## 6 Virtual Sensor Design
To facilitate the implementation on an on-board device, the system (32) was discretized with the sampling time \(T_{s}=0.01\) [s] using the Euler methods, which leads to:  
\[\mathbf{G}_{k}\mathbf{x}_{k+1}=\mathbf{G}_{k}\mathbf{x}_{k}+\mathbf{B}_{k}\mathbf{u}_{k}+\mathbf{E}_{k}\mathbf{ d}_{k}+\mathbf{W}\mathbf{w}_{k}, \tag{40}\]  
with  
\[\mathbf{G}_{k} =\mathbf{G}(\delta_{f,k},\delta_{r,k}),\quad\mathbf{B}_{k}=T_{s}\mathbf{B}( \delta_{f,k},\delta_{r,k}), \tag{41}\] \[\mathbf{E}_{k} =T_{s}\mathbf{E}(\delta_{f,k},\delta_{r,k}), \tag{42}\]  
where \(\mathbf{w}_{k}\) stands for an exogenous disturbance vector (which includes the discretization error) with a known distribution matrix \(\mathbf{W}\), while \(\mathbf{G}(\delta_{f,k},\delta_{r,k})\), \(\mathbf{B}(\delta_{f,k},\delta_{r,k})\) and \(\mathbf{E}(\delta_{f,k},\delta_{r,k})\) are obtained by substituting \(\delta_{f,k}\) and \(\delta_{r,k}\) into Equations (36)-(38), respectively. It should be pointed out that the proposed approach was tested with the Runge-Kutta discretization framework of various orders. However, the results were almost identical as those provided by the Euler method. As a consequence, it was used as the dedicated approach. The reason behind such a situation can be associated with the fact that the discretized system described in Equation (32) is a linear-time varying one composed of a set of first order differential equations. For the purpose of further deliberations, it is necessary to underline the fact that all state variables of the system descibed in Equation (40) are measured. Thus, contrarily to the approaches present in the literature (see, e.g., [26] and the references therein), the attention is focused on estimating \(\mathbf{d}_{k}\) only. Indeed, as the estimation of the state vector is unnecessary, it will simplify the proposed design procedure. Moreover, as can be deduced from Equation (38), depending on the steering angles \(\delta_{f,k}\) and \(\delta_{r,k}\), the matrix \(\mathbf{G}_{k}\) can be either nonsingular or singular. This means that the system described in Equation (40) can be either a linear time-varying system or a descriptor time-varying linear system (see, e.g., [27] and the references therein). Contrarily to the Kalman-filter-based approaches to descriptor systems, the system (40) has an unknown input \(\mathbf{d}_{k}\), which has to be estimated. Another class of approaches is dedicated to descriptor linear time-varying systems with unknown inputs described within linear-parameter-varying framework (see, e.g., [28]). Unfortunately, all these approaches inherit one common drawback, which is related to \(\mathbf{G}_{k}\) being a constant matrix \(\mathbf{G}_{k}=\mathbf{G}\), which is of course a singular one. This is however not the case in system (40). Thus, to the authors' knowledge, there is no approach present in the literature, which can be used to settle the problem of estimating \(\mathbf{d}_{k}\) in system (40). In the light of the above discussion, there is a need for developing a new estimator structure capable of:  
* estimation of \(\mathbf{d}_{k}\) without unnecessary estimation of the system state \(\mathbf{x}_{k}\),
* handling the issue of time-varying matrix \(\mathbf{G}_{k}\) of system (40). Finally, to tackle the virtual sensor design problem, the following novel adaptive estimator structure is proposed:  
\[\mathbf{\hat{d}}_{k+1}=\mathbf{\hat{d}}_{k}+\mathbf{L}(\mathbf{G}_{k}\mathbf{x}_{k+1}-\mathbf{G}_{k} \mathbf{x}_{k}-\mathbf{B}_{k}\mathbf{u}_{k}-\mathbf{E}_{k}\mathbf{\hat{d}}_{k}), \tag{43}\]  
where \(\mathbf{\hat{d}}_{k}\) stands for an estimate of \(\mathbf{d}_{k}\) and \(\mathbf{L}\) is the estimator gain matrix. Substituting system (40) into Equation (43) yields:  
\[\mathbf{\hat{d}}_{k+1}=\mathbf{\hat{d}}_{k}+\mathbf{L}(\mathbf{E}_{k}\mathbf{e}_{d,k}-\mathbf{W}\mathbf{w} _{k}), \tag{44}\]where \(\mathbf{e}_{d,k}=\mathbf{d}_{k}-\mathbf{\hat{d}}_{k}\) is an unknown input estimation error.