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 $x\in {ℝ}^n$ is the state vector, $u\in {ℝ}^{n_u}$ is the input and $y\in {ℝ}^{n_y}$ is the output.

The Data

$x\in {ℝ}^n$ is the state vector, $u\in {ℝ}^{n_u}$ is the input and $y\in {ℝ}^{n_y}$ 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

• $\left[\begin{array}{cc}A(x)& B_u(x)\\ C_y(x)& D_{yu}(x)\end{array}\right]$ = $\left[\begin{array}{cc}A& B_u\\ C_y& D_{yu}\end{array}\right]$ + $\left[\begin{array}{c}{B}_p\\ D_{yp}\end{array}\right]△(x)\left[\begin{array}{c}I-D_{qp}△(x)\end{array}\right]6-1$$\left[\begin{array}{cc}{C}_q& D_{qu}\end{array}\right]$