A Class of ${L}_{1}$ -to-${L}_{1}$ and ${L}_{\infty}$ -to-${L}_{\infty}$ Interval Observers for (Delayed) Markov Jump Linear Systems

Abstract

We exploit recent results on the stability and performance analysis of positive Markov jump linear systems (MJLS) for the design of interval observers for MJLS with and without delays. While the conditions for the L 1 performance are necessary and sufficient, those for the L ∞ performance are only sufficient. All the conditions are stated as linear programs that can be solved very efficiently. Two examples are given for illustration.

Necessary and Sufficient Conditions for Stochastic Stability and Stabilization of Positive Systems With Two Independent Markov Processes

“…However, for positive MJLSs, necessary and sufficient conditions of mean stability, which is equivalent to first-moment stochastic stability, are achieved by linear programming (LP) method in [3]- [6]. Departing from the definitions of stochastic stability and performance, some results of positive MJLSs have been elegantly developed in recent years by the LP method to investigate the delay-dependent stochastic stability [7], [8], and to design a mode-dependent fuzzy controller [9], a robust non-fragile controller [10], an L ∞ controller [11], an l 1 /l − filter [12], a network-based l 1 -gain filter [13], L 1 -to-L 1 and L ∞ -to-L ∞ interval observers [14], and so on. It should be remarked that these results are only focused on positive systems with one Markov jumping parameter.…”

Stability andL1× ℓ1-to-L1× ℓ1performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

“…Over the past recent years, this problem witnessed an increase in its popularity and various methodologies for their design in different setups have been proposed. To cite a few, those observers have been obtained for systems with inputs [3,4], linear systems [5][6][7], timevarying systems [8], delay systems [4,9], impulsive systems [10], uncertain/LPV systems [11][12][13], discrete-time systems [4,14], systems with samplings [15,16], impulsive systems [10,17,18], switched systems [18][19][20] and Markovian jump systems [21].…”