Abstract
This paper is concerned with stability of interconnected systems with time delays. We develop a self-contained approach to stability analysis for linear and nonlinear systems in a unified framework. New lemmas are established on matrix properties and used as the key to make negativity of the derivative of the Lyapunov function. The scalar and simple analytical stability conditions are given. Unlike the majority of the literature on stability of delay systems, no matrix equations/inequalities are involved in our conditions, which is true even for large-scale systems and nonlinear subsystems with delayed interconnections. They are applicable to the more general nonlinear, time-varying, and/or interconnected systems than the relevant results reported in the literature. The examples are presented for illustration of the new results.
Highlights
INTRODUCTIONThe controller receives all sensor data available from the subsystems and determines all input signals of the plant where all information is assumed to be available for a single unit that designs and applies the controller to the plant
In this paper, the stability of interconnected nonlinear systems with delays is analyzed from a new perspective n−1 n−1 y(jn)(t) + fj,i (.) y(ji)(t) + gj,i(.)y(ji)(t − τj(t)) = 0, (1)i=0 i=0 with the initial time at t = t0 and the initial conditions: y(ji)(t) = φj,i(t), t ∈ [t0 − τm, t0], i = 0, . . . , n − 1, where yj(t) is the system output, and τj(t) is the time delay in the system
As the interconnected Lur’e Postnikov systems with time delays is one of the most important connected systems, we investigate, in this paper, their stability analysis
Summary
The controller receives all sensor data available from the subsystems and determines all input signals of the plant where all information is assumed to be available for a single unit that designs and applies the controller to the plant This procedure becomes incorrect when there is time delay in the interconnection links. It is well known that the majority of the literature on stability of delay systems studied stability conditions in terms of linear matrix inequalities (LMIs) [50]–[53] This observation remains true until now in a huge volume of publications and the size of LMIs increases with order/complexity of the systems. 3) Theorem 1, applied in our paper, enables stability analysis for a general system, where all the elements of an arrow-form matrix can be nonlinear or time-varying including both the system’s coefficient functions, fi and gi; and artificially introduced parameters, αi.
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