User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Conic Sector Lemma

Conic Sector Lemma

For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.

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 $$A \in \R^{n\times n}, B \in \R^{n\times m}, C \in \R^{m\times n},$$ and $$ D \in \mathcal{R}^{m \times m} $$. The state-space representation is:



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

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$u(t)\in \R^m$$ are the system state, output, and the input vector respectively.

The Data
The system coefficient matrices $$ (A,B,C,D) $$ are required. Optionally, the parameters to define a cone, either in the form of $$ [a,b] $$ where $$ a,b \in \R, a < b $$ or a radius $$ r \in \R_+ $$ and ceter $$ c \in \R $$.

The Feasibility LMI
The system $$ \mathcal{G} $$ is inside the given cone $$[a,b]$$ if the following is feasible:



\begin{align} \text{Find: } & P \\ \text{subj. to: } & P > 0 \\ & \begin{bmatrix} PA+A^\top P+C^\top C & PB - \frac{a+b}{2} C^\top + C^\top D \\ (PB - \frac{a+b}{2} C^\top + C^\top D)^\top & D^\top D - \frac{a+b}{2}(D+D^\top) + abI \end{bmatrix} \leq 0. \end{align} $$

The above LMI can be used to also determine the cone parameters by setting $$ a $$ as a variable along with the condition $$ a < b $$, and use the bisection method to find $$ b $$.

If the given cone is represented by a center $$ c $$ and radius $$ r $$, then the following feasibility problem can be evaluated to check if $$ \mathcal{G} $$ is inside the given cone:



\begin{align} \text{Find: } & P \\ \text{subj. to: } & P > 0 \\ & \begin{bmatrix} PA+A^\top P+C^\top C & PB - c C^\top + C^\top D \\ (PB - cC^\top + C^\top D)^\top & D^\top D - c(D+D^\top) + (c^2-r^2)I \end{bmatrix} \leq 0. \end{align} $$

In order to also find the cone parameters, substituting $$ \gamma = r^2 $$ as a decision variable with additional constraint $$ \gamma \geq 0 $$ and then solving for $$ c $$ via the bisection method will give the cone in which the system $$ \mathcal{G} $$ resides if the problem is feasible.

Conclusion:
The aforementioned LMIs can be utilized to either check if $$ \mathcal{G} $$ is in the specified cone or not, or can be used to check the stability of $$ \mathcal{G} $$ by finding if a feasible cone can be obtained that encloses $$ \mathcal{G} $$. An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:

$$ Q = -1, S = frac{a+b}{2}I = cI, R = -abI = (r^2 - c^2)I $$.

Implementation
To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Conic_sector_example.m

Related LMIs
Exterior Conic Sector Lemma.

KYP Lemma