LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Non-Strict Schur Complement

Nonstrict Schur Complement

Consider $A\in \mathbb {S} ^{n},B\in \mathbb {R} ^{n\times m},$ and $C\in \mathbb {S} ^{m}.$ The following statements are equivalent.

a) ${\begin{bmatrix}A&B\\B^{T}&C\end{bmatrix}}\leq 0$

b) $A-BC^{+}B^{T}<0,C\leq 0,B(\mathbf {1} -CC^{+})=\mathbf {0} ,$ where $C^{+}$ is the Moore-Penrose inverse of $C.$

c) $C-B^{T}A^{+}B<0,A\leq 0,B^{T}(\mathbf {1} -AA^{+})=\mathbf {0}$ where $A^{+}$ is the Moore-Penrose inverse of $A.$

External Links

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