L1-gain control with adjustable convergence rate for impulsive positive systems: A geometric constraint approach to pole configuration

This paper investigates the L1-gain control problem based on convergence rate constraints for impulsive positive systems. The main contribution is the proposal of a novel L1-gain performance analysis framework, which transforms the convergence rate constraints of the system into pole placement constraints within a specific band-shaped region (see Fig. 1) in the open left-half s-plane. For the exact placement of the closed-loop system matrix eigenvalues in the specified band-shaped region, while guaranteeing the positivity, asymptotic stability, and L1-gain performance of the impulsive positive system, two auxiliary systems are introduced and a copositive Lyapunov function is constructed, yielding a new L1-gain control criterion. The criterion is formulated as a set of solvable linear programming inequalities. Compared to traditional stabilization methods that primarily require the eigenvalues of the closed-loop system matrix to lie in the open left-half s-plane, the proposed L1-gain control criterion enforces strict geometric constraints on the eigenvalue distribution, enabling more precise regulation of the system convergence rate. Specifically, by flexibly adjusting the boundaries of the band-shaped region, corresponding controller parameters can be obtained, allowing the closed-loop impulsive positive system to converge to equilibrium at different rates while ensuring stability and L1-gain performance. The effectiveness of the proposed method is validated through numerical simulations.