Reduced state estimation with biased and out-of-sequence measurements from multiple sensors
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=0, 1, 2, . . . , which parameters λ
are unknown, arbitrarily time-varying, but bounded, and driven by the input function u(x(k), λ
), which may be nonlinear, and expressed by the state equation
x(k+1)=Φ
x(k)+Γ
u(x(k),λ
)
(73)where Φ
, Γ
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 expressed by the measurement equation
z(k)=Hx(k)+Jb+n(k)
(74)for 1≦
k≦
k0, where, if no measurements are used in the initialization of the filter, k0=0, where b is an unknown arbitrarily time-varying, but bounded, measurement bias vector with covariance B, whose components correspond to the different sensors, and where the sensor selector matrix J selects the appropriate components of sensor bias, and where n(k) is the measurement noise with covariance N and measurement matrix H at time tk;
initializing state estimates {circumflex over (x)}(k0|k0) and the matrices M(k0|k0), D(k0|k0), E(k0|k0) using a priori information and the initial measurements (D(k0|k0)=0 if the initial state estimates do not depend on the parameters λ
;
E(k0|k0)=0 if there are no initial measurements),wherevector {circumflex over (x)}(t|k) is defined as the estimate of the state of the system at time t after processing k 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 random errors in the k measurements z(i) for 1≦
i≦
k and a priori initial information that is independent of the parameter uncertainty and measurement bias 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 measurements z(i) for 1≦
i≦
k;
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, . . . ;
matrix E(t|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the sensor measurement bias, at time t after processing k measurements z(i) for 1≦
i≦
k;
matrix E(tj|k) is denoted as E(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;
determining the system transition matrices Φ
and Γ
using the update interval T=tk+1−
t;
determining the mean value λ
of unknown but bounded parameters λ
, and the input vector u({circumflex over (x)}(t|k), λ
);
measuring aspects of the state of the system expressed by the measurement equation
z(k)=Hx(k)+Jb+n(k)
(75)where b is an unknown arbitrarily time-varying, but bounded, measurement bias vector with covariance B, whose components correspond to the different sensors, and where the sensor selector matrix J selects the appropriate components of sensor bias, and 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-latedetermining F,G as follows 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), E(t|k) to {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), E(k+1|k) as in
{circumflex over (x)}(k+1|k)=Φ
{circumflex over (x)}(t|k)+Γ
u({circumflex over (x)}(t|k), λ
)
(78)
M(k+1|k)=FM(t|k)F′
(79)
D(k+1|k)=FD(t|k)+G
(80)
E(k+1|k)=FE(t|k)
(81) and calculating P(k+1|k) as in
P(k+1|k)=M(k+1|k)+D(k+1|k)Λ
D(k+1|k)′
(82)determining covariance of the residual Q as in
V=HE(k+1|k)+J
(83)
Q=HP(k+1|k)H′
+VBV′
+N
(84)determining the filter gain matrix K as in
A=M(t|k)F′
H′
+D(t|k)Λ
D(k+1|k)H′
+E(t|k)BV′
(85)
K=AQ−
1
(86)determining the matrix L as in
L=I−
KHF
(87)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)]
(88)updating the matrices M(t|k), D(t|k), E(t|k) to yield M(t|k+1), D(t|k+1), and E(t|k+1) as in
M(t|k+1)=LM(t|k)L′
+KNK′
(89)
D(t|k+1)=D(t|k)−
KHD(k+1|k)
(90)
E(t|k+1)=E(t|k)−
KV
(91)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)′
+E(t|k+1)BE(t|k+1)
(92)(b) and if the measurement is not time-latedetermining F,G using 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), and E(k|k), to {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), and E(k+1|k) as
{circumflex over (x)}(k+1|k)=Φ
{circumflex over (x)}(k|k)+Γ
u({circumflex over (x)}(k|k), λ
)
(95)
M(k+1|k)=FM(k|k)F′
(96)
D(k+1|k)=FD(k|k)+G
(97)
E(k+1|k)=FE(k|k)
(98) determining covariance of the residual Q as
P(k+1|k)=M(k+1|k)+D(k+1|k)Λ
D(k+1|k)′
(99)
V=HE(k+1|k)+J
(100)
Q=HP(k+1|k)H′
+VBV′
+N
(101) determining the filter gain matrix K as
A=P(k+1|k)H′
+E(k+1|k)BV′
(102)
K=AQ−
1
(103) determining the matrix L as
L=I−
KH
(104) 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)]
(105) updating the matrices M(k+1|k), D(k+1|k), E(k+1|k) as
M(k+1|k+1)=LM(k+1|k)L′
+KNK′
(106)
D(k+1|k+1)=LD(k+1|k)
(107)
E(k+1|k+1)=LE(k+1|k)−
KJ
(108) 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)′
+E(k+1|k+1)BE(k+1|k+1)′
(109)
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Abstract
This invention relates to state estimation after processing time-delayed measurements with unknown biases that may vary arbitrarily in time within known physical bounds. These biased measurements are obtained from systems characterized by state variables and by multidimensional parameters, for which the latter are also unknown and may vary arbitrarily in time within known physical bounds. If a measurement is time-late, apply the measurements to an out-of-sequence filter using a mean square optimization criterion that accounts for all sources of uncertainty and delay time, to produce estimates of the true states of the system. If the measurement is not time-late, apply the measurements to an in-sequence filter using a mean square optimization criterion that accounts for all sources of uncertainty to produce estimates of the true states of the system. The estimates are applied to one of (a) making a decision, (b) operating a control system, and (c) controlling a process.
21 Citations
6 Claims
-
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=0, 1, 2, . . . , which parameters λ
are unknown, arbitrarily time-varying, but bounded, and driven by the input function u(x(k), λ
), which may be nonlinear, and expressed by the state equation
x(k+1)=Φ
x(k)+Γ
u(x(k),λ
)
(73)where Φ
, Γ
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 expressed by the measurement equation
z(k)=Hx(k)+Jb+n(k)
(74)for 1≦
k≦
k0, where, if no measurements are used in the initialization of the filter, k0=0, where b is an unknown arbitrarily time-varying, but bounded, measurement bias vector with covariance B, whose components correspond to the different sensors, and where the sensor selector matrix J selects the appropriate components of sensor bias, and where n(k) is the measurement noise with covariance N and measurement matrix H at time tk;initializing state estimates {circumflex over (x)}(k0|k0) and the matrices M(k0|k0), D(k0|k0), E(k0|k0) using a priori information and the initial measurements (D(k0|k0)=0 if the initial state estimates do not depend on the parameters λ
;
E(k0|k0)=0 if there are no initial measurements),where vector {circumflex over (x)}(t|k) is defined as the estimate of the state of the system at time t after processing k 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 random errors in the k measurements z(i) for 1≦
i≦
k and a priori initial information that is independent of the parameter uncertainty and measurement bias 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 measurements z(i) for 1≦
i≦
k;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, . . . ; matrix E(t|k) is defined as the matrix of bias coefficients, which linearly relates state estimation errors to the sensor measurement bias, at time t after processing k measurements z(i) for 1≦
i≦
k;matrix E(tj|k) is denoted as E(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; determining the system transition matrices Φ
and Γ
using the update interval T=tk+1−
t;determining the mean value λ of unknown but bounded parameters λ
, and the input vector u({circumflex over (x)}(t|k),λ );measuring aspects of the state of the system expressed by the measurement equation
z(k)=Hx(k)+Jb+n(k)
(75)where b is an unknown arbitrarily time-varying, but bounded, measurement bias vector with covariance B, whose components correspond to the different sensors, and where the sensor selector matrix J selects the appropriate components of sensor bias, and 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 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), E(t|k) to {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), E(k+1|k) as in
{circumflex over (x)}(k+1|k)=Φ
{circumflex over (x)}(t|k)+Γ
u({circumflex over (x)}(t|k),λ )
(78)
M(k+1|k)=FM(t|k)F′
(79)
D(k+1|k)=FD(t|k)+G
(80)
E(k+1|k)=FE(t|k)
(81)and calculating P(k+1|k) as in
P(k+1|k)=M(k+1|k)+D(k+1|k)Λ
D(k+1|k)′
(82)determining covariance of the residual Q as in
V=HE(k+1|k)+J
(83)
Q=HP(k+1|k)H′
+VBV′
+N
(84)determining the filter gain matrix K as in
A=M(t|k)F′
H′
+D(t|k)Λ
D(k+1|k)H′
+E(t|k)BV′
(85)
K=AQ−
1
(86)determining the matrix L as in
L=I−
KHF
(87)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)]
(88)updating the matrices M(t|k), D(t|k), E(t|k) to yield M(t|k+1), D(t|k+1), and E(t|k+1) as in
M(t|k+1)=LM(t|k)L′
+KNK′
(89)
D(t|k+1)=D(t|k)−
KHD(k+1|k)
(90)
E(t|k+1)=E(t|k)−
KV
(91)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)′
+E(t|k+1)BE(t|k+1)
(92)(b) and if the measurement is not time-late determining F,G using 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), and E(k|k), to {circumflex over (x)}(k+1|k), M(k+1|k), D(k+1|k), and E(k+1|k) as
{circumflex over (x)}(k+1|k)=Φ
{circumflex over (x)}(k|k)+Γ
u({circumflex over (x)}(k|k),λ )
(95)
M(k+1|k)=FM(k|k)F′
(96)
D(k+1|k)=FD(k|k)+G
(97)
E(k+1|k)=FE(k|k)
(98)determining covariance of the residual Q as
P(k+1|k)=M(k+1|k)+D(k+1|k)Λ
D(k+1|k)′
(99)
V=HE(k+1|k)+J
(100)
Q=HP(k+1|k)H′
+VBV′
+N
(101)determining the filter gain matrix K as
A=P(k+1|k)H′
+E(k+1|k)BV′
(102)
K=AQ−
1
(103)determining the matrix L as
L=I−
KH
(104)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)]
(105)updating the matrices M(k+1|k), D(k+1|k), E(k+1|k) as
M(k+1|k+1)=LM(k+1|k)L′
+KNK′
(106)
D(k+1|k+1)=LD(k+1|k)
(107)
E(k+1|k+1)=LE(k+1|k)−
KJ
(108)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)′
+E(k+1|k+1)BE(k+1|k+1)′
(109)- View Dependent Claims (2, 3)
- in addition to state variables x(k) at time tk for k=0, 1, 2, . . . , which parameters λ
-
4. A method for estimating the state of a system, said method comprising the steps of:
-
observing a system having state variables and also having unknown, multidimensional, arbitrarily time-varying parameters, but which are subject to known bounded values; measuring certain aspects of the state of the system in the presence of sensor measurement biases and random errors to produce initial measurements; initializing state estimates and matrices using a priori information and the initial measurements; using the update interval in determining the system transition matrices and the mean value of unknown but bounded parameters and the input vector; determining if the measurement is time-late by testing the sign of the update interval; if the measurement is time-late applying the measurements to an out-of-sequence estimating filter that explicitly uses a mean square optimization criterion that separately accounts for measurement biases and errors and said bounding values, as well as the delay time, to produce estimates of the true state of the system; if the measurement is not time-late applying the measurements to an in-sequence estimating filter that explicitly uses a mean square optimization criterion that separately accounts for measurement biases and errors and said bounding values, to produce estimates of the true state of the system; applying said estimates to one of (a) make a decision, (b) operate a control system, and (c) control a process.
-
-
5. A method for state estimation by processing time-delayed measurements with unknown biases that may vary arbitrarily in time within known physical bounds, said method comprising the steps of:
-
obtaining measurements from systems characterized by state variables and by multidimensional parameters, which multidimensional parameters are also unknown and may vary arbitrarily in time within known physical bounds; if a measurement is time-late, the measurements are applied to an out-of-sequence filter using a mean square optimization criterion that nominally accounts for all sources of uncertainty and delay time, to produce estimates of the true states of the system; if the measurement is not time-late, the measurements are applied to an in-sequence filter using a mean square optimization criterion that nominally accounts for all sources of uncertainty to produce estimates of the true states of the system. - View Dependent Claims (6)
-
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Current AssigneeLockheed Martin Corporation (Martin Marietta Corporation)
-
Original AssigneeLockheed Martin Corporation (Martin Marietta Corporation)
-
InventorsMookerjee, Purusottam, Reifler, Frank J.
-
Primary Examiner(s)Sotomayor; John B.
-
Application NumberUS11/204,630Time in Patent Office1,008 DaysField of Search342/195, 342/192, 342/90, 342 95- 97, 342160-162US Class Current342/195CPC Class CodesG01S 13/723 by using numerical dataG01S 7/295 Means for transforming co-o...
