LMIs in Control/pages/Modified Exterior Conic Sector Lemma

The Concept
The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

The System
Consider a square, contiuous-time linear time-invariant (LTI) system, $$\mathcal{G} : \mathcal{L}_{2e} \rightarrow \mathcal{L}_{2e}$$, with minimal state-space relization (A, B, C, D), where $$\mathcal{E, A} \in \mathcal{R}^{n\times n}, \mathcal{B} \in \mathcal{R}^{n\times m}, \mathcal{C} \in \mathcal{R}^{p\times n},$$ and $$\mathcal{D} \in \mathcal{R}^{p\times m} $$.



\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t)\\ \end{align}$$

The Data
The matrices The matrices $$ A,B,C $$ and $$D $$

LMI : Modified Exterior Conic Sector Lemma
The system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c (i.e. $$\mathcal{G} \in $$exconer(c)), where $$ r \in \mathcal{R}_{>0} $$ and $$ \in \mathcal{R}$$, under either of the following sufficient conditions.
 * 1. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that


 * $$\begin{bmatrix}

PA + A^{T}P &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix} \le 0.$$


 * Proof. The term $$ -C^{T}C $$ in the Actual Exterior Conic Sector Lemma makes the matrix inequality more neagtive definite.

Therefore,
 * $$\begin{bmatrix}

PA + A^{T}P -C^{T}C &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix} \le \begin{bmatrix} PA + A^{T}P &  PB - C^{T}(D - CI)  \\ (PB - C^{T}(D - CI))^{T} &  r^{2}I - (D - cI)^{T}(D - cI) \end{bmatrix}$$
 * 2. There exists P $$ \in \mathcal{S}^{n}$$, where P $$\ge 0$$, such that


 * $$\begin{bmatrix}

PA + A^{T}P &  PB - C^{T}(D - CI) & 0  \\ (PB - C^{T}(D - CI))^{T} &  - (D - cI)^{T}(D - cI) & rI \\ 0 & (rI)^{T} & -I \end{bmatrix} \le 0.$$
 * Proof. Applying the Schur complement lemma to the $$ r^{2}I $$ terms in (1) gives (2).

Conclusion:
If there exist a positive definite $$P$$ matrix satisfying above LMIs then the system $$\mathcal{G}$$ is in the exterior cone of radius r centered at c.

Implementation
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIs
KYP Lemma State Space Stability Exterior Conic Sector Lemma