*3.5. Application of Multi-Innovation in Kalman Filter Framework Algorithm*

In the traditional KF framework algorithm, only the error of the current time is used to update the state of the next time. The model is simple and easy to calculate, but it also brings problems. For highly complex nonlinear time-varying battery operating conditions, the state of the next time is likely to be related not only to the current time but also to several times before the current time, resulting in a decline in accuracy. In order to solve the problem and further improve the estimation progress, the multi-innovation identification theory is introduced into the measurement equation. The calculation formula of multiinnovation identification is as follows.

Expand a single innovation *ek* into an innovation matrix *ep*,*k*.

$$\begin{aligned} \mathcal{C}\_{p,k} &= \begin{bmatrix} \mathcal{C}\_k \\ \mathcal{C}\_{k-1} \\ \mathcal{C}\_{k-2} \\ \vdots \\ \mathcal{C}\_{k-p+1} \end{bmatrix} \end{aligned} \tag{34}$$

At the same time, the gain *kk* is extended to the gain matrix *kp*,*<sup>k</sup>*

$$k\_{p,k} = \begin{bmatrix} k\_{k'}k\_{k-1}\cdots \ \_ \prime k\_{k-p+1} \end{bmatrix} \tag{35}$$

Therefore, the status measurement update needs to be modified as follows:

$$\mathbf{y}\_k = \mathbf{y}\_k + \left[\mathbf{k}\_k, k\_{k-1}, \dots, k\_{k-\mathbf{p}+1}\right] \mathbf{e}\_{p,k} \tag{36}$$

Namely,

$$y\_k = \hat{y}\_k + \sum\_{i=0}^p \gamma\_i k\_{i,k} e\_{k-i} \tag{37}$$

where

$$\begin{cases} \gamma\_1 = 1\\ \gamma\_2 = \gamma\_3 = \dots \cdot \gamma\_p = \frac{a}{M-1}, 0 \le a \le 1, M \ge 2 \end{cases} \tag{38}$$

where *M* is the innovation length and *a* is the adjustable coefficient. This paper takes *M* as 22and takes *a* as 0.5. A detailed discussion about the selection of the two parameters will be conducted in Section 5.3.1.
