2020/05/04 by Corentin Briat, Briat, Corentin
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · #Adaptive Dynamic Programming Control #Control and Stability of Dynamical Systems #FOS: Electrical engineering #FOS: Mathematics #Gene Regulatory Network Analysis #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2005.01771
openalex publication_date 2020/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recent works have shown that the L1 and L_\∞-gains are natural\nperformance criteria for linear positive systems as they can be characterized\nusing linear programs. Those performance measures have also been extended to\nlinear positive impulsive and switched systems through the concept of hybrid\nL1\×\ℓ1-gain. For LTI positive systems, the L_\∞-gain is known\nto coincide with the L1-gain of the transposed system and, as a consequence,\none can use linear copositive Lyapunov functions for characterizing the\nL_\∞-gain of LTI positive systems. Unfortunately, this does not hold in\nthe time-varying setting and one cannot characterize the hybrid\nL_\∞\×\ℓ_\∞-gain of a linear positive impulsive system in terms\nof the hybrid L1\×\ℓ1-gain of the transposed system. An alternative\napproach based on the use of linear copositive max-separable Lyapunov functions\nis proposed. We first prove very general necessary and sufficient conditions\ncharacterizing the exponential stability and the L_\∞\×\ℓ_\∞-\nand L1\×\ℓ1-gains using linear max-separable copositive and linear\nsum-separable copositive Lyapunov functions. Results characterizing the\nstability and the hybrid L_\∞\×\ℓ_\∞-gain of linear positive\nimpulsive systems under arbitrary, constant, minimum, and range dwell-time\nconstraints are then derived from the previously obtained general results.\nThese conditions are then exploited to yield constructive convex stabilization\nconditions via state-feedback. By reformulating linear positive switched\nsystems as impulsive systems with multiple jump maps, stability and\nstabilization conditions are also obtained for linear positive switched\nsystems. It is notably proven that the obtained conditions generalize existing\nones of the literature.\n