Preliminary results on positizable systems

Great achievements have been made in the field of positive systems in the past two decades and many elegant properties of positive systems have been revealed. Among all the nice properties, stability of delayed positive systems is of great interest: It is not affected by the magnitude of delays. Though positive systems have very nice properties, these properties cannot be applied widely since positive systems are just a little portion among all dynamic systems. This paper tries to consider a more broad class of systems-positizable systems whose stability is also not affected by the size of delays. Positizable systems means a class of systems which are non-positive themselves but can be transformed into positive systems by means of an invertible linear transformation. Generally speaking, both judging if a system is positizable and realizing positization are very difficult. This paper first discusses these two questions and focuses on some preliminary cases. Some necessary and sufficient conditions for judging if a system is diagonally positizable are established, several cases are also proposed where systems are not positizable.