Abstract
This article presents the study of the stability of single-input and multiple-input systems with point or distributed state delay and input delay and input saturation. By a linear transformation applied to the initial system with delay, a system is obtained without delay, but which is equivalent from the point of view of stability. Next, using sufficient conditions for the global asymptotic stability of linear systems with bounded control, new sufficient conditions are obtained for global asymptotic stability of the initial system with state delay and input delay and input saturation. In addition, the article presents the results on the instability and estimation of the stability region of the delay and input saturation system. The numerical simulations confirming the results obtained on stability were performed in the MATLAB/Simulink environment. A method is also presented for solving a transcendental matrix equation that results from the process of equating the stability of the systems with and without delay, a method which is based on the computational intelligence, namely, the Particle Swarm Optimization (PSO) method.
Highlights
The time-delay systems are challenging since they involve delay differential equations that are infinite-dimensional functional differential equations, which are more difficult to handle than finite-dimensional ordinary differential equations [1,2]
In the last decades, considerable attention has been devoted to the problem of stability analysis and controller design for time-delay systems
We follow the results from the previous section for both type of delay: point and distributed
Summary
The time-delay systems are challenging since they involve delay differential equations that are infinite-dimensional functional differential equations, which are more difficult to handle than finite-dimensional ordinary differential equations [1,2]. While in the approaches where the stability of the delayed systems is based on fulfilling a matrix inequality like the one presented above, in this article, by using an Arstein type transformation (or generalized to systems with both state and input delay), the difficulty of the numerical computation is reduced to solving a transcendental matrix equation such as A = A0 + e−Ah A1. In this respect, the article presents methods based on the calculation of their own values and methods based on a computational-intelligence algorithm, namely, the PSO.
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