Some new subclasses of systems having a common quadratic Lyapunov function and comparison of known subclasses

A common quadratic Lyapunov function (CQLF) guarantees the asymptotic stability of a set of systems. A complete characterization of the set of systems with such a property have been unsuccessful (except for second-order systems). Thus, for both the continuous-time and discrete-time cases, several subsets of linear systems which have a CQLF are known. Some results indicate that there is a parallelism between the continuous-time case and the discrete-time case. In this paper, we show a new subclass for continuous-time systems which have a CQLF by using a property of M-matrices. We also show the discrete-time counterpart of the above new subclass. Next, it is shown that the whole class of continuous-time linear systems having a CQLF is connected directly with its discrete-time counterpart by using a bilinear transformation. For some known subclasses of systems having a CQLF, the transformation gives a one-to-one correspondence between the continuous-time and discrete-time cases. We further show relationships among the obtained results and other, known results.