A class of L1L_1-to-L1L_1 and L∞L_\infty-to-L∞L_\infty interval observers for (delayed) Markov jump linear systems

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 $L_\infty$ performance are only sufficient. All the conditions are stated as linear programs that can be solved very efficiently. Two examples are given for illustration.Comment: 11 pages; 2 figure

Stability and L1/ℓ1L_1/\ell_1-to-L1/ℓ1L_1/\ell_1 performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

Solutions to the interval observation problem for delayed impulsive and switched systems with $L_1$-performance are provided. The approach is based on first obtaining stability and $L_1/\ell_1$-to-$L_1/\ell_1$ performance analysis conditions for uncertain linear positive impulsive systems in linear fractional form with norm-bounded uncertainties using a scaled small-gain argument involving time-varying $D$-scalings. Both range and minimum dwell-time conditions are formulated -- the case of constant and maximum dwell-times can be directly obtained as corollaries. The conditions are stated as timer/clock-dependent conditions taking the form of infinite-dimensional linear programs that can be relaxed into finite-dimensional ones using polynomial optimization techniques. It is notably shown that under certain conditions, the scalings can be eliminated from the stability conditions to yield equivalent stability conditions on the so-called "worst-case system", which is obtained by replacing the uncertainties by the identity matrix. These conditions are then applied to the special case of linear positive systems with delays, where the delays are considered as uncertainties. As before, under certain conditions, the scalings can be eliminated from the conditions to obtain conditions on the worst-case system, coinciding here with the zero-delay system -- a result that is consistent with all the existing ones in the literature on linear positive systems with delays. Finally, the case of switched systems with delays is considered. The approach also encompasses standard continuous-time and discrete-time systems, possibly with delays and the results are flexible enough to be extended to cope with multiple delays, time-varying delays, distributed/neutral delays and any other types of uncertain systems that can be represented as a feedback interconnection of a known system with an uncertainty.Comment: 35 pages; 13 figures. arXiv admin note: text overlap with arXiv:1801.0378

Hybrid L∞×ℓ∞L_\infty\times\ell_\infty-Performance Analysis and Control of Linear Time-Varying Impulsive and Switched Positive Systems

Recent works have shown that the $L_1$ and $L_\infty$-gains are natural performance criteria for linear positive systems as they can be characterized using linear programs. Those performance measures have also been extended to linear positive impulsive and switched systems through the concept of hybrid $L_1\times\ell_1$-gain. For LTI positive systems, the $L_\infty$-gain is known to coincide with the $L_1$-gain of the transposed system and, as a consequence, one can use linear copositive Lyapunov functions for characterizing the $L_\infty$-gain of LTI positive systems. Unfortunately, this does not hold in the time-varying setting and one cannot characterize the hybrid $L_\infty\times\ell_\infty$-gain of a linear positive impulsive system in terms of the hybrid $L_1\times\ell_1$-gain of the transposed system. An alternative approach based on the use of linear copositive max-separable Lyapunov functions is proposed. We first prove very general necessary and sufficient conditions characterizing the exponential stability and the $L_\infty\times\ell_\infty$- and $L_1\times\ell_1$-gains using linear max-separable copositive and linear sum-separable copositive Lyapunov functions. Results characterizing the stability and the hybrid $L_\infty\times\ell_\infty$-gain of linear positive impulsive systems under arbitrary, constant, minimum, and range dwell-time constraints are then derived from the previously obtained general results. These conditions are then exploited to yield constructive convex stabilization conditions via state-feedback. By reformulating linear positive switched systems as impulsive systems with multiple jump maps, stability and stabilization conditions are also obtained for linear positive switched systems. It is notably proven that the obtained conditions generalize existing ones of the literature.Comment: 62 pages, 9 Figure