Robust properties of systems with negative degree of homogeneity with respect to delay

Abstract

The paper considers robustness problems of a class of weighted homogeneous systems with negative homogeneity degree in relation to the delay. It is shown that in the case of global asymptotic stability of a nonlinear weighted homogeneous system with negative homogeneity degree, in the presence of a delay in the system, all the trajectories converge asymptotically to some compact set containing the origin. In the absence of delay, such systems reach their equilibrium position in a finite time. The robustness analysis also covers cases of variable and multiple delays. The presented analysis is based on the application of the Lyapunov methods for delayed systems (the Lyapunov-Razumikhin function method) and the theory of weighted homogeneous systems. Computer simulation was performed to verify the analysis of system robustness with a negative degree of homogeneity in relation to the delay. A stabilizing system that represents a double integrator is used as an example. This system is weighted homogeneous with a negative degree when using a nonlinear state feedback control law that ensures that the system achieves its equilibrium position for the desired finite time. During computer simulation, the state vector was available for measurement with some delays. The computer simulation has confirmed the effectiveness of the presented theoretical results.