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Models for Control
Published in Raghunathan Rengaswamy, Babji Srinivasan, Nirav Pravinbhai Bhatt, Process Control Fundamentals, 2020
Raghunathan Rengaswamy, Babji Srinivasan, Nirav Pravinbhai Bhatt
In general, from a control viewpoint, nonlinear systems are more difficult to deal with than linear systems. As a result, a nonlinear state-space system is usually linearized to derive a linear state-space system that is amenable to simpler control analysis techniques. Linearization is used to derive transfer function models using Laplace transforms from linearized state-space models. Linearization is performed around an operating steady-state of the system. Since with good control one would expect to maintain controlled variables around their steady-states, linearization works in operational control systems. Formally, a function f(x) is linear if it satisfies the following two properties: f(x + y) = f(x) + f(y)f(ζx) = ζf(x)
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
In the last 20 years nonlinear control has reached the level of a mature discipline, both in its theoretical developments and in engineering applications. Examples of successful application of modern nonlinear control theory are widespread, and range from aerospace to robotics, and from electrical and mechanical to biomedical engineering. The advantage of nonlinear control systems design versus more conventional (i.e., linear) design methodologies lies in the fact that the fundamentally nonlinear nature of the plant to be controlled is taken directly into account (and, sometimes, exploited) rather than neglected or ignored. The price to pay, however, is the lack of a general design methodology that is applicable to all nonlinear systems, and a substantial complexity of the mathematical tools required for the analysis and the synthesis of nonlinear control systems. The first drawback is intrinsic to the nature of the problem, and it is alleviated by the fact that most nonlinear design techniques indeed apply to entire classes of nonlinear systems sharing a common structure and properties. The second requires the modern control engineer to acquire the mathematical tool necessary to master the discipline, and meet the challenges for more demanding applications.
Systems Theory and Optimal Control
Published in Larry W. Mays, Optimal Control of Hydrosystems, 1997
The outputs from a static system only depend on the current values of the inputs, whereas outputs from a dynamic system depend on the current and the previous values of the inputs. The inputs and outputs for a linear system satisfy the superposition theorem, whereas a system is nonlinear if it does not satisfy the superposition theorem. The superposition theorem includes two parts, additivity and homogeneity. Additivity is that the output due to the sum of inputs is equal to the sum of the outputs due to each of the inputs or ϕ(∑n=1Nun)=(∑n=1Nϕun). Homogeneity is ϕ(cun) = c(ϕun) in which c is a constant.
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).
Dissipation of energy analysis approach for vehicle plane motion stability
Published in Vehicle System Dynamics, 2022
Fanyu Meng, Shuming Shi, Minghui Bai, Boshi Zhang, Yunxia Li, Nan Lin
To study the stability of nonlinear systems, another classical method besides the phase plane analysis method, the Lyapunov function method is used [13]. Johnson and Huston [17] established two Lyapunov functions using standard methods and improved kinetic energy functions based on a 2-DOF nonlinear vehicle model, and analysed the lateral stability of the vehicle while keeping straight-line motion. However, because the tire force calculation uses a cubic term expression, the obtained stability region is a conservative elliptical region. On the basis of [17], Samsundar and Huston [14] used the same vehicle model to derive the analytical expression of the equilibrium point and used the Lyapunov function method, the tangent point method, and the trajectory reversal approach to numerically estimate the lateral stability region of the vehicle, in which the trajectory reversal approach has obtained better estimation results. Sobhan Sadri and Christine Wu [18] proposed two new Lyapunov functions based on the research in [14] and [17]. These two functions do not depend explicitly on the vehicle parameters, and compared with previous work, a larger stability region is obtained. However, there is no general method for constructing the Lyapunov function currently. For complex strongly nonlinear systems, to construct the form of the Lyapunov function, it is usually necessary to simplify the model to a certain extent. Meanwhile, the solution result of the stability region is also limited by the function expression. It is difficult to obtain the accurate stability region and generalise to a high degree of freedom systems.
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.