Exponential Regulation of Uncertain Nonlinear Triangular Impulsive Systems: A Logic-Based Switching Gain Approach

This article addresses the exponential regulation problem for uncertain nonlinear impulsive systems with generalized triangular structures. Different from the existing works, the nonlinear growth rates under consideration are allowed to contain unknown constants and state-input-dependent functions, and the impulse effects are allowed to have unknown sizes. This problem has remained unsolved because it exceeds the coverage of common assumptions in impulsive systems. To this end, we propose a novel logic-based switching gain approach. Specifically, the well-designed logic-based switching mechanism detects the Lyapunov function in real time to adjust the candidate gain online such that the gain grows large enough to effectively dominate the strong nonlinearities and serious uncertainties. Interestingly, the proposed approach removes the restriction on impulse frequency (reflected by the average impulsive interval constant) within the lower-triangular structural framework, and executes only one switching action to obtain the upper bound of the required gain within the upper-triangular structural framework. It is shown that the logic-based switching adaptive controller ensures that all the closed-loop signals are bounded and the system states converge to the origin at an exponential rate. Finally, the effectiveness of the presented results is demonstrated by two representative examples.