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E
7
m
/
(2.18)
m
T h e class of functions satisfying the c o n d i t i o n 2
/ ' w(t)f (t)dt •'to
< oo
is k n o w n as square integrable over the i n t e r v a l [ i , * / ] 0
E x a m p l e 2.1
S i n e - C o s i n e F u n c t i o n s The set of
functions
{ 1 , cos 6, sinff, cos 20, sin 2 0 , . . . } ore orthogonal
over the interval
[ 0 , 2 t t ] with the weighting
function
w(6) = 1.
Indeed,
we have J
/ a
cos mddff = 7r; / o
2
f' /
sin m8d9 = n\
J
ft* cos m 0 cos r6d6 = 0; /
sin m0 sinrSdO
= 0;
cos mO sin r9d6 = 0; u where r ^ m.But
the set {
1
sin0
cos 26
cos 6 sin 26
is orthonormal. The above sets are called Fourier each of them is a complete set of functions . E x a m p l e 2.2 H a a r F u n c t i o n s
har(r,n,t)
sine-cosine
The set of functions
har(0,0,t)
=1,
|
2f,
==i < t
{i) is a p o l y n o m i a l of degree r differentiating Eq.2.24, ( r + 1) times we have r
d
T+1
\
1
r
d u {t) — T
= 0
(2.25)
2.2 Orthogonal
Polynomial
Approx.:
A Generalized
Approach
w h i c h is a d i f f e r e n t i a l e q u a t i o n for the new f u n c t i o n u .
37
As g _ i ( t ) is a r b i t r a r y , for
T
r
Eq.2.23 t o be satisfied we must have u (t )
=
u (t )
= i i ( l ) = • • • = u
0
. /
fw{t)dt
where r is a p o s i t i v e integer, exists. T h u s we have shown t h a t given the w e i g h t i n g f u n c t i o n w{t) and the i n t e r v a l [ t o , * / ] how we can generate t h e o r t h o g o n a l p o l y n o m i a l s (t) = /3 T
r0
+ 0 it r
2
+ • • • + T f~ r
+ a i r
r _ 1
+ ji f.
(2.29)
r
I f we choose (2.30) then
is a p o l y n o m i a l of degree at most r and cosequently can be expressed as a linear c o m b i n a t i o n of d> (i), d>i(t), • • • ,&•( (t)
- a td> (t)
r+l
T
=
r
2:Least
Squares
Approximation
of
Signals
b {t) + c _!(t) + e _ _ (t) + • • • r
r
r
r
r
2
r
2
+ £ ^ . ( t ) + £o {t) m
e
m
is a p o l y n o m i a l of degree less t h a n r.
As 7 ' s are positive, we must have m
= 0 , m = 0 , 1 , . . . , r - 2. Hence the relation of Eq.2.28 follows f r o m E q . 2 . 3 1 . B y v i r t u e of Eq.2.29 we can evidently w r i t e -Z~Mi)
* (t) 's, r > 1 possess
r distinct
r
real
zeros
0
Proof. Since (t) is a constant and w(t) 0
is nonnegative, for the o r t h o g o n a l r e l a t i o n
to be t r u e , 4> (t) must change sign at least once in [r-o>*/] and consequently 4> {t) must T
r
have at least one real zero i n [ i , i / ] . Let 0
4> (t) =
f} {t-t )---{t-t )
T
where 0
T
i
r
is the coefficient of the leading t e r m of ^> (z), see Eq.2.29. Factors of 4> (t)
T
r
T
associated w i t h complex conjugate zeros o c c u r r i n g in pairs and factors associated w i t h real zeros o f even m u l t i p l i c i t y do not change for t v a r y i n g i n [ t , i / ] . T h e r e m a i n i n g 0
factors c o n t a i n real zeros o f o d d m u l t i p l c i t y in [ t , * / ] w h i c h we denote by ( i — t i ) , ( i — 0
t ), 2
. . . , ( < — < ) where t\,t ,. m
2
.. , t
(t>r(t)
where g(t),
=
m
are d i s t i n c t and m < r. Let »(*)(
(t)dt m
r
= 0
or
Since the i n t e g r a n d is nonnegative, i t does n o t change sign w i t h t in [ i , i / ] and the 0
left h a n d side of the last r e l a t i o n is positive . T h i s c o n t r a d i c t i o n ensures t h a t d> (t) T
has r d i s t i n c t zeros i n [ < , t / ] . 0
Eq.2.1 does n o t give a convenient expression
for c o m p u t i n g the n o r m a l i z a t i o n
factor 7 . A suitable expression is o b t a i n e d using Eq.2.29, Eq.2.20 and E q . 2 . 2 1 . T h u s r
we get
(2.35)
40
Chapter
2:Least
Squares
Approximation
of
Signals
where the last step follows f r o m i n t e g r a t i o n by p a r t s . F r o m the foregoing discussion t i o n w(t)
we observe t h a t f r o m t h e given w e i g h t i n g func-
and the i n t e r v a l [ i , * / ] we can d e t e r m i n e the associated set of o r t h o g o n a l 0
p o l y n o m i a l s d> (t)'s w i t h the help of Eq.2.24-Eq.2.27.
A n y f u n c t i o n f(t)
r
p r o x i m a t e d by Eq.2.7 where / ' s
is t h e n ap-
are d e t e r m i n e d by Eq.2.15 and Eq.2.35.
r
above b a c k g r o u n d , we shall now see how different selections of w(t)
W i t h the
a n d the i n t e r v a l
[*o,*/] give rise t o the different classes of o r t h o g o n a l p o l y n o m i a l s .
2.3
Legendre
Polynomials
I n t h e case of Legendre p o l y n o m i a l s we assume t h a t w(t)
= 1 and [t , t/\ = [—1,1]. By 0
the linear t r a n s f o r m a t i o n (a — 6 ) r = — 2t + (a + 6), we can always change t belonging t o any a r b i t r a r y i n t e r v a l [a, 6] to r € [—1,1]. B y v i r t u e of Eq.2.25-Eq.2.27 we have to solve 2r+l
d u {t) r
— m)
ui (-l)
= u
1 we note t h a t cos(r + 1)8 + cos(r — 1)9 = 2 cos r 0 cos 8 We now p u t t = cos 9 and substitute from Eq.2.80 to get T (t)
= 2tT (t)
T+1
- T _,(t)
r
(2.83)
r
w h i c h is the three-term recurrence f o r m u l a for the Tchebycheff p o l y n o m i a l s of the first k i n d . P u t t i n g r = 0 , 1 , . . . i n Eq.2.81 and Eq.2.83 we can generate recursively all the Tchebycheff polynomials of the first k i n d , the first eight o f w h i c h are included in Table 2.4 T h e Rodrigues' f o r m u l a for X ( t ) ' s is given in Section 2.9. Tchebycheff polynomials of the first k i n d satisfy a second-order differential equat i o n . To derive i t , we note t h a t i f T = cosr9,t = cos 8 t h e n r
T
2
- 1 .
(2.98)
To compute the o r t h o g o n a l polynomials 0 ( t ) ' s given by Eq.2.24, we have to solve r
r
1 dt^
1
d u {t) r
w(t)
=
0
(2.99)
dV
subject to the conditions u (-l) T
r
=
u ( - l ) = ••• = u < - ° ( - l ) = 0
=
u ( l ) = • • • = u[ - (\)
r
(2.100) u (l) r
r l)
= 0
r
2
Based on our discussion in Section 2.3, we conclude t h a t u (t) w o u l d contain (1 — i ) as a factor. Moreover, to satisfy the Rodrigues' f o r m u l a , v i z . ,
r
T
4>Jt) v
(1 - « ) ° ( 1 + tY
w h i c h is a p o l y n o m i a l of degree r, u (t) r
=
-
d u (t) — T
dr
(2.101)
should be of the form
T
u (i)
T
1
=
;
2
d (l -
+ 0"(1 - * ) '
r
(2.102)
since in this case, the r i g h t hand side of Eq.2.101 w o u l d be a p o l y n o m i a l of degree r after r times of differentiation as can be verified by L e i b n i t z f o r m u l a Eq.2.47. Let us write v(t)
= u (t)
r +
=
r
d (l - (t) in Eq.2.101 are defined as the Jacobi polynoT
mials of degree r and are represented by
if d = r
(-l) z r!
r K
—r~
(2.104)
2.8 Jacobi
57
Polynomials
T h e Rodrigv.es'
formula
for Jacobi p o l y n o m i a l s is therefore given by
2"r!
(1 -
i)P
+
^
dt
T
J
I n v i e w of E q . 2 . 1 0 1 , Eq.2.103 a n d the L e i b n i t z f o r m u l a of Eq.2.47, we have the relation r
r
2 r!(l - i)°(l + t f P ^ \ t ) =
( - l )
r
£
m
C) ^
[(1 + 0
= (-l) D 1
r +
r
r +
[(1 + i )
[(1 -
t)
r +
r+a
"(l -
t) ]
"]
m = 0
=
r
("l) £
( L ) ( r + 0){r
+ /? - 1) • • • ( r + /? - m + 1)(1 + « )
r +
"-
m
m = 0 r
m
( - l ) ~ ( r + o;)(r + a - 1) • • • ( r + a - r + m + 1)(1 a
Hence t h e J a c o b i P o l y n o m i a l s P^ ^(t) r
t + l \
« -
are represented by
t
(fi
a + m
( a + 1) • • • (a + m)(a + m + 1 ) • • • (a + r )
m (
t)
(a + 1 ) • • • (a + m)r!
+ r - m + ! ) • • •(/? + ;•)
1
f E - 1
(—y V
m!
'
(
r
^ -
r
(r - m ) ! E + 1 '
_ i ) . . . (
( +l). Q
r
_
m
+
i)
-(a + m)
/ i U
n -
' W+ l '
t - ) ( ! ± l ) ' F ( - , ^ - f t .
+
l
!
(2.105)
£ ^ )
where F(a,6;c;r)
:=
(
£ m
=
0
~
m
a
1-2
(a)
m
(a)o
m
(2.106)
••••m
1) 6 ( 6 + ! ) • • - ( 6 + m - 1)
m
c(c + 1) • - • (c + m - 1)
1 + ~
(-J»)(l)
)
( a + 1 ) • • - (a + m -
a ( q + ! ) • • - (a + m - 1) 6 ( 6 + 1) • • - ( 6 + m - 1)
' ~ F
6
m
~~ =
(
°\ "« (c) m!
=
1-2
m
m
c(c + 1) • • • (c + m - 1)
(;+»)
=
a(a + 1) • • • (a + m - 1)
=
1
T h e f u n c t i o n F ( a , 6 ; c ; t ) is defined as the h y p e r g e o m e t r i c f u n c t i o n . r e p r e s e n t a t i o n of J a c o b i p o l y n o m i a l s is derived in Section 2.10.
(2.107)
A n alternative
58
Chapter
2.9
2:Least
Squares
Approximation
G a m m a , Beta andHypergeometric
of
Signals
Functions
To derive an a l t e r n a t i v e representation o f Jacobi p o l y n o m i a l s we must k n o w w h a t are G a m m a and Beta functions. A brief description of these functions w i l l now be given. T h e G a m m a f u n c t i o n T ( z ) is defined by the Euler's i n t e g r a l oo 1
e"'t*" dc,Re z > 0
/
(2.108)
u
I n t e g r a t i n g by parts we get r(z + l )
z
=
/
e'*t dt,
=
zT(z).
(2.109)
O b v i o u s l y for an integer m we can easily derive t h a t T ( m + 1)
=
m!
(2.110)
T(a + m )
=
(a + m - l ) T ( a + m - 1)
=
(o + m - 1) • • • o r ( a )
=
(a) T(a)
(2.111)
m
where a is any number, real or complex and we have used the r e l a t i o n Eq.2.107. For two numbers a and b p r o p e r l y restricted as i n the definition of a G a m m a function, the p r o d u c t of t w o G a m m a functions T ( a ) and T(6) is foo
oo
(i
/
/
u
e-
+
T)
1
t"- T'-'dtdT
•'O
Setting t = x ;r = y , 2
2
where x and y are two a u x i l i a r y variables, the above p r o d u c t becomes
foo
oo
/ u
(
/
e" * +» V ' - y - ' d x d y
•'o
I n t r o d u c i n g another variable 0 defined by x = r cos 9,
y = r sin 6
and replacing the elemental area dx dy w i t h r dO dr, we get oo
/
u
/-ir/2
/ •'O
e-
r
r
2 a + 2 4
2
- cos
2 a
-' 9sin
2 4
- 0rd0dr
2.9 Gamma,
Beta and Hypeigeometric
Functions
59
F i n a l l y using a last change of variable by s e t t i n g u = r the f o r m
r(o)r(6)
2 r
=
u
a+b
r •* n
l
e- u - du ft
J
the above p r o d u c t assumes
2
2
cos -
1
2
6 s i n ' - ' 0 d6 (2.112)
T h e second i n t e g r a l o f the above expression is, in fact, equal to
2
2/ where t =
- 1
2 6
1
c o s " 9 s i n - Bd8
=
/
(1 - t )
a _ 1
i
t _ 1
0, Re 6 > 0.
(2.114)
Therefore the relations of Eq.2.108 and Eq.2.11 l-Eq.2.113 give us l
k
i
f (i-ty- t - dt
=
[
t—\\
-
tf-'di
0 =
5 ( 6 , a)
r(q)r(6)
(2.115)
r(a + 6) T h e h y p e r g e o m e t r i c f u n c t i o n F(a,b;c;z)
rpf
i
\
f-
can be w r i t t e n as, see Eq.2.106
(«)»(*) m!(c)
m = 0
m
r(c) S Z^r E r(a)
m = 0
/ \ T(a + m ) „ ("') - l ) ^ f(c + m) k
m
m
(2.116)
w h i c h follows f r o m E q . 2 . I l l and the fact t h a t
(->) ( - i r = -
6
M
-
1
}
- ! -
6
-
r
o
+
1
)
( - i r
=
I n view o f Eq.2.115, we have T(a + m ) T(c + m )
1 — / r r ( c - a) •'o
+
m
-'(i -
c
t) -°-'dt,
^
Chapter
60
2:Least
Squares
Approximation
of
Signals
w h i c h when inserted i n Eq.2.116 gives us
F(a,6;c; ) 2
I
=
f
1
°° / \ - t ) — £ ("') e
- /
r(c)
/
r ( o ) r ( c - a) Q
1
l
c
( - z ^ d t
_ 1
t — ( l - < ) - ° ( l - zt)-' {t) +
t * + i ( * ) = (a t r
(a)
T
T
Crfc^'t)
r
Legendre
Polynomials:
to = " I
tj = 1
Pr{t)
6 = 0 P
00
Laguerre
Polynomials:
L (Ql)
io = 0
tj = oo 6 _ ('+"•)
i u ( t ) = exp( - a t )
Polynomials:
r = -(7+7) tf (at)
t
i / = oo b = 0
= exp( - a t ) c = —2a r
Polynomials Kind: t, = 1
w(t)
= -
6
c =
-1
= — oo
0
a (d)
r
= 2a
2
r
Tchebycheff of F i r s t to = - 1 l , a = 2(r > 1) a Tchebycheff r
0
(e)
of Second
r
0
Jacobi to
= -1 =
(g)
r
tf (t) w(0 = v/(i - < )
b = 0
c =
Polynomials:
rwi(tj
t, = 1
i»( Jt) ^ — l j t ~ ^ -{{a-0) dt'
da>Jt) + {2 + a + 0)t] dt
r ( l + a + 0 + r)4> (t) = 0
(2.140)
r
Assigning p a r t i c u l a r values to a and 0 we get (a) Legendre differential equation: a = 0 = 0 (b) Tchebycheff differential equation of the First K i n d : a = 0 = — i (c) Tchebycheff differential equation of the Second K i n d : ct = 0 =
|
Moreover, as a result of Eq.2.101 and Eq.2.102, the Rodrigues' formulae for generating the Tchebycheff p o l y n o m i a l s of the first k i n d T ( i ) ' s and the Tchebycheff polynomials of the second k i n d f / ( t ) ' s take the f o r m r
r
2
1
2
T,(*)
=
d,(l - t ) ' ^ -
tf ( + !/+-)
(2.165)
Moreover, we have
=
1 3 1 1 (r + i / - - ) ( r + i / - - ) • • • ( ! / + - ) r ( i / + - )
=
2 " ( 2 r + 2u-
r
l ) ( 2 r + 2i/ - 3) • • • (2v + \)V(v
+
1
(2r + 2t/ - l ) ( 2 r + 2 i / - 2) • • • (2t/ + l ) 2 i / r ( » / + f ) r
2 ( 2 r + 2 f - 2) • • • ( 2 i / + 2 ) 2 i /
^ ^ I > 2r
+ i)
V
2 (v)
2
r
r(2r + 2 i / + l )
(2.166)
;
=
(2i/)
2 r + 1
r(2i/)
(2.167)
Hence
r(r
+
i
>+i)
r(i)
T ( 2 r + 2i/ + l ) 2.11.2
2
2 r + 2
"(K) (// + r)r(^)
Normalization Factor of Gegenbauer
To derive the n o r m a l i z a t i o n factor 7
r
(2.168)
r
Polynomials
of the Gegenbauer p o l y n o m i a l s we use the
r e l a t i o n Eq.2.35 and t a k i n g i n t o consideration of Eq.2.150, Eq.2.156 a n d Eq.2.166Eq.2.168, we get T
(-l) r!A. /
u (t)dt T
2
+
(-iy \/3 c f\i-t y "r
T
r
1 / 2
dt
2.12 Rodrigues'
Formula
2 r + 2
73 r
=
2
=
a ^ » .
r +
"(-l) r!/? c f r
_
(
V
i
r
r
W
r
e
r
T
r t r
+
l l 2
"'
(l
- r)
-i/a)r(r
y
'
r +
+
"y
1 / 2
d Y , i f 2r = 1 + t
- i / 2 )
T ( 2 r + 2 i / + 1)
( 2 i / ) r ( i / 2 ) r ( » + i/2) r
r!(i/ + r ) I V ) w h i c h gives the n o r m a U z a t i o n factor o f the Gegenbauer p o l y n o m i a l s . Using Eq.2.30, Eq.2.34 a n d Eq.2.33 we now get 2
r
=
^r+i ft 0,
c,
=
ft ft._i —-
6
r
0 + ) r + 1 '
=
7r
2i/ + r - 1
7r-l
r + 1
+1
;
respectively. I n view o f Eq.2.28, the t h r e e - t e r m recurrence f o r m u l a for Gegenbauer p o l y n o m i a l s becomes
(r + l)C " (f) = 2( / + r)iC "( -1
r
c;(t)
1
0
1
Signals
F o r m u l a , N o r m a l i z a t i o n Factor d
Polynomials
a
T
of
(-i) W »*>! ( » + | ) '
2
(1 - i ) " - J
and
2.13 Differential
Recurrence
Relations
75
Table 2.9: Parameters a,6 ,ri T
and Function
T
g(t)
in the D i f f e r e n t i a l Recurrence R e l a t i o n g(t) (t) = ( r a t + 6 ) (t) + T) d> _,(t) of T
Orthogonal
Pr(t)
i)
u
K
»7r
1
0
—r
L {at)
t
0
r
—r
B (at)
i
0
0
2a r
T (t)
( i - t)
2
-1
0
r
U (t)
(i
2
-1
0
1 + r
pK/*)(t)
( i - t)
2
-1
a+B + 2r
c:(t)
(i
-1
0
r
T
r
r
2.13
T
Differential
- t)
2
- t)
T
T
T
Polynomials
2
r(o-/9) a-r3+2r
2v + r - 1
Recurrence
Relations
T h e class of a l l o r t h o g o n a l p o l y n o m i a l s satisfy a differential recurrence r e l a t i o n of the form g(t)d\ (t)
=
T
(rat + 6 ) {t) + Vrr-i{t) r
(2.172)
r
where o, (t) T
l
-
-rg{t)t4> (t) T
is a p o l y n o m i a l o f degree r , as this follows f r o m Eq.2.29 and Eq.2.171 a n d the fact t h a t the coefficient o f t
r+1
i n h(t)
is i d e n t i c a l l y zero. I n fact, this coefficient is
1 ar0
r
r2a(3
r
=
0
Chapter
76
2:Least
Squares
Approximation
of
Signals
Therefore m a k i n g use of T h e o r e m 1 we can w r i t e g{t)4> {t)
- art {t)
T
r
=
(2.173)
£«* (t)w(t)<j> (t)dt r
k
[ ' 4> {t)—
[g{t)ct> {t)w{t)\dt
r
J
f
k
Hi
4.
4>r{t)
w{t)—{g{t)4> {t)) dt
g{t) (t)w(t) dt
+
k
k
«"W— dr (ff(0 (t)dt r
=
k
0, k < (r -
2)
as a result again of Eq.2.20. Hence m u l t i p l y i n g b o t h sides of Eq.2.173 by we have i n t e g r a t i n g f r o m t
0
to
bklk
= 0,
T
k < (r — 2)
and the r e l a t i o n of Eq.2.172 follows f r o m Eq.2.173 where r/ = 6 _ . r
r
T
r/ and S can be c o m p u t e d as follows. E q u a t i n g coefficients of t r
w(t)
+ r W + r ) ( a + ^ + 2r)
(g) Gegenbauer:
+,J + 2r)
r
r)U ^(t) T
/*"«(*)
* > - !
W
-rtC?(t) (2i/ + r - l ) C " _ ( t ) r
l
78
2.14
Chapter
2:Least
Squares
Approximation
of
Signals
O r d i n a r y Differential R e c u r r e n c e Relations
T h e class of o r t h o g o n a l p o l y n o m i a l s also satisfies ordinary differential recurrence relations w h i c h w i l l now be established. These relations are of p r i m e i m p o r t a n c e to us as the i n t e g r a t i o n o p r a t i o n a l matrices are derived from t h e m . (a) Legendre
polynomials
For Legendre p o l y n o m i a l s , the three-term recurrence f o r m u l a of Eq.2.44, the differential recurrence r e l a t i o n and the differential equation Eq.2.48 are respectively Pr+dt)
=
r
^~rtPr(t)
—P - (t)
2
(i -l)P (r) r
2
(\-t )P {t) r
(2.176)
r l
r + 1
r + 1
=
r t P ^ - r P ^ t )
-
2tP (t)
(2.177)
+ r(r + l)P (i) = 0
r
r
(2.178)
We differentiate b o t h sides of Eq.2.177 w i t h respect to t and add the r e s u l t i n g equation to Eq.2.178 and simplify to get rP (t) r
+ 4_t(
d
r
>
1
0
{t)dt =
E4{t)
(2.201)
ifhere 4>(t)
=
\4> {t)
•••
0
0m-l(*)]
(2.202)
T
and E is the i n t e g r a t i o n o p e r a t i o n a l m a t r i x h a v i n g the f o r m : Bo
Ao
D
Si c
2
2
0 A B
l
2
0
0
0
0
0
0
0
0
0
0
0
A
2
•
(2.203) D.
m 2
0
0
0
0
0
0
Bm0
2
Cm-l
A-
m 2
B
M
- L
T h u s i t is shown t h a t any a r b i t r a r y signal can be a p p r o x i m a t e d as a series of o r t h o g o n a l f u n c t i o n s . I f the n u m b e r of terms i n the series is f i n i t e , t h e n the i n t e g r a l of the square o f the error is a m i n i m u m i n t h i s a p p r o x i m a t i o n . T h e p o l y n o m i a l d> (t) r
of degree r i n any system of o r t h o g o n a l p o l y n o m i a l s can be generated by solving a differential e q u a t i o n of order 2r + 1. F o r t u n a t e l y , a t h r e e - t e r m recurrence f o r m u l a is available by means of w h i c h the o r t h o g o n a l p o l y n o m i a l s i n any system can be generated recursively, and consequently, one should not go for the s o l u t i o n of the differential e q u a t i o n . B y choosing the w e i g h t i n g f u n c t i o n w(t) and the i n t e r v a l [ t , tA, any system 0
of o r t h o g o n a l p o l y n o m i a l s e.g., Legendre, Laguerre, H e r m i t e , Tchebycheff first a n d second k i n d s , Jacobi, and Gegenbauer p o l y n o m i a l s can be generated. Each o f these o r t h o g o n a l p o l y n o m i a l s is shown to satisfy a differential e q u a t i o n and an ( o r d i n a r y ) differential recurrence r e l a t i o n , the l a t t e r is found to be useful i n the d e r i v a t i o n of the integration operational matrix.
Chapter
84
2.16
2:Least
Squares
Approximation
Problems
l.Show t h a t r
M
2
m
=
f _ T
_ l)f
m
m!(r — 2m)!
0
m
W
^(o=E(-i) -^—-!—(2ty-*" m
=o
m!(r — 2m)!
2.Verify the following generating functions (a)
, I V I - 2«x + x l
W
2
= E r
=
S (t)x
r
r
0
X
~ r = T (t) + 2 E 2 t x + x^
T (r)x
0
1
—
r
1 ( C )
r
r
=
1
°°
l - ^ x +
2
x
^
^
1
(d)
2
2
V T - 2 i x + x { l - x + V l - 2rx + x } ° 2
a +
" = E
2
{ 1 + x + V I - 2tx + x } '
expjtxjx - 1) ~ j = 2^ L (t)x 1-x OO j exp(2rx - x ) = E -H (t)x
(e)
^ ( t ) .
r
2
(/)
r
T
3. Prove the following special values: T ( l ) = 1;
(7 (l) = r + 1
T (-1) = (-1)'; T (0) = ( - ! ) ' ; T (0) = 0;
(7 (_l) = ( - i r ( r + t) 2
^ - u ( * ) fc!M i_* r+
of
Signals
C hapter
3
Signal
Processing
Time
D o m a i n
in
Continuous
T h e o r t h o g o n a l p o l y n o m i a l s developed i n C h a p t e r 2 were seen t o be o r t h o g o n a l over a specified range. To carry o u t the analysis of signals available over an a r b i t r a r y i n t e r v a l , shifted o r t h o g o n a l p o l y n o m i a l s are i n t r o d u c e d w h i c h are capable of describing signals over any i n t e r v a l of o u r interest. However, for i n f i n i t e range p o l y n o m i a l s , the t e r m shift has been used i n a different sense. A c o m p a r a t i v e s t u d y of the signal representations v i a different classes of o r t h o g o n a l systems is i n c l u d e d considering the effect of noise, a n d the
filtering
properties of these o r t h o g o n a l systems are also s t u d i e d . T h e
t w o - d i m e n s i o n a l o r t h o g o n a l functions and t h e i r a p p l i c a t i o n t o the representation of two d i m e n s i o n a l signals are also o u t l i n e d . T h e i n t e g r a l a n d the derivative o p e r a t i o n a l m a t r i x are i n t r o d u c e d as the t w o t i m e - d o m a i n operators t o reduce the i n t e g r a l a n d the d e r i v a t i v e operations t o algebraic operations i n the sense of least squares.
For
each o r t h o g o n a l system, the error i n t r o d u c e d by the i n t e g r a l o p e r a t o r is analyzed. A n i n t e g r a l f r a m e w o r k is p r o v i d e d for the d e r i v a t i o n o f the i n t e g r a t i o n o p e r a t i o n a l m a t r i x v i a generalized o r t h o g o n a l p o l y n o m i a l s .
3.1
Shifted Orthogonal
In Chapter
Polynomials
2, we have studied i n details how t o represent an a r b i t r a r y square-
i n t e g r a b l e f u n c t i o n i n a series of an o r t h o g o n a l system of functions { < / > ( t ) } , r r
0 , 1 , 2 . . . over the i n t e r v a l [ t o , * / ] thogonality
T h i s set of o r t h o g o n a l functions satisfies t h e
= or-
relation
(3.1) where j
T
is k n o w n as the normalization
factor.
For a complete system of o r t h o g o n a l
f u n c t i o n s , t h e m i n i m u m i n t e g r a l square error is zero. T h i s result can be s u m m a r i z e d i n the f o l l o w i n g t h e o r e m .
Chapter
88
3: Signal
T h e o r e m 3 . 1 Let { (t)} be a complete r
to the weighting function
function
w(t)
in this interval.
Processing
system
in the interval
Then f(t)
in Continuous
of orthogonal
[ i o , * / ] and f(t)
can be represented
Time
functions be any
as an infinite
Domain
with
respect
square-integrable series
represented
by OO
/(*)=
E/rM*), r=0
(3.2)
f
(3.3)
where f
r
7 r
In other words,
if s (t)
=
-
=
f
represents
m
w(t)f(t)4>At)dt, w{t)l{t)dt.
(3.4)
an approximation
of f(t)
described
by
m - l
[£ (r) - £ _,(r)l,
(3.16)
=
C (T)
(3.17)
r
r
r
T
C {t)dt
f
r
T
- £
r + 1
(r),
TO
dC (r)
(3.18)
r
A square-integrable f u n c t i o n / ( r ) i n r
0
< r < ry can be a p p r o x i m a t e d i n fintie series
of shifted Laguerre p o l y n o m i a l s as m - l
f(r)
*
E
fkC (r) k
T
= f £(r),
(3.19)
92
Chapter
3: Signal
Processing
in Continuous
Time
Domain
where fk
C(t)
=
f
*>{t)f(t)C„{t)dt,
=
[£„(*) £,(*) •••
(3.20)
C - (t)f, rn
1
and f is given by Eq.3.10. I n Eq.3.20, i f 77 = 00 a n d / ( t ) is e x p l i c i t l y k n o w n i n this i n f i n i t e range, t h e n Gauss-Laguerre open q u a d r a t u r e f o r m u l a can be used t o compute f
k
efficiently.
T h e m a i n drawback of these i n f i n i t e range p o l y n o m i a l s is t h a t
the
representation of a f u n c t i o n defined over a finite range is, i n general, n o t exact.
3.1.2
Shifted
Hermite
Polynomials
A system of shifted H e r m i t e p o l y n o m i a l s ( H e P ) { W ( r ) } is called o r t h o g o n a l on —00 < r
T < 00 w i t h respect t o the w e i g h t i n g f u n c t i o n w ( r ) : [
w(r) =
-
e
( T
T
- °'
2 1
(3.21)
if
/
°°
f n w(0««(t)« (i)* =
jf m J
f -
r
v
.
i
T
r '
m
(3-22)
Shifted H e r m i t e p o l y n o m i a l s are e x p l i c i t l y denned as 1
W - n E ' to
^ - ^
( 3
.
2 3 )
k>.(r-2k)\
These p o l y n o m i a l s have the special value , , J (—l) ^ K,(r,)= 1 „ tO, r
2
r t
i f r is zero or even, ., . it r is o d d ,
( r / 2 ) !
(3.24)
and the i m p o r t a n t relations: f
n (t)dt T
= [n (T)
- W
R+L
r + 1
( r ) ] / [ 2 ( r + 1)]
(3.25)
= 2r?i _ (r)
(3.26)
0
TO
dri {T) T
dr
r
A square-integrable f u n c t i o n / ( r ) i n T < T < r 0
f
1
can be a p p r o x i m a t e d i n a
finite
series of shifted H e r m i t e p o l y n o m i a l s as m - l
fir)
«
£
r
/*W*(r) = f W ( r ) ,
(3.27)
/J' "W/(0W*(0*.
(3-28)
k=0
where /* =
3.1 Shifted
Orthogonal
93
Polynomials = [H (T)
W(r)
T
W , ( r ) ••• « _ , ( r ) ] ,
0
and f is given by Eq.3.10.
m
I n the above e q u a t i o n , i f To = —oo,T
F
= oo a n d f(r)
is
e x p l i c i t l y k n o w n i n t h i s i n f i n i t e range, t h e n G a u s s - H e r m i t e open q u a d r a t u r e f o r m u l a m a y be used t o c o m p u t e f .
These p o l y n o m i a l s also have t h e same d r a w b a c k of
k
Laguerre p o l y n o m i a l s i n representing t h e functions over a finite range. I n general,
i n f i n i t e range p o l y n o m i a l s should be avoided t o represent square-
integrable functions i n the
finite
interval, r
0
< r
< r,
as i t necessitates a large
f
n u m b e r of basis functions t o represent an a r b i t r a r y f u n c t i o n w i t h a resonably g o o d degree of accuracy.
3.1.3
Sine-Cosine
Functions
T h e Fourier series representation i.e., t h e representation i n terms of sine-cosine funct i o n s ( S C F ) of a square-integrable f u n c t i o n / ( r ) defined over t h e i n t e r v a l t
0
< t < tj
is given by /(*)
~
/ o F o ( i ) + H [fkhW
=
f F(i),
+
f F {t)] k
k
T
(3.29)
where for k = 0 , 1 , 2, • • • , m — 1, F (t)
= cos{A;7r[2i - {t
k
f
+ i )]c},
(3.30)
0
w h i l e for fc = 1, 2, • • • , m — 1 F (t)
= sin{fc7r[2t - {t, + t )]c},
k
(3.31)
0
and t h e Fourier coefficients fo,f
and f
k
are
k
h = cf ' w h i c h is t h e mean of / ( < ) over t
0
f(t)dt,
(3.32)
< t < t
f
A
=
2c f ' f(t)F (t)dt,
(3.33)
A
=
2 c / ' f(t)F 't)dt
(3.34)
k
k
with 1 (3.35)
(t,-t )' 0
f
=
r
[/o,A,---,/m-i./i,---,/ -i] , m
(3.36)
Chapter
94
3: Signal
Processing
in Continuous
Time
Domain
w h i c h is a ( 2 m — l ) - d i m e n s i o n a l spectral vector of / ( t ) , and T
F(0 = [F„(0, A ( < ) , • • ' . £ » - i ( * ) , A ( 0 , • • • , F _ ( < ) ] , m
1
(3.37)
w h i c h is a ( 2 m — 1)— d i m e n s i o n a l Fourier basis vector . T h e series i n Eq.3.29 w i t h coefficients given i n Eq.3.32-Eq.3.34 converges t o (a) / ( « ) i f t is a p o i n t o f continuity, and t o ( b ) [f(t
+ 0) + f(t
— 0 ) ] / 2 i f t is a p o i n t of d i s c o n t i n u i t y .
c o m p u t a t i o n of Fourier s p e c t r u m of f(t)
T h e numerical
is n o t a difficult p r o b l e m . T h e sine-cosine
functions have the special values: F (< ) = (-l)*;Ft(«o) = 0 t
(3.38)
0
and the recurrence relations for k = 1,2, • • • , m — 1: dF {t) k
= -2kircF {t),
(3.39)
k
dt dF {t) k
2kivcF (t),
(3.40)
= - J — F (t), 2kirc
(3.41)
k
dt
F {r)dT k
and
/ i
t
F (r)dr = [(-1)* - F (*)]—— 2K7TC t
0
(3.42)
t
T h e results of the signal analysis may be compared w i t h those o b t a i n e d by piecewise constant basis functions. T h e o r t h o g o n a l system of block-pulse f u n c t i o n s ( B P F ) i n t r o d u c e d i n E x a m p l e 2.5 w i l l be taken for this purpose. T h i s is because, the representation of any system of functions i n this f a m i l y is piecewise constant and the set of block-pulse functions is a good representative of t h i s f a m i l y .
Moreover, the
c o m p u t a t i o n w i t h i t is simpler t h a n the other systems i n this f a m i l y .
3.1.4
Block-Pulse
Functions
A set of m block-pulse f u n c t i o n s ( B P F ) , o r t h o g o n a l over i € [t ,t/), 0
b it) 1
I
J
o + i A t < t < i + (j + l ) A t ; otherwise; 0
I 0,
for j = 0 , 1 , 2, • • • , m — 1, where At
=
(t -t )/m f
0
These functions are disjoint and o r t h o g o n a l i.e.,
I bj(t),
if i = j ;
is defined as
3.1 Shifted
Orthogonal
Polynomials
95
and = { °> *\* :< I At if i = j . i n M,(t)dxdt (Xf - X ) ( t - t ) ''io ' . i l functions are the Laguerre p o l y n o m i a l s , t h e n we have +
+
0
fi, = J ' J
/
0
' w {x)w (t)f{x,t)ip (x) it)dxdt l
(3.58)
0
2
i
J
(3.59)
112
Chapter
3: Signal Processing
in Continuous
Time
Domain
where
(
( t - T) = E ( - l ) V ' # ( < ) / i ! tt)
where u(t) is a u n i t step f u n c t i o n . S u b s t i t u t i n g Eq.4.2 i n t h e above e q u a t i o n we get CO
*(.*-
r)u(t
- T ) =
[J2(-l) T D /j\}^(t) i=o J
j
j
= Ll,(t)
(4.3)
where t > t + r a n d 0
CO
L = £(-l)V'l>Vjl
(4.4)
i=o is the delay operational Eq.4.4 reduces t o
matrix of V (*)> the basis vector. For o r t h o g o n a l polynomials m - l
Tti-iW&m
(4.5)
i=o w h i c h is exact and is given i n terms of r a n d D. However, for sine-cosine functions, L is given by the i n f i n i t e series of Eq.4.4 w h i c h s h o u l d be t r u n c a t e d p r o p e r l y for the purpose of c o m p u t a t i o n . I t is interesting t o note t h a t the recursive formulae of Tchebycheff polynomials of the first k i n d t o generate L developed b y H o r n g a n d C h o u , 1985 [89] a n d that of Legendre p o l y n o m i a l s , developed by Lee a n d K u n g , 1985 [174] can be obtained d i r e c t l y f r o m Eq.4.5. C h a n g et a l , 1985 [13] suggested a new delay o p e r a t i o n a l m a t r i x for modified Laguerre p o l y n o m i a l s w h i c h c o u l d be used over t h e w h o l e i n t e r v a l of orthogonality.
U n l i k e the one derived above, i t is never exact i n its representation
and hence i t always introduces an error i n t h e delay o p e r a t i o n .
4.3 Dela.y-Integra.tion
4.3
Operational
131
Matrix
Delay-Integration Operational
Matrix
I n t e g r a t i n g Eq.4.3 w i t h respect to t from t + T t o t we get 0
I
it> ()-
Therefore «
. . *=* , / F„(i) + £
/ ' F»d +
- 2 e - ( ' - ° - )
for 0.75 < t < 1.
W i t h m = 6, t h e response x ( t ) c o m p u t e d v i a Laguerre a n d H e r m i t e p o l y n o m i a l s are shown i n T a b l e 4.3 a n d Fig.4.2.
Table 4.3: T h e response x(t) i n E x a m p l e 4.2 v i a Laguerre a n d H e r m i t e p o l y n o m i a l s t
Actual x(t)
Lap x ( t )
HeP x ( t )
0
0
0
0
0.125
0
0
0
0.25
0
0
0
0.375
0.23501
0.23486
0.59456
0.5
0.44293
0.40827
1.6414
0.625
0.59666
0.54855
2.2788
0.75
0.68094
0.65916
2.387
0.875
0.71194
0.74333
1.8849
1.0
0.71174
0.80414
0.74092
C o n t r a r y t o the last e x a m p l e , Laguerre p o l y n o m i a l s have offered satisfactory results w h i l e H e r m i t e p o l y n o m i a l s n o t . These examples suggest t h a t t h e i n f i n i t e range p o l y n o m i a l s do n o t always gurantee acceptable accuracy w h e n applied t o t h e analysis of delay systems. T h e response x(t) o b t a i n e d v i a Laguerre approach by Lee a n d Tsay [179] is f o u n d t o be inferior t o t h e one presented i n T a b l e 4.3. T h i s is perhaps due t o t h e Pade a p p r o x i m a t i o n made by t h e m i n t h e analysis of t h i s delay-differential equation.
140
Figure 4.2: Response of i ( < ) = Hermite polynomials.
Chapter
4: Analysis
of Time-Delay
Systems
- 2x(t - 0.25) + 2 u ( i - 0.25) v i a Laguerre and
4.6 Time-Partition
4.6
Method
141
Time-Partition
Method
To i m p r o v e the accuracy of the response c o m p u t e d for the state vector x ( t ) o f the delay systems i n t h e E x a m p l e s 4.1 and 4.2, p a r t i c u l a r l y over a large t i m e i n t e r v a l , t h e n u m b e r of f u n c t i o n s m i n the basis vector must be increased.
T h e r e is a l i m i t a t i o n
t o this a p p r o a c h , because w i t h an increase i n the value of m , the order of the mat r i x t o be i n v e r t e d increases, thereby i n v i t i n g a n u m b e r of c o n c o m i t a n t disadvantages. Therefore, using the above approach v i a o r t h o g o n a l p o l y n o m i a l s and sine-cosine funct i o n s , t h e c o m p u t a t i o n a l accuracy for the response of a time-delay system cannot be increased
indefinitely.
A careful s c r u t i n y o f the a n a l y t i c a l solutions i n Examples 4.1 and 4.2 reveals t h a t the expressions of the response for a time-delay system over different subintervals are different. T h e o r t h o g o n a l f u n c t i o n representation of such a set o f different functions over t h e whole i n t e r v a l is, i n general, poor. However, the f u n c t i o n over each subint e r v a l can be represented w i t h a reasonably good degree of accuracy by means of o r t h o g o n a l f u n c t i o n s . T h i s suggests t h a t , for a good a p p r o x i m a t i o n of the response x{t)
over the whole i n t e r v a l of our interest, the i n t e r v a l under consideration has to
be d i v i d e d i n t o some convenient n u m b e r of subintervals, over each of w h i c h x(t)
is
t o be c a l c u l a t e d recursively w i t h the i n i t i a l value taken to be the final value of the previous s u b i n t e r v a l . T h e single-term piecewise constant basis f u n c t i o n ( P C B F ) approach proposed i n [257] is based on this p r i n c i p l e . T h i s approach requires a t i m e scaling and n o r m a l i z a t i o n of each s u b i n t e r v a l . Since o n l y the first t e r m of P C B F is used i n t h i s approach, a higher n u m b e r o f divisions to be called time-partitions
of the
i n t e r v a l are required for b e t t e r accuracy. Moreover, the c o m p u t e d response is i n the f o r m of a staircase f u n c t i o n due to the n a t u r e o f the a p p r o x i m a t i o n made by these basis f u n c t i o n s . To get a r o u n d the above difficulties, the time partition the carry-over
spectrum
a < t < b is i n t r o d u c e d . I f {ip (t)}
defined over the i n t e r v a l
is a complete system o f o r t h o g o n a l functions, t h e n
T
f(t)
method w h i c h is based on
of a square-integrable f u n c t i o n f(t)
can be a p p r o x i m a t e d as
m - l
f(t)
n
T
£ / , 0 , ( O = f *(O, r= 0
where m n u m b e r o f functions t/> (t), Vi( • • • > V \ n - i M 0
the set {f
0
f(t)
f,
• • • f -i} m
is called the spectrum
of f(t)
a
r
e
used for the a p p r o x i m a t i o n ,
and f, the spectral vector of
is given by f = [fo fl
•••
/—if.
and
^(t) = [tMi(0 ••• t i - i ( * ) f m
I t is evident f r o m the expression for / ( i ) t h a t the s p e c t r u m of f(t — r ) i n 6 < t < 2b —a is the same as t h a t of f(t)
in a < t < b if T = b — a. T h i s shows t h a t the s p e c t r u m
142
Chapter
4: Analysis
of Time-Delay
Systems
of / ( < ) over o < t < b is carried over to define the s p e c t r u m o f t h e delayed function f(t — r ) over the next i n t e r v a l b < t < 2b — a and is called the carry-over spectrum. T h e t i m e - p a r t i t i o n technique i n t r o d u c e d above w i l l now be applied t o t h e analysis of three different scalar systems, v i z . , ( i ) a delay system excited by a piecewise continuous signal, ( i i ) a m u l t i - d e l a y system a n d ( i i i ) a piecewise constant delay systyem. E x a m p l e 4.3 It is required scribed by
to compute
x(t)
=
x(t - T) +
x{t ) 0
=
c,
x(t)
=
x (t)
if r = 1,< — 0, c = x (t) 0
the response
u(t) K
'
system
de-
u(t), (4.24)
for t —
b
a
= 1, and the input
b
of the time-delay
T
< t < t. 0
is
J - 2 . 1 + 1.05r, = < I -1.05,
0 < t < 1 1 < t < 2
T h e exact response o f the system is given by J l - l . l i - t - 0.525i , ' ~ I - 0 . 2 5 + 1.575< - 1.075i + 0 . 1 7 5 t , 2
X (
2
3
0 < I < 1 l < t < 2
^ '
'
To c o m p u t e the response x ( i ) over t < t < tf for tf > r , we first check whether or not (tf — t ) is equal to r t i T , where is an integer. I f i t is n o t , tf is increased such t h a t (tf — r ) = r t j r as the final t i m e of our interest w i l l not be affected by such a choice. N o w the delay r is d i v i d e d i n t o n equal number of subintervals such that T = n h, where h is the l e n g t h of the subinterval. T h e n any s u b i n t e r v a l i n t 6 [*o,'/j can be represented as a
0
0
2
2
t + k,r + k h 0
o(*) = l , 0 i ( t ) = 2 1 — 1 and xb (t) 2
x(t)
2
= 2(2< - l ) - 1 we have
2
,0)
= 1 - l . l t + 0 . 5 2 5 i w h i c h is the exact s o l u t i o n . Also x ° f
For 1 < t < 2,fc, = 1 and k
2
X
S
1,0)
= 0,x
(1.0)
(0,0)
'
T
1
= [0.425,0,0,0] ,u( '
0 )
T
= x ( l ) = 0.425.
Therefore
= [-1.05,0,0,0]'
4.6 Time-Partition
145
Method
F i g u r e 4.3: Response of x(t)
= x(t — 1) + u ( t ) v i a block-pulse
functions.
Chapter
146
4: Analysis
of Time-Delay
Systems
T h e n , f r o m Eq.4.31 ( 1
x '
0 )
3
= [0.2484375,-0.2179687,-0.0359375,5.46875 2
Since 0 ( < ) = l,ti(t) = 2 i - 3 , ^ ( t ) = 2 ( 2 i - 3 ) - 1 and jp (t) i n this i n t e r v a l , the response is O
2
x(t)
3
x 10~ ]
T
3
= 4 ( 2 i - 3 ) - 3(2r - 3)
2
= - 0 . 2 5 0 0 0 0 3 + 1.57500021 - 1.075i + 0 . 1 7 5 i
3
w h i l e the exact response x(t) is given by Eq.4.25. T h e very s m a l l difference between the a c t u a l x(t) and t h e c o m p u t e d x(t) v i a shifted Tchebycheff p o l y n o m i a l s of the first k i n d is due t o the t r u n c a t i o n and round-off errors i n the c o m p u t a t i o n . ( i i i ) S h i f t e d T c h e b y c h e f f p o l y n o m i a l s of t h e s e c o n d k i n d T h e response x(t) c o m p u t e d v i a shifted Tchebycheff p o l y n o m i a l s o f the second k i n d for 0 < t < 1, is s i m p l y the a c t u a l response. B u t , for 1 < t < 2, x(t)
= - 0 . 2 4 9 9 9 6 2 + 1.5749926r - 1.0749956*
(iv) Shifted Legendre
2
3
-I- 0.174999t .
polynomials
T h e response x(t) c o m p u t e d v i a shifted Legendre p o l y n o m i a l s for 0 < t < 1 is the same as the a c t u a l response, while for 1 < t < 2 x(t)
= -0.2499992 + 1.5749988* - 1.0749996*
2
3
-I- 0 . 1 7 5 i .
(v) Shifted L a g u e r r e polynomials Whenever i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s are used t o analyse time-delay systems v i a the t i m e - p a r t i t i o n m e t h o d , the i n i t i a l f u n c t i o n x (t) and the control f u n c t i o n u(t) are assumed t o be k n o w n i n the whole i n t e r v a l of o r t h o g o n a l i t y for a very accurate evaluation of the spectra. T h e functions k n o w n over a finite interval may be s u i t a b l y e x t r a p o l a t e d for this purpose. t
For the p r o b l e m under consideration, ki = k = 0 over the i n t e r v a l 0 < t < 1. A l t h o u g h x (t) = 1 and u(t) = —2.1 + 1.051 are respectively k n o w n over the intervals 2
b
— 1 < t < 0 and 0 < t < 1, i t is assumed t h a t t h e y are given over — 1 < t < oo and 0 < t < oo respectively. Using the shifted Laguerre p o l y n o m i a l s defined over — 1 < t < oo, the carry-over s p e c t r u m is given by 1
0
x'- ' ' = [1,0,0,0]
T
and 0
T
u< '°) = - [ 1 . 0 5 , 1 . 0 5 , 0 , 0 ] ; x < ° ' 1 - 1 0 E.
0
o )
= [1,0,0,0] 0
1 - 1 0
0
0
L 0
0
1 - 1 0
1
T
4.6 Time-Partition
147
Method
Therefore f r o m Eq.4.31 [0.95,-1,1.05, Since ip (t)
= 1,
0
f u n c t i o n x(t)
T
0] . 2
= 1 - i and i> {t)
= 1 - 2 i + 0 . 5 i i n the i n t e r v a l t € [0, oo], the
2
t u r n s o u t to be the exact response given by Eq.4.25. Also x (t) = l,ipi(t) 0
1 - 3(< -
0 )
= [0.325,-0.9,2.05,-1.05]
= 1 - ( i - l),ip (t)
= 1 - 2(< - 1) 4- 0.5(t - l )
2
2
T
2
and tp (t)
1) + 1.5(i - l ) - ( t - l ) / 6 for 1 < t < oo, the c o m p u t e d x(t)
the same as the a c t u a l response of the system.
=
3
3
is e x a c t l y
T h e power of the shifted
Laguerre
p o l y n o m i a l s defined i n Section 3.1.1 is w e l l demonstrated i n this example. (vi) Shifted H e r m i t e polynomials F o l l o w i n g a s i m i l a r k i n d of approach as for the shifted Laguerre p o l y n o m i a l s discussed above, the c o m p u t e d response x(t)
v i a shifted H e r m i t e p o l y n o m i a l s ( o r i g i n is
shifted to the desired p o i n t s , see Section 3.1.2) is completely i n agreement w i t h the a c t u a l response. (vii) Sine-cosine functions I n the t i m e - p a r t i t i o n m e t h o d , the s o l u t i o n f u n c t i o n over any s u b i n t e r v a l is cont i n u o u s at the end p o i n t s .
As the Fourier series cannot produce the t r u e value of
the f u n c t i o n at the end points of a d i s c o n t i n u i t y , the t i m e - p a r t i t i o n m e t h o d v i a sine1
cosine functions cannot be a p p l i e d .
2
T h i s is because, the c o m p u t a t i o n of x^* '* ' is
never precise and therefore recursion is not possible. E x a m p l e 4 . 4 It is required described
to compute
the response
x(t)
of a multi-delay
x( ( ) i i n t e g r a t e d once w i t h respect t o r and the result is expressed i n t e r m s of the o r i g i n a l set of basis functions, see Section 3.4. I n other words T
l where E
i,(r)dr
T
x
= E,. I f / ( r ) « f ip{r) f
f(r)dT T
=
V
.(r)« £
l
¥
(r)
(5.1)
= /*(r),then = i
T
J\
(r)dr
T
» t Erf
(r)
(5.2)
TO
0
T h e error due t o a p p r o x i m a t i o n i n the i n t e g r a t i o n process is given by ei(r) = f
T
f\r)dr-i E^{T). TO
N o w , let the r e l a t i o n i n Eq.5.2 be i n t e g r a t e d once more w i t h respect t o r , t h e n we have f r o m Eq.5.1
fP TO
f'(h)dt
2
TD
dt, «
T
f E,J
T
y> ( * , ) * !
Kt E E 4,(T) l
l
TO T
2
= f £ V(r)
(5.3)
160
Chapter
5: Identification
of Lumped
Parameter
Systems
T h e error i n the process o f two i n t e g r a t i o n s i n Eq.5.3 is given by
T
0
To
A l t e r n a t i v e l y , i t is also possible t o w r i t e t h a t J TO
= ^ j ( r ) R*
J ' ^(t^dt^dt, To
E rl>{r) 2
so t h a t f
T
X
f
T
f'(t )dhdt rV 2
t E i>(r)
1
(5.4)
i
T h e error i n the process of i n t e g r a t i o n of Eq.5.4 is given by
e
r
2 < )
=
/
/ TO
' f ' ^ )
d
t
*
d t
i
~
{
T
e
^
r
( )
T
0
T h e expression £ ( r ) is the accumulated error at the end of t w o i n t e g r a t i o n s , each 2
one w i t h an error of £ i ( r ) . B u t , a l t h o u g h £ ( r ) also represents t h e error at the end of 2
two i n t e g r a t i o n s , this error is due t o the a p p r o x i m a t i o n i n t r o d u c e d at t h e final stage of t h e i n t e g r a t i o n only. N a t u r a l l y £ ( r ) is greater t h a n e j ( r ) . 2
I n general, for k times repeated integrations of V ( T ) we have
(5.5) where E
k
is called one shot o p e r a t i o n a l m a t r i x for repeated i n t e g r a t i o n s ( O S O M R I ) .
T h e expression for O S O M R I for a l l classes of piecewise constant o r t h o g o n a l functions is available in the l i t e r a t u r e [258]. T h i s concept of O S O M R I w i l l now be extended to a l l classes of o r t h o g o n a l p o l y n o m i a l s and sine-cosine f u n c t i o n s .
For t h e sake of
completeness, we include below the O S O M R I of block-pulse f u n c t i o n s .
5.2.1
O S O M R I
The O S O M R I E
k
E
k
=
of Block-Pulse
Functions
for block-pulse functions can be represented as [258] (Tf ~ r„)*
i n w h i c h I is an m x m i d e n t i t y m a t r i x and
0
:
fy»-i)x(™-.ij
A = L 0
0'
5.2 One Shot
Operational
Matrix
for Repeated
161
Integrations
For m — 4 a n d k = 2 we have
E
=
2
E x a m p l e 5.1 O S O M R I
16
1/4
1
2
3
0
1/4
1
2
0
0
1/4
0
0
0
1 1/4
1/6
1
2
0 0
1/6 0
1 1/6
1
0
0
0
1/6
3 2
of Block-Pulse Functions
/ ( T ) = 1,0 < T < 1 twice with respect to r from
It is required
to
2
T h e exact value o f the given f u n c t i o n u p o n twice i n t e g r a t i o n is g(r) m = 4,r
0
integrate
zero to r. = T / 2 . For
= 0 and Tj = 1, we have t h e block-pulse spectra for / ( r ) and g(r)
as
f = [1,1,1,1,f g = [0.0104166,0.0729166,0.1979166,0.3854166] g
T
1
v i a E] = fE]
= [0.015625,0.078125,0.203125,0.390625] "
and g
via E
2
= fE
= [0.0104166,0.0729166,0.1979166,0.3854166]
2
5
I t m a y be observed t h a t the spectral vector g o b t a i n e d v i a O S O M R I E
2
is exactly
t h e same as t h e a c t u a l g .
5.2.2
O S O M R I First
of Shifted
Tchebycheff
Polynomials
Kind:
F r o m t h e development o f Section 3.4 we have, 2
( r - r ) [ 0 . 7 5 V o ( r ) + t/, (r) + /
/
f
*
Mt )dt dt 2
2
l
0
1
0.25t/> (r)]/4 2
/
J>,(i )di u
( m )
m
(r) + 6 _,u
( m
-
m
i n w h i c h U(T) and y(r)
1 )
w
( r ) + • • • + b,u (r)
+ 6 u(r)
(5.12)
0
are respectively the i n p u t and o u t p u t of the system assumed
to be available over an a r b i t r a r y b u t active p e r i o d r e [ r , r ^ ] ; n is the order of the 0
sysem assumed to be k n o w n ; a _ n
l t
a „ _ , • • • , a ,6 , t> _i, • • • , b 2
0
TO
m
a
are the parameters
of the system t o be identified and m < n. Let the i n i t i a l conditions of Eq.5.12 be = Pi Although a
=
(i)
y {To), «
( , )
i = 0 , 1 , 2 , - • • ,n - 1 )
(TO),
i = 0 , 1 , 2 , - • ,m - 1
/'
and /J are a c t u a l l y k n o w n from the o u t p u t and i n p u t records t h e y w i l l
0
0
be t r e a t e d as u n k n o w n s along w i t h the other i n i t i a l conditions i n the i d e n t i f i c a t i o n process as t h e y m a y n o t represent t r u e values i n the noisy s i t u a t i o n [258]. T h e i n p u t and o u t p u t signals are now represented i n a finite series o f o r t h o g o n a l functions as: t-i
u(r)
«
r
T,u,Mr)
= u V(r)
(5.13)
j= 0
t-1
y(r)
as
Let us i n t e g r a t e Eq.5.12 f r o m r
0
£ y ^ ( r ) j=o
T
= y ^l>(r)
(5.14)
t o r successively rt times w i t h respect t o r to get
an i n t e g r a l e q u a t i o n , i n t r o d u c e Eq.5.13 and Eq.5.14 i n t o the i n t e g r a l e q u a t i o n , make use o f Eq.5.5 for o r t h o g o n a l p o l y n o m i a l s and simplify t o o b t a i n Qp = y
(5.15)
where Q
=
[ - E * y \ - E < _
m
+ 1
T 2
y \ - - - \ - E
u | ••• | £ £ _ i " I E I M
n
^ y \ - E
T
I e | Eje
n
y \ E l _
m
u \
T
| •• • | E_e 2
| ^ . e ] ,
Chapter
168 P
5: Identification
[o»-i,n»-2i' ••
=
Parameter
Systems
,a ,a ,b ,b . , l
/o • / l
I " ' " 1 f n - 2 , / n - 1f
[1,
f ^ J ) ]
7
of Lumped
0
m
m 1
,
"
( £ - 1 ) term* Ao -
B/3
[/(J,
' " ' I /TI-2»
/ l i
« 0 , O h
f
/?m-2, /?m-
f
' 1 « n - l
„
/ n - l ]
• • • >« n - 2 . « n - l
[A>,y3„
fl
(5.16)
—2
a?
• • ,
0 1 u
0 0
• • • •
0 0
0 0
1
••
0
0
3
a
4
••
1
2
a
3
••
0 1
n
- I
. «i
a a
' 0
0
... o
0
0
0 K
... o ... o ... o
0 0 0
&2
b
• •
b
. 6,
b
•••
b_i
0
3
2
0
m
b
m
m
.
T h e m a t r i x A is always nonsingular. T h e m a t r i x B does n o t exist i f b alone is present i n Eq.5.12. Since Q is an I x ( 2 n + m + 1) m a t r i x , t m u s t be at least equal to (2n + m + 1) to solve the linear algebraic system i n Eq.5.15. T h e least squares estimate of the augmented parameter vector p is given by 0
P =
T
1
T
[Q Qr [Q y]
(5.17)
T h u s , i t is always possible to estimate a l l the parameters o f the system. Once the system parameters are e s t i m a t e d , the i n i t i a l conditions a, t o o can be e s t i m a t e d from Eq.5.17, p r o v i d e d fl is a n u l l m a t r i x . Otherwise, i n i t i a l c o n d i t i o n i d e n t i f i c a t i o n is not possible.
This
polynomials, Remark:
algorithm
can be applied
Walsh functions
or Haar
in conjuction
with
any
class
of
orthogonal
functions.
T h e model considered by H w a n g and S h i h , 1982 [136] can be obtained
f r o m Eq.5.12 by p u t t i n g m = n — 1, For this m o d e l , Q a n d p i n Eq.5.15 are shown t o be [136]: Q
=
l-Efy
| -Ely
T
| • • • | -E y n
| Efu | £
r 2
u | ••• | £
r n
u | e |
5.3 Identification
of Lumped
< e
Parameter
Systems
r
I --- I g - g j
~£, e | -£
n — l cola P
*7n-l , Co> Cl I t m a y be observed
e | •• • |- E ^ e ]
n— 1 e o i «
," n - j r' ' .08(fra-l,*n-2i
=
r 2
i ' ' ' i d - 2 ]
T
' ' ' fi j il O i
'
1 1
>
-
t h a t t h e m a t r i x Q does not have f u l l r a n k as t h e last ( n — 1)
columns i n i t are repeated i n t h e ( n — l ) columns preceding t h e m and so t h e inverse T
of [Q Q]
does n o t exist.
Therefore, t h e i r a l g o r i t h m i n this f o r m cannot be a p p l i e d
for t h e i d e n t i f i c a t i o n of parameters and i n i t i a l c o n d i t i o n s for systems o f order higher t h a n one. i) B l o c k - P u l s e F u n c t i o n A p p r o a c h : I n t h e a l g o r i t h m proposed above, w i t h e=
[1,
T
1 ^ _ M (*-l) iefma
we can also a p p l y the block-pulse functions for the i d e n t i f i c a t i o n of systems defined by
Eq.5.12. i i ) S i n e - C o s i n e F u n c t i o n A p p r o a c h [213]: I n t h e case of sine-cosine f u n c t i o n s , we can also d i r e c t l y a p p l y t h e proposed algo-
rithm with e=
T
[ 1 , 0, • • • 0 ] ,
a (2£ — 1)— vector. I n t h e present case, u and y are also (2£ — 1)— vectors.
Therefore,
to solve Eq.5.17, (2£ — 1) m u s t be at least equal t o (2rt + m + 1).
5.3.1
Examples
of L u m p e d
Parameter
System
Identifica-
tion T h e a p p l i c a b i l i t y o f the i d e n t i f i c a t i o n a l g o r i t h m v i a different classes o f o r t h o g o n a l systems is now d e m o n s t r a t e d
i n d e t a i l by considering a few n u m e r i c a l examples.
I n each case, t h e p r a c t i c a l difficulties t h a t are n o r m a l l y encountered i n p a r a m e t e r e s t i m a t i o n are discussed. E x a m p l e 5.6
L u m p e d P a r a m e t e r System Identification V i a Orthogonal Sys-
tems It is required y ( l ) and y ( l )
to estimate
of the
the parameters
H>{.T) + a,y(r) from
its unit
the system
ramp
are a
x
a i , a o , and b as also the initial Q
conditions
system
response
+ a y{r)
data available
= 3 , a „ = 2 and b = 1. B
0
= b u(r) 0
(5.18)
over 1 < T < 2. The actual parameters
of
170
Chapter
5: Identification
of Lumped
Parameter
T h i s example, once again, emphasizes the i m p o r t a n c e of shifting orthogonal polynomials.
Systems
i n i n f i n i t e range
I n order to proceed for the i d e n t i f i c a t i o n of the system i n Eq.5.18, first the u n i t r a m p response d a t a is generated by s i m u l a t i n g Eq.5.18 w i t h U(T)
= T applied at
r = 0 and zero i n i t i a l c o n d i t i o n s . F r o m this d a t a , the i n i t i a l conditions are f o u n d to be i / ( l ) = 0.0840456 and y(l)
= 0.1997882.
Since there are altogether five u n k n o w n s to be e s t i m a t e d , to start w i t h we take £ = 5 for block-pulse functions and o r t h o g o n a l p o l y n o m i a l s b u t I — 3 for the sinecosine functions. For all finite range o r t h o g o n a l functions, we take r and a p p l y the i d e n t i f i c a t i o n a l g o r i t h m .
0
I n each case the e s t i m a t e d
= 1 and Tj = 2 results are as
shown in Table 5 . 1 . I t may be seen from this table, t h a t the e s t i m a t e d results are i n close agreement w i t h the a c t u a l results and i t is more so as the value o f £ is increased. T h e system i d e n t i f i c a t i o n v i a the infinite range o r t h o g o n a l p o l y n o m i a l s is not as t r i v i a l as i t m a y appear. T h e i d e n t i f i c a t i o n v i a Laguerre and H e r m i t e polynomials w i t h r = 1 and Tj = 2 w i l l not be successful i f carried o u t i n the usual manner. I t is because, the signal characterization over r € [1,2] w i t h a finite n u m b e r of infinite range o r t h o g o n a l p o l y n o m i a l s is not posible as discussed i n C h a p t e r 3. Unless the signal characterization is accurate, one cannot proceed w i t h the i d e n t i f i c a t i o n . The signal characterization v i a the infinite range o r t h o g o n a l p o l y n o m i a l s is possible only when the signal i n f o r m a t i o n is available over the whole i n t e r v a l of o r t h o g o n a l i t y . For the Laguerre p o l y n o m i a l s , i t is not a serious m a t t e r as the system under identification is excited for T > 0. B u t for H e r m i t e p o l y n o m i a l s , the signal characterization w i t h a finite n u m b e r of p o l y n o m i a l s is difficult as the i n p u t and the o u t p u t signals are simply zero for t < 0. T h i s p r a c t i c a l difficulty, makes the system i d e n t i f i c a t i o n v i a Hermite p o l y n o m i a l s not reliable. I t is because of t h i s difficulty, the H e r m i t e polynomials for the system i d e n t i f i c a t i o n are not pursued a l t h o u g h i t has the advantages of (i) a simple s t r u c t u r e o f the i n t e g r a t i o n o p e r a t i o n a l m a t r i x and ( i i ) the nonexistence of OSOMRI. 0
However, to study the c a p a b i l i t y of the i d e n t i f i c a t i o n a l g o r i t h m v i a a l l classes of o r t h o g o n a l functions i n c l u d i n g the i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s , i t is assumed t h a t the i n p u t signal « ( r ) = r and the o u t p u t signal j / ( r ) = —0.75 + 0.5r + e x p ( — r ) — 0 . 2 5 e x p ( — 2 r ) are k n o w n i n their a n a l y t i c a l f o r m so t h a t a l l the pract i c a l difficulties in signal characterization are removed.
T h i s a s s u m p t i o n makes the
spectra evaluation of the i n p u t and the o u t p u t signals faster and q u i t e accurate as we employ Gaussian open q u a d r a t u r e formulae for this purpose. I n order to estimate the i n i t i a l conditons along w i t h the system parameters the shifted m i a l s , o r t h o n o r m a l over r € [ l , o o ] and the shifted
Laguerre polyno-
H e r m i t e p o l y n o m i a l s , orthogonal
over T € [—oo, oo] w i t h the o r i g i n shifted from zero to one are employed to o b t a i n the spectra of the i n p u t and the o u t p u t signals. W i t h t h i s m o d i f i c a t i o n , the identification via the Laguerre and H e r m i t e p o l y n o m i a l s is carried o u t and i n each case, the estim a t e d results, seem to be excellent, are as shown i n Table 5 . 1 . T h i s means t h a t the proposed i d e n t i f i c a t i o n a l g o r i t h m works efficiently v i a the i n f i n i t e range o r t h o g o n a l
5.3 Identification
of Lumped
Parameter
Systems
171
T a b l e 5 . 1 : Parameter estimates i n E x a m p l e 5.6. Appr-
t
OQ
6o
3
2
1
0.0840456
0.1997882
5 6
3.0284429
2.0085034
1.0068576
0.0860669
0.1940037
3.0199119
2.0059606
1.0048031
0.0854443
0.1957967
7
3.0147001
2.0044038
1.0035470
0.0850710
0.1968671
8
3.0112901
2.0033839
1.0027247
0.0848296
0.1975574
5
3.0193772
2.0032223
1.0043256
0.0840459
0.1997680
6
2.9999753
1.9999920
0.9999939
0.0840455
0.1997883
5
3.0127620
2.0021398
1.0028555
0.0840470
0.1997688
6
3.0000380
2.0000062
1.0000085
0.0840456
0.1997880
5
2.0026052
1.0034843
0.0840465
0.1997682
6
3.0155869 3.0000532
2.0000085
1.0000118
0.0840456
0.1997881
LaP
5
3.0000369
2.0000226
1.0000115
0.0840459
0.1997859
HeP
5
2.9999999
1.9999999
0.9999999
0.0730514
0.2229334
6
»?
11
»»
0.0871496
0.1935069
11
)i
>>
)) 3.0002314
)» 2.0000388
>) 1.0000518
0.0833944
0.2010942
0.1018564
0.1382217
2.0000490
1.0000659
0.0968457
0.1555389
8
3.0002945 3.0004122
2.0000654
1.0000917
0.0900502
0.1790287
16
3.0004404
2.0000454
1.0000945
0.0869543
0.1897318
32
3.0001227
1.9997955
0.9999955
0.0854767
0.1948439
2/(1)
oach I actual - * BPF
TPl TP2 LeP
7
1 8 SCF
3 4
systems also, p r o v i d e d the signal c h a r a c t e r i z a t i o n is done accurately. I n t h e present and also i n t h e subsequent examples, t h e signal c h a r a c t e r i z a t i o n is made possible by t h e above a s s u m p t i o n w i t h a view t o s t u d y i n g t h e c a p a b i l i t y of t h e i d e n t i f i c a t i o n algorithm via Hermite polynomials. E x a m p l e 5.7 L u m p e d P a r a m e t e r S y s t e m I d e n t i f i c a t i o n v i a O r t h o g o n a l S y s t e m s i n P r e s e n c e of N o i s e w i t h Z e r o I n i t i a l C o n d i t i o n s T h i s e x a m p l e shows the parameter e s t i m a t i o n i n noisy e n v i r o n m e n t . I n C h a p t e r 3 i t is already seen t h a t t h e o r t h o g o n a l functions have inherent f i l t e r i n g properties. T h e r e f o r e , i n t h i s example we estimate the parameters a i , a
0
and b
given b y Eq.5.18 f r o m its u n i t step response j / * ( r ) , defined as y'(r)
0
of t h e system = y(r)
+ n(r)
w h e r e j / ( f ) is t h e a c t u a l response of the system a n d r)(r) is the w h i t e noise w i t h a c e r t a i n a m o u n t of noise-to-signal r a t i o ( N S R ) , available over r 6 [ 0 , 2 0 ] . D u e t o t h e
172
Chapter
5: Identification
of Lumped
Table 5.2: Parameter estimates i n E x a m p l e Approach J
No of
a
NSR
0
Parameter
Systems
5.7 bo
terms t A c t u a l —>
BPF
3 4
0 0
8
0
16 32
0
TP2
1
1
1.2791912
-0.5093126
3.2877246 2.0734912
1.2331763
1.5240077
1.0387510
1.0953028
1.8613561 1.8149477
1.0090934
1.0215492
1.0022410 1.0020818
1.0052606 1.0038849 1.0030674
32
0 0.05
32
0.10
1.8065789
32
0.15
1.8019165
1.0019513 1.0018504
32
0.20 0.25
1.7969300 1.7916158
1.0017798 1.0017402
1.0021621
3 4
0
11.7549262
-8.7944016
3.6787687
0 0
2.2477140 1.7992371
0.9414090 1.0012397
1.1184758
5 3 4
0 0
0.9868161
1.1596184
0.7883851
1.7562783
0.9789322
0.9795571
5 6 6
0 0
1.7790485 1.7977851
1.0002769
0.9934729
1.0000240
0.9993069
1.7970406 1.7993084
0.9989611 0.9964522
0.9994610
6
0.05 0.10
6
0.15
1.8015234
0.9939547
1.0008901
6
0.20
1.8036743
1.0015852
6
0.25
1.8057898
0.9914690 0.9889954
32 TPl
1.8 -3.6632494
1.8109213
1.0046156
1.0011679
1.0002749
1.0001819
1.0022670
c o m p u t a t i o n a l difficulties associated w i t h the evaluation of t h e Tchebycheff first kind s p e c t r u m of a noisy signal y'(r),
see C h a p t e r 3, the parameter e s t i m a t i o n i n noisy
e n v i r o n m e n t v i a the Tchebycheff p o l y n o m i a l s of the first k i n d is n o t carried out in this and in the next example. S i m i l a r l y for the same p r a c t i c a l difficulties discussed in the previous example i n connection w t i h the e v a l u a t i o n of the H e r m i t e s p e c t r u m of the i n p u t and o u t p u t signals, the parameter e s t i m a t i o n v i a the H e r m i t e polynomials i n a noisy e n v i r o n m e n t w i l l not be considered. B u t , the noise-free case is studied i n the same way as i n the last example, as the system response y(r)
= 1 -
2.2941573 x e x p ( - 0 . 9 r ) sin (0.4358898r + 0.4510268)
in the present example is assumed to be k n o w n in its a n a l y t i c a l f r o m over t h e whole i n t e r v a l of o r t h o g o n a l i t y i.e., T
€
[—00,00].
T h e Laguerre approach w o r k e d here
5.4 Transfer
Function
Matrix
Identification
173
effectively for t h e obvious reason t h a t the i n p u t - o u t p u t d a t a is collected over r
€
[ 0 , 2 0 ] . T h e p a r a m e t e r estimates o b t a i n e d v i a each system of o r t h o g o n a l functions is as s h o w n i n Tables 5.2 and 5.3. T h e t r u e parameters are a, = 1.8, o
0
= 1 and 6 = 1. 0
Here no a t t e m p t is made t o estimate the i n i t i a l conditions as t h e y are s i m p l y zero. I t is clear f r o m t h e Tables 5.2 and 5.3 t h a t a l l the parameter estimates o b t a i n e d v i a t h e finite range o r t h o g o n a l p o l y n o m i a l s are very near t o the a c t u a l parameters. T h e estimates based on t h e Laguerre p o l y n o m i a l s are also q u i t e satisfactory. B u t due to t h e p o o r representation of y(r)
or y * ( r ) v i a sine-cosine functions, t h e parameter
estimates are not as a t t r a c t i v e as t h a t o b t a i n e d v i a o r t h o g o n a l p o l y n o m i a l s .
Of
course, t h e result w i l l i m p r o v e , i f more n u m b e r o f terms i n t h e series expansion of sine-cosine f u n c t i o n are i n c l u d e d .
E x a m p l e 5.8 L u m p e d P a r a m e t e r S y s t e m I d e n t i f i c a t i o n v i a O r t h o g o n a l S y s t e m s i n P r e s e n c e of N o i s e w i t h N o n - z e r o I n i t i a l C o n d i t i o n s .
I n the last e x a m p l e , zero i n i t i a l conditions were considered and the parameters were only estimated.
B u t , now t h e i n i t i a l c o n d i t i o n s along w i t h t h e parameters w i l l be
e s t i m a t e d . T o do t h i s , a system described by V(T)
+ a y(r) 0
=
b u(r) 0
w i t h an i n i t i a l c o n d i t i o n 2/(0) is considered. T h e a i m is t o estimate the parameters and b a n d t h e i n i t i a l c o n d i t i o n s 2/(0) f r o m its u n i t step response y'(r) 0
*" 6 [ 0 , 1 8 ] , where y'(r) i n i t i a l c o n d i t i o n are: a are as s h o w n i n Tables
a
0
available over
is as defined i n the last example. T h e t r u e parameters and t h e 0
= b = 2 and y(0) = 0.25 respectively. T h e e s t i m a t e d results 0
5.4 and 5.5. T h e system response y(r)
= 1 — 0.75 exp( — 2 r )
is assumed t o be k n o w n i n its a n a l y t i c a l f o r m to carry o u t t h e system i d e n t i f i c a t i o n v i a H e r m i t e p o l y n o m i a l s . T h e e s t i m a t e d results i n each case are f o u n d t o be q u i t e satisfactory.
5.4
Transfer Function Matrix
Identification
Let t h e M I M O system be described by
(
E«,rf, "(t) = E E *=0
j' = l
b uf\t),i=\,2,...,p ijk
(5.19)
*=0
where r is t h e n u m b e r o f i n p u t s and p the n u m b e r of o u t p u t s of t h e system. I n terms of transfer f u n c t i o n m a t r i x , t h e system described by Eq.5.19 can be w r i t t e n as Y(s)
=
G(s)U(s)
(5.20)
C h a p t e r 5: Identification
174
of Lumped
Parameter
Table 5.3: Parameter estimates i n E x a m p l e 5.7 A p p r o a c h J.
No of
a
NSR
0
bo
Terms I 3 4 5 6 LeP
LaP
SCF
1.3677901
0.9917605
0.8758327
0 0
2.1181219
1.0349396
1.7887488 1.7965880
0.9971963
1.1022000 0.9957780
0.9995188 0.9989527
0.9988587 0.9994654
6
0 0.05
6
0.10
1.7975398 1.7983462
6
0.15
1.7990034
6 6
0.20 0.25
3 3 3 3 3 3 HeP
0
0.9983741
1.0000241 1.0005334
1.7995076
0.9977758 0.9971574
1.7998550
0.9965185
1.0013988
0 0.05
1.8000040
1.0000011 0.9978397
0.10
1.7935757 1.7903232 1.8156052
0.15 0.20 0.25
1.7986808
1.0091820 1.0137390
1.0009920 1.0000015 0.9970551 1.0060926 1.0091220 1.0004232
1.8195439
0.9997469 0.9996819
1.6511276
0.1063884
0.5342819
0
1.7931580
1.0665975
1.0272499
0
1.9448625
1.2995514
1.1505024
0
1.8015404
1.1165289
1.0415885
8
0
1.8056741
1.0429993
1.0182453
8
0.05
1.8105321
1.0345909
1.0180840
8
1.7930093
1.0428776
1.0134564
8
0.10 0.15
1.7853223
1.0435507
1.0108116
8
0.20
1.8207498
1.0641069
1.0277333
8
0.25
1.8239079
1.0697146
1.0299966
3 4
0
2 4
1.0005337
Systems
5.4 Transfer
Function
Matrix
Identification
175
Table 5.4: Estimates i n E x a m p l e 5.8 Approach
[
No of
NSR
a
0
bo
y(0)
Terms t A c t u a l —»
BPF
TPl
TP2
LeP
2
2
0.25
3 4
0
1.9816849
1.9816849
0.2568681
0
1.9896480
1.9896480
0.2538819
8
0
1.9973998
0.2509750
8 8
0.05 0.10
1.9895927 1.9817949
1.9973998 1.9946382
8
0.15
8
0.20
8
0.25
3
1.9918770
0.2512235 0.2514749
1.9740068
1.9891162
0.2517204
1.9662286
1.9863561
0.2519689
1.9584605
1.9835968
0.2522173
2.0000140
2.0000044
0.2499999
3 4
0
1.8527362
1.8733323
0.2844763
0
1.9907627
1.9920546
0.2537868
5
0
1.9996903
0.2504322
5
1.9945020
0.2496433
5
0.05 0.10
1.9996400 1.9938018 2.0021133
2.0032929
0.2529298
5
0.15
2.0075617
2.0034957
0.2484985
5
0.20
1.9584288
1.9612405
5
0.25
2.0403448
2.0384496
0.2536887 0.2483324
3 4
0
1.8055322
1.8343827
0.2761781
0
1.9841179
1.9864740
0.2521379
5
0
1.9992390
1.9993519
0.2501024
5
0.05
1.9999312
2.0010270
0.2503351
5
0.10
2.0005978
2.0026808
0.2505713
5
0.15
2.0012385
2.0043133
0.2508109
5
0.20
2.0018536
2.0059245
0.2510539
5
0.25
2.0024428
2.0075142
0.2513004
176
Chapter
5: Identification
of Lumped
Parameter
Systems
Table 5.5: Estimates i n E x a m p l e 5 . 8 Approach j
No of Terms
NSR
h
f(0)
I 2
2
0.25
3 3 3
0 0.05
1.9999963 1.9927142
1.9999965 1.9961582
0.2500006 0.2482502
0.10
1.9854071
1.9922835
0.2465185
3
1.9782191 1.9711474
1.9885047
0.2447809
3
0.15 0.20
1.9848194
3
0.25
1.9641890
1.9812253
0.2430375 0.2412885
3 4
0
2 2
1.9999999 1.9999999
2
1.9999999
-0.0193556 0.2354832
A c t u a l —»
LaP
HeP
0 0
8 12
0
2
1.9999999
0.2496409
2
0
1.9999968
1.9999966
0.2548170
4
0 0
1.9999983 1.9999981 2.0011819 2.0023524
0.2521212
0.05 0.10
1.9999959 1.9999955 1.9999049 1.9997821
8
0.15
1.9996274
2.0035097
0.2512858 0.2514314
8
0.20
1.9994408
2.0046537
0.2515772
8
0.25
1.9992226
2.0057845
0.2517232
8 8 8
SCF
0.9999999
0.2509951 0.2511404
where Di(.)
G(s)
£> (.)
D,(S)
D (.)
D,(.)
2
= I
R
£>,(«) *^(.) D (.) 2
'•"
D„(.) J
n,
Di(s)
=
5Z«i*A«lin. =
M
=
l,2,...,p
t = 0 n , - l
=
E
= 1,2,....p,j = 1 , 2 , . . . , r
k=0 Di(s) is the least c o m m o n d e n o m i n a t o r of the i t h row o f G ( s ) h a v i n g t h e degree n,-, and the Laplace t r a n s f o r m of a t i m e function / ( < ) is represented by t h e corresponding c a p i t a l l e t t e r F(s).
5.4 Transfer To
Function
Matrix
Identification
177
get E q . 5 . 2 0 f r o m E q . 5 . 1 9 by t a k i n g the Lapalace t r a n s f o r m o f b o t h sides o f
E q . 5 . 1 9 , we have considered the i n i t i a l c o n d i t i o n s t o be zero. I n fact, i f we consider any r e c o r d o f t h e o u t p u t due t o a given i n p u t , the response s h o u l d also have c o m p o n e n t s due t o these i n i t i a l c o n d i t i o n s . Therefore the system i d e n t i f i c a t i o n requires t h e determ i n a t i o n of t h e parameters { o , * , 6 , 7 * } , * = 1,2,...,p;j
= l,2,...,r;fc = 0,l,...,n,-
a n d t h e i n i t i a l c o n d i t i o n s f r o m a t r a n s i e n t record o f t h e system i n p u t s a n d o u t p u t s . I n t e g r a t i n g Eq.5.19 rt, times f r o m t
"i
t o t we have
0
f
t=o
- E
hnk {
i=u=o
vrXto)-———
,=0
l
= E E
1
k-l
E m {in.-kivm
(n