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}}$