In the article Controller Design using Bifurcation Map for Aircraft Spin Recovery, the sliding surface derivative is obtained using a numerical method.
The steady space system is defined by the following equations:
$$\dot{\underline{x}}=A(\underline{x},t)+B(\underline{x},t)\underline{U}, \\ \underline{x}=[V,\alpha,\beta,p,q,r,\phi,\theta], \\ s=\left[\begin{array}{c} \dfrac{d}{dt}(\phi-\phi_d)+\lambda_1(\phi-\phi_d) \\ \dfrac{d}{dt}(\alpha-\alpha_d)+\lambda_2(\alpha-\alpha_d) \\ \dfrac{d}{dt}(\beta-\beta_d)+\lambda_3(\beta-\beta_d) \\ \end{array}\right]$$
where $\lambda_1$, $\lambda_2$ and $\lambda_3$ are positive real numbers.
The article states:
The control law from reaching law in Eq. (6) can be derived as: $$\dot{s}=\dfrac{\partial{s}}{\partial{x}}\dot{x}=\dfrac{\partial{s}}{\partial{x}}(A+Bu)=-Qsgn(s)+Kf(s).\tag{8}$$
By solving the above equation (8) for u (3 x 1), one gets expressions for u as:
$$u=-\left(\dfrac{\partial{s}}{\partial{x}}B\right)^{-1}\left[\dfrac{\partial{s}}{\partial{x}}A+Qsgn(S)+Kf(s)\right].\tag{9}$$
The matrix $\frac{\partial{s}}{\partial{x}}$ is computed numerically. The control inputs computed are subject to position and rate constraints on control surface deflections as listed in the Table 1.
Which numerical method is used to compute $\dfrac{\partial s}{\partial x}$ in $u$?And also how would be the exact form of matrix of $\dfrac{\partial s}{\partial x}$ (element of $\dfrac{\partial s}{\partial x}$ in columns and raws)?
