**5. Vehicle simulation results**

In this section, the computer testing displays a critical driving situation. Initially, the system states and actuator faults will be estimated. Next, to demonstrate the effectiveness of the proposed FTC, the states of the faulty system will be simulated considering the fault-tolerant control law.

Simulations are performed with the forward steering angle profile given in **Figure 4**. This shows a sequence of right and left turns. The vehicle data is shown in **Table 3** [22].

Each *Ai* ð Þ ,*Ci* is observable. The resolution of the LMIs of the theorem above, using the LMI toolbox [23] and selecting *ρ* ¼ 4, results in the following matrices:

**Figure 4.** *Steering wheel angle δ <sup>f</sup>*ð Þ*t by* [rad]*.*


#### **Table 3.**

*Values of the vehicle parameters used in the simulations.*

*T-S Fuzzy Observers to Design Actuator Fault-Tolerant Control for Automotive Vehicle Lateral… DOI: http://dx.doi.org/10.5772/intechopen.92582*

*T*

*<sup>P</sup>* <sup>¼</sup> <sup>10</sup><sup>3</sup> *:*3145 0*:*0416 0*:*0057 0*:*0696 *:*0416 0*:*0197 0*:*0026 �0*:*0219 *:*0057 0*:*0026 0*:*0844 �0*:*3410 *:*0696 �0*:*0219 �0*:*3410 3*:*5255 , *F*<sup>1</sup> ¼ *:*2112 �0*:*8970 �0*:*0188 �0*:*0094 *<sup>Q</sup>*<sup>2</sup> <sup>¼</sup> <sup>10</sup><sup>3</sup> *:*0844 �0*:*1222 0*:*0146 �0*:*0014 0*:*0250 �0*:*1222 1*:*5054 �0*:*0286 �0*:*0433 �0*:*0480 *:*0146 �0*:*0286 3*:*2562 0*:*0068 �0*:*0039 �0*:*0014 �0*:*0433 0*:*0068 0*:*0464 0*:*0001 *:*0250 �0*:*0480 �0*:*0039 0*:*0001 1*:*3162 , *F*<sup>2</sup> ¼ *:*2642 �1*:*1348 �0*:*0227 �0*:*0117 *T L*<sup>1</sup> ¼ �2*:*0916 14*:*8942 �4*:*8065 �1*:*8332 �0*:*1687 1*:*0810 *:*9900 �6*:*3497 , *L*<sup>2</sup> ¼ *:*4158 �43*:*4496 *:*8851 �1*:*4886 *:*5984 0*:*3418 *:*2517 �11*:*5319 *G*<sup>1</sup> ¼ ½ � 40*:*1735 39*:*7987 , *G*<sup>2</sup> ¼ ½ � 5*:*8133 6*:*7343

To see the effectiveness of the proposed scheme, we compare the states of the zero-fault T-S model with the states of the faulty system both with and without the FTC strategy.

**Figure 5** illustrates the time evolution of the fault *f t*ð Þ and its estimate ^*f t*ð Þ, which shows that the fault estimate followed closely the fault, whereas **Figures 6**–**9** show the response of the automotive lateral dynamics reference system states simultaneously with its states in the presence of faults, in two scenarios: with and without the application of the FTC approach.

**Figure 5.** *Fault f t*ð Þ *and its estimate* ^*f t*ð Þ*.*

**Figure 6.**

*Comparison of the lateral velocity state vy*ð Þ*t response in the presence of actuator faults with the reference.*

**Figure 7.** *Comparison of the yaw rate state ψ*\_ ð Þ*t response in the presence of actuator faults with the reference.*

In **Figures 6**–**9**, simulations of faulty system states in the absence of FTC clearly show that the performances of the vehicle has been lost and that its states reached unsustainable levels immediately after the actuator became faulty, but applying fault-tolerant control law, the vehicle stays stable throughout the simulation

*T-S Fuzzy Observers to Design Actuator Fault-Tolerant Control for Automotive Vehicle Lateral… DOI: http://dx.doi.org/10.5772/intechopen.92582*

**Figure 9.** *Comparison of the roll rate state <sup>ϕ</sup>*\_ð Þ*<sup>t</sup> response in the presence of actuator faults with the reference.*

without losing performances despite the presence of faults, demonstrating the success of the proposed FTC approach.
