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

∞−

−∞

∞

−∞

=

=

=

=

∫

∫