## 8 Experimental Results and Discussion  
Let us start with an _Offline_ phase of the proposed algorithm, which involves (65) under constraints eqrefeq:theoremst2. As a result, the optimal gain matrix of the virtual sensor (43) is:  
\[L=\begin{bmatrix}-2.9501&-27.9812&-28.3581&-28.3751&-28.4762\\ -16.3261&940.2042&-311.4692&-311.2939&-311.1626\\ -16.3437&-311.0201&939.9038&-311.4558&-311.1203\\ -16.3236&-311.0833&-311.4491&939.9822&-311.1366\\ -16.3238&-310.8944&-311.2802&-311.2160&939.6778\end{bmatrix}, \tag{74}\]  
along with \(\alpha=0.9\).  
Thus, the remaining objective of this section is to provide experimental results regarding the application of the developed AGV virtual sensors and their application to fault diagnosis according to the principles detailed in Section 6.2.  
Following the above defined driving scenario, Figures 4-7 present the longitudinal forces calculated with the model (nominal case) and their counterparts obtained with a set of measurements available from the AGV sensors. Note also that the plots show uncertainty intervals calculated according to (63) and (64). Analogous results are obtained for the torque, which are presented in Figure 8.  
As it was mentioned in Section 6, the proposed estimation strategy can be also realized with an alternative approach, which employs the descriptor-like form [28] of the system (40). For that purpose, it is necessary to use the small angle approximation for which parameter \(p_{r}\) in (38) is set to zero. This means that this matrix is constant \(\mathbf{G}_{k}=\mathbf{G}\) and a singular one. Figure 9 presents a comparison between the nominal force and the descriptor-based one. Note that the results are presented for \(F_{x,fr}\); however, a similar effect can be observed for the remaining forces as well as the torque. Indeed, it can be observed that the descriptor-based approach does not perform satisfactorily. Taking into account the results obtained with the proposed approach, it can be concluded that the developed strategy can be perceived as a very good tool for estimating AGV longitudinal forces as well as the torque.  
Figure 4: Nominal and estimated \(F_{x,fl}\).  
Figure 5: Nominal and estimated \(F_{x,fr}\).  
Figure 6: Nominal and estimated \(F_{x,rl}\).  
Figure 7: Nominal and estimated \(F_{x,rr}\).  
It can be observed that the estimates are consistent with the longitudinal forces and torques obtained by the model and the forces are equally distributed among the wheels. Indeed, Figure 10 presents the residual \(z_{x,ft}\), which clearly indicates that it is close to zero for the fault-free case except for some transient phases corresponding to fast changes in the steering angle. As can be seen in Figure 10, the residual centers nicely close to zero, clearly indicating the fault-free, nominal case. The same situation occurs for the remaining wheels as well as the torque, which are omitted.  
Figure 8: Nominal and estimated \(T_{r}\).  
Figure 9: Estimate of \(F_{x,ft}\) with the descriptor-LPV (Linear Parameter-Varying) approach (dashed line).  
If the virtual sensors operate properly in the fault-free case, then their performance can be evaluated in the fault case. For that purpose, the AGV was steered to two overlapping surfaces in a manner that one of the wheels was hanging in the air. In particular, the front right wheel had no contact with the surface, and, hence, it did not generate the longitudinal force appropriately. This unappealing phenomenon was immediately indicated by residual \(z_{x,fr}\), which is shown in Figure 11.  
Figure 11: Residual \(z_{x,fr}\) for the faulty case.  
Figure 10: Residual \(z_{x,fl}\) for the fault-free case.