# Combined Identification of Vehicle Parameters and Road Surface Roughness Using Vehicle Responses
### Identification of Unknown Vehicle Physical Parameters, State, and Unknown Force
\right\}=\mathbf{h}(\mathbf{Z}_{k+1},\mathbf{f}_{k+1}^{u})+\bm {v}_{k+1} \tag{6}\] \[=\mathbf{h}\Big{(}\widetilde{\mathbf{Z}}_{k+1|k},\vec{\mathbf{f}}_{k|k}^{u} \Big{)}+\mathbf{H}_{k+1|k}\Big{(}\mathbf{Z}_{k+1}-\dot{\mathbf{Z}}_{k+1|k}\Big{)}+\mathbf{D}_{k +1|k}^{u}\Big{(}\vec{\mathbf{f}}_{k+1}^{u}-\vec{\mathbf{f}}_{k|k}^{u}\Big{)}+\mathbf{v}_{k+1}\]  
where \(\mathbf{v}\) is mean-zero Gaussian white noise, \(\mathbf{H}_{k+1|k}=\frac{\partial\mathbf{h}(\mathbf{Z},\mathbf{f}^{u})}{\partial\mathbf{Z}}\Big{|} _{\mathbf{Z}_{k}=\widetilde{\mathbf{Z}}_{k+1|k}\vec{\mathbf{f}}^{u}=\vec{\mathbf{f}}_{k|k}^{ u}}\); \(\mathbf{D}_{k+1|k}=\frac{\partial\mathbf{h}(\mathbf{Z},\mathbf{f}^{u})}{\partial\mathbf{f}^{u}}\Big{|}_{ \mathbf{Z}_{k}=\widetilde{\mathbf{Z}}_{k|k}\vec{\mathbf{f}}^{u}=\vec{\mathbf{f}}_{k|k}^{u}}\). Based on the state and observation equations, the extended Kalman filtering with unknown input process is as follows:  
1. The predicted values for the first step of the state vector of the vehicle equation are expressed as follows: \[\widetilde{\mathbf{Z}}_{k+1|k}=\mathbf{\hat{Z}}_{k|k}+\int_{k\Delta t}^{(k+1)\Delta t} \mathbf{g}\Big{(}\mathbf{\hat{Z}}_{k|k},\vec{\mathbf{f}}_{k|k}^{u}\Big{)}\mathbf{dt}\] (7) The error covariance of the predicted values is as follows: \[\widetilde{\mathbf{P}}_{k+1|k}^{\mathbf{Z}}=\Big{[}\mathbf{\phi}_{k|k}^{\mathbf{Z}}\quad \Delta t\mathbf{B}_{k|k}^{u}\Bigg{]} \begin{bmatrix}\mathbf{\hat{P}}_{k|k}^{\mathbf{Z}}& \widetilde{\mathbf{P}}_{k|k}^{\mathbf{Z}}\\ \widetilde{\mathbf{P}}_{k|k}^{\mathbf{Z}}&\widetilde{\mathbf{P}}_{k|k}^{\mathbf{Z}}\end{bmatrix} \begin{bmatrix}\mathbf{\hat{P}}_{k|k}^{\mathbf{Z}T}\\ \mathbf{\hat{\Delta t}}\mathbf{B}_{k|k}^{\mathbf{Z}T}\end{bmatrix} +\mathbf{Q}_{k}\] (8)
2. The corrected values for the second step of the state vector of the vehicle equation are expressed as follows: \[\hat{\mathbf{Z}}_{k+1|k+1}=\widetilde{\mathbf{Z}}_{k+1|k}+\mathbf{K}_{k+1}\Big{[}\mathbf{y}_{k +1}-\mathbf{h}\big{(}\widetilde{\mathbf{Z}}_{k+1|k},\vec{\mathbf{f}}_{k|k}^{u}\big{)}-\mathbf{D} _{k+1|k}^{u}\big{(}\vec{\mathbf{f}}_{k+1|k+1}^{u}-\vec{\mathbf{f}}_{k|k}^{u}\big{)}\Big{]}\] (9) of which \[\mathbf{K}_{k+1}=\widetilde{\mathbf{P}}_{k+1|k}^{\mathbf{Z}}\mathbf{H}_{k+1|k}^{T}(\mathbf{H}_{k+1 }\widetilde{\mathbf{P}}_{k+1|k}^{\mathbf{Z}}\mathbf{H}_{k+1|k}^{T}+\mathbf{R}_{k+1})^{-1}\]The error covariance of the predicted values is as follows:  
\[\mathbf{P}_{k+1|k+1}^{\mathbf{Z}}=\Big{(}\mathbf{I}+\mathbf{K}_{k+1}\mathbf{D}_{k+1|k}^{u}\mathbf{S}_{k+1 }\mathbf{D}_{k+1}^{uT}\mathbf{R}_{k+1}^{-1}\mathbf{H}_{k+1|k}^{T}\Big{)}\times\Big{(}\mathbf{I}- \mathbf{K}_{k+1}\mathbf{H}_{k+1|k}\Big{)}\widetilde{\mathbf{P}}_{k+1|k}^{\mathbf{Z}} \tag{10}\]  
* Estimates of unknown forces for the vehicle equation are as follows:  
\[\widetilde{\mathbf{f}}_{k+1|k+1}^{u}=\mathbf{S}_{k+1}\mathbf{D}_{k+1|k}^{uT}\mathbf{R}_{k+1}^{- 1}(\mathbf{I}_{m}-\mathbf{H}_{k+1|k}\mathbf{K}_{k+1})\Big{[}\mathbf{y}_{k+1}-\mathbf{h}(\widetilde{ \mathbf{Z}}_{k+1|k}\mathbf{j}_{k|k}^{\mu})+\mathbf{D}_{k+1|k}^{u}\widetilde{\mathbf{f}}_{k|k}^{ u}\Big{]} \tag{11}\]  
* of which \(\mathbf{S}_{k+1}=\Big{[}\mathbf{D}_{k+1|k}^{uT}\mathbf{R}_{k+1}^{-1}(\mathbf{I}_{m}-\mathbf{H}_{k +1|k}\mathbf{K}_{k+1})\mathbf{D}_{k+1|k}^{u}\Big{]}^{-1}\). The error covariance of the unknown force is as follows:  
\[\hat{\mathbf{P}}_{k+1|k+1}^{\mathbf{f}}=\mathbf{S}_{k+1} \tag{12}\]  
* The error covariance of the cross terms is as follows:  
\[\hat{\mathbf{P}}_{k+1|k+1}^{\mathbf{Z}\mathbf{f}}=\left(\hat{\mathbf{P}}_{k+1|k+1}^{\mathbf{Z}\mathbf{ f}}\right)^{T}=-\mathbf{K}_{k+1}\mathbf{D}_{k+1|k}^{u}\mathbf{S}_{k+1} \tag{13}\]  
This completes the estimation of the partial parameters \(\mathbf{\theta}=\big{\{}m_{\mathbf{v}}\;\;I_{\mathbf{v}}\;\;k_{s1}\;\;k_{s2}\;\;c_{s1}\;\; c_{s2}\;\;a_{1S}\;\;a_{2S}\big{\}}\). and the vehicle state \(\mathbf{X}_{\mathbf{v}}(\mathbf{t}),\;\dot{\mathbf{X}}_{\mathbf{v}}(\mathbf{t})\), as well as the estimation of the unknown force timescales \(\mathbf{f}^{u}(t)\) applied to the front and rear axles.