ESTIMATION THEORY - ocw.snu.ac.krocw.snu.ac.kr/sites/default/files/NOTE/7068.pdfآ  Estimation...

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Transcript of ESTIMATION THEORY - ocw.snu.ac.krocw.snu.ac.kr/sites/default/files/NOTE/7068.pdfآ  Estimation...

  • ESTIMATION THEORY

    September 2010 Instructor: 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=

    ( ) 212 1zx t σσ = 2Estimation Theory (10_2)

  • Introduction (continued)

    3Estimation Theory (10_2)

    2 2 1(2). The second measurement: , z t t≅

    ( )[ ] [ ] 22221222 )/(/ 211212 zz zzzzzz σσσσσσμ +++= ( ) ( )222

    21 /1/1/1 zz σσσ +=

    ( ) ( )[ ] [ ] 222212222 )/(/ 211212 zztx zzzzzz σσσσσσμ +++== ( )[ ]( )122221 211 / zzz zzz −++= σσσ

    ( ) ( ) ( )1 2 2 1ˆ =Predictor + Correctorx t K t z x t⎡ ⎤= + −⎢ ⎥⎣ ⎦ ( ) ( ) ( ) ( )1221222 ttKtt xxx σσσ −=

  • Introduction (continued)

    Estimation Theory (10_2) 4

    ( ) ( )3 3 3(3). The third measurement: ; dx

    z x t w t u w dt

    = + = +

    ( ) ( ) ( )3 2 3 2x̂ t x t u t t− = + − ( ) ( ) ( )2322232 tttt wxx −+=− σσσ

    ( ) ( ) ( ) ( )[ ]−−+ −+= 33333 txztKtxtx ( ) ( ) ( ) ( )−−+ −= 3233232 ttKtt xxx σσσ

    ( ) ( ) ( )[ ]232323 3/ zxx tttK σσσ += −−

    ( )

    ( ) ( ) 3

    2 3

    2 2 3 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 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

    ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

  • 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

    i i

    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 B A 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 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 T T

    x A A A x

    x AAx A A

    x x

    Tr ABA Tr ABA AB AB AB B B

    A A

    = =

    ∂∂ = =

    ∂ ∂

    ∂ ∂ = + = =

    ∂ ∂

  • Continuous, Deterministic Linear Systems

    Estimation Theory (10_2) 10

    ( ) ( ) ( )

    ( )

    ( )

    0

    0 0

    0

    11

    Models

    Solution

    ( ) ( )

    !

    = .

    t A t t A t

    t

    j At

    j

    x Ax Bu

    y Cx

    x t e x t e Bu d

    At e

    j

    sI A

    τ τ τ− −

    =

    −−

    = +

    =

    = +

    =

    ⎡ ⎤−⎢ ⎥⎣ ⎦

    L

  • Nonlinear Systems

    Estimation Theory (10_2) 11

    ( )

    ( )

    ( ) ( )

    ( )

    2 3 2 3

    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

    ⎛ ⎞⎟⎜ +⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

    ⎛ ⎞∂ ∂ ⎟⎜ + + +⎟⎜ ⎟⎜ ⎟⎜ ∂ ∂⎝ ⎠

    ⎛ ⎞∂ ∂ ⎟⎜ + + +⎟⎜ ⎟⎜ ⎟⎜ ∂ ∂⎝ ⎠

  • Nonlinear Systems (continued)

    Estimation Theory (10_2) 12

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( )

    ( )

    2 3

    1

    1 1 2! 3!

    (1.90)

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

    , ,

    , ,

    kk k

    x x x x i i i

    x

    x

    x

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

    f f 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 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

    ∂ ∂ ∂ + − + − + −

    ∂ ∂ ∂

    = + + +

    = + + =

    ∂ ∂ = +

    ∂ ∂

    = +

  • Simulation/Trapezoidal Integration

    Estimation Theory (10_2) 13

    ( )

    ( ) ( ) ( ) ( )[ ]

    ( ) ( ) ( )[ ]

    [ ] ( ) ( )

    0

    1

    0

    0 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 t

    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

    +

    =

    +

    =

    = +

    = + = =

    ∈ −