Recursive state estimation by matrix factorization

• US 6,721,770 B1
• Filed: 10/25/1999
• Issued: 04/13/2004
• Est. Priority Date: 10/25/1999
• Status: Expired due to Fees

• Current Assignee: Honeywell Incorporated (Honeywell International Inc.)
• Original Assignee: Honeywell Incorporated (Honeywell International Inc.)
• Inventors: Elgersma, Michael Ray, Morton, Blaise Grayson
• Primary Examiner(s): Ngo, Chuong Dinh
• Assistant Examiner(s): DO, CHAT C
• Application Number:
  US09/425,814
• Time in Patent Office: 1,632 Days
• Field of Search: 708/300, 702/104, 702/150, 702/183, 702/190, 703/2, 375/355
• US Class Current: 708/300
• CPC Class Codes: G06F 17/11 for solving equations , e.g. nonlinear equations, general mathematical optimization problems optimization specially adapted for a specific administrative, business or logistic context G06Q10/04

Abstract

A filter is disclosed for recursive state estimation of a process by matrix factorization. The filter preferably takes the general form of P=XY, where P is a matrix of previous state and/or current observations, Y is a matrix of functions that are used to model the process, and X is a coefficient matrix relating the functions in matrix Y to the previous state and/or current observations of matrix P. Given the previous state and/or current observations, a value for matrix Y can be computed from which a current state estimate can be recovered.

Claims

1. A method for recursive state estimation of a process, the method comprising the steps of:

obtaining a number of current observations of the process from one or more sensors;

providing the current observations to a filter that has a general form P=XY, where P is a matrix that includes the number of current observations, Y is a matrix of functions that model the process, and X is a coefficient matrix relating the functions in the Y matrix to the current observations of the P matrix; and

calculating the current state of the process by determining a value of selected functions in the Y matrix.

2. A method according to claim 1, wherein said obtaining, providing and calculating steps are repeated.

3. A method according to claim 1, wherein said filter is a non-linear filter that takes the general form of P=XY, where P is a matrix that includes the number of current observations, Y is a matrix of functions, including non-linear functions, that model the non-linear process, and X is a coefficient matrix relating the functions in the Y matrix to the current observations of the P matrix.

4. A method according to claim 3, wherein said non-linear filter calculates the value of the functions in the Y matrix using the current observations of the P matrix and selected coefficients of the X matrix.

5. A method according to claim 4, further comprising the steps of:

identifying a current process condition by examining the current observations of the P matrix;

providing an X matrix that corresponds to the current process condition; and

scheduling the X matrix for use during the calculating step of claim 4.

6. A method according to claim 4, further comprising the steps of:

providing an X matrix for each of a number of preselected process conditions;

identifying the current process condition by examining the current observations of the P matrix;

identifying three nearest preselected process conditions to the current process condition;

computing an interpolated X matrix for the current process condition from the three nearest predefined process conditions; and

scheduling the interpolated X matrix for use during the calculating step of claim 4.

7. A method according to claim 3, wherein the P matrix also includes a number of parameters reflecting a previous state of the non-linear process.

8. A method for recursive state estimation of a non-linear process, the method comprising the steps of:

providing a matrix P that includes current observation data of the process, the current observation data provided by one or more sensors;

providing a matrix Y that includes one or more functions, including one or more non-linear functions, that model the non-linear process;

providing a matrix X that includes coefficients that relate functions in the Y matrix to the current observation data of the P matrix;

determining a value for at least selected functions in the Y matrix using relation P=XY; and

estimating the current state of the non-linear process using selected functions in the Y matrix.

9. A method according to claim 8, wherein said estimating step estimates the current state of the non-linear process by estimating the current value of selected process parameters of the non-linear process.

10. A method according to claim 9, wherein said estimating step estimates the current value of the selected process parameters using a singular value decomposition algorithm.

11. A method according to claim 9, wherein said non-linear process is a flight control process.

12. A method according to claim 11, wherein said flight control process includes identifying the static pressure, dynamic pressure, attack angle and sideslip of an airplane.

13. A method according to claim 12, wherein the current observations of the P matrix include pressure measurements at various locations on the airplane.

14. A method according to claim 13, wherein the one or more functions in the Y matrix include static pressure, dynamic pressure, dynamic pressure times attack angle, dynamic pressure times sideslip, dynamic pressure times the square of attack angle (q-bar*α

²), dynamic pressure times attack angle times sideslip, and dynamic pressure times the square of sideslip (q-bar*β

²).

15. A method according to claim 14, wherein said estimating step estimates the current value of the static pressure, dynamic pressure, attack angle and sideslip of the airplane using selected values of the Y matrix.

16. A method according to claim 15, wherein said estimating step estimates the current value of the static pressure, dynamic pressure, attack angle and sideslip of the airplane using a singular value decomposition algorithm.

17. A method according to claim 9, wherein the current observation data entries in the P matrix are generated by one or more sensors.

18. A method according to claim 17, further comprising the step of determining if any of the sensors have failed.

19. A method according to claim 18, wherein said determining step of claim 18 includes the steps of:

calculating a matrix Z, such that Z^TX=0;

calculating a scalar err1 by multiplying the matrix Z^T times the matrix P; and

determining if the magnitude of err1 is larger than a predetermined threshold, and if so, concluding that one or more of the sensors failed.

20. A method according to claim 19, wherein if said determining step of claim 19 determined that one or more of the sensors failed, the method further comprises the steps of:

providing a system of equations P(i)=X(i)Y(i), which corresponds to the relation P=XY with the i^th row of the P and X matrix removed;

determining the value of the functions in the Y(i) matrix using the relation P(i)=X(i)Y(i);

providing a complete set of algebraic generators of the algebraic relations among the entries of the Y(i) matrix; and

identifying the one or more sensors that correspond to the i^th row of the P matrix as the failed sensors if all of the algebraic generators that correspond to the Y(i) matrix are satisfied within a predetermined range.

21. A method according to claim 20, wherein if said identifying step of claim 20 determines that the algebraic generators for each of the Y(i) matrices are not satisfied within a predetermined range, the method further comprises the steps of:

providing a system of equations P(i,j)=X(i,j)Y(i,j), which corresponds to the relation P=XY with the i^th row and the j^th row of the P and X matrix removed;

determining the value of the functions in the Y(i,j) matrix using the relation P(i,j)=X(i,j)Y(i,j);

providing a complete set of algebraic generators of the algebraic relations among the entries of the Y(i,j) matrix; and

identifying the one or more sensors that correspond to the i^th row and the j^th row of the P matrix as the failed sensors if all of the algebraic generators are satisfied within a predetermined range.

22. A method according to claim 19, wherein if said determining step of claim 19 determined that one or more of the sensors failed, the method further comprises the steps of:

providing a system of equations P(i)=X(i)Y(i), which corresponds to the relation P=XY with the i^th row of the P and X matrix removed;

determining a family of solutions for the functions in the Y(i) matrix using the relation P(i)=X(i)Y(i), wherein the family of solutions have the form Y(i)=A+λ

B, where matrix A is a particular solution for Y(i), matrix B satisfies the homogeneous equation X(i)B=0, and λ

is a scalar;

providing a complete set of algebraic generators of the algebraic relations among the entries of the Y(i) matrix;

performing a one parameter search over λ

to identify if all of the algebraic generators for any Y(i) are satisfied within a predetermined range; and

identifying the one or more sensors that correspond to the i^th row of the P matrix as the failed sensors if the performing step identifies that all of the algebraic generators for Y(i) are satisfied within the predetermined range.

23. A method according to claim 18, wherein said determining step of claim 18 includes the steps of:

calculating a first matrix Z₁, such that Z₁^TX=0;

calculating a second matrix Z₂, such that Z₂^TX=0;

calculating a first scalar err1 by multiplying the matrix Z₁^T times the matrix P;

calculating a second scalar err2 by multiplying the matrix Z₂^T times the matrix P; and

determining if the magnitude of either err1 or err2 is larger than a predetermined threshold, and if so, concluding that one or more of the sensors failed.

24. A method according to claim 8, further comprising the steps of:

identifying a current process condition by examining the current observations of the P matrix;

providing an X matrix that corresponds to the current process condition; and

scheduling the X matrix for use during the determining step of claim 8.

25. A method according to claim 8, wherein said determining step includes the steps of:

providing an X matrix for each of a number of preselected process conditions;

identifying the current process condition by examining the current observations of the P matrix;

identifying the three nearest preselected process conditions to the current process condition;

computing an interpolated X matrix for the current process condition from the three nearest predefined process conditions; and

scheduling the interpolated X matrix for use during the determining step of claim 8.

26. A method according to claim 25, wherein said computing step computes the interpolated X matrix using barycentric interpolation.

27. A method according to claim 8, further comprising the steps of:

providing an X matrix for each of a number of preselected process conditions;

providing a diagonal matrix W for each of the number of preselected process conditions, wherein each diagonal matrix has the selected functions in the Y matrix on its diagonal;

setting selected functions in the W matrix to approximate expected values for the corresponding process condition;

multiplying the diagonal matrix W by the corresponding X matrix to compute a normalized X matrix;

identifying the largest entry in each row of the normalized X matrix;

identifying those entries in each row of the normalized X matrix that fall below a predetermined threshold value relative to the largest entry of the row; and

setting the coefficients of the corresponding X matrix that correspond to the entries in the normalized X matrix that fall below the predetermined threshold value to zero.

28. A data processing system for recursive state estimation of a process, the data processing system comprising:

storage means for storing a number of current observations; and

a filter coupled to said storage means, said filter having a general form P=XY, where P is a matrix that includes the number of current observations, Y is a matrix of functions that model the process, and X is a coefficient matrix relating the functions in the Y matrix to the current observations of the P matrix, said filter receiving said current observations from said storage means and calculating the current state of the process by determining selected functions in the Y matrix.

29. A data processing system for recursive state estimation of a non-linear process, the data processing system comprising:

storage means for storing a number of current observations; and

a non-linear filter coupled to said storage means, said non linear filter receiving said current observations from said storage means and calculating the current state of the non-linear process using matrix factorization.

30. A data processing system according to claim 29, wherein said non-linear filter calculates the current state of the non-linear process using the relation P=XY, where P is a matrix that includes the number of current observations, Y is a matrix of functions including non-linear functions that model the non-linear process, and X is a coefficient matrix relating the functions in the Y matrix to the current observations of the P matrix.

31. A data processing system according to claim 30, wherein said non-linear filter determines the value of the functions in the Y matrix using the relation P=XY, and estimates the current state of the non-linear process using selected functions in the Y matrix.

32. A data processing system according to claim 31, wherein said non-linear filter estimates the current state of the non-linear process by estimating the current value of selected process parameters of the non-linear process.

33. A data processing system according to claim 32, wherein said non-linear filter estimates the current value of the selected process parameters of the non-linear process by using a singular value decomposition algorithm.

34. A data processing system according to claim 29, wherein said non-linear filter comprises a microprocessor.