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{align} \dot x(t)&=A(\delta(t))x(t)\\

\end{align}$$

The Data
The system coefficient matrix takes the form of



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

where $$ A_0 \in \mathbb{R} $$is a known matrix, which represents the nominal system matrix, while $$ \begin{align} \Delta A(\delta(t))x(t) = \delta_1(t))A_1 + \delta_2(t))A_2 +...+\delta_k(t))A_k \end{align}$$ 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{align} (A_0 + \Delta A(\delta(t))x(t))^T + P(A_0 + \Delta A(\delta(t))x(t)) <0   \delta(t) \in \Delta \end{align}$$

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{align} \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{align}$$

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



\begin{align} (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{align}$$

Case 2: Polytope
Consider the case where the set of perturbation parameters is defined by a polytope as



\begin{align} \Delta = { \delta(t) = [\delta_1 (t) \delta_2(t)... \delta_k(t)] \in \mathbb{R}^k \mid \delta_i(t) \in \mathbb{R}_{\geq0}}, \sum_{i=1}^k \delta_i(t)=1 \end{align}$$

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



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

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

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