**4. Fault-tolerant control strategy based on T-S fuzzy observers**

The purpose of this section is to design a faulty T-S fuzzy system for the vehicle model. Then, observers are trained to estimate system states and actuator faults. Next, a fault-tolerant control law is developed from the information provided by the observers.

#### **4.1 Faulty vehicle lateral dynamics system description**

From the T-S zero fault model (10), the system in the presence of actuator faults *f t*ð Þ is described as follows:

*<sup>x</sup>*\_ *<sup>f</sup>*ðÞ¼ *<sup>t</sup>* <sup>P</sup> 2 *i*¼1 *hi*ð Þ *<sup>ξ</sup>*ð Þ*<sup>t</sup> Aix <sup>f</sup>*ðÞþ*<sup>t</sup> Bi*ð Þ *uFTC*ðÞþ*<sup>t</sup> f t*ð Þ � � *<sup>y</sup> <sup>f</sup>*ðÞ¼ *<sup>t</sup>* <sup>P</sup> 2 *i*¼1 *hi*ð Þ *ξ*ð Þ*t Cix <sup>f</sup>*ð Þ*t* 8 >>>>>< >>>>>: (11)

where *x <sup>f</sup>*ð Þ*t* is the faulty system state vector, *y <sup>f</sup>*ð Þ*t* is the measured output vector of the faulty system, and *f t*ð Þ are the actuator faults. *uFTC*ð Þ*t* is the control law to be conceived thereafter; it is considered to be equivalent to a fault-tolerant control law added to the front wheel steering angle given by the driver.

#### **4.2 T-S fuzzy observers design**

In this subsection, the T-S fuzzy observers are constructed to simultaneously estimate states and actuator faults. Consider the following T-S observers:

$$\begin{cases} \dot{\hat{\mathbf{x}}}\_f(t) = \sum\_{i=1}^2 h\_i(\xi(t)) \left( A\_i \hat{\mathbf{x}}\_f(t) + B\_i \left( u\_{FTC}(t) + \hat{f}(t) \right) + L\_i \left( \mathbf{y}\_f(t) - \hat{\mathbf{y}}\_f(t) \right) \right) \\\\ \hat{\mathbf{y}}\_f(t) = \sum\_{i=1}^2 h\_i(\xi(t)) \mathbf{C}\_i \hat{\mathbf{x}}\_f(t) \\ \dot{\hat{f}}(t) = \sum\_{i=1}^2 h\_i(\xi(t)) \mathbf{G}\_i \left( \mathbf{y}\_f(t) - \hat{\mathbf{y}}\_f(t) \right) \end{cases} \tag{12}$$

where *<sup>x</sup>*^ *<sup>f</sup>*ð Þ*<sup>t</sup>* and ^*f t*ð Þ are the estimates of the state vector and the faults, respectively. ^*y <sup>f</sup>*ð Þ*t* is the estimate of the output vector. *Li* and *Gi* are the gain matrices with the appropriate dimensions to be resolved.

The output error between faulty T-S fuzzy systems (11) and T-S fuzzy observers (12) is given by

$$e\_{\mathcal{Y}}(t) = \mathcal{Y}\_f(t) - \hat{\mathcal{Y}}\_f(t) \tag{13}$$

$$=\sum\_{i=1}^{2} h\_i(\xi(t)) \mathbf{C}\_i(\mathbf{x}\_f(t) - \hat{\mathbf{x}}\_f(t)) \tag{14}$$

$$\mathbf{x} = \sum\_{i=1}^{2} h\_i(\xi(t)) \mathbf{C}\_i \mathbf{e}\_x(t) \tag{15}$$

The dynamics of the estimation error*ex*ðÞ¼ *t x <sup>f</sup>*ðÞ�*t x*^ *<sup>f</sup>*ð Þ*t* can be written as follows:

$$\dot{e}\_x(t) = \sum\_{i=1}^{2} \sum\_{j=1}^{2} h\_i(\xi(t)) h\_j(\xi(t)) \left( (A\_i - L\_i C\_j) e\_x(t) + B\_i e\_f(t) \right) \tag{16}$$

with *ef*ðÞ¼ *<sup>t</sup> f t*ð Þ� ^*f t*ð Þ. The tracking error *et*ð Þ*t* is expressed by

$$e\_t(t) = \mathbf{x}(t) - \mathbf{x}\_f(t) \tag{17}$$

and its dynamics is given by

$$\dot{e}\_{i}(t) = \sum\_{i=1}^{2} h\_{i}(\xi(t)) (A\_{i}\mathbf{x}(t) + B\_{i}\mathbf{u}(t)) - \sum\_{i=1}^{2} h\_{i}(\xi(t)) \left( A\_{i}\mathbf{x}\_{f}(t) + B\_{i}(\mathbf{u}\_{FTC}(t) + f(t)) \right) \tag{18}$$

$$=\sum\_{i=1}^{2} h\_i(\xi(t)) (A\_i e\_t(t) + B\_i(u(t) - u\_{FTC}(t)) - B\_i f(t)) \tag{19}$$

#### **4.3 Design fault-tolerant control based on T-S fuzzy observers**

A fault-tolerant control is proposed in this section to ignore the impact of faults and to maintain the stability of the faulty vehicle dynamics system. The suggested control procedure is outlined in **Figure 3** [20].

Consider the following control law:

$$u\_{\rm FTC}(t) = -\sum\_{i=1}^{2} h\_i(\xi(t)) F\_i(\mathbf{x}(t) - \hat{\mathbf{x}}\_f(t)) + u(t) - \hat{f}(t) \tag{20}$$

where *Fi* are the controller gains. Substituting Eq. (20) in Eq. (19) gives:

$$\dot{e}\_{t}(t) = \sum\_{i=1}^{2} \sum\_{j=1}^{2} h\_{i}(\xi(t))h\_{j}(\xi(t)) \left( A\_{i}e\_{t}(t) + B\_{i}F\_{j}\left(\mathbf{x}(t) - \hat{\mathbf{x}}\_{f}(t)\right) - B\_{i}e\_{f}(t) \right) \tag{21}$$

$$\hat{\mathbf{e}} = \sum\_{i=1}^{2} \sum\_{j=1}^{2} h\_i(\xi(t)) h\_j(\xi(t)) \left( \left( A\_i + B\_i F\_j \right) \mathbf{e}\_l(t) + B\_i F\_j \mathbf{e}\_x(t) - B\_i \mathbf{e}\_f(t) \right) \tag{22}$$

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

Assume that \_ *f t*ðÞ¼ 0, so the dynamics of the fault estimation error *ef*ð Þ*t* is given as follows:

$$\dot{e}\_f(t) = \dot{f}(t) - \sum\_{i=1}^{2} h\_i(\xi(t)) G\_i \left( \mathcal{Y}\_f(t) - \dot{\mathcal{Y}}\_f(t) \right) \tag{23}$$

$$=-\sum\_{i=1}^{2}\sum\_{j=1}^{2}h\_{i}(\xi(t))h\_{j}(\xi(t))G\_{i}C\_{j}e\_{x}(t)\tag{24}$$

Consider the following augmented system:

$$\dot{\overline{e}}(t) = \sum\_{i=1}^{2} \sum\_{j=1}^{2} h\_i(\xi(t)) h\_j(\xi(t)) \overline{A}\_{ij} \overline{e}(t) \tag{25}$$

$$\begin{aligned} \text{where } \overline{\boldsymbol{\pi}} &= \begin{bmatrix} \boldsymbol{e}\_{t}(t) \\ \boldsymbol{e}\_{\overline{\boldsymbol{x}}}(t) \\ \boldsymbol{e}\_{f}(t) \end{bmatrix} \text{ and }\\ \overline{\boldsymbol{A}}\_{\overline{\boldsymbol{y}}} &= \begin{bmatrix} \boldsymbol{A}\_{i} + \boldsymbol{B}\_{i}\boldsymbol{F}\_{j} & \boldsymbol{B}\_{i}\boldsymbol{F}\_{j} & -\boldsymbol{B}\_{i} \\ \mathbf{0} & \boldsymbol{A}\_{i} - \boldsymbol{L}\_{i}\mathbf{C}\_{j} & \boldsymbol{B}\_{i} \\ \mathbf{0} & -\mathbf{G}\_{i}\mathbf{C}\_{j} & \mathbf{0} \end{bmatrix} \end{aligned}$$

**Theorem 1.1.** The state tracking error *et*ð Þ*t* , the system state estimation error *ex*ð Þ*t* , and the fault estimation errors *ef*ð Þ*t* converge asymptotically toward zero if there exist positive scalar *ρ*> 0, symmetrical matrices *P*> 0, *Q*<sup>2</sup> >0, *Y* >0, and ϒ*ii* ð Þ *i* ¼ 1, 2 , as well as other matrices with appropriate dimensions *Ui*, *Vi* and ϒ*ij* ð Þ *i*, *j* ¼ 1, 2&*i*<*j* , such that the following conditions are satisfied:

$$
\Delta\_{\vec{n}} < \Upsilon\_{\vec{n}}, \quad \vec{n} = \mathbf{1}, \mathbf{2}. \tag{26}
$$

$$
\Delta\_{\vec{i}\vec{j}} + \Delta\_{\vec{j}\vec{i}} \le \mathbf{Y}\_{\vec{i}\vec{j}} + \mathbf{Y}\_{\vec{i}\vec{j}}^{T}, \qquad i, j = \mathbf{1}, 2, \quad i < j \tag{27}
$$

$$
\begin{bmatrix}
\mathbf{Y}\_{11} & \mathbf{Y}\_{12} \\
\\ \* & \mathbf{Y}\_{22}
\end{bmatrix} < \mathbf{0} \tag{28}
$$

where

$$
\Delta\_{\overline{i}} = \begin{bmatrix}
A\_i \mathbf{P} + P \mathbf{A}\_i^T + \mathbf{B}\_i \mathbf{V}\_j + \mathbf{V}\_j^T \mathbf{B}\_i^T & \mathbf{B}\_i \overline{\mathbf{X}}\_j & \mathbf{0} \\
& \ast & -2\rho Y & \rho I \\
& \ast & \mathbf{Q}\_2 \overline{\mathbf{A}}\_i + \overline{\mathbf{A}}\_i^T \mathbf{Q}\_2 - \mathbf{U}\_i \overline{\mathbf{C}}\_j - \overline{\mathbf{C}}\_j^T \mathbf{U}\_i^T \\
& & \ddots & \vdots \\
\end{bmatrix} \tag{29}
$$

$$\text{with } \overline{X}\_j = \begin{bmatrix} V\_j & -Z \end{bmatrix} \text{ and } Y = \begin{bmatrix} P & \mathbf{0} \\ \mathbf{0} & Z \end{bmatrix}.$$

The gains of T-S fuzzy observers *Li* and *Gi* and T-S fuzzy controllers *Fi* are calculated from

$$F\_i = V\_i P^{-1}, \ \overline{E}\_i = \begin{bmatrix} L\_i \\ G\_i \end{bmatrix} = \mathbf{Q}\_2^{-1} U\_i \tag{30}$$

**Proof:** the gains *Li*, *Gi*, and *Fi* are calculated by analyzing the system stability outlined in differential Eq. (25) by using the Lyapunov method with a quadratic function.

Let us select the following quadratic Lyapunov function:

$$V(\overline{e}(t)) = \overline{e}^T(t)Q\overline{e}(t) \tag{31}$$

where *Q* is divided as follows:

$$\mathbf{Q} = \begin{bmatrix} \mathbf{Q}\_1 & \mathbf{0} \\ \mathbf{0} & \mathbf{Q}\_2 \end{bmatrix}$$

The time derivative of *V t*ðÞ¼ *V*ð Þ *e t*ð Þ can be shown to be

$$
\dot{V}(t) = \overline{e}^T(t)Q\dot{\overline{e}}(t) + \dot{\overline{e}}^T(t)Q\overline{e}(t) \tag{32}
$$

$$\overline{\mathbf{c}} = \sum\_{i=1}^{2} \sum\_{j=1}^{2} h\_i(\xi(t)) h\_j(\xi(t)) \overline{\mathbf{c}}^T(t) Q \overline{A}\_{i\overline{j}} \overline{\mathbf{c}}(t) + \left( \overline{A}\_{i\overline{j}} \overline{\mathbf{c}}(t) \right)^T(t) Q \overline{\mathbf{c}}(t) \tag{33}$$

$$\overline{\mathbf{c}} = \sum\_{i=1}^{2} \sum\_{j=1}^{2} h\_i(\xi(t)) h\_j(\xi(t)) \overline{\mathbf{c}}^T(t) \left( Q \overline{A}\_{\overline{\mathbf{y}}} + \overline{A}\_{\overline{\mathbf{y}}}^T Q \right) \overline{\mathbf{c}}(t) \tag{34}$$

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

*Aij* can be defined as follows:

$$
\overline{A}\_{ij} = \begin{bmatrix} A\_i + B\_i F\_j & B\_i \overline{F}\_j \\\\ \mathbf{0} & \overline{A}\_i - \overline{E}\_i \overline{C}\_j \end{bmatrix} \tag{35}
$$

where

$$
\overline{F}\_j = \begin{bmatrix} F\_j & -I \end{bmatrix}, \ \overline{A}\_i = \begin{bmatrix} A\_i & B\_i \\ & \mathbf{0} \end{bmatrix}, \ \overline{\mathbf{C}\_i} = \begin{bmatrix} C\_i & \mathbf{0} \end{bmatrix}, \ \text{and} \ \overline{E}\_i = \begin{bmatrix} L\_i \\ G\_i \end{bmatrix}
$$

Inequality (34) is negative if the following conditions are satisfied:

$$\dot{V}(t) = \sum\_{i=1}^{2} \sum\_{j=1}^{2} h\_i(\xi(t)) h\_j(\xi(t)) \Lambda\_{ij} < 0 \tag{36}$$

with

$$
\Lambda\_{\overline{i}} = \begin{bmatrix}
\mathbf{Q}\_1 \mathbf{A}\_i + \mathbf{A}\_i^T \mathbf{Q}\_1 + \mathbf{Q}\_1 \mathbf{B}\_i \mathbf{F}\_j + \mathbf{F}\_j^T \mathbf{B}\_i^T \mathbf{Q}\_1 & \mathbf{Q}\_1 \mathbf{B}\_i \overline{\mathbf{F}}\_j \\\\ \mathbf{B}\_i \overline{\mathbf{F}}\_j^T \mathbf{Q}\_1 & \mathbf{Q}\_2 \overline{\mathbf{A}}\_i + \overline{\mathbf{A}}\_i^T \mathbf{Q}\_2 - \mathbf{Q}\_2 \overline{\mathbf{E}}\_i \overline{\mathbf{C}}\_j - \overline{\mathbf{C}}\_j^T \overline{\mathbf{E}}\_i^T \mathbf{Q}\_2
\end{bmatrix} \tag{37}
$$

Using the lemma of congruence, we have

$$
\Lambda\_{\vec{\eta}} < 0 \Leftrightarrow X \\
\Lambda\_{\vec{\eta}} X^T < 0 \tag{38}
$$

with

$$X = \begin{bmatrix} Q\_1^{-1} & \mathbf{0} \\ \mathbf{0} & Y \end{bmatrix}, \quad Y = \begin{bmatrix} Q\_1^{-1} & \mathbf{0} \\ \mathbf{0} & Z \end{bmatrix}, \ Z = Z^T > \mathbf{0}$$

Then, this inequality is obtained

$$\begin{bmatrix} A\_i Q\_1^{-1} + Q\_1^{-1} A\_i^T + B\_i F\_j Q\_1^{-1} + Q\_1^{-1} F\_j^T B\_i^T & B\_i \overline{F}\_j Y \\ & Y B\_i \overline{F}\_j^T & Y S\_{\overline{\eta}} Y \end{bmatrix} < 0 \tag{39}$$

The negativity of (39) enforces that

*Sij* <0

with *Sij* <sup>¼</sup> *<sup>Q</sup>*2*Ai* <sup>þ</sup> *AT <sup>i</sup> <sup>Q</sup>*<sup>2</sup> � *<sup>Q</sup>*2*EiCj* � *<sup>C</sup><sup>T</sup> j ET <sup>i</sup> Q*2. which can be analyzed using the following property:

$$\rho \left( \mathbf{Y} + \rho \mathbf{S}\_{\vec{\eta}}^{-1} \right)^{T} \mathbf{S}\_{\vec{\eta}} \left( \mathbf{Y} + \rho \mathbf{S}\_{\vec{\eta}}^{-1} \right) \le \mathbf{0} \Leftrightarrow \mathbf{Y} \mathbf{S}\_{\vec{\eta}} \mathbf{Y} \le -\rho \left( \mathbf{Y} + \mathbf{Y}^{T} \right) - \rho^{2} \mathbf{S}\_{\vec{\eta}}^{-1} \tag{40}$$

Accordingly, (39) can then be delineated as follows:

$$
\begin{bmatrix}
A\_i Q\_1^{-1} + Q\_1^{-1} A\_i^T + B\_i F\_j Q\_1^{-1} + Q\_1^{-1} F\_j^T B\_i^T & \mathbf{B}\_i \overline{F}\_j Y & \mathbf{0} \\
\* & -2\rho Y & \rho I \\
\* & \* & Q\_2 \overline{A}\_i + \overline{A}\_i^T Q\_2 - Q\_2 \overline{E}\_i \overline{C}\_j - \overline{C}\_j^T \overline{E}\_i^T Q\_2
\end{bmatrix} < \mathbf{0} \tag{41}$$

Using lemma 1, and with some manipulations, we can obtain easily (26)–(28). This completes the proof.

**Remark 1.** *The calculation of the observer and controller gains is done independently in* [11, 21]*, which is restrictive. Therefore, in this study, the resolution of the LMIs is carried out in one step.*
