LMIs in Control/Click here to continue/Applications of Non-Linear Systems/LMI

The System
Consider a nonlinear, continuous-time system $$\dot{y} = A(x) + B_u(x)u,$$ $$y = C_y(x) +D_{yu}(x)u,$$ where $$x \in \mathbb{R}^{n}$$ is the state vector, $$u \in \mathbb{R}^{n_u}$$ is the input and $$y \in \mathbb{R}^{n_y}$$ is the output.

The Data
$$x \in \mathbb{R}^{n}$$ is the state vector, $$u \in \mathbb{R}^{n_u}$$ is the input and $$y \in \mathbb{R}^{n_y}$$ is the output. $$A, B_u, C_y, D_{yu}$$ are multivariable functions of x. $$A(0) = 0$$ (that is, 0 is an equilibrium point of the unforced system associated with the system). $$C_y(0) = 0$$ and $$B_u, D_{yu}$$ have no singularities at the origin.

Proof

 * $$\begin{bmatrix} A(x) & B_u(x) \\ C_y(x) & D_{yu}(x) \end{bmatrix}$$ = $$\begin{bmatrix} A & B_u \\ C_y & D_{yu} \end{bmatrix}$$ +  $$\begin{bmatrix} B_p \\ D_{yp} \end{bmatrix}\bigtriangleup(x)\begin{bmatrix} I - D_{qp}\bigtriangleup(x) \end{bmatrix}6-1$$$$\begin{bmatrix} C_q & D_{qu} \end{bmatrix}$$