LMIs in Control/pages/Generalized KYP Lemma Conic Sector

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}$$

Also consider $$ \pi_{c}(a,b) \in \mathcal{S}^{m} $$, which is defined as


 * $$ \pi_{c}(a,b) = \begin{bmatrix}

-\tfrac{1}{b}I & \tfrac{1}{2}(1+\tfrac{a}{b})I  \\ (\tfrac{1}{2}(1+\tfrac{a}{b})I)^{T} &  -aI \end{bmatrix}$$,

where $$ a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a < b $$.

The Data
The matrices The matrices $$ A,B,C $$ and $$D$$. The values of a and b

LMI : Generalized KYP (GKYP) Lemma for Conic Sectors
The following generalized KYP Lemmas give conditions for $$\mathcal{G}$$ to be inside the cone $$ [a,b] $$ within finite frequency bandwidths.
 * 1. (Low Frequency Range) The system $$\mathcal{G}$$ is inside the cone $$ [a,b] $$ for all $$ \omega \in {\omega \in \mathcal{R} | |\omega| < \omega_{1}, det(j\omega I - A) \neq 0} $$, where $$\omega_{1} \in \mathcal{R}_{>0}, a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a0}$$, where $$Q \ge 0$$, such that


 * $$\begin{bmatrix}

A &  B  \\ I &  0 \end{bmatrix}^{T}\begin{bmatrix} -Q & P \\ P^{T} & (\omega_{1} - \overline{\omega}_{1})^{2}Q \end{bmatrix} \begin{bmatrix} A &  B  \\ I &  0 \end{bmatrix} - \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix}^{T} \pi_{c}(a,b) \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix} < 0 $$.


 * If $$\omega_{1} \rightarrow \infty, P>0. $$ and Q = 0, then the traditional Conic Sector Lemma is recovered. The parameter $$\overline{\omega}_{1}$$ is incuded in the above LMI to effectively transform $$|\omega| \le (\omega_{1} - \overline{\omega}_{1})$$ into the strict inequality $$|\omega| < \omega_{1}$$


 * 2. (Intermediate Frequency Range) The system $$\mathcal{G}$$ is inside the cone $$ [a,b] $$ for all $$ \omega \in {\omega \in \mathcal{R} | \omega_{1} \le |\omega| < \omega_{2}, det(j\omega I - A) \neq 0} $$, where $$\omega_{1},\omega_{2} \in \mathcal{R}_{>0}, a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a0}$$ and $$\hat{\omega}_{2} = (\omega_{1} + \tfrac{(\omega_{2} - \hat{\omega}_{2})}{2}),$$ where $$ P^{H} = P, Q^{H} = Q $$ and $$Q \ge 0$$, such that


 * $$\begin{bmatrix}

A &  B  \\ I &  0 \end{bmatrix}^{T}\begin{bmatrix} -Q & P + \mathcal{j\hat{\omega}_{2}}Q\\ P - \mathcal{j\hat{\omega}_{2}}Q & \omega_{1}(\omega_{2} - \hat{\omega} - 2)Q \end{bmatrix} \begin{bmatrix} A &  B  \\ I &  0 \end{bmatrix} - \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix}^{T} \pi_{c}(a,b) \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix} < 0 $$.


 * The parameter $$\overline{\omega}_{2}$$ is incuded in the above LMI to effectively transform $$ \omega_{1} \le |\omega| \le (\omega_{2} - \overline{\omega}_{2})$$ into the strict inequality $$\omega_{1} \le |\omega| < \omega_{2}$$.


 * 3. (High Frequency Range) The system $$\mathcal{G}$$ is inside the cone $$ [a,b] $$ for all $$ \omega \in {\omega \in \mathcal{R} | \omega_{2} < |\omega|, det(j\omega I - A) \neq 0} $$, where $$\omega_{2} \in \mathcal{R}_{>0}, a \in \mathcal{R}, b \in \mathcal{R}_{>0} $$ and $$ a<b $$, if there exist $$ P,Q \in \mathcal{S}^{n} $$, where $$Q \ge 0$$, such that


 * $$\begin{bmatrix}

A &  B  \\ I &  0 \end{bmatrix}^{T}\begin{bmatrix} -Q & P \\ P^{T} & \omega_{2}^{2}Q \end{bmatrix} \begin{bmatrix} A &  B  \\ I &  0 \end{bmatrix} - \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix}^{T} \pi_{c}(a,b) \begin{bmatrix} C &  D  \\ 0 &  I \end{bmatrix} < 0 $$.

If (A, B, C, D) is a minimal realization, then the matrix inequalities in all of the above LMI, then it can be nostrict.

Conclusion:
If there exist a positive definite $$q$$ matrix satisfying above LMIs for the given frequency bandwidths then the system $$\mathcal{G}$$ is inside the cone [a,b].

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 Modified Exterior Conic Sector Lemma