control//state estimation//a priori and a posteriori

The notation every book on estimation uses once time enters. An index \(k\) marks the instant (0, 1, 2, one per tick of the system clock), and a superscript minus marks *before looking at the sensor*. Three objects then speak at each instant, and each has a voice:

The notation every book on estimation uses once time enters. An index kkk marks the instant (0, 1, 2, one per tick of the system clock), and a superscript minus marks before looking at the sensor. Three objects then speak at each instant, and each has a voice:

The a priori estimate x^k−\hat x_k^-x^k−​, the prediction, is the Physics speaking: what I believe the state is now, using only what I knew before and my model.

The measurement zkz_kzk​ is the Sensor speaking: what the sensor says at instant kkk.

The a posteriori estimate x^k\hat x_kx^k​, the corrected estimate, is the Judge's decision: what I decide to believe after hearing both.

?Why a superscript minus? It looks like a subtraction.

It is only a label: I have not listened to the sensor yet. Some books write x^k∣k−1\hat x_{k|k-1}x^k∣k−1​ (the estimate at kkk using data up to k−1k-1k−1) and x^k∣k\hat x_{k|k}x^k∣k​; exactly the same thing with more ink. In Python they would be x_pred and x_post.

The corrected estimate becomes the starting point of the next prediction, x^k+1−=Fx^k\hat x_{k+1}^-=F\hat x_kx^k+1−​=Fx^k​, and that is fine: the estimate is not the prediction, the two stay separate at every instant, and so do the model that predicts and the belief it is applied to. The same pair exists for the uncertainty, Pk−P_k^-Pk−​ before the sensor and PkP_kPk​ after, and how the Judge weighs the two voices is the Kalman gain of the Kalman filter. The three objects without time are in estimate.