ESTIMATION THEORY

(1)

ESTIMATION THEORY

September 2010

Instructor: Jang Gyu Lee

(2)

Introduction

1

( )

1

Find the best estimate from the measurements of the formˆ where is a rando

Problem:

(1). The fi

m proc rst me

ess.

asurement:

x z

t w

z w

x w

= x

=

+ +

( )

₁ ₁

ˆ

x t = z

( )

²

2 t σ

σ =

(3)

Introduction (continued)

2 2 1

(2). The second measurement: , z t ≅ t

( )

[ ] [ ]

2

2 2

2 1

2 2

2 / /( )

2 1

1 2

1

2 _z _z z _z _z _z z

z σ σ σ σ σ

σ

μ = + + +

(

²

) (

²

)

2

1 1/

/ 1 /

1 σ = σ_z + σ_z

( ) [ ( ) ] [ ]

2

2 2

2 1

2 2

2

2 / /( )

2 1

1 2

1

2 z z

t

x = μ = σ_z σ_z +σ_z + σ_z σ_z +σ_z

( )

[ ] (

2 1

)

2 2

2

1 ₁ / ₁ ₂ z z

z + _z _z + _z −

= σ σ σ

( )₁ ( )₂ ₂ ( )₁

ˆ =Predictor + Corrector

x t K t ⎡z x t ⎤

= + ⎢⎣ − ⎥⎦

( )

₂ ²

( )

₁

( ) ( )

₂ ² ₁

2 t _x t K t _x t

x σ σ

σ = −

(4)

Introduction (continued)

( ) ( )

3 3 3

(3). The third measurement: ; dx

z x t w t u w

= + dt = +

( )

³

( )

²

(

³ ²

)

ˆ

x t ⁻ = x t +u t −t

( ) ( ) (

3 2

)

2

2 2 3

2 t _x t _w t t

x ⁻ =σ +σ −

σ

( ) ( )

t3⁺ ⁼ x t3⁻ ⁺ K

( )

t3

[

z3 ⁻ x

( )

t3⁻

]

x

( )

⁺ ⁼

( )

⁻ ⁻

( ) ( )

3⁻ 2 3 3

2 3

2 t _x t K t _x t

x σ σ

σ

( ) ( ) [ ( )

3 ²

]

2 3

2

3 _x t / _x t _z₃

t

K =σ ⁻ σ ⁻ +σ

( )

( ) ( )

3

2

3

2 2

As , 0.

As , and 1.

z K t

t K t

σ

σ σ ⁻

→ ∞ =

→ ∞ → ∞ =

(5)

Chapter 1

Linear Systems Theory

(6)

Matrix Algebra

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

2 2

1 1

; 1, , ; 1, , . (1.13) In general, . (no commutability)

Inner Product:

(2). Vector Product

.

Outer Prod

s

r

ij ik kj

k

T

n n

n

C A B i n j p

AB BA

x

x x x x x x

x

=

= = =

≠

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥

= ⎢⎣ ⎥⎦ ⎢ ⎥⎢ ⎥ = + +

⎢ ⎥⎣ ⎦

∑

^" ^"

" # "

2

1 1 1

1

2

1 1

uct: . (1.14)

n T

n

n n

x x x x

xx x x

x x x x

⎡ ⎤

⎡ ⎤ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎡ ⎤ ⎢ ⎥

= ⎢ ⎥ ⎢⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎥⎦ = ⎢⎢⎢⎣ ⎥⎥⎥⎦

"

# " # % #

"

(7)

Matrix Algebra (continued)

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

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

i i

A

Ax b x

Tr A

b A

Tr AB Tr BA AB

A

B A

• ≠

• =

•

=

∑

( ) ¹ ¹ ¹

AB ⁻ = B A⁻ ⁻ .

(8)

Matrix Algebra (continued)

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 B

A D

C D B

⎡ ⎤

⎢ ⎥

⎣ ⎦

(

¹

)

¹ ¹ 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⁻

(9)

Matrix Calculus

( )

( ) ( ) ( ) [ ] ( )

( ) ( )

1

11 1

1

1 1

(1). .

/ /

(2). ; , 1, , , 1, , .

/ /

(3). / / ; .

(4). ;

n

ij

m mn

T T

T T T T

n n

T T

T T T

f x f f

x x x

f A f A

f A A A i m j n

A f A f A

x y x y

x y x x y x y y y x

x y

x Ax x Ax

x A x A x

⎡ ⎤

∂∂ = ⎢⎢⎣∂∂ ∂∂ ⎥⎥⎦

⎡∂ ∂ ∂ ∂ ⎤

⎢ ⎥

∂ ⎢ ⎥

= ⎢ ⎥ = = =

⎢ ⎥

∂ ⎢⎢⎣∂ ∂ ∂ ∂ ⎥⎥⎦

∂ ∂ = ∂⎡⎢⎣ ∂ ∂ ∂ ⎤⎥⎦ = = ∂ ∂ =

∂ ∂

= +

∂ ∂

"

# % # " "

"

" "

( )

( ) ( )

2 if .

(5). ; .

(6). ; 2 , if .

T T

T

T T

x A A A x

Ax x A

A A

x x

Tr ABA Tr ABA

AB AB AB B B

A A

= =

∂ ∂

= =

∂ ∂

= + = =

∂ ∂

(10)

Continuous, Deterministic Linear Systems

( )

⁽ ⁾

( )

0

0 1 1

Models

Solution

( ) ( )

!

= .

A t t t A t

t j

At j

x Ax Bu y Cx

x t e x t e Bu d

e At

j

sI A

τ τ τ

− −

∞

=

− −

= +

=

= +

=

⎡ − ⎤

⎢ ⎥

⎣ ⎦

∫

∑

L

(11)

Nonlinear Systems

( )

( ) ( ) ( )

2 3

1 1

Models

Linearized Models Employing the Taylor Series Expansio , ,

, (1.83)

1 1

2! 3!

n

x x x

n

x f x u w y h x v

f f f

f x f x x x x

x x x

f x x x

x x

=

∂ ∂ ∂

= + + + +

∂ ∂ ∂

∂ ∂

= + + +

∂ ∂

"

" ( )

( )

2 1

1

3 1

1

1 2!

1 3!

n x

n

n x

n

n x

f x

x x f x

x x

x x f x

x x

⎛ ⎞⎟

⎜ ⎟ +

⎜ ⎟

⎜⎝ ⎠

⎛ ∂ ∂ ⎟⎞

⎜ + + ⎟ +

⎜ ⎟

⎜ ∂ ∂

⎝ ⎠

⎛ ∂ ∂ ⎟⎞

⎜ + + ⎟ +

⎜ ⎟

⎜ ∂ ∂

⎝ ⎠

"

" "

(12)

Nonlinear Systems (continued)

( ) ( ) ( )

( )

2 3

1

1 1

2! 3!

(1.90)

Applying Eq. (1.90) into Eq. (1.83) , ,

, ,

k k k

x x x x i

i i

x

f x f x D f D f D f D f x f x

x f x D f f x f 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 f

x x u u w w

x u w

x Ax Bu Lw

x Ax Bu Lw w

h h

y x v

x v

Cx Dv

∂ ∂ ∂

+ − + − + −

∂ ∂ ∂

= + + +

= + + =

∂ ∂

= +

∂ ∂

= +

(13)

Simulation/Trapezoidal Integration

( )

( ) ( )

^[ ^{( ) ( )} ^]

( )

[ ^{( ) ( )} ]

[ ]

( ) ( )

0

1

0

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

f t

L t

k f

k t

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

0

0 1

1 0

0

, ,

, , . (1.110)

k k

k

k n n

n t

n k t

n t

k t

n t

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

+

= + +

=

= +

∑ ∫

∫

(14)

Simulation/Trapezoidal Integration (continued)

( )

(

( ) ( )

) (

^{( ) ( )}

) (

^{( ) ( )}

)

( ) [ ]

( ) ( )

(

( ) ( )

) (

^{( ) ( )}

) (

^{( ) ( )}

)

1 1 1

1

1 1 1

1

Approximate the integral in Eq. (1.110) as a trapezoid

, , , ,

, , , , for ,

, , , ,

, ,

n n n n n n

n n n n n

f x t u t t f x t u t t

f 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 t

x t x t f x t u t t

T

+ + +

+

+ + +

+

⎛ − ⎞⎟

⎜ ⎟

≈ +⎜⎜⎜⎜⎝ ⎟⎟⎟⎠ − ∈

⎛ − ⎞⎟

⎜ ⎟

≈ + +⎜⎜⎜⎜⎝ ⎟⎟⎠( )

( )

(

( ) ( )

) (

^{( ) ( )}

)

( )

( (

( ) ( )

) (

^{( ) ( )}

) )

( ) ( )

( )

1

1 1 1

1

, , , ,

2

1 , , , , (1.114)

2 Defining

, ,

n n

t t n

n n n n n n

n

n n n n n n n

n n n

t t dt

f x t u t t f x t u t t

x 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

f x t u t t T

f x t x u t t T

x t x t x x

+ + +

+ +

+

=

≈ + +

+

+ +

(15)

Simulation/Trapezoidal Integration (continued)

( )

( ) ( )

( )

( ) ( )

( )

0 0

1

2 1

1 2

Assume that is given

for : :

, ,

, , ,

( ) ( ) 1 2 e

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

= −

=

= + + + +

+ = + +

+

+ +

(16)

Observability and Controllability

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 p

k R k R k R

A B H

∈ ∈ ∈

1.157) can be solved exactly for x , z for 0.

Deterministic asymptotic estimation problem: Design an estimator whose output x converges with k to the actual state x of Eq. (1.157)ˆ when the initial s

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

x x z xˆ u

as shown in Fig. 2.1-1.

k k

k₊ = A k +L k −H k +B k

(17)

Observability and Controllability (continued)

(18)

Observability and Controllability (continued)

0

1 1 1

ˆ

To 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

x z ˆx 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

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.

(19)

Observability and Controllability (continued)

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

x Ax Bu

y Hx

Q H

HA Q

HA

− −

−

= +

=

⎡ ⎤

⎢ ⎥

⎢⎢

= ⎢⎢

⎢⎢

⎢⎣ ⎦

# .

The system is observable if and only if ( )ρ Q n.

⎥⎥

⎥

=

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 ( )ρ P = n.

(20)

Chapter 2

Probability Theory

(21)

Probability

{ }

{ } ^{ ^}

{ }

[ ]

[

{ }

]

^[^{ ^}^] ^{[ ]}

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

=

= =

= ⊂ =

= = = = =

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

≥

=

∪ ∪ = + +

"

" "

(22)

Random Variables

s

.

x X(s)=x

S

real line

( )

( ) (

⁽ ⁾

)

⁽ ⁾

( )

( ) [0,1]

( ) 0

P

( ) 1

( ) ( ) if

robability Distribution Function (

( ) ( )

| | ,

( ) PDF)

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 A F x A P X x A

P A

= ≤

∈

−∞ =

∞ =

≤ ≤

< ≤ = −

= ≤ = ≤

(23)

Random Variables (continued)

( )

( ) ( )

( )

²

( )

²

2

( ) ( )

( ) 0

( ) 1

( )

| |

( ) ( )

( ) Probability Density Function (pdf):

Expected Value:

Variance:

X X

x

X X

X

b a X

X X

X

X X

f x dF x

dx

F x f z dz f x

f x dx

P a X b f x dx dF x A

f x A

dx E X xf x dx

E X X x X f x dx

σ

−∞

∞

−∞

∞

−∞

∞

−∞

=

≥

=

< ≤ =

=

⎡ ⎤

= ⎢⎣ − ⎥⎦ = −

∫

(24)

Random Variables (continued)

[ ] [ ] ( )

( )

[ ] [ ]

2 2

2

/2

2

( ) 1

2 ; 12

( ) , 0

2

;

Uniform Random Variable:

Gaussian (Normal) Random Variable:

X

x X

X X

X

f x a x b

b a

b a a b

E X VAR X

f x e x

E X X VAR X

σ

πσ σ

σ

− −

= ≤ ≤

−

= + =

= − ∞ < < ∞ >

= =

(25)

Multiple Random Variables

( )

( , ) , ; ( , ) [0,1]

( , ) ( , ) 0; ( , ) 1

( , ) ( , ) if and

, ( , ) ( , ) ( , ) ( , )

( , ) ( ); ( , ) ( )

Joint Probability Distribution Function:

Joint Probabi

FXY 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:

XY XY

x y

d b

c a

F x y f 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

−∞ −∞

∞ ∞

−∞ −∞

∞ ∞

−∞ −∞

= ∂

∂ ∂

=

≥ =

< ≤ < ≤ =

= =

∫ ∫

(26)

Multiple Random Variables (continued)

( )

( ) ( )

( )( ) ( )

( )

1 1 1

1 1

1 /2 1/ 2

( , )

1 1

( ) exp

(2 ) 2

Correlation Matrix:

Covariance Matrix:

Gaussian Random Vector:

m T

XY

n m

T T T

XY

T

X n X

X

E X Y E X Y R x y E XY

E X Y E X Y

C x y E X X Y Y E XY XY

f x X X C

π C

−

⎡ ⎤

⎢ ⎥

= =

⎢ ⎥

⎣ ⎦

⎡ ⎤

= ⎢⎣ − − ⎥⎦ = −

= − −

"

# #

"

( )

( ) ( ) ( )

( ) ,

( , ) ( ) ( ) ( , ) ( ) ( )

( ) ( ) (uncorrelatedness) Statistical Independence:

XY X Y

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

⎡ ⎤

⎢ − ⎥

⎢ ⎥

⎣ ⎦

≤ ≤ = ≤ ≤

=

= =

(27)

Stochastic Processes

Conceptual Representation of Stochastic Process

(1) Ensemble Average (2) Time Average

(28)

Stochastic Processes (continued)

( , ),

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 S

t s X

t X

s

t X

∈ ∈

=

= =

=

= = a random variable

variable

(4) variable a random process (a family of time functions) variable

s

t X

s

=

= =

=

(29)

Stochastic Processes (continued)

Stationary Process:

(30)

Stochastic Processes (continued)

Wide-Sense Stationary (WSS):

(0) 2( )

( ) ( )

( ) (0)

Properties of WSS:

X

X X

E X t

R

R R

R

τ τ

τ

⎡ ⎤

= ⎣ ⎦

− =

≤

(31)

White Noise and Colored Noise

( ) ( )

( ) 1 ( )

2

( ) (0) for all ( ) (0) ( )

Power Spectrum:

White Noise:

j

X X

j

X X

S R e d

S e d

S

R

R R

ωτ

ω τ τ

τ ω ω

π

ω ω

τ δ τ

∞ −

−∞

∞

−∞

=

∫