LMIs in Control/Stability Analysis/Kharitonov-Bernstein-Haddad

Kharitonov-Bernstein-Haddad

Consider the set of matrices A = { ${\begin{aligned}A={\begin{bmatrix}0_{(n-1)\times 1}&1_{(n-1)\times (n-1)}\\-a_{0}&-a_{n-1}\\\end{bmatrix}}\qquad |\qquad {\underline {a}}_{j}\leq {\underline {a}}_{j}\leq {\overline {a}}_{j},\qquad j=1,2,3,4,...,n-1\end{aligned}}$ }

Every matrix in the set A is Hurwitz if and only if there exist $P_{i}\in {\text{S}}^{n}$, i = 1, 2, 3, 4, where P_i > 0, i = 1, 2, 3, 4, such that

${\begin{aligned}\qquad P_{i}A_{i}+A_{i}^{T}P_{i}<0\qquad i=1,2,3,4,\end{aligned}}$

where

${\begin{aligned}\qquad A_{i}={\begin{bmatrix}[0_{(n-1)\times 1}&1_{(n-1)\times (n-1)}]\\\qquad a_{i}\\\end{bmatrix}}\end{aligned}},\qquad i=1,2,3,4,$

${\begin{aligned}\qquad a_{1}=-{\begin{bmatrix}{\underline {a}}_{0}&{\underline {a}}_{1}&{\overline {a}}_{2}&{\overline {a}}_{2}&...&{\underline {a}}_{n-4}&{\underline {a}}_{n-3}&{\overline {a}}_{n-2}&{\overline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$

${\begin{aligned}\qquad a_{2}=-{\begin{bmatrix}{\underline {a}}_{0}&{\overline {a}}_{1}&{\overline {a}}_{2}&{\underline {a}}_{2}&...&{\underline {a}}_{n-4}&{\overline {a}}_{n-3}&{\overline {a}}_{n-2}&{\underline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$

${\begin{aligned}\qquad a_{3}=-{\begin{bmatrix}{\overline {a}}_{0}&{\underline {a}}_{1}&{\underline {a}}_{2}&{\overline {a}}_{2}&...&{\overline {a}}_{n-4}&{\underline {a}}_{n-3}&{\underline {a}}_{n-2}&{\overline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$

${\begin{aligned}\qquad a_{4}=-{\begin{bmatrix}{\overline {a}}_{0}&{\overline {a}}_{1}&{\underline {a}}_{2}&{\underline {a}}_{2}&...&{\overline {a}}_{n-4}&{\overline {a}}_{n-3}&{\underline {a}}_{n-2}&{\underline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$

Equivalently, every matrix in the set A is Hurwitz if and only if there exist $Q_{i}\in {\text{S}}^{n}$, i = 1, 2, 3, 4, where Q_i > 0, i = 1, 2, 3, 4, such that

${\begin{aligned}\qquad A_{i}Q_{i}+Q_{i}A_{i}^{T}<0\qquad i=1,2,3,4,\end{aligned}}$

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.