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Section Wrap-Up
Published in Niket S. Kaisare, Computational Techniques for Process Simulation and Analysis Using MATLAB®, 2017
The first three sections will use specific examples of nonlinear dynamical systems to introduce the readers to nonlinear analysis techniques. Nonlinear systems display very rich dynamic behavior. Strogatz’s book Nonlinear Dynamics and Chaos is an excellent introductory text on this topic. The linear stability analysis (Section 5.2) and the examples in the next three sections cover some basic and important concepts of nonlinear analysis. Specifically, this involves studying the stability, dynamics, and bifurcation behavior of nonlinear systems. The examples of nonisothermal continuous stirred tank reactor (CSTR) and chemostat (from Chapter 3) that show multiple steady state behavior will be discussed at length to introduce the turning point and transcritical bifurcations, respectively. The third section will study systems with cyclic dynamics (limit cycle).
Systems of First-Order Ordinary Differential Equations
Published in David V. Kalbaugh, Differential Equations for Engineers, 2017
is asymptotically stable (damped) or unstable. Nonlinearities impose limits where linear systems do not. And finally, nonlinear systems can exhibit behavior such as harmonic distortion, limit cycles and virtual unpredictability (chaos) that linear systems do not. Engineers have developed effective methods for designing and analyzing nonlinear systems but they tend to be ad hoc. This chapter gave one example, the concept of equivalent gain, which enables engineers to employ on nonlinear systems the powerful methods available for design and analysis of linear systems.
Surrogation
Published in Nicholas Stergiou, Nonlinear Analysis for Human Movement Variability, 2018
Before getting into a discussion of methods to identify possible nonlinearity in data, let us first define a nonlinear system. A nonlinear system is defined as a system that does not have a linear origin. This includes a system that may contain nonlinearity, but the underlying dynamics are linear. In this case, the presence of nonlinearity is caused by some measurement distortion, but it is originally generated by a linear stochastic process. We will look at such a case in detail later in this chapter. Furthermore, discussion in this chapter will be limited only to a stationary time series and will not include nonstationary stochastic processes. A time series is considered nonstationary if its distribution changes across time. In other words, the mean and variance of the time series change over different time intervals. Applying surrogate methods to nonstationary time series can lead to problems regarding the proper interpretation of results (Breakspear and Terry 2002; Palus 1996). For example, when the null hypothesis is rejected in a surrogate analysis, there is no way of knowing whether nonlinearity exists in the data or whether the data were generated by a nonstationary stochastic process. This problem was noted by Breakspear and Terry (2002) in their study of electroencephalographic (EEG) data (Breakspear and Terry 2002). The problem of nonstationarity was also highlighted by Peng et al. (1995) in the analysis of heart rate variability. Specifically, nonstationarity makes it difficult to determine whether the structure of the time series is the result of the dynamics of the system or from changes in the external environment. Therefore, we will restrict our discussion to time series that are stationary.
Test-based model-free adaptive iterative learning control with strong robustness
Published in International Journal of Systems Science, 2023
Nonlinear systems occupy the majority of systems in practical engineering applications, and the control of nonlinear systems is a widespread problem in engineering research (Cheng et al., 2012; Xu & Tan, 2003). Among the nonlinear control problems, a common problem is the control of time-varying nonlinear systems with finite lengths (Barton & Alleyne, 2010; Chien & Tayebi, 2008; Ouyang et al., 2012; Song et al., 2022). Nonlinear time-varying systems are systems that satisfy the characteristics of both nonlinear and time-varying, and when the value of a parameter in the system changes with time, the fundamental characteristic also changes with time. Iterative learning control (ILC) is one of the effective control methods available to achieve good control effects of time-varying nonlinear systems with finite lengths (Chien, 1998).
Global asymptotic stability analysis of discrete-time stochastic coupled systems with time-varying delay
Published in International Journal of Control, 2021
Hou Rui, Jiayi Liu, Yanbin Qu, Shujian Cong, Huihui Song
It is well known that Lyapunov method is an effective method to study the stability of nonlinear systems. While for coupled systems, which are derived by coupling some nonlinear subsystems, it is usually hard to construct a global Lyapunov function straightly. Fortunately, in Li and Shuai (2010), Li et al. firstly combined the graph-theoretic technique with Lyapunov method to explore the global stability of CSNs. Besides, the influence of the topological structure of the underlying networks was also studied. Then, on the basis of Li and Shuai (2010), many extensions of the model and method have been investigated extensively (Liu, Li, & Feng, 2018; Su et al., 2012, 2016; Wang et al., 2018; Wang, Jin, & Su, 2018; Wang, Zhang, & Su, 2018; Wu, Chen, & Li, 2017). And in Su et al. (2012, 2016), Wang et al. (2018), the authors used the method in Li and Shuai (2010) to investigate the stability of discrete-time CSNs. However, they did not consider the influence of time delay. To the best of our knowledge, there are few results on the stability of DSCSTD by using the combination of graph-theoretic technique and Lyapunov method. This paper will aim to explore the influence of the topological structure of underlying networks, the upper and the lower bounds of time-varying delay, and the strength of stochastic disturbance on the stability of DSCSTD.
Event-triggered fault detection for T-S fuzzy systems subject to data losses
Published in International Journal of Systems Science, 2020
Ziran Chen, Baoyong Zhang, Yijun Zhang, Yongmin Li, Zhengqiang Zhang
Moreover, the systems in practical are almost with nonlinearities, so the study of nonlinear systems is more realistic (Liu & Tong, 2015; Min et al., 2019a, 2019b; Zhu et al., 2014; Zuo et al., 2018). Useful methods have been developed to stabilise the nonlinear system, such as adaptive control, sliding mode control and fuzzy control. In existing methods for analysing nonlinear systems, T-S fuzzy methods are very effective and widely used (Lam & Lo, 2013; Tanaka & Wang, 2004; Zhang et al., 2019; Zhang, Qiao, et al., 2017). And many relevant results have been developed. To mention a few, Xie et al. (2015) employed a novel multi-instant homogeneous polynomial approach to lower the conservatism of stability conditions for T-S fuzzy systems. In the process of applying T-S fuzzy method to networked systems (Pan & Yang, 2017a; Zhang et al., 2015), the asynchronous membership function induced by the imperfect communication channel is the main problem that should be considered.