LMIs in Control/Click here to continue/LMIs in system and stability Theory/Interval Quadratic Stability

An LMI to determine the quadratic stability of a system with parametric, interval uncertainties.

The System

Consider the system with Affine Time-Varying uncertainty

${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta (t))x(t)\\\end{aligned}}$

where

${\begin{aligned}\Delta (t)=A_{1}\delta _{1}(t)+...+A_{k}\delta _{k}(t)\\\end{aligned}}$

where $\delta _{i}(t)$ lies in the intervals

${\begin{aligned}\delta _{i}(t)\in [\delta _{i}^{-},\delta _{i}^{+}]\\\end{aligned}}$

where $x\in \mathbb {R} ^{m}$ and $A\in \mathbb {R} ^{mxm}$.

The Data

The matrices A and $\Delta$ are known.

The Optimization

This optimization problem ensures quadratic stability of the system with k interval uncertainties using $2^{k}$ LMI constraints.

$\delta (t)$ lies in the hypercube. The vertices of the hypercube define the vertices of the uncertainty set

$V:={\Biggl \{}A_{0}+\sum _{i}A_{i}\delta _{i},\ \delta _{i}\in \{\delta _{i}^{-},\delta _{i}^{+}\}{\Biggl \}}$

$(A+\Delta ,{\boldsymbol {\Delta }})$ is quadratically stable over ${\boldsymbol {\Delta }}:=Co(V)$ if and only if there exists a P > 0 such that

${\begin{aligned}{\Biggl (}A_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}^{T}P+P{\Biggl (}A_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}<0\quad for\ every\quad \delta _{i}\in \{\delta _{i}^{-},\delta _{i}^{+}\}^{k}\end{aligned}}$

The LMI

${\begin{aligned}P>0,\quad {\Biggl (}A_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}^{T}P+P{\Biggl (}A_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}<0\quad for\ every\quad \delta _{i}\in \{\delta _{i}^{-},\delta _{i}^{+}\}^{k}\end{aligned}}$

Conclusion

Quadratic Stability Implies Stability of trajectories for any $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$ for all $t\geq 0$
Quadratic stability is conservative.
Stability does not imply quadratic stability.
Interval uncertainty is a special case of polytopic uncertainty.

Implementation

Example of implementation of the LMI
https://github.com/MichaelDobos/LMI/blob/main/intervalquadraticstability.m

Related LMIs

• Polytopic Quadratic Stability
• Discrete-Time Quadratic Stability

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Control Systems: Analysis, Design, and Applications - A textbook on LMIs in control by Duan and Yu.