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.
Consider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization , where and . The state-space representation is:
where , and are the system state, output, and the input vector respectively.
The system coefficient matrices are required. Optionally, the parameters to define a cone, either in the form of where or a radius and ceter .
The Feasibility LMIEdit
The system is inside the given cone if the following is feasible:
The above LMI can be used to also determine the cone parameters by setting as a variable along with the condition , and use the bisection method to find .
If the given cone is represented by a center and radius , then the following feasibility problem can be evaluated to check if is inside the given cone:
In order to also find the cone parameters, substituting as a decision variable with additional constraint and then solving for via the bisection method will give the cone in which the system resides if the problem is feasible.
The aforementioned LMIs can be utilized to either check if is in the specified cone or not, or can be used to check the stability of by finding if a feasible cone can be obtained that encloses . 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:
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:
A list of references documenting and validating the LMI.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.