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
 * $$ \begin{align} \dot x(t)&=(A_0+\Delta (t))x(t)\\ \end{align} $$

where
 * $$ \begin{align} \Delta (t) = A_1 \delta_1 (t)+...+A_k \delta_k (t)\\ \end{align} $$

where $$ \delta_i (t) $$ lies in the intervals
 * $$ \begin{align} \delta_i (t) \in [\delta_i^-,\delta_i^+]\\ \end{align} $$

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


 * $$ \begin{align} \Biggl(A_0 + \sum_{i} A_i \delta_i \Biggl)^TP + P\Biggl(A_0 + \sum_{i} A_i \delta_i \Biggl) < 0 \quad for \ every \quad \delta_i \in \{\delta_i^-,\delta_i^+\}^k \end{align} $$

The LMI
$$ \begin{align} P>0, \quad \Biggl(A_0 + \sum_{i} A_i \delta_i \Biggl)^TP + P\Biggl(A_0 + \sum_{i} A_i \delta_i \Biggl) < 0 \quad for \ every \quad \delta_i \in \{\delta_i^-,\delta_i^+\}^k \end{align} $$

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