# LMIs in Control/pages/Output Energy Bounds for Lure System

## The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q(t)&=C_{q}x(t),\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}}

## The Data

The matrices ${\displaystyle A,B_{p},B_{w},C_{q},C_{z},x(0)}$.

## The Optimization Problem:

The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.

{\displaystyle {\begin{aligned}&\min _{P\succ 0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0}x^{\top }(0)(P+C_{q}^{\top }\Lambda C_{q})x(0)\\&\quad \quad \quad \quad \quad \quad \quad {\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\preceq 0\\\end{aligned}}}

## Conclusion

The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., ${\displaystyle J=\int _{0}^{\infty }z^{\top }z\ dt}$ with initial conditions ${\displaystyle x(0)}$.

## Remark

The key step in the proof is to satisfy ${\displaystyle {\frac {d}{dt}}V(x)+z^{\top }z\leq 0}$, where ${\displaystyle V(.)}$ is Lyapunov function in a special form.