User:Margav06/sandbox/Click here to continue/Controller synthesis/Hinf Optimal Output Controllability for Systems With Transients

$$H_\infty$$ Optimal Output Controllability for Systems With Transients

This LMI provides an $$H_\infty$$ optimal output controllability problem to check if such controllers for systems with unknown exogenous disturbances and initial conditions can exist or not.

The System


\begin{align} \dot x&=Ax+B_1 v + B_2 u, x(0) = x_0, \\ z&=C_1 x+D_{11} v+D_{12} u, \\ y&= C_2x + D_{21}v, \end{align}$$

where $$ x \in \R^n $$ is the state, $$ v \in \R^r $$ is the exogenous input, $$ u\in \R^m $$ is the control input, $$ y \in \R^p $$ is the measured output and $$ z \in \R^s $$ is the regulated output.

The Data
System matrices $$ (A,B_1,B_2,C_1,C_2,D_{11},D_{12},D_{21},D_{22}) $$ need to be known. It is assumed that $$ v \in L_2[0,\infty) $$. $$ N_1, N_2 $$ are matrices with their columns forming the bais of kernels of $$ C_2D_{21} $$ and $$ C_2D_{12} $$ respectively.

The Optimization Problem
For a given $$ \gamma $$, the following $$H_\infty $$ condition needs to be fulfilled:

$$ \gamma_w = sup_{\|v\|^2_\infty+x_0^\top R x_0 \neq 0} \frac{\|z\|_\infty}{(\|v\|_\infty^2+x_0^\top R x_0)^{1/2}} < \gamma_w, $$

The LMI: $$H_\infty$$ Output Feedback Controller for Systems With Transients


\begin{align} &\text{min}_{\gamma, X_{11}, Y_{11}}: \gamma \\ & \text{subj. to: } X_{11} > 0, Y_{11} > 0, \\ & \quad \begin{bmatrix} N_1 & 0 \\ 0 & I \end{bmatrix}^\top \begin{bmatrix} A^\top X_{11} + X_{11}A & X_{11}B_1 & C_1^\top \\ * & -\gamma^2I & D_{11}^\top \\ * & * & -I \end{bmatrix} \begin{bmatrix} N_1 & 0 \\ 0 & I \end{bmatrix} < 0, \\ & \quad \begin{bmatrix} N_2 & 0 \\ 0 & I \end{bmatrix}^\top \begin{bmatrix} A Y_{11} + Y_{11}A^\top & Y_{11}C_1^\top & B_1 \\ * & -I & D_{11} \\ * & * & -\gamma^2I \end{bmatrix} \begin{bmatrix} N_2 & 0 \\ 0 & I \end{bmatrix} < 0, \\ & \quad \begin{bmatrix} X_{11} & I \\ I & Y_{11} \end{bmatrix} \geq 0, X_{11} < \gamma^2 R, \end{align}$$

Conclusion:
Solution of the above LMI gives a check to see if an $$H_\infty$$ optimal output controller for systems with transients can exist or not.

Implementation
A link to CodeOcean or other online implementation of the LMI

Related LMIs
Links to other closely-related LMIs