×

Reduced state estimation with multisensor fusion and out-of-sequence measurements

  • US 7,009,554 B1
  • Filed: 03/30/2005
  • Issued: 03/07/2006
  • Est. Priority Date: 03/30/2005
  • Status: Active Grant
First Claim
Patent Images

1. A method for recursively estimating the state of a system having multidimensional parameters λ

  • in addition to state variables x(k) at time tk for k=1,2,3, . . . , which parameters λ

    are unknown, arbitrarily time-varying, but bounded, and driven by the input function u(x(k),λ

    ) and governed by the state equation
    x(k+1)=Φ

    x(k)+Γ

    u(x(k),λ

    )where Φ



    are system matrices dependent on the discrete time interval T=tk+1

    tk;

    said method comprising the following steps;

    measuring aspects of the state of the system to produce initial measurements;

    initializing state estimates {circumflex over (x)}(k0|k0) and the matrices M(k0|k0), D(k0|k0) using a priori information and the initial measurements,wherevector {circumflex over (x)}(t|k) is defined as the estimate of the state of the system at time t after processing the measurements z(i) for 1≦

    i≦

    k;

    vector {circumflex over (x)}(tj|k) is denoted as {circumflex over (x)}(j|k) when the time t=tj is the time of the jth measurement for j=1,2,3, . . . ;

    matrix M(t|k) is defined as the covariance of the state estimation errors at time t due only to the errors in the measurements z(i) for 1≦

    i≦

    k and a priori initial information that is independent of the parameter uncertainty;

    matrix M(tj|k) is denoted as M(j|k), when the time t=tj is the time of the jth measurement for j=1,2,3, . . . ;

    matrix D(t|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the parameter errors, at time t (after processing k=0,1,2, . . . measurements);

    matrix D(tj|k) is denoted as D(j|k), when the time t=tj is the time of the jth measurement for j=1,2,3, . . . ;

    determining the time tk+1 of a new measurement and the time t when the filter was last updated;

    using the update interval T=tk+1

    t, determine the system transition matrices Φ and

    Γ

    ;

    determining the mean value {overscore (λ

    )} of unknown but bounded parameters λ

    , and the input vector u({circumflex over (x)}(t|k),{overscore (λ

    )});

    measuring aspects of the state of the system governed by the measurement equation
    z(k)=Hx(k)+n(k) 

    where n(k) is the measurement noise with covariance N and measurement matrix H at time tk for k=1,2,3, . . . ;

    determining if the measurement is time-late by testing T<

    0;

    (a) if the measurement is time-late determining F,G as follows F=Φ

    +Γ







    u


    x


    x=x^

    (t

    k
    )
    ,λ

    =λ

    _
    G=Γ







    u


    λ



    x=x^

    (t

    k
    )
    ,λ

    =λ

    _
    generating a parameter matrix Λ

    , representing physical bounds on the parameters λ

    that are not state variables of the system;

    extrapolating said state estimates {circumflex over (x)}(t|k) and matrices M(t|k), D(t|k), to {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), and calculating S(k+1|k) as in
    {circumflex over (x)}(k+1|k)=Φ

    {circumflex over (x)}(t|k)+Γ

    u({circumflex over (x)}(t|k),{overscore (λ

    )})
    M(k+1|k)=FM(t|k)F′


    D(k+1|k)=FD(t|k)+G
    S(k+1|k)=M(k+1|k)+D(k+1|k

    D(k+1|k)′

    determining covariance of the residual Q as in
    Q=HS(k+1|k)H′

    +N
    determining the filter gain matrix K as in
    K=[M(t|k)F′

    +D
    (t|k

    D(k+1|k)′

    ]H′

    Q


    1
    determining the matrix L as in
    L=I−

    KHF


    where I is the identity matrix;

    updating the state estimate {circumflex over (x)}(t|k) as
    {circumflex over (x)}(t|k+1)={circumflex over (x)}(t|k)+K[z(k+1)−

    H{circumflex over (x)}(k+1|k)]updating the matrices M(t|k) and D(t|k) as
    M(t|k+1)=LM(t|k)L′

    +KNK′


    D(t|k+1)=D(t|k)−

    KHD(k+1|k) 

    respectively, and generating the total mean square error S(t|k+1) as in
    S(t|k+1)=M(t|k+1)+D(t|k+1)Λ

    D(t|k+1)′

    (b) and if the measurement is not time-late determining F,G using F=Φ

    +Γ







    u


    x


    x=x^

    (k

    k
    )
    ,λ

    =λ

    _
    and, G=Γ







    u


    λ



    x=x^

    (k

    k
    )
    ,λ

    =λ

    _
    generating a parameter matrix Λ

    , representing physical bounds on those parameters that are not state variables of the system;

    extrapolating said state estimates {circumflex over (x)}(k|k) and matrices M(k|k), D(k|k), S(k|k) to {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), and S(k+1|k) as in
    {circumflex over (x)}(k+1|k)=Φ

    {circumflex over (x)}(k|k)+Γ

    u({circumflex over (x)}(k|k),{overscore (λ

    )})
    M(k+1|k)=FM(k|k)F′


    D(k+1|k)=FD(k|k)+G
    S(k+1|k)=M(k+1|k)+D(k+1|k)Λ

    D(k+1|k)′

    determining covariance of the residual Q as in
    Q=HS(k+1|k)H′

    +N
    determining the filter gain matrix K as in
    K=S(k+1|k)H′

    Q


    1
    determining the matrix L as in
    L=I−

    KH


    where I is the identity matrix;

    updating the state estimate {circumflex over (x)}(k+1|k) as
    {circumflex over (x)}(k+1|k+1)={circumflex over (x)}(k+1|k)+K[z(k+1)−

    H{circumflex over (x)}(k+1|k)]updating the matrices M(k+1|k) and D(k+1|k) as
    M(k+1|k+1)=LM(k+1|k)L′

    +KNK′

    and,
    D(k+1|k+1)=LD(k+1|k)respectively, and generating the total mean square error S(k+1|k+1) as
    S(k+1|k+1)=M(k+1|k+1)+D(k+1|k+1)Λ

    D(k+1|k+1)′

    after

View all claims
  • 1 Assignment
Timeline View
Assignment View
    ×
    ×