**3. Fuzzy switching control development**

In this section, a fuzzy switching control will be developed for the ADMIRE fighter aircraft. **Figure 2** illustrates a schematic of the control structure. Here, linear quadratic integral (LQI) control computes an optimal state feedback gain for the regulating closed-loop system. The control law consists of the solution of the Riccati equation in the linear-quadratic regulatory framework with the integral of the output variable. The linearized dynamics of the aircraft at a trim condition with state-space realization are given as:

$$
\dot{\mathbf{x}}(t) = A\mathbf{x}(t) + Bu(t)
$$

$$
\mathbf{y}(t) = \mathbf{C}\mathbf{x}(t) + Du(t)\tag{2}
$$

The objective of the LQI control is to find the state feedback control law, such as

$$\mu(t) = -K[\varkappa(t)e\_l(t)]^T \tag{3}$$

where *K* is the feedback gain matrix, and *eI*ð Þ*t* is the integral state for the output variable. The optimal feedback law minimizes the quadratic performance index.

$$J = \int\_0^\infty (\mathbf{x}^T Q \mathbf{x} + \mathbf{x}^T R \,\mathbf{u}) dt \tag{4}$$

**Figure 2.** *Schematic of the control structure.*

In which *Q* is a positive semi-definite weight matrix, and *R* is a positive-definite weight matrix.

Then, this control law guarantees that the output *y t*ð Þ tracks the demand signal *r t*ð Þ. In fact, *eI*ð Þ*t* is

$$
\sigma\_I(t) = \int\_0^t (r(\tau) - \jmath(\tau))d\tau \tag{5}
$$

The state-space presentation of augmented dynamic is written as:

$$
\begin{vmatrix}
\dot{\mathbf{x}}(t) \\
\dot{e}\_I(t)
\end{vmatrix} = \begin{vmatrix}
A(t) & \mathbf{0} \\
\end{vmatrix} \begin{vmatrix}
\mathbf{x}(t) \\
e\_I(t)
\end{vmatrix} + \begin{vmatrix}
B(t) \\
\end{vmatrix} u(t) \tag{6}
$$

To cover the flight envelope, the flight envelope is divided into some cells. Augmented switched state-space model is given as:

$$
\begin{vmatrix}
\dot{\boldsymbol{x}}\_{\sigma(t)}(t) \\
\dot{\boldsymbol{e}}\_{I\sigma(t)}(t)
\end{vmatrix} = \begin{vmatrix}
\boldsymbol{A}\_{\sigma(t)}(t) & \mathbf{0} \\
\end{vmatrix} \begin{vmatrix}
\boldsymbol{\varkappa}\_{\sigma(t)}(t) \\
\boldsymbol{e}\_{I\sigma(t)}(t)
\end{vmatrix} + \begin{vmatrix}
\boldsymbol{B}\_{\sigma(t)}(t) \\
\end{vmatrix} \mathbf{u}(t) \tag{7}
$$

The system matrices of Eq. (7) are rewritten as:

$$
\begin{vmatrix} A\_{\sigma(t)} & B\_{\sigma(t)} \\ C\_{\sigma(t)} & D\_{\sigma(t)} \end{vmatrix} = \begin{vmatrix} A\_i & B\_i \\ C\_i & D\_i \end{vmatrix} \mathbf{i} = \mathbf{1}, \dots, \mathbf{M} \tag{8}
$$

where *σ*ð Þ*t* is a switching rule that takes values {**1**, … , *M*}, *M* is the number of subsystems. The switched control scheme is

$$u(t) = -K\_{\sigma(t)}[\mathbf{x}(t)e\_I(t)]^T\tag{9}$$

To design a fuzzy switching controller, ADMIRE flight envelope has been divided into four overlapping cells as shown in **Figure 3** with the dotted lines showing the boundaries between cells. Here, the fuzzy switching control law is

$$u(t) = -K\_{fuxxy\sigma(t)}[\mathbf{x}(t)\mathbf{e}\_I(t)]^T\tag{10}$$

The controller gains are designed using the data from each related cell center, and the fuzzy switching controller is computed as follows, based on the fuzzy logic rule:

$$K\_{\text{fuxz}}\rho(t) = \begin{cases} K\_1 & \text{Alt} \le 1550 \text{ and } \text{Mach} \le 0.6, \\ K\_2 & \text{Alt} \le 1550 \text{ and } \text{Mach} \ge 1.1, \\ K\_3 & \text{Alt} \ge 4500 \text{ and } \text{Mach} \ge 1.1, \\ K\_4 & \text{Alt} \ge 4500 \text{ and } \text{Mach} \le 0.6, \\ \rho\_1 K\_1 + \rho\_2 K\_2 & \text{Alt} \le 1550 \text{ and } \text{Mach} \in (0.6, 1.1), \\ \rho\_2 K\_2 + \rho\_3 K\_3 & \text{Alt} \in (1550, 4500) \text{ and } \text{Mach} \ge 1.1, \\ \rho\_3 K\_3 + \rho\_4 K\_4 & \text{Alt} \ge 4500 \text{ and } \text{Mach} \in (0.6, 1.1), \\ \rho\_1 K\_1 + \rho\_4 K\_4 & \text{Alt} \in (1550, 4500) \text{ and } \text{Mach} \le 0.6, \\ \rho\_1 K\_1 + \rho\_2 K\_2 + \rho\_3 K\_3 + \rho\_4 K\_4 & \text{Alt} \in (1550, 4500) \text{and } \text{Mach} \in (0.6, 1.1). \end{cases}$$

*Performance Improvement for Fighter Aircraft Using Fuzzy Switching LQI Controller DOI: http://dx.doi.org/10.5772/intechopen.107032*

**Figure 3.** *Flight envelope with the four overlapping cells.*

**Figure 4.** *Fuzzy controller rules between two cells.*

where *ρi*,*i* ½ � **1 4** are multipliers for the related controllers as given in **Figures 4** and **5**. **Figure 4** illustrates fuzzy controller rules between two cells. One can see that multiplier of the controller change linearly between active two cells, also multipliers of passive cells remain zero. In addition, the change of the controller multipliers for overlapping four cells is given in **Figure 5**.
