16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 6
Lecture 14
Last time: (, ) ( )wt wtτ τ??
0
() ( ) ( )
'
Let:
() ( ) ( )
t
yt wt x d
t
dd
yt w xt d
τ ττ
ττ
ττ
τ ττ
?∞
∞
=?
=?
?
?
′?=
?
′′′=?
∫
∫
For the differential system characterized by its equations of state, specialization
to invariance means that the system matrices ,,ABC are constants.
x Ax Bu
yCx
=+
=
For ,,ABC constant:
0
00
() ()
() ( ) ( ) ( ) ( )
t
t
yt Cxt
x tttxt tBudτ ττ
=
=Φ ? + Φ ?
∫
The transition matrix can be expressed analytically in this case.
( , ) ( , ), where ( , )
d
tAt I
dt
ττ τΦ=Φ Φ=
This is a matrix form of first order, constant coefficient differential equation. The
solution is the matrix exponential.
()
() 2 2
(, )
11
( ) ( ) ... ( ) ...
2!
At
At k k
te
eIAt At At
k
τ
τ
τ
ττ τ
?
?
Φ=
=+ ? + ? ++ ? +
Useful for computing ()tΦ for small enough t τ? .
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 2 of 6
The solution is
0
0
() ()
0
() ()
() ( ) ()
t
At t At
t
yt Cxt
x te xt e Butd
τ
τ
? ?
=
=+
∫
For
0
t →∞:
()
0
() ( )
()
t
At
A
x teBud
eBut d
τ
τ
τ τ
τ τ
?
?∞
∞
′
=
′′=?
∫
∫
and for a single input, single output (SISO) system,
()
TAt
wt ce b=
If ()
jt
x te
ω
= for all past time
()
0
0
() ( )
()
()()
jt
jjt
yt w e d
we de
Fxt
ωτ
ωτω
ττ
ττ
ω
∞
?
∞
?
=
??
=
??
??
=
∫
∫
Since () 0w τ = for 0τ < for a realizable system, we see that the steady state
sinusoidal response function, ()F ω , for a system is the Fourier transform of the
weighting function – where the coefficient unity must be used.
() ()
j
Fwed
ωτ
ω ττ
∞
?
?∞
=
∫
and ()w τ for a stable system is Fourier transformable.
Then
1
() ( )
2
jt
wt F e d
ω
ω ω
π
∞
?∞
=
∫
You can compute the response to any input at all, including transient responses,
having defined ()F ω for all frequencies.
The static sensitivity of the system is the zero frequency gain, (0)F , which is just
the integral of the weighting function.
0
(0) ( )Fwdτ τ
∞
=
∫
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 3 of 6
Stationary statistics
Invariant output statistics implies more than stationary inputs and invariant
systems; it also implies that the system has been in operation long enough under
the present conditions to have exhausted all transients.
Input-Output Relations for Correlation and Spectral Density
Functions
Derive autocorrelation of output in terms of autocorrelation of input
111
0
11
0
11
0
() ( ) ( )
()
()
yt w xt d
ywxd
xw d
τ ττ
ττ
ττ
∞
∞
∞
=?
=
=
∫
∫
∫
11 2 2 1 2
00
11 2 2 12
00
() ()( )
() ()( )( )
() () ( )
yy
xx
Rytyt
dw dw xt xt
dw dw R
ττ
τ τττ τ ττ
ττ ττ τττ
∞∞
∞∞
=+
=?+?
=+
∫∫
∫∫
2
11 2 2 12
00
() () ( )
xx
ydwdwRτ τττ ττ
∞∞
=?
∫∫
111
0
111
0
() ()( )
() ( ) ( )
() ( )
xy
xx
Rxtyt
x tw xt d
wR d
ττ
τ ττ τ
ττττ
∞
∞
=+
=+?
=?
∫
∫
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 4 of 6
Transform to get power density spectrum of output.
1
12 1 2
0
11 2 2 12
00
()
12 11 2 2
00
first integral
()
(0)
() ()
)()( )
() () ()
In first integral o
j
yy yy
j
xx
jjj
xx
yxwd
Fx
SRed
ddw dw R e
dR e d w e d w e
ωτ
ωτ
ω ττ τ ωτ ωτ
ττ
ωττ
τττ ττ τττ
ττττ ττ ττ
∞
∞
?
?∞
∞∞ ∞
?
?∞
∞
?+? ?
?∞
=
=
=
=( +?
=+?
∫
∫
∫∫ ∫
∫
12
12
11 2 2
2
=+
nly, let
() () () ( )
()( )()
() ()
jjj
yy xx
xx
xx
dd
SdRedwedwe
SFF
FS
ωτ ωτωτ
τττ τ
ττ
ωττ ττ ττ
ωωω
ωω
∞∞∞
′ ??
?∞ ?∞ ?∞
′ ?
?
?
′=
?
′′=
=?
=
∫∫∫
The power spectral density thus does not depend upon phase properties.
The cross-spectral density function can be derived similarly, to obtain:
() () ()
xy xx
SFSω ωω=
Mean squared output in time and frequency domain
2
11 2 2 12
00
(0) ( ) ( ) ( )
1
()
2
1
()( ) ()
2
yy xx
yy
xx
yR dw dwR
Sd
FF S d
τ τττ ττ
ωω
π
ωωωω
π
∞∞
∞
?∞
∞
?∞
== ?
=
=?
∫∫
∫
∫
Generally speaking, with linear invariant systems it is easier to work in the
transform domain than the time domain – so we shall commonly use the last
expression to calculate the mean squared output of a system. However, control
engineers are more accustomed to working with Laplace transforms than with
Fourier transforms. By making the change of variables sjω= we can cast this
expression in that form.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 5 of 6
2
1
2
1
() ( ) ()
2
j
xx
j
xx
sssds
yFFS
j jjj
FsF sS sds
j
π
π
∞
?∞
∞
?∞
??? ? ??
=?
??? ? ??
??? ? ??
′′ ′=?
∫
∫
We know that ()
xx
S ω is even. If it is a rational function of ω , and we will work
exclusively with rational spectra, it is then a rational function of
2
ω . So only
even powers of ω appear in ()
xx
S ω and thus
xx
s
S
j
??
??
??
which we may call ()
xx
Ss is
derived from ()
xx
S ω by replacing
2
ω by
2
s? .
()Fs′ is the ordinary transfer function of the system – the Laplace transform of its
weighting function. Because () 0, 0wt t= < .
We shall drop the primes from now on.
2
22
44
1
() ( ) ()
2
in ( )
j
xx
j
xx
yFsFsSsd
j
s
Ss
s
π
ω
ω
∞
?∞
=?
?=?
?
?
=
?
?
∫
Integrating the output spectrum
General method
Cauchy Residue Theorem
()( ) 2 residues of ( ) at the poles enclosed in the contour C
C
Fsds j Fsπ=
∑
∫ >
If ()Fs has a pole of order m at za= ,
()
1
1
1
Res( ) ( ) ( )
1!
m
m
m
s a
d
asaF
mds
?
?
=
??
??=?
??
??
?
??
()Fs has a pole of order m at sa= if m is the smallest integer for which
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 6 of 6
()lim ( )
m
sa
saFs
→
??
?
??
is finite.
If ()Fs is rational and has a 1
st
order pole at a ,
()
()
()
()
( )( )...
Ns
Fs
Ds
Ns
sasb
=
=
??
then
[ ]
1
Res( ) lim ( ) ( )
()
( )( )...
m
sa
asaFs
Na
abac
=
→
=?
=
??