Stable and verifiable state estimation methods and systems with spacecraft applications
First Claim
1. A method of enhancing the stability of a recursive estimator process that generates estimates X*(tn+) of state parameters X(tn) of a system from measurements Y(tn) wherein tn represents successive times and tn+ represents times that each lag a respective one of said successive times tn, said method comprising the steps of:
- generating at least one measurement Y(tn);
with an error covariance P(tn) and a measurement-noise covariance R(tn) of said process, calculating a gain K(tn);
processing said measurement Y(tn) with the aid of said gain K(tn) to generate at least one estimate X*(tn+);
recursively repeating said generating, calculating and processing steps to reduce said error covariance P(tn) to operational values Po that are not substantially reduced with further recursion; and
periodically resetting said error covariance P(tn) to a reset value Pr that exceeds said operational values Po to thereby reset said calculating step and enhance said stability.
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Abstract
The stability of a recursive estimator process (e.g., a Kalman filter is assured for long time periods by periodically resetting an error covariance P(tn) of the system to a predetermined reset value Pr. The recursive process is thus repetitively forced to start from a selected covariance and continue for a time period that is short compared to the system'"'"'s total operational time period. The time period in which the process must maintain its numerical stability is significantly reduced as is the demand on the system'"'"'s numerical stability. The process stability for an extended operational time period To is verified by performing the resetting step at the end of at least one reset time period Tr whose duration is less than the operational time period To and then confirming stability of the process over the reset time period Tr. Because the recursive process starts from a selected covariance at the beginning of each reset time period Tr, confirming stability of the process over at least one reset time period substantially confirms stability over the longer operational time period To.
18 Citations
30 Claims
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1. A method of enhancing the stability of a recursive estimator process that generates estimates X*(tn+) of state parameters X(tn) of a system from measurements Y(tn) wherein tn represents successive times and tn+ represents times that each lag a respective one of said successive times tn, said method comprising the steps of:
-
generating at least one measurement Y(tn);
with an error covariance P(tn) and a measurement-noise covariance R(tn) of said process, calculating a gain K(tn);
processing said measurement Y(tn) with the aid of said gain K(tn) to generate at least one estimate X*(tn+);
recursively repeating said generating, calculating and processing steps to reduce said error covariance P(tn) to operational values Po that are not substantially reduced with further recursion; and
periodically resetting said error covariance P(tn) to a reset value Pr that exceeds said operational values Po to thereby reset said calculating step and enhance said stability. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9)
with a state transition Φ
of said process, extrapolating a delayed version of an estimate X*(tn−
1) into a state prediction X*(tn−
) and a measurement prediction Y*(tn−
);
differencing said measurement Y(tn) and said measurement prediction Y*(tn−
) to form a residue Y(tn)−
Y*(tn−
); and
summing said state prediction X*(tn−
) with a correction K(tn){Y(tn)−
Y*(tn−
)} that is the product of said gain K(tn) and said residue Y(tn)−
Y*(tn−
) to thereby generate said attitude estimate X*(tn+);
wherein tn−
represents times that each lead a respective one of said successive times tn.
-
-
7. The method of claim 6, wherein said extrapolating step includes the step of multiplying said state prediction X*(tn−
- ) by a measurement converter H(tn) of said process to realize said measurement prediction Y*(tn−
).
- ) by a measurement converter H(tn) of said process to realize said measurement prediction Y*(tn−
-
8. The method of claim 6, further including the step of facilitating said calculating step with said measurement converter H(tn) and a process-noise covariance Q(tn) of said process.
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9. The method of claim 1, wherein said system is an attitude-control system of a spacecraft and said measurement Y(tn) includes measurements of at least one of the attitude and attitude rate of said spacecraft.
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10. A method of verifying the stability of a recursive estimator process for an operational time period To of a system wherein said process generates estimates X*(tn+) of state parameters X(tn) of said system from measurements Y(tn) wherein tn represents successive times and tn+ represents times that each lag a respective one of said successive times tn, said method comprising the steps of:
-
generating at least one measurement Y(tn);
with an error covariance P(tn) and a measurement-noise covariance R(tn) of said process, calculating a gain K(tn);
processing said measurement Y(tn) with the aid of said gain K(tn) to generate at least one estimate X*(tn+);
recursively repeating said generating, calculating and processing steps to reduce said error covariance P(tn) to operational values Po that are not substantially reduced with further recursion;
continuing said repeating step for at least one reset time period Tr whose duration is less than that of said operational time period To;
at the end of each reset time period Tr, resetting said error covariance P(tn) to a reset value Pr that exceeds said operational values Po to thereby reset said calculating step; and
confirming stability of said process over at least one reset time period Tr;
said continuing, resetting and confirming steps thereby verifying stability over said operational time period To. - View Dependent Claims (11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
with a state transition Φ
of said process, extrapolating a delayed version of an estimate X*(tn−
1) into a state prediction X*(tn−
) and a measurement prediction Y*(tn−
);
differencing said measurement Y(tn) and said measurement prediction Y*(tn−
) to form a residue Y(tn)−
Y*(tn−
); and
summing said state prediction X*(tn−
) with a correction K(tn){Y(tn)−
Y*(tn−
)} that is the product of said gain K(tn) and said residue Y(tn)−
Y*(tn−
) to thereby generate said attitude estimate X*(tn+);
wherein tn−
represents times that each lead a respective one of said successive times tn.
-
-
17. The method of claim 16, wherein said extrapolating step includes the step of multiplying said state prediction X*(tn−
- ) by a measurement converter H(tn) of said process to realize said measurement prediction Y*(tn−
).
- ) by a measurement converter H(tn) of said process to realize said measurement prediction Y*(tn−
-
18. The method of claim 16, further including the step of facilitating said calculating step with said measurement converter H(tn) and a process-noise covariance Q(tn) of said process.
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19. The method of claim 10, wherein said error covariance P(tn) is expressed as a matrix and said confirming step includes the step of verifying that said matrix remains a positive definite matrix.
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20. The method of claim 10, wherein said system is an attitude-control system of a spacecraft and said at least one measurement Y(tn) includes measurements of the attitude and attitude rate of said spacecraft.
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21. An attitude-controlled spacecraft, comprising:
-
a spacecraft body;
at least one star tracker that is coupled to said body for providing attitude signals;
at least one gyroscope that is coupled to said body for providing attitude-rate signals;
a data processor in said body that is programmed to perform a recursive estimator process that generates attitude estimates X*(tn+) from attitude measurements Y(tn) wherein tn represents successive times and tn+ represents times that each lag a respective one of said successive times tn, said process including the steps of;
a) generating at least one measurement Y(tn);
b) with an error covariance P(tn) and a measurement-noise covariance R(tn) of said process, calculating a gain K(tn);
c) processing said measurement Y(tn) with the aid of said gain K(tn) to generate at least one estimate x*(tn+);
d) recursively repeating said generating, calculating and processing steps to reduce said error covariance P(tn) to operational values Po that are not substantially reduced with further recursion; and
e) periodically resetting said error covariance P(tn) to a reset value Pr that exceeds said operational values Po to thereby reset said calculating step;
an attitude controller in said spacecraft that generates torque generation signals in response to an attitude difference between a commanded attitude and said attitude estimate X*(tn+); and
p1 a torque generation system that is coupled to generate torques in said body in response to said torque generation signals to thereby reduce said attitude difference;
said resetting step thereby resetting said calculating step and enhancing said stability. - View Dependent Claims (22, 23, 24, 25, 26, 27, 28, 29, 30)
with a state transition Φ
of said process, extrapolating a delayed version of an estimate X*(tn−
1) into a state prediction X*(tn) and a measurement prediction Y*(tn−
);
differencing said measurement Y(tn) and said measurement prediction Y*(tn−
) to form a residue Y(tn)−
Y*(tn−
); and
summing said state prediction X*(tn−
) with a correction K(tn){Y(tn)−
Y*(tn−
)} that is the product of said gain K(tn) and said residue Y(tn)−
Y*(tn−
) to thereby generate said attitude estimate X*(tn+);
wherein tn−
represents times that each lead a respective one of said successive times tn.
-
-
25. The spacecraft of claim 24, wherein said extrapolating step includes the step of multiplying said state prediction X*(tn−
- ) by a measurement converter H(tn) of said process to realize said measurement prediction Y*(tn−
).
- ) by a measurement converter H(tn) of said process to realize said measurement prediction Y*(tn−
-
26. The spacecraft of claim 24, further including the step of facilitating said calculating step with said measurement converter H(tn) and a process-noise covariance Q(tn) of said process.
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27. The spacecraft of claim 21, wherein said at least one star tracker comprises three star trackers directed along three mutually-orthogonal axes of said spacecraft and said at least one gyroscope comprises three gyroscopes arranged to detect spacecraft rotation about said axes.
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28. The spacecraft of claim 21, further including solar wings coupled to said body to generate power.
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29. The spacecraft of claim 21, further including antennas coupled to said body to transmit and receive communication signals.
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30. The spacecraft of claim 21, wherein said torque generation system includes thrusters coupled to said body.
Specification