LMIs in Control/pages/Peak to Peak norm

Peak-to-peak norm performance of a system

The System
Considering the following system:



\begin{align} \dot{x} = A x + B u\\ z     = C x + D u\\ \end{align}$$

Where $$ x(t) \in \mathbb{R}^{n} $$ is the state signal, $$ u(t) \in \mathbb{R}^{m} $$ is the input signal, and $$ z(t) \in \mathbb{R}^{p} $$ is the output. When given an initial condition $$ x(0) = 0 $$, the system can be defined to map the output and input signals for the peak-to-peak performance.



\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The Data
The matrices $$ A $$, $$ B $$, $$ C $$, and $$ D $$ are the only data sets required for this optimization problem.

The Optimization Problem
Consider a continuous-time LTI system, $$ G: L_{2e} \to L_{2e} $$, given that: $$ A $$, $$ B $$, $$ C $$, and $$ D $$ $$ A \in \mathbb{R}^{n \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$. Given that the matrix $$ A $$ is Hurwitz,The peak-to-peak norm of $$ G $$ is given as:



\begin{align} ||T||_{\infty,\infty} := \sup\limits_{0<||w||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}}\\ \end{align}$$

The LMI: Peak-to-Peak norm
There exists a matrix $$ P \in \mathbb{S}^{n} $$ and $$ \gamma $$, $$ \epsilon $$, $$ \mu \in \mathbb{R}_{>0} $$, where the following constraints are used: $$ \min \mu $$



\begin{align} P < 0 \\ \begin{bmatrix} A^T P + P A + \gamma P & P B \\ B' P & \epsilon I\end{bmatrix}&<0\\ \begin{bmatrix} \gamma P & 0 & C^T \\ 0 & (\mu - \epsilon) I & D^T \\ C & D & \mu I\end{bmatrix}&>0\\ \end{align}$$

Since this optimization has $$ \gamma P $$ in the constraints, this does make this optimization bi-linear. attempting to solve this LMI is not feasible unless some type of substitute is implemented to the variables $$ \gamma P $$.

Conclusion:
The results from this LMI will give the peak to peak norm of the system:

\begin{align} \sup\limits_{0<||u||_{\infty}<\infty} \frac{||z||_{\infty}}{||u||_{\infty}} < \mu \end{align}$$

Implementation

 * Peak-to-Peak Norm Example