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 $t\to \mathrm{\infty }$. 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

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

The Data

The system coefficient matrix takes the form of

$\begin{array}{cc}\dot{x}(t)& =A_0+\mathrm{\Delta }A(\delta (t))x(t)\\ \end{array}$

where $A_0\in ℝ$is a known matrix, which represents the nominal system matrix, while $\begin{array}{c}\mathrm{\Delta }A(\delta (t))x(t)={\delta }_1(t))A_1+{\delta }_2(t))A_2+...+{\delta }_k(t))A_k\end{array}$ is the system matrix perturbation, where

$A_i\in {ℝ}^{n\times n},i=1,2,..,k,$ 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 : : $\mathrm{\Delta }$ that is

$\delta (t)=[{\delta }_1(t){\delta }_2(t)...\delta {]}^T\in \mathrm{\Delta }$

The LMI: Continuous-Time Quadratic Stability

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

$\begin{array}{c}(A_0+\mathrm{\Delta }A(\delta (t))x(t){)}^T+P(A_0+\mathrm{\Delta }A(\delta (t))x(t))<0\delta (t)\in \mathrm{\Delta }\end{array}$

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

$\begin{array}{c}\mathrm{\Delta }=\delta (t)=[{\delta }_1(t){\delta }_2(t)...{\delta }_k(t)]\in {ℝ}^k\mid {\delta }_i(t),\underset{\_}{{\delta }_i}(t),\overline{{\delta }_i}(t),\underset{\_}{{\delta }_i}\le {\delta }_i(t)\le \overline{{\delta }_i)}\end{array}$

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

$\begin{array}{c}(A_0+\mathrm{\Delta }A(\delta (t))x(t){)}^T+P(A_0+\mathrm{\Delta }A(\delta (t))x(t))<0{\delta }_i(t)\in \underset{\_}{{\delta }_i},\overline{{\delta }_i},i=1,2,...k.\end{array}$

Case 2: Polytope

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

$\begin{array}{c}\mathrm{\Delta }=\delta (t)=[{\delta }_1(t){\delta }_2(t)...{\delta }_k(t)]\in {ℝ}^k\mid {\delta }_i(t)\in {ℝ}_{\ge 0},\sum _{i=1}^k{\delta }_i(t)=1\end{array}$

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

$\begin{array}{c}(A_0+A_i{)}^{T}P+P(A_0+A_i)<0,i=1,2...,k.\end{array}$

Conclusion:

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

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.