$L_1/\ell_1$-to-$L_1/\ell_1$ analysis of linear positive impulsive systems with application to the $L_1/\ell_1$-to-$L_1/\ell_1$ interval observation of linear impulsive and switched systems
This work addresses the problem of interval observer design for impulsive and switched systems, which is important for safety-critical applications requiring guaranteed state bounds, but the results are incremental as they extend existing L1/ℓ1 analysis to positive impulsive systems with dwell-time constraints.
The paper provides sufficient conditions for asymptotic stability and hybrid L1/ℓ1-gain of linear positive impulsive systems under dwell-time constraints, formulated as infinite-dimensional linear programs solvable via sum-of-squares programming. These conditions are then used to design L1/ℓ1-to-L1/ℓ1 interval observers for linear impulsive and switched systems, with observer gains extracted from a suboptimal solution minimizing the L1/ℓ1-gain from disturbances to weighted observation errors.
Sufficient conditions characterizing the asymptotic stability and the hybrid $L_1/\ell_1$-gain of linear positive impulsive systems under minimum and range dwell-time constraints are obtained. These conditions are stated as infinite-dimensional linear programming problems that can be solved using sum of squares programming, a relaxation that is known to be asymptotically exact in the present case. These conditions are then adapted to formulate constructive and convex sufficient conditions for the existence of $L_1/\ell_1$-to-$L_1/\ell_1$ interval observers for linear impulsive and switched systems. Suitable observer gains can be extracted from the (suboptimal) solution of the infinite-dimensional optimization problem where the $L_1/\ell_1$-gain of the system mapping the disturbances to the weighted observation errors is minimized. Some examples on impulsive and switched systems are given for illustration.