Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the stability under arbitrary switching of a discrete-time linear switched system. A porful approach for addressing this problem is based on studying the ''most unstable'' switching law (MUSL). If the solution of the switched system corresponding to the MUSL converges to the origin, then the switched system is stable for any switching law. The MUSL can be characterized using optimal control techniques. This variational approach leads to a Hamilton-Jacobi-Bellman equation describing the behavior of the switched system under the MUSL. The solution of this equation is sometimes referred to as a Barabanov norm of the switched system. Although the Barabanov norm was studied extensively, it seems that there are few examples where it was actually computed in closed-form. ,  consider a special class of positive planar discrete-time linear switched systems and provide a closed-form expression for a corresponding Barabanov norm and a MUSL. The unit circle in this norm is a parallelogram.
['Ron Teichner', 'Michael Margaliot']
I am writing a research paper on a idea Brief paper: Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems
[{"from": "human", "value": "I am writing a research paper on a idea Brief paper: Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the stability under arbitrary switching of a discrete-time linear switched system. A porful approach for addressing this problem is based on studying the ''most unstable'' switching law (MUSL). If the solution of the switched system corresponding to the MUSL converges to the origin, then the switched system is stable for any switching law. The MUSL can be characterized using optimal control techniques. This variational approach leads to a Hamilton-Jacobi-Bellman equation describing the behavior of the switched system under the MUSL. The solution of this equation is sometimes referred to as a Barabanov norm of the switched system. Although the Barabanov norm was studied extensively, it seems that there are few examples where it was actually computed in closed-form. ,  consider a special class of positive planar discrete-time linear switched systems and provide a closed-form expression for a corresponding Barabanov norm and a MUSL. The unit circle in this norm is a parallelogram."}]