Dynamics and control of the active control system with the state-dependent actuation time delay
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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-900148-9
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Dynamics and control of the active control system with the state-dependent actuation time delay Lijun Pei1,a and Huifang Jia1,2 1
2
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, P.R. China Basic Teaching Department, Zhengzhou University of Industry Technology, Zhengzhou, Henan 451100, P.R. China Received 21 July 2019 / Accepted 8 June 2020 Published online 28 September 2020 Abstract. In this paper, a model of active control system with statedependent actuation time delay is investigated. Galerkin projection scheme is used to obtain its low dimensional approximation system. State-dependency and effect on dynamics and performance of actuation delay in this active control system are considered. The following results are obtained. Firstly, a state-dependent actuation delay is proposed, which has never been reported in the previous works. It displays that actuation delay is state-dependent. Secondly, employing Galerkin projection scheme, low dimensional approximation system of this delayed active control system is obtained. At last, its stability and Hopf bifurcation are investigated by Routh-Hurwitz Criterion and Hopf bifurcation theory. Using software WinPP, numerical simulation is executed and supports the theoretical results in this paper.
1 Introduction Time delay is universal and ubiquitous in both the natural world and humankind world. It reflects that the tendency of the object depends on not only its current state but also its past state. This kind of dynamical systems with time delay are usually called delayed dynamical systems [1–3]. Delay differential equations (DDEs) have applications in manufacturing process, control system, robotics, biology, controlling chaos, secure communication via chaotic synchronization, economics, chemical kinetics and other areas [4–7]. The delayed dynamical systems are very different from those without the time delay (ODEs). Since the former’s characteristic equation is usually transcendental and has infinitely many characteristic roots, thus correspondingly their solution spaces are infinite-dimensional [1–3,8]. This basic character induces the co-dimension 1 bifurcations such as Hopf bifurcation, the higher co-dimensions bifurcations such as the double Hopf bifurcation, Double Zero bifurcation and Zero-Hopf a
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The European Physical Journal Special Topics
bifurcation, etc., and the complex dynamic phenomena such as chaos extremely easily. Thus due to this character, the study on the dynamics of the delayed dynamical systems becomes very difficult [9]. Perhaps studies of differential equations with state-dependent delay were started earlier by Poisson [10], but as an object of a broader mathematical activity this area is rather young. State-dependent delays were addressed earlier in survey papers on the area of functional differential equations, notably by Halanay and
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