LMIs in Control/pages/Continuous time Quadratic stability

LMIs in Control/pages/Continuous time Quadratic stability

To study stability of a LTI system, we first ask whether all trajectories of system converge to zero as $t\rightarrow \infty$. A sufficient condition for this is the existence of a quadratic function $V(\xi )=\xi ^{T}P\xi$, $P>0$ that decreases along every nonzero trajectory of system . If there exists such a P, we say the system is quadratically stable and we call $V$ a quadratic Lyapunov function.

The System

${\begin{aligned}{\dot {x}}(t)&=A(\delta (t))x(t)\\\end{aligned}}$

The Data

The system coefficient matrix takes the form of

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

where $A_{0}\in \mathbb {R}$is a known matrix, which represents the nominal system matrix, while ${\begin{aligned}\Delta A(\delta (t))x(t)=\delta _{1}(t))A_{1}+\delta _{2}(t))A_{2}+...+\delta _{k}(t))A_{k}\end{aligned}}$ is the system matrix perturbation, where

$A_{i}\in \mathbb {R} ^{n\times n},i=1,2,..,k,$ are known matrices, which represent the perturbation matrices.

$\delta _{i}(t),i=1,2,...,k,$ which represent the uncertain parameters in the system.

$\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]^{T}$ is the uncertain parameter vector, which is often assumed to be within a certain compact and convex set : : $\Delta$ that is

$\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta ]^{T}\in \Delta$

The LMI: Continuous-Time Quadratic Stability

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+\Delta A(\delta (t))x(t))^{T}+P(A_{0}+\Delta A(\delta (t))x(t))<0\delta (t)\in \Delta \end{aligned}}$

The following statements can be made for particular sets of perturbations.

Case 1: Regular Polyhedron

Consider the case where the set of perturbation parameters is defined by a regular polyhedron as

${\begin{aligned}\Delta ={\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]\in \mathbb {R} ^{k}\mid \delta _{i}(t),{\underline {\delta _{i}}}(t),{\overline {\delta _{i}}}(t),{\underline {\delta _{i}}}\leq \delta _{i}(t)\leq {\overline {\delta _{i})}}}\end{aligned}}$

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+\Delta A(\delta (t))x(t))^{T}+P(A_{0}+\Delta A(\delta (t))x(t))<0\delta _{i}(t)\in {{\underline {\delta _{i}}},{\overline {\delta _{i}}}},i=1,2,...k.\end{aligned}}$

Case 2: Polytope

Consider the case where the set of perturbation parameters is defined by a polytope as

${\begin{aligned}\Delta ={\delta (t)=[\delta _{1}(t)\delta _{2}(t)...\delta _{k}(t)]\in \mathbb {R} ^{k}\mid \delta _{i}(t)\in \mathbb {R} _{\geq 0}},\sum _{i=1}^{k}\delta _{i}(t)=1\end{aligned}}$

The uncertain system is quadratically stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0,$ such that

${\begin{aligned}(A_{0}+A_{i})^{T}P+P(A_{0}+A_{i})<0,i=1,2...,k.\end{aligned}}$

Conclusion:

If feasible, System is Quadratically stable for any $x\in \mathbb {R} ^{n}$

Implementation

https://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities

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.