Sliding Mode Observer-Based Control for Uncertain Singular Biological Economic System With Invasion of Alien Species

Abstract

In this paper, the controller based on sliding mode observer (SMO) is studied for a singular biological economic model with stage structure and uncertain parameters. First, a biological economic system is established by the singular system with uncertain parameters for the invasion of alien species. And SMO is designed by use of control inputs and measurement outputs. Then, the integral sliding surface is constructed for the error system and the SMO system. Second, by constructing an augmented system, it is ensured that the system is regular, free of impulse, and stable by use of the linear matrix inequality technique and Lyapunov stability theory. Third, a sliding mode control (SMC) law is designed to guarantee the considered system reaching the sliding mode surface. Finally, the effectiveness of the proposed method is illustrated by a simulation example. INDEX TERMS Stage structure, singular biological economic system, sliding mode observer. 91548 VOLUME 7, 2019 Z. Shi et al.: SMO-Based Control for Uncertain Singular Biological Economic System Let z 1 (t) = −(a 1 + b + b) − c 1 x 1 (t), z 2 (t) = −a 2 − c 2 x 2 (t), z 3 (t) = −x 2 (t), z 4 = pE 1 (t). So max z 1 (t) = −(a 1 + b + b) − c 1 x 1 min (t), min z 1 (t) = −(a 1 + b + b) − c 1 x 1 max (t), max z 2 (t) = −a 2 − c 2 x 2 min (t), min z 2 (t) = −a 2 − c 2 x 2 max (t), max z 3 (t) = −x 2 min (t), min z 3 (t) = −x 2 max (t), max z 4 = pE 1 max (t), min z 4 = pE 1 min (t). Using the maximum and minimum values, z 1 (t) , z 2 (t) , z 3 (t) and z 4 (t) can be represented by z 1 (t) = M 11 (z 1 (t)) max z 1 (t) + M 12 (z 1 (t)) min z 1 (t) , z 2 (t) = M 21 (z 2 (t)) max z 2 (t) + M 22 (z 2 (t)) min z 2 (t) , z 3 (t) = M 31 (z 3 (t)) max z 3 (t) + M 32 (z 3 (t)) min z 3 (t) , z 4 (t) = M 41 (z 4 (t)) max z 4 (t) + M 42 (z 4 (t)) min z 4 (t) .