## 3 Cubature-Principle-Assisted IMM-Adaptive UKF Algorithm (CPIMM-NAUKF)
### New Adaptive Unscented Kalman Filter (NAUKF)  
Motivated by the proposed UKF algorithm, we propose a new adaptive UKF (NAUKF) consisting of an adaptive matrix gene. \(\mathbf{P}_{k/k-1}\), \(\mathbf{P}_{(XZ)k/k-1}\) and \(\mathbf{P}_{(ZZ)k/k-1}\) represent the corresponding covariance computed by the standard UKF, and they are changed into \(\mathbf{P}_{k/k-1}^{e}\), \(\mathbf{P}_{(XZ)k/k-1}^{e}\) and \(\mathbf{P}_{(ZZ)k/k-1}^{e}\) in the NAUKF.  
**Theorem 1**.: _On the basis of standard UKF, set up a new adaptive matrix gene\(\mathbf{\Delta}_{k}\) as_  
\[\mathbf{\Delta}_{k}=\Bigg{\{}\mathbf{P}_{(\widetilde{Z}\widetilde{Z})k/k-1}-\sum_{i=0 }^{2n}W_{i}^{(c)}[\gamma_{k/k-1}^{(i)}-\hat{\mathbf{Z}}_{k/k-1}][\gamma_{k/k-1}^{ (i)}-\hat{\mathbf{Z}}_{k/k-1}]^{T}\Bigg{\}}\mathbf{R}_{k}{}^{-1}. \tag{27}\]  
_Let_  
\[\mathbf{P}_{(ZZ)k/k-1}=\sum_{i=0}^{2n}W_{i}^{(c)}[\gamma_{k/k-1}^{(i)}-\hat{\mathbf{Z }}_{k/k-1}][\gamma_{k/k-1}^{(i)}-\hat{\mathbf{Z}}_{k/k-1}]^{T}+\mathbf{\Delta}_{k}\mathbf{ R}_{k} \tag{28}\]  
_where \(\mathbf{\gamma}_{k/k-1}^{(i)}\) is calculated from the Sigma points in the standard UKF; \(\hat{\mathbf{Z}}_{k/k-1}\) is the innovation; \(W_{i}^{(c)}\) denotes the weighted coefficient [3]; \(\widetilde{\mathbf{Z}}_{k/k-1}=\mathbf{Z}_{k}-\hat{\mathbf{Z}}_{k/k-1}\); and \(\mathbf{P}_{(\widetilde{Z}\widetilde{Z})k/k-1}\) is the covariance of \(\widetilde{\mathbf{Z}}_{k/k-1}\); Then the NAUKF can address the filtering problem in the presence of sensor faults shown by Equation (2)._  
Proof.: Because model mismatches exist in the system shown by Equation (2), based on Theorem 1, the result is  
\[\mathbf{P}_{k/k-1}^{e}=\mathbf{P}_{k/k-1}=\mathbf{P}_{(\widetilde{X}\widetilde{X})k/k-1} \tag{29}\]  
\[\mathbf{P}_{(XZ)k/k-1}^{e}=\mathbf{P}_{(XZ)k/k-1}=\mathbf{P}_{(\widetilde{X}\widetilde{Z} )k/k-1} \tag{30}\]  
where \(\widetilde{\mathbf{X}}_{k/k-1}=\mathbf{X}_{k}-\hat{\mathbf{X}}_{k/k-1}\).  
After setting up \(\mathbf{\Delta}_{k}\) shown by Equation (27), we have  
\[\mathbf{P}_{(ZZ)k/k-1}^{e}=\mathbf{P}_{(ZZ)k/k-1}=\mathbf{P}_{(\widetilde{Z}\widetilde{Z} )k/k-1}. \tag{31}\]  
From Equations (29) to (31), we know that the third-order precision conditions can be satisfied, which means that the NAUKF can be used for the system that has sensor faults. This completes the proof.  
To avoid the divergence problem of NAUKF, a covariance matching criterion is used:  
\[\text{tr}(\widetilde{\mathbf{Z}}_{k/k-1}\widetilde{\mathbf{Z}}_{k/k-1}{}^{\mathsf{T}} )\leq\Psi\cdot\text{tr}(\mathbf{P}_{(ZZ)k/k-1}) \tag{32}\]  
where \(\Psi\) (\(\Psi\geq 1\)) is a preset adjustable coefficient.  
If Equation (32) is satisfied, the NAUKF has been convergent; \(\mathbf{P}_{k/k-1}\) should be updated with an adaptive weighted coefficient \(\zeta_{k}\) by increasing the contribution of \(\mathbf{Z}_{k}\) on the filtering solutions.  
\[\overline{\mathbf{P}}_{k/k-1}=\zeta_{k}\sum_{i=0}^{2n}W_{i}^{(c)}[\mathbf{\chi}_{k/k-1} ^{(i)}-\hat{\mathbf{X}}_{k/k-1}][\mathbf{\chi}_{k/k-1}^{(i)}-\hat{\mathbf{X}}_{k/k-1}]^{ \mathsf{T}}+\mathbf{Q}_{k-1} \tag{33}\]  
where \(\zeta_{k}\) is set by \[\zeta_{k}=\left\{\begin{array}{cc}\zeta_{0}&\zeta_{0}\geq 1\\ 1&\zeta_{0}<1\end{array}\right. \tag{34}\]  
\[\zeta_{0}=\frac{\text{tr}(\mathbf{P}_{(ZZ)k/k-1}-\mathbf{R}_{k})}{\text{tr}(\sum \limits_{i=0}^{2n}W_{i}^{(c)}[\mathbf{\chi}_{k/k-1}^{(i)}-\mathbf{\hat{X}}_{k/k-1}][ \mathbf{\chi}_{k/k-1}^{(i)}-\mathbf{\hat{X}}_{k/k-1}]^{T})}\ . \tag{35}\]