Obtaining alternative LMI constraints with applications to discrete-time MJLS and switched systems

Discrete-time Markov jump linear systems (MJLS) and switched linear systems (SLS) stability, H2 and H∞ performance conditions can be very similar, and we explore those similarities in this paper. First of all, starting from the fact that MJLS second moment stability can be checked through four different linear matrix inequalities (LMIs), we show how one LMI condition can be obtained from the other using only LMI manipulations. Then, we show the Lyapunov-Metzler stability condition of SLS may also be checked through equivalent matrix inequalities and apply the same steps for H2 and H∞ performances, obtaining new conditions for both MJLS and SLS. Special attention is given to the case where the transition probabilities are independent of the mode, which is equivalent to consider the Metzler matrices on SLS framework to have identical columns. Finally, we propose a method to design a switching rule based on a randomly generator with given probability distribution.