# 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{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}} ## 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 $\geq 0$ , such that
${\begin{bmatrix}PA+A^{T}P&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq 0.$ Proof. The term $-C^{T}C$ in the Actual Exterior Conic Sector Lemma makes the matrix inequality more neagtive definite.

Therefore,

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