System, method, and article of manufacture for determining an estimated battery parameter vector

1. A method for determining an estimated battery parameter vector indicative of a parameter of a battery at a first predetermined time, the method comprising:

determining a first plurality of predicted battery parameter vectors that are indicative of the parameter of the battery and an uncertainty of the parameter of the battery at the first predetermined time;

determining a battery state vector having at least one value indicative of a battery state at the first predetermined time;

determining a second plurality of predicted battery output vectors that are indicative of at least one output variable of the battery and an uncertainty of the output variable at the first predetermined time based on the first plurality of predicted battery parameter vectors and the battery state vector;

determining a first battery output vector having at least one measured value of a battery output variable obtained at the first predetermined time;

determining a first estimated battery parameter vector indicative of the parameter of the battery at the first predetermined time based on the first plurality of predicted battery parameter vectors, the second plurality of predicted battery output vectors, and the first battery output vector; and

storing the first estimated battery parameter vector in a memory device.

2. The method of claim 1 , wherein determining the first plurality of predicted battery parameter vectors comprises:

retrieving from a memory an estimated battery parameter vector indicative of the parameter of the battery at a second predetermined prior to the first predetermined time;

retrieving an estimated battery parameter vector covariance matrix indicative of an uncertainty of the parameter of the battery at the second predetermined time from the memory;

determining an estimated battery parameter noise covariance matrix indicative of an uncertainty of a parameter noise at the second predetermined time; and

calculating the first plurality of predicted battery parameter vectors based on the estimated battery parameter vector, the estimated battery parameter vector covariance matrix, and the estimated battery parameter noise covariance matrix.

3. The method of claim 2 , wherein calculating the first plurality of predicted battery parameter vectors comprises:

setting one member of the first plurality of predicted battery parameter vectors equal to the estimated battery parameter vector;

determining a predicted battery parameter vector covariance matrix indicative of an uncertainty of the parameter of the battery at the first predetermined time based on the estimated battery parameter vector covariance matrix and the estimated battery parameter noise covariance matrix;

setting a first additional set of L members of the first plurality of predicted battery parameter vectors equal to the first estimated battery parameter vector added to a constant value multiplied by respective columns extracted from a matrix square-root of the first predicted battery parameter vector covariance matrix, where L is the length of the first estimated battery parameter vector; and

setting a second additional set of L members of the first plurality of predicted battery parameter vectors equal to the constant value multiplied by respective columns extracted from a matrix square-root of the first predicted battery parameter vector covariance matrix subtracted from the first estimated battery parameter vector.

4. The method of claim 1 , wherein determining the first plurality of predicted battery parameter vectors comprises:

retrieving an estimated battery parameter vector indicative of the parameter of the battery at the second predetermined time from the memory device;

retrieving an estimated battery parameter vector square-root covariance matrix indicative of an uncertainty of the parameter of the battery at the second predetermined time from the memory device;

determining an estimated battery parameter noise square-root covariance matrix indicative of an uncertainty of a parameter noise at the second predetermined time; and

calculating the first plurality of predicted battery parameter vectors based on the estimated battery parameter vector, the estimated battery parameter vector square-root covariance matrix, and the estimated battery parameter noise square-root covariance matrix.

5. The method of claim 4 , wherein calculating the first plurality of predicted battery parameter vectors comprises:

setting one member of the first plurality of predicted battery parameter vectors equal to the estimated battery parameter vector;

determining a predicted battery parameter vector square-root covariance matrix indicative of an uncertainty of the parameter of the battery at the first predetermined time based on the estimated battery parameter vector square-root covariance matrix and the estimated battery parameter noise covariance matrix;

setting a first additional set of L members of the first plurality of predicted battery parameter vectors equal to the first estimated battery parameter vector added to a constant value multiplied by respective columns extracted from the first predicted battery parameter vector square-root covariance matrix, where L is the length of the first estimated battery parameter vector; and

setting a second additional set of L members of the first plurality of predicted battery parameter vectors equal to the constant value multiplied by respective columns extracted from the first predicted battery parameter vector square-root covariance matrix subtracted from the first estimated battery parameter vector.

6. The method of claim 1 , wherein determining the first estimated battery parameter vector comprises:

determining a first predicted battery output vector indicative of at least one output variable of the battery at the first predetermined time based on the second plurality of predicted battery output vectors;

determining a gain matrix based on the first predicted battery parameter vector, the first predicted battery output vector, the first plurality of predicted battery parameter vectors, and the second plurality of predicted battery output vectors; and

calculating the first estimated battery parameter vector based on the first predicted battery parameter vector, the first predicted battery output vector, the gain matrix, and the first battery output vector.

7. The method of claim 6 , wherein determining the first predicted battery output vector comprises calculating a weighted average of the second plurality of predicted battery output vectors.

8. The method of claim 6 , wherein determining the gain matrix comprises:

determining a first predicted battery output vector covariance matrix based on the first predicted battery output vector and the second plurality of predicted battery output vectors;

determining a first predicted cross-covariance matrix between a battery output vector and a battery parameter vector based on the first predicted battery output vector, the second plurality of predicted battery output vectors, the first predicted battery parameter vector, and the first plurality of predicted battery parameter vectors; and

calculating the gain matrix based on the first predicted cross-covariance matrix and the first predicted battery output vector covariance matrix.

9. The method of claim 8 , wherein determining the first predicted battery output vector covariance matrix comprises:

determining a first predicted battery output vector square-root covariance matrix based on the first predicted battery output vector and the second plurality of predicted battery output vectors; and

calculating the first predicted battery output vector covariance matrix based on the first predicted battery output vector square-root covariance matrix and the first predicted battery output vector square-root covariance matrix.

10. The method of claim 6 , wherein calculating the first estimated battery parameter vector comprises:

determining an innovation vector based on the first battery output vector and the first predicted battery output vector;

determining an update vector based on the gain matrix and the innovation vector; and

calculating the first estimated battery parameter vector based on the predicted battery parameter vector and the update vector.

11. The method of claim 6 , further comprising determining an estimated battery parameter vector covariance matrix at the first predetermined time based on the first predicted battery parameter vector, the first plurality of predicted battery parameter vectors, the gain matrix, and the first predicted battery output vector covariance matrix.

12. The method of claim 6 , further comprising determining an estimated battery parameter vector square-root covariance matrix at the first predetermined time based on the first predicted battery parameter vector, the first plurality of predicted battery parameter vectors, the gain matrix, and a predicted battery output vector square-root covariance matrix.