ESTIMATION THEORY - ocw.snu.ac.krocw.snu.ac.kr/sites/default/files/NOTE/7068.pdf · Estimation...
Transcript of ESTIMATION THEORY - ocw.snu.ac.krocw.snu.ac.kr/sites/default/files/NOTE/7068.pdf · Estimation...
ESTIMATION THEORY
September 2010Instructor: Jang Gyu Lee
Introduction
( )1 1
ˆ Find the best estimate from the measurements of the form
where is a rando
Problem:
(1). The fi
m proc
rst me
ess.
asurement:
x
z
t
w
z w
x
w
x=
=
+
+
( )1 1x̂ t z=
( ) 21
21zx t σσ =
2Estimation Theory (10_2)
Introduction (continued)
3Estimation Theory (10_2)
2 2 1(2). The second measurement: , z t t≅
( )[ ] [ ] 2222
1222 )/(/
211212zz zzzzzz σσσσσσμ +++=
( ) ( )22221
/1/1/1 zz σσσ +=
( ) ( )[ ] [ ] 2222
1222
2 )/(/211212
zztx zzzzzz σσσσσσμ +++==
( )[ ]( )12222
1 211/ zzz zzz −++= σσσ
( ) ( ) ( )1 2 2 1ˆ =Predictor + Correctorx t K t z x t⎡ ⎤= + −⎢ ⎥⎣ ⎦
( ) ( ) ( ) ( )12
212
22 ttKtt xxx σσσ −=
Introduction (continued)
Estimation Theory (10_2) 4
( ) ( )3 3 3(3). The third measurement: ;dx
z x t w t u wdt
= + = +
( ) ( ) ( )3 2 3 2x̂ t x t u t t− = + −
( ) ( ) ( )232
22
32 tttt wxx −+=− σσσ
( ) ( ) ( ) ( )[ ]−−+ −+= 33333 txztKtxtx
( ) ( ) ( ) ( )−−+ −= 32
332
32 ttKtt xxx σσσ
( ) ( ) ( )[ ]23
23
23 3
/ zxx tttK σσσ += −−
( )
( ) ( )3
23
2 23 3
As , 0.
As , and 1.
z
w x
K t
t K t
σ
σ σ −
→ ∞ =
→ ∞ → ∞ =
Chapter 1
Linear Systems Theory
Matrix Algebra
Estimation Theory (10_2) 6
Suppose that is an matrix and is an matrix. Then the product of
and is written as . Each element in the matrix p
(1). Matrix Multiplica
roduct is computed as
n
tio
A n r B r p
A B C AB C
× ×
=
1
1
2 21 1
; 1, , ; 1, , . (1.13)
In general, . (no commutability)
Inner Product:
(2). Vector Product
.
Outer Prod
s
r
ij ik kjk
Tn n
n
C A B i n j p
AB BA
x
x x x x x x
x
=
= = =
≠
⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤= = + +⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦
∑
21 1 1
1
21 1
uct: . (1.14)
n
Tn
nn
x x x x
xx x x
x x x x
⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦
Matrix Algebra (continued)
Estimation Theory (10_2) 7
1
is nonsingular.
exists.
The rank of is equal to .
The rows of are linearly independent.
The columns of are linearly
(3). Rank and Nonsingularity of
i
( )
A
A
A
A
A
A
n
n
n
−
•
•
•
×
•
=
•
( )
ndependent.
0.
has a unique solution for all .
0 is not an eigenvalue of .
(4). Trace of a square matrix:
Note that ( ) ( ),
)
,
(
T T
i
T
ii
A
Ax b x
Tr A
b
A
Tr AB Tr BA
AB
A
B A
• ≠
• =
•
=
=
=∑
( ) 1 1 1 .AB B A− − −=
Matrix Algebra (continued)
Estimation Theory (10_2) 8
is:
Positive definite if 0 for all nonzero 1 vectors . This is equivalent
to
(5). Defini
saying that all of the eigenvalues of are o
teness of a Symmet
isutu
ric matrix
ve r
T
n n
A
x A x
A
x n
A
• > ×
×
eak numbers. If is
positive definite, then is also positive definite.
Positive semidefinite if 0.
Negative definite if 0.
Negative semidefinite if 0.
Indefinite
T
T
T
A
x Ax
x Ax
x Ax
• ≥
• <
• ≤
• if it does not fit into any of the above four caregories.
Suppose we have the partitioned matrix where and are invertible
square matrices, and
(6). Matrix
the a
Inversion Le
n
mma
A BA D
C D
B
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
( ) ( )1 11 1 1 1 1
d matrices may or may not be square. Then,
. (1.38)
C
A BD C A A B D CA B CA− −− − − − −− = + −
Matrix Calculus
Estimation Theory (10_2) 9
( )
( ) ( )
( ) ( ) ( ) [ ]( )
( ) ( )
1
11 1
1
1 1
(1). .
/ /
(2). ; , 1, , , 1, , .
/ /
(3). / / ; .
(4). ;
n
n
ij
m mn
T TT T T T
n n
T TT T T
f x f fx x x
f A f Af A
A A i m j nA
f A f A
x y x yx y x x y x y y y x
x y
x Ax x Axx A x A
x
⎡ ⎤∂ ∂ ∂⎢ ⎥= ⎢ ⎥∂ ∂ ∂⎣ ⎦⎡ ⎤∂ ∂ ∂ ∂⎢ ⎥
∂ ⎢ ⎥= = = =⎢ ⎥
⎢ ⎥∂⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦
∂ ∂⎡ ⎤= ∂ ∂ ∂ ∂ = = =⎢ ⎥⎣ ⎦∂ ∂
∂ ∂= +
∂ ∂
( ) ( )
( ) ( )
2 if .
(5). ; .
(6). ; 2 , if .
T T
T
T TT T
x A A Ax
x AAxA A
x x
Tr ABA Tr ABAAB AB AB B B
A A
= =
∂∂= =
∂ ∂
∂ ∂= + = =
∂ ∂
Continuous, Deterministic Linear Systems
Estimation Theory (10_2) 10
( ) ( ) ( )
( )
( )
0
00
0
11
Models
Solution
( ) ( )
!
= .
tA t t A t
t
jAt
j
x Ax Bu
y Cx
x t e x t e Bu d
Ate
j
sI A
τ τ τ− −
∞
=
−−
= +
=
= +
=
⎡ ⎤−⎢ ⎥⎣ ⎦
∫
∑
L
Nonlinear Systems
Estimation Theory (10_2) 11
( )
( )
( ) ( )
( )
2 32 3
2 3
11
Models
Linearized Models Employing the Taylor Series Expansio
, ,
, (1.83)
1 12! 3!
n
x x x
n
x f x u w
y h x v
f f ff x f x x x x
x x x
f x x xx x
=
=
∂ ∂ ∂= + + + +
∂ ∂ ∂
∂ ∂= + + +
∂ ∂( )
( )
( )
2
11
3
11
1
2!
1
3!
n x
nn
x
nn
x
f x
x x f xx x
x x f xx x
⎛ ⎞⎟⎜ +⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠
⎛ ⎞∂ ∂ ⎟⎜ + + +⎟⎜ ⎟⎜ ⎟⎜ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ⎟⎜ + + +⎟⎜ ⎟⎜ ⎟⎜ ∂ ∂⎝ ⎠
Nonlinear Systems (continued)
Estimation Theory (10_2) 12
( ) ( ) ( )
( ) ( ) ( )
( )
( )
2 3
1
1 12! 3!
(1.90)
Applying Eq. (1.90) into Eq. (1.83)
, ,
, ,
kkk
x x x x ii i
x
x
x
f x f x D f D f D f D f x f xx
ff x D f f x x f x Ax
x
x f x u w
f x u w
=
⎛ ⎞⎛ ⎞ ⎟⎜ ∂ ⎟⎟⎜ ⎜ ⎟= + + + + = ⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎜ ⎟∂⎝ ⎠⎜ ⎟⎜⎝ ⎠
∂≈ + = + = +
∂
=
≈
∑
( ) ( ) ( )
( )
.
. Say, 0 (1.93)
Similarly,
. (1.94)
x u w
x v
f f fx x u u w w
x u w
x Ax Bu Lw
x Ax Bu Lw w
h hy x v
x v
Cx Dv
∂ ∂ ∂+ − + − + −
∂ ∂ ∂
= + + +
= + + =
∂ ∂= +
∂ ∂
= +
Simulation/Trapezoidal Integration
Estimation Theory (10_2) 13
( )
( ) ( ) ( ) ( )[ ]
( ) ( ) ( )[ ]
[ ] ( ) ( )
0
1
0
00
1
We want to numerically solve the state equation, , , .
, ,
, , where =k for 0, , and / .
For some 0, 1 , we can write and as
f
k
k
t
ft
L t
k ft
k
n n
x f x u t
x t x t f x t u t t dt
x t f x t u t t dt t T k L T t L
n L x t x t
x t
+
=
+
=
= +
= + = =
∈ −
∫
∑∫
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ]
( ) ( ) ( )[ ]
1
1
1
00
1
1 00
, ,
, ,
, , . (1.110)
k
k
k
k
n
n
n t
nt
k
n t
nt
k
t
nt
x t f x t u t t dt
x t x t f x t u t t dt
x t f x t u t t dt
+
+
+
=
+
+=
= +
= +
= +
∑∫
∑∫
∫
Simulation/Trapezoidal Integration (continued)
Estimation Theory (10_2) 14
( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )
( ) [ ]
( ) ( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )
1 1 11
1 1 11
Approximate the integral in Eq. (1.110) as a trapezoid
, , , ,, , , , for ,
, , , ,, ,
n n n n n nn n n n n n
n n n n n nn n n n n
f x t u t t f x t u t tf x u t f x t u t t t t t t t
T
f x t u t t f x t u t tx t x t f x t u t t
T
+ + ++
+ + ++
⎛ ⎞− ⎟⎜ ⎟⎜≈ + − ∈⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠
⎛ ⎞− ⎟⎜ ⎟⎜≈ + + ⎟⎜ ⎟⎜⎜⎝ ⎠( )
( )( ) ( )( ) ( ) ( )( )
( ) ( ) ( )( ) ( ) ( )( )( )
( ) ( )( )
1
1 1 1
1 1 1
1
, , , ,
2
1 , , , , (1.114)
2Defining
, ,
n
n
t
nt
n n n n n nn
n n n n n n n
n n n
t t dt
f x t u t t f x t u t tx t T
x t f x t u t t T f x t u t t T
x f x t u t t T
x
+
+ + +
+ + +
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪−⎨ ⎬⎪ ⎪⎟⎪ ⎪⎪ ⎪⎩ ⎭⎛ ⎞+ ⎟⎜ ⎟⎜= + ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠
= + +
=
∫
( ) ( )( )( ) ( )( )
( ) ( ) ( )
2 1 1 1
1 1 1
1 1 2
, ,
, , ,
Eq. (1.114) may be expressed by
1. (1.115)
2
n n n
n n n
n n
f x t u t t T
f x t x u t t T
x t x t x x
+ + +
+ +
+
=
≈ + +
≈ + +
Simulation/Trapezoidal Integration (continued)
Estimation Theory (10_2) 15
( )
( ) ( )( )
( ) ( )( )
( )
0
0
1
2 1
1 2
Assume that is given
for : :
, ,
, , ,
1 ( ) ( )
2e
Trapezoidal Integration Algorit
nd
hm
f
x t
t t T t T
x f x t u t t T
x f x t x u t T t T T
x t T x t x x
= −
=
= + + + +
+ = + +
Observability and Controllability
Estimation Theory (10_2) 16
1
Consider the following time-invariant system and the deterministic asymptotic
estimation
x x u
z x (1.157)
wh
k k k
k k
A B
H
+ = +
=
0
ere state x , control input u , output z ;
and , , and are known constant matrices of appropriate dimension.
All variables are deterministic, so that if initial state x is known then Eq. (
n m pk k kR R R
A B H
∈ ∈ ∈
1.157)
can be solved exactly for x , z for 0.
Deterministic asymptotic estimation problem: Design an estimator
ˆwhose output x converges with k to the actual state x of Eq. (1.157)
when the initial s
k k
k k
k ≥
( )
0
1
tate x is unknown, but u and z are given exactly.
An estimator of observer which solves this problem has the form
ˆ x x z x u
as shown in Fig. 2.1-1.
k k
k k k k kA L H B+ = + − +
Observability and Controllability (continued)
Estimation Theory (10_2) 17
Observability and Controllability (continued)
Estimation Theory (10_2) 18
0
1 1 1
ˆTo Choose in Eq. (1.157) so that the estimation error x x x goes to zero with
for all x , it is necessary to examine the dynamics of x . Write
ˆ x x x
x u
k k k
k
k k k
k k
L k
A B A
+ + +
= −
= −
= + − ( )
( ) ( )
0
ˆx z x u
ˆ x x x x
( )x
It is now apparent that in order that x go to zero with for any x ,
observer gain must be selected so that ( ) is s
k k k k
k k k k
k
k
L H B
A L H H
A LH
k
L A LH
⎡ ⎤+ − +⎣ ⎦= − − −
= −
− table. can be chosen
so that x 0 if and only if ( , ) is detectable which is defined in the sequel.
(1). ( , ) is observable if the poles of ( ) can be arbitrarily assigned
by appropriate cho
k
L
A H
A H A LH
→
−
ice of the output injection matrix .
(2). ( , ) is detectable if ( ) can be made asymptotically stable by some matrix .
(If ( , ) is observable, then the pair is detectable; but the reverse is
L
A H A LH L
A H
−
not necessarily true.)
(3). (A, B) is controllable (reachable) if the poles of (A-BK) can be arbitrarily assigned by
appropriate choice of the feedback matrix K.
(4). (A, B) is stabilizable if (A-BK) can be made asymptotically stable by some matrix K.
(If (A, B) is controllable, then (A, B) is stabilizable; but the reverse is not necessarily true.)
Observability and Controllability (continued)
Estimation Theory (10_2) 19
1 1
1
: The -state discrete linear time-invariant system
has
Theor
the observability matrix defined by
em (Observability)
k k k
k k
n
n
x Ax Bu
y Hx
Q
H
HAQ
HA
− −
−
= +
=
⎡ ⎤⎢ ⎥⎢⎢
= ⎢⎢⎢⎢⎢⎣ ⎦
.
The system is observable if and only if ( ) .Q nρ
⎥⎥⎥⎥⎥⎥⎥
=
1 1
1
The n-state discrete linear time-invariant system
has the controllability matrix P defined by
Theore
.
The system is controllable if an
m (Controllability):
d
k k k
n
x Ax Bu
P B AB A B
− −
−
= +
⎡ ⎤= ⎢ ⎥⎣ ⎦only if ( ) .P nρ =
Chapter 2
Probability Theory
Probability
Estimation Theory (10_2) 21
{ }
{ }
{ } { }{ }
[ ] { }[ ] { }[ ] [ ]
1 2 3 4 5 6
, ,
sample space, e.g., , , , , ,
event space, , , , ,
probability assigned to ev
Probability Space
Probability Axi
ents, e.g., 0, 1/2, 1
Axiom 1 For any eve
oms
S A P
S S f f f f f f
A A S A odd even S
P P P odd P even P S
φ
φ
=
= =
= ⊂ =
= = = = =
P
[ ]
[ ]
[ ] [ ] [ ]
1 2
1 2 1 2
nt , 0.
Axiom 2 1.
Axiom 3 For any countable collection , , of
mutually exclusive events
.
A P A
P S
A A
P A A P A P A
≥
=
∪ ∪ = + +
Random Variables
Estimation Theory (10_2) 22
.s
x
X(s)=x
Sreal line
( )
( )
( ) ( )( ) ( )
( )
( ) [0,1]
( ) 0
P
( ) 1
( ) ( ) if
robability Distribution Function (
( ) ( )
,| |
( )
PDF)
X
X
X
X
X X
X X
X
F x P X x
F x
F
F
F a F b a b
P a X b F b F a
P X x AF x A P X x A
P A
= ≤
∈
−∞ =
∞ =
≤ ≤
< ≤ = −
≤= ≤ =
Random Variables (continued)
Estimation Theory (10_2) 23
( )
( )( )
( ) ( )2 22
( )( )
( ) ( )
( ) 0
( ) 1
( )
||
( ) ( )
( )
Probability Density Function (pdf):
Expected Value:
Variance:
XX
x
X X
X
X
b
Xa
XX
X
X X
dF xf x
dx
F x f z dz
f x
f x dx
P a X b f x dx
dF x Af x A
dx
E X xf x dx
E X X x X f x dxσ
−∞
∞
−∞
∞
−∞
∞
−∞
=
=
≥
=
< ≤ =
=
=
⎡ ⎤= − = −⎢ ⎥⎣ ⎦
∫
∫
∫
∫
∫
Random Variables (continued)
Estimation Theory (10_2) 24
[ ] [ ]( )
( )
[ ] [ ]
2 2
2
/2
2
1( )
;2 12
( ) , 02
;
Uniform Random Variable:
Gaussian (Normal) Random Variable:
X
X
x X
X XX
X
f x a x bb a
b aa bE X VAR X
ef x x
E X X VAR X
σ
σπσ
σ
− −
= ≤ ≤−
−+= =
= −∞ < < ∞ >
= =
Multiple Random Variables
Estimation Theory (10_2) 25
( )
( )
( , ) , ; ( , ) [0,1]
( , ) ( , ) 0; ( , ) 1
( , ) ( , ) if and
, ( , ) ( , ) ( , ) ( , )
( , ) ( ); ( , ) ( )
Joint Probability Distribution Function:
Joint Probabi
XYF x y P X x Y y F x y
F x F y F
F a c F b d a b c d
P a x b c y d F b d F a c F a d F b c
F x F x F y F y
= ≤ ≤ ∈
−∞ = −∞ = ∞ ∞ =
≤ ≤ ≤
< ≤ < ≤ = + − −
∞ = ∞ =
( )
( )
( ) ( )
2
1 2 1 2
( , )( , )
( , ) ,
( , ) 0; , 1
, ,
( ) ( , ) ; ( ) ( , )
lity Density Function:
XYXY
x y
d b
c a
F x yf x y
x y
F x y f z z dz dz
f x y f x y dxdy
P a x b c y d f x y dxdy
f x f x y dy f y f x y dx
−∞ −∞∞ ∞
−∞ −∞
∞ ∞
−∞ −∞
∂=
∂ ∂
=
≥ =
< ≤ < ≤ =
= =
∫ ∫∫ ∫
∫ ∫∫ ∫
Multiple Random Variables (continued)
Estimation Theory (10_2) 26
( )( ) ( )
( ) ( )
( )( ) ( )
( )
1 1 1
1 1
11/2/2
( , )
( , )
1 1( ) exp
2(2 )
Correlation Matrix:
Covariance Matrix:
Gaussian Random Vector:
m
TXY
n m
T T TXY
T
X XnX
E XY E XY
R x y E XY
E XY E X Y
C x y E X X Y Y E XY XY
f x X X CCπ
−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤= − − = −⎢ ⎥⎣ ⎦
= − − ( )
( ) ( ) ( )
( )
,
( , ) ( ) ( )
( , ) ( ) ( )
( ) ( ) (uncorrelatedness)
Statistical Independence:
XY X Y
XY X Y
XY
X X
P X x Y y P X x P Y y
F x y F x F y
f x y f x f y
R E XY E X E Y
⎡ ⎤−⎢ ⎥
⎢ ⎥⎣ ⎦
≤ ≤ = ≤ ≤
=
=
= =
Stochastic Processes
Estimation Theory (10_2) 27
Conceptual Representation of Stochastic Process
(1) Ensemble Average(2) Time Average
Stochastic Processes (continued)
Estimation Theory (10_2) 28
1( , ), ,(1) , = fixed a single number (an outcome of an experiment)(2) variable a time function fixed(3) fixed
X t s t R s St s Xt Xst X
∈ ∈=
= === = a random variable
variable(4) variable a random process (a family of time functions) variable
st Xs
== ==
Stochastic Processes (continued)
Estimation Theory (10_2) 29
Stationary Process:
Stochastic Processes (continued)
Estimation Theory (10_2) 30
Wide-Sense Stationary (WSS):
2(0) ( )
( ) ( )
( ) (0)
Properties of WSS:
X
X X
X X
E X t
R
R
R R
R
τ τ
τ
⎡ ⎤= ⎣ ⎦− =
≤
White Noise and Colored Noise
Estimation Theory (10_2) 31
( ) ( )
1( ) ( )
2
( ) (0) for all
( ) (0) ( )
Power Spectrum:
White Noise:
jX X
jX X
X X
X X
S R e d
S e d
S
R
R
R R
ωτ
ωτ
ω τ τ
τ ω ωπ
ω ω
τ δ τ
∞−
−∞
∞
−∞
=
=
=
=
∫
∫