Linear matrix inequality for nonlinear system Considering the nonlinear control system:
$$ \dot{x}=Ax(t)+B\phi(y) $$
where
$$ \phi(.): \mathbb{R} \mapsto \mathbb{R} $$ is a scalar sector-bounded nonlinearity, viz
$$ \phi \in Sector[\alpha,\beta] $$
How would I begin to obtain linear matrix inequality conditions which guarantee stability of the networked system? And how can I do this on MATLAB?
Any help would be greatly appreciated, thank you
Edit1: Editing the question for clarity.
 A: There are many ways to deal with such a problem. I will develop a simple one that relies on the use of a quadratic Lyapunov function and the use of the S-procedure.
Let us consider the case where $y=Cx$ where $y\in\mathbb{R}^m$ and consider the following sector condition
$$(\phi- K_1y)^T(K_2y-\phi)\ge0$$ for some $y,\phi\in\mathbb{R}^m$ where $K_1$ and $K_2$ are diagonal matrices such that $K_2-K_1$ has nonnegative entries.
It can be reformulated as
$$\begin{bmatrix}
x\\
\phi
\end{bmatrix}^T\begin{bmatrix}
-2C^TK_1^TK_2C & C^T(K_2-K_1)^T\\
\star & -2I
\end{bmatrix}\begin{bmatrix}
x\\
\phi
\end{bmatrix}\ge0$$
which holds for some $x\in\mathbb{R}^n,\ \phi\in\mathbb{R}^m$. Define the set
$$\mathcal{S}:=\left\{(x,\phi)\in\mathbb{R}^n\times\mathbb{R}^m:\begin{bmatrix}
x\\
\phi
\end{bmatrix}^T\begin{bmatrix}
-2C^TK_1^TK_2C & C^T(K_2-K_1)^T\\
\star & -2I
\end{bmatrix}\begin{bmatrix}
x\\
\phi
\end{bmatrix}\ge0\right\}$$
Consider now the Lyapunov function $V(x)=x^TPx$ whose derivative along the trajectories of the system is given by
$$\dot{V}(x,\phi) =\begin{bmatrix}
x\\
\phi
\end{bmatrix}^T\begin{bmatrix}
A^TP+PA & PB\\
\star & 0
\end{bmatrix}\begin{bmatrix}
x\\
\phi
\end{bmatrix}.$$ Invoking now the S-procedure, we have that $(x,\phi)\in\mathcal{S}$ implies that $\dot{V}(x,\phi)$ is negative definite if and only if there exist a scalar $\tau\ge0$ such that
$$\begin{bmatrix}
A^TP+PA & PB\\
\star & 0
\end{bmatrix}+\tau\begin{bmatrix}
-2C^TK_1^TK_2C & C^T(K_2-K_1)^T\\
\star & -2I
\end{bmatrix}$$
is negative definite. This condition is an LMI condition that can be checked using any semidefinite programming solvers such as SeDuMi using Matlab or Python.
