16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 5
Lecture 18
Last time: Semi-free configuration design
This is equivalent to:
Note ,ns enter the system at the same place. F is fixed. We design C (and
perhaps )B . We must stabilize F if it is given as unstable.
()
()
1()()()
Cs
Hs
CsFsBs
=
+
so that having the optimum H , we determine C from
()
()
1()()()
Hs
Cs
HsFsBs
=
?
We do not collect H and F together because if F is non-minimum phase, we
would not wish to define H by
()
opt
HF
H
F
=
This leads to an unstable mode which is not observable at the output – thus
cannot be controlled by feeding back.
Associate weighting functions with the given transfer functions.
() ()
() ()
() ()
H
F
D
Hs w t
Fs w t
Ds w t
→
→
→
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 2 of 5
If ()Fs is unstable, put a stabilizing feedback around it, later associate it with the
rest of the system.
Error Analysis
We require the mean squared error.
111
222
22 11 12
333
22 2
() ( )( )
() ( )( )
() ()( )
() ( )( )
() () ()
() () 2 () () ()
H
F
FH
D
ct w it d
ot w ct d
dw dw it
dt w st d
et ot dt
et ot otdt dt
τττ
τττ
τ τττ ττ
τττ
∞
?∞
∞
?∞
∞∞
?∞ ?∞
∞
?∞
=?
=?
?
=?
=?
=? +
∫
∫
∫∫
∫
2
22 11 12 44 33 34
11 22 33 44 12 34
11 22 33
() ( ) ( )( ) ( ) ( )( )
() () () ()( )( )
() () ()
FH FH
HFHF
HFH
ot dw dw it dw dw it
dw dw dw dw it it
dw dw dw d
τ τττ ττ ττττ ττ
ττ ττ ττ ττ ττ ττ
ττ ττ ττ
∞∞ ∞∞
?∞ ?∞ ?∞ ?∞
∞∞∞∞
?∞ ?∞ ?∞ ?∞
∞∞
?∞ ?∞
????
? ?
????
????
=
∫∫ ∫∫
∫∫∫∫
∫∫
44 1234
()( )
Fii
wRττττττ
∞∞
?∞ ?∞
+??
∫∫
22 11 12 33 3
11 22 33 12 3
11 22 33 123
() () ( ) ( )( ) ( )( )
() () ()( )( )
() () () ( )
FH D
HFD
HFDis
otdt d w d w it d w st
dw dw dw it st
dw dw dw R
τ τττ ττ ττ τ
ττ ττ ττ ττ τ
ττ ττ τττττ
∞∞ ∞
?∞ ?∞ ?∞
∞∞∞
?∞ ?∞ ?∞
∞∞∞
?∞ ?∞ ?∞
????
=??
????
????
?
=+
∫∫ ∫
∫∫∫
∫∫∫
We shall not require
2
()dt in integral form.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 3 of 5
The problem now is to choose ()
H
wt so as to minimize this
2
()et , for which we
use variational calculus.
Let:
0
() () ()
H
wt wt wtδ=+
where
0
()wt is the optimum weighting function (to be determined) and ()wtδ is
an arbitrary variation – arbitrary except that it must be physically realizable.
Calculate the optimum
2
e and its first and second variations.
22 222
0
22 2
() 2 () () ()
ee e e
eot otdtdt
δδ=+ +
=+ +
The optimum
2
e (
2
e for () 0wtδ = ):
2
010122303441234
2
10 1 2 2 3 3 1 2 3
() ( ) ( ) ( ) ( ) ( )
2() () ()( )()
FFi
FDis
et dw dw dw dw R
dw dw dw R dt
τ τττττττττττ
ττ ττ ττ τττ
∞∞ ∞∞
?∞ ?∞ ?∞ ?∞
∞∞ ∞
?∞ ?∞ ?∞
?
?++
∫∫∫∫
∫∫∫
The first variation in
2
()et is
2
11 22 303 44 1234
10 1 2 2 3 3 4 4 1 2 3 4
11 22 33 123
() ( ) ( ) ( ) ( ) ( )
() () () () ( )
2()()()( )
FFi
FDis
et d w dw dw dw R
dw dw d w dw R
dw dw dw R
δ τδ τ τ τ τ τ τ τ τ τ τ τ
τ ττττδτττττττ
τδ τ τ τ τ τ τ τ τ
∞∞∞∞
?∞ ?∞ ?∞ ?∞
∞∞∞∞
?∞ ?∞ ?∞ ?∞
∞∞
?∞ ?∞ ?∞
=+?
++
?+
∫∫∫∫
∫∫∫∫
∫∫∫
In the second term, let:
13
24
31
42
τ τ
τ τ
τ τ
τ τ
′=
′=
′=
′=
and interchange the order of integration.
2
nd
term
11 22 303 44 3412
() () () ()( )
FFi
d w dw dw dw Rτ δτττ ττ ττττττ
∞∞∞∞
?∞ ?∞ ?∞ ?∞
′′ ′′ ′′′ ′′′′′′=+?
∫∫∫∫
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 4 of 5
but since
3412 1234
()()
ii ii
RRτ τττ ττττ′′′′ ′′′′+?? = +?? we see that the second term is
exactly equal to the first term. Collecting these terms and separating out the
common integral with respect to
1
τ gives
2
11 22 303 44 1234
22 33 123
() 2 ( ) ( ) ( ) ( ) ( )
() () ( )
FFi
FDis
et d w dw dw dw R
dw dw R
δ τδ τ τ τ τ τ τ τ τ τ τ τ
ττ τττττ
∞∞∞∞
?∞ ?∞ ?∞ ?∞
∞∞
?∞ ?∞
?
=+?
?
?
?
?+?
?
?
∫∫∫∫
∫∫
The second variation of
2
()et is
22
11 22 33 44 1234
() ( ) ( ) ( ) ( ) ( )
FFi
et d w dw d w dw Rδ τδ τ τ τ τδ τ τ τ τ τ τ τ
∞∞∞∞
?∞ ?∞ ?∞ ?∞
=+?
∫∫∫∫
By comparison with the expression for
2
()ot , this is seen to be the mean squared
output of the system
()
2
22
output ( ) 0, non-zero inputetδ=>
This second variation must be greater than zero, so the stationary point defined
by the vanishing of the first variation is shown to be a minimum.
In the expression for the first variation,
1
() 0wδ τ = for
1
0τ < by the requirement
that the variation be physically realizable. But
1
()wδ τ is arbitrary for
1
0τ ≥ , so
we can be assured of the vanishing of
2
()etδ only if the { } term vanishes almost
everywhere for
1
0τ ≥ . The condition which defines the minimum in
2
()et is then
22 303 44 1234
22 33 123
() () ()( )
() () ( ) 0
FFi
FDis
dw dw dw R
dw dw R
τ τττττττττ
ττ τττττ
∞∞∞
?∞ ?∞ ?∞
∞∞
?∞ ?∞
+??
?+?=
∫∫∫
∫∫
for all
1
τ , non-real-time.
Using this condition in the expression for
2
0
()et and remembering that
0
() 0wt=
for 0t < gives the result
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 5 of 5
222
00
() () ()et dt ot=?
which is convenient for the calculation of
2
0
()et .
Also since
222
00
() () ()ot dt et=?, this says the optimum mean squared output is
always less than the mean squared desired output.
Autocorrelation Functions
We have arrived at an extended form of the Wiener-Kopf equation which defines
the optimum linear system under the ground rules stated before.
Recall that:
() () () () ()
() () ()
ii ss sn ns nn
is ss ns
RRRRR
RRR
τ ττττ
τττ
=+++
=+
since isn=+.
The free configuration problem is a specialization of the semi-free configuration.
In this expression we would take () 1Fs= , or () ()
F
wt tδ= . In that case we have
22 303 44 1234
22 3 3 123
03 133 3 133 1
() () ()( )
() () ( )
()() ()()0 for 0
ii
Dis
ii D is
dd dw d R
ddwR
wR d wR d
ττ τ τ τδτ ττττ
τδτ τ τ τ τ τ
ττττ ττττ τ
∞∞ ∞
?∞ ?∞ ?∞
∞∞
?∞ ?∞
?∞ ?∞
+??
?+?=
?? ?= ≥
∫∫∫
∫∫