LMIs in Control/Matrix and LMI Properties and Tools/Quadratic Inequalities

Weighted Norm

Required data

Consider $W\in {𝕊}^n$, $x,y\in {ℝ}^n$, and $\gamma \in {ℝ}_{⩾0}$, where $W>0$.

Matrix inequality

The inequality $(x-y{)}^{T}W(x-y)\le \gamma$ is equivalent to the matrix inequality given by

$\left[\begin{array}{cc}\gamma & (x-y{)}^T\\ *& W^{-1}\end{array}\right]\ge 0$.

Quadratic Inequality

Required data

Consider $W\in {𝕊}^n$, $A\in {ℝ}^{n\times m}$, $x$, $c\in {ℝ}^m$, $b\in {ℝ}^n$, and $d\in ℝ$, where $W>0$.

Matrix inequality

The quadratic inequality $(Ax+b{)}^{T}W(Ax+b)-c^{T}x-d\le 0$ with $W>0$ is equivalent to the matrix inequality given by

$\left[\begin{array}{cc}{W}^{-1}& Ax+b\\ *& c^{T}x+d\end{array}\right]\ge 0$.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.