 2a

b3() = N Z 2a

b4() = N M 2a


72/106


73/106


74/106


75/106


76/106

BZ


77/106

G(s, t)|t=t0 = b1(t0) s

2 +b2(t0) s

s3 +a1(t0) s2 +a2(t0) s+a3(t0)


78/106


79/106

JId

na nb N b0 JId

6


80/106

a

6

m

k= cos(x)

m

k= sen(x)


81/106

JId JV al

a


k= cos(x) 5 5

k= sen(x)

4

3


82/106

JId JV al



83/106


84/106


85/106


86/106


87/106

m

u

y

yk

na nb N

na nb N

a

yk uk

Gn

n 1


88/106

Y


89/106


90/106

2


91/106


92/106


93/106


94/106


95/106


96/106


97/106


98/106


99/106

M

K=SEN(X)

m

k= sen(x)

10 10 30

11 11 31

12 12 32

13 13 33

14 14 34

15 15 35

16 16 36

20 20

21 21

22 22

23 23

24 24

25 25

26 26

M

K=SEN(X)

m

k = sen(x)

10 40 70

11 41 71

12 42 72

13 43 73

14 44 74

15 45 75

20 50 10

21 51 11

22 52 12

23 53 13

24 54 14

25 55 15

30 60 20

31 61 21

32 62 22

33 63 23

34 64 24

35 65 25


100/106

M

K=C OS(X)

m

k= cos(x)

10 10 40

11 11 41

12 12 42

13 13 43

14 14 44

15 15 45

16 16 46

20 20 50

21 21 51

22 22 52

23 23 53

24 24 54

25 25 55

26 26 56

30 30 60

31 31 61

32 32 62

33 33 63

34 34 64

35 35 65

36 36 66

M

K=C OS(X)

m

k= cos(x)

10 50 90 20

11 51 91 21

12 52 92 22

13 53 93 23

14

54

94

24

15 55 95 25

16 56 96 26

17 57 97 27

18 58 98 28

19 59 99 29

110 510 910 210

20 60 100 30

21 61 101 31

22 62 102 32

23 63 103 33

24 64 104 34

25 65 105 35

26 66 106 36

27 67 107 37

28 68 108 38

29 69 109 39

210 610 1010 310

30 70 00 40

31 71 01 41

32 72 02 42

33 73 03 43

34 74 04 44

35 75 05 45

36 76 06 46

37 77 07 47

38 78 08 48

39 79 09 49

310 710 010 410

40 80 10

41 81 11

42 82 12

43 83 13

44 84 14

45 85 15

46 86 16

47 87 17

48 88 18

49 89 19

410 810 110


101/106

M

K=C OS(X)

m k=cos(x)

10 50 90 20

11 51 91 21

12 52 92 22

13 53 93 23

14 54 94 24

15 55 95 25

16 56 96 26

20 60 100 30

21 61 101 31

22 62 102 32

23 63 103 33

24 64 104 34

25 65 105 35

26 66 106 36

30 70 00 40

31 71 01 41

32 72 02 42

33 73 03 43

34 74 04 44

35 75 05 45

36 76 06 46

40 80 10 50

41 81 11 51

42 82 12 52

43 83 13 53

44 84 14 54

45 85 15 55

46 86 16 56

M

K=SEN(X)

m

k= sen(x)

10 40 70 100

11 41 71 101

12 42 72 102

13 43 73 103

14 44 74 104

15 45 75 105

20 50 80 10

21 51 81 11

22 52 82 12

23

53

83

13

24 54 84 14

25 55 85 15

30 60 90 20

31 61 91 21

32 62 92 22

33 63 93 23

34 64 94 24

35 65 95 25


102/106

M1

6

m1 x1

10 40 2

20

11 41 2

21

12 42 2

22

13 43 2

23

14 44 2

24

15 45 2

25

20 1

10 2

30

21

1

11

2

31

22 1

12 2

32

23 1

13 2

33

24 1

14 2

34

25 1

15 2

35

30 2

10 1

40

31 2

11 1

41

32 2

12 1

42

33 2

13 1

43

34 2

14 1

44

35 2

15 1

45

M1 4

m1 v1

10 1

00 1

50 2

40

11 1

01 1

51 2

41

12 1

02 1

52 2

42

13 1

03 1

53 2

43

20 1

10 2

00 2

50

21 1

11 2

01 2

51

22 1

12 2

02 2

52

23 1

13 2

03 2

53

30 1

20 2

10

31 1

21 2

11

32 122 212

33 1

23 2

13

40 1

30 2

20

41 1

31 2

21

42 1

32 2

22

43 1

33 2

23

50 1

40 2

30

51 1

41 2

31

52 1

42 2

32

53 1

43 2

33


103/106

M2

m2 x2

10 1

00 1

40 2

20

11 1

01

1

41

2

21

12 1

02 1

42 2

22

20 1

10 1

50 2

30

21 1

11 1

51 2

31

22 1

12 1

52 2

32

30 1

20 2

00 2

40

31 1

21 2

01 2

41

32 1

22 2

02 2

42

40 1

30 2

10 2

50

41 1

31 2

11 2

51

42 1

32 2

12 2

52

M2

10

m2 v2 10 40

2

10 2

40

11 41 2

11 2

41

12 42 2

12 2

42

13 43 2

13 2

43

14 44 2

14 2

44

15 45 2

15 2

45

16 46 2

16 2

46

17 47 2

17 2

47

18 48 2

18 2

48

20 1

10 2

20

21 111 221

22 1

12 2

22

23 1

13 2

23

24 1

14 2

24

25 1

15 2

25

26 1

16 2

26

27 1

17 2

27

28 1

18 2

28

30 1

20 2

30

31 1

21 2

31

32 1

22 2

32

33 1

23 2

33

34 1

24 2

34

35 1

25 2

35

36 1

26 2

36

37 1

27 2

37

38 1

28 2

38


104/106

10 60 30 80

11 61 31 81

12 62 32 82

13 63 33 83

14 64 34 84

15 65 35 85

16 66 36 86

17 67 37 87

18 68 38 88

20 70 40 90

21 71 41 91

22 72 42 92

23 73 43 93

24 74 44 94

25 75 45 95

26 76 46 96

27 77 47 97

28 78 48 98

30 80 50

31 81 51

32 82 52

33 83 53

34 84 54

35 85 55

36 86 56

37 87 57

38 88 58

40 10 60

41 11 61

42 12 62

43 13 63

44 14 64

45 15 65

46 16 66

47 17 67

48 18 68

50 20 70

51 21 71

52 22 72

53 23 73

54 24 74

55 25 75

56 26 76

57 27 77

58 28 78


105/106

10 40 70

11 41 71

12 42 72

20 50 10

21 51 11

22 52 12

30 60

31 61

32 62

10 30 00

11 31 01

20 40 10

21 41 11

10 40 00

11 41 01

12 42 02

13 43 03

14 44 04

20 50 10

21 51 11

22 52 12

23 53 13

24 54 14

30 60 20

31 61 21

32 62 22

33 63 23

34 64 24


106/106

10

30

00

11 31 01

20 40 10

21 41 11

10 40 00

11 41 01

12 42 02

13 43 03

14 44 04

20 50 10

21 51 11

22 52 12

23 53 13

24 54 14

30 60 20

31 61 21

32 62 22

33 63 23

34 64 24

Documents

Identificaçao Linear a Parametros Variantes No Tempo de Sistemas Nao-lineares