Equivalent LMI constraints: 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 are very similar. 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. Then, we show the stability 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 switched linear systems framework to have identical columns.