4. Sliding mode control

In this section, the main work is to find a state feedback control law based on the sliding mode control technique for the class of multimotor systems.

Nonlinear systems are described as follows:

$$\frac{d}{dt}\mathbf{x} = A\mathbf{x} + B(\boldsymbol{\mu} + \boldsymbol{\mu}\_d(\mathbf{x}, t)) \tag{9}$$

where udð Þ x; t is the nonlinear term in system. Lemma 2 [10]: The sliding mode controller is described as follows

$$\mu = -\left[\mathbb{S}.A\mathbf{x} + \beta \text{sgn}(\sigma)\right] \tag{10}$$

based on the sliding surface:

$$\begin{aligned} \{\mathbf{x} : \sigma = \mathbf{S} \mathbf{x} = \mathbf{0}\}, \\ \mathbf{S} = \left(\mathbf{B}^T \mathbf{X}^{-1} \mathbf{B}\right)^{-1} \mathbf{B}^T \mathbf{X}^{-1} \mathbf{X} \end{aligned} \tag{11}$$

with X is satisfied, the LMI problem as follows:

$$\text{II}^T \{ AX + XA^T \} \text{II} < 0, X > 0,\\ \text{II} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}^T \tag{12}$$

Remark 3. We obtain the sliding mode control for multimotor systems (2) based on Lemma 3 because it belongs to the class of systems (9).
