1. A system for determining an estimated battery state vector indicative of a state of a battery at a first predetermined time, the system comprising: a sensor configured to generate a first signal indicative of an output variable of the battery; and a computer operably coupled to the sensor, the computer configured to determine a first plurality of estimated augmented battery state vectors that are indicative of a state of the battery, a battery input noise, a sensor noise, an uncertainty of the state of the battery, an uncertainty of the battery input noise, and an uncertainty of the sensor noise at a second predetermined time prior to the first predetermined time, the computer further configured to determine a second plurality of predicted battery state vectors that are indicative of the state of the battery and an uncertainty of the state of the battery at the first predetermined time based on the first plurality of estimated augmented battery state vectors, the computer further configured to determine a third plurality of predicted battery output vectors that are indicative of an 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 estimated augmented battery state vectors and the second plurality of predicted battery state vectors, the computer further configured to determine a first battery output vector based on the first signal, the computer further configured to determine a first estimated battery state vector indicative of the state of the battery at the first predetermined time based on the second plurality of predicted battery state vectors, the third plurality of predicted battery output vectors, and the first battery output vector.