Abstract
In the design of robustly stable feedback systems, the key step is to analyze the stability of closed loop systems in case nominal plant is perturbed. In this paper, robust stability of time delay systems is discussed. We propose a method to check the robust stability for time delay systems in case that the parameter perturbations occur in both lumped parameter part and distributed delay element. It is known that the properties of Lyapunov type operator equation of time delay systems are almost similar to those of finite dimensional case; for example, the time delay system is asymptotically stable if and only if there exists a unique positive definite solution for the Lyapunov type operator equation.
By using this property of Lyapunov type operator equation, we first derive an operator type condition of robust stability, in which the parameter perturbations are described as a bounded operator. Secondly, we derive a matrix type condition of robust stability, which is equivalent to the operater type condition. By using these preliminary results, we lastly propose a norm condition of robust stability in case that parameter perturbations are characterized with structural variables.
The method we proposed needs a positive definite self-adjoint solution to Lyapunov type operator equation. Averaging method can be applied for the approximate calculation of Lyapunov type operator equation, and the computational work needed to obtain the solution is shown to be quite moderate.
