Nonlinearity
By Steven A. Frank. Encyclopedia of Social Network Analysis and Mining. 2nd Ed., 2018
548 citations recorded by the source corpus.
Real systems are nonlinear. Before discussing nonlinear theory, I review three reasons why the core theory of control focuses on linear analysis. First, feedback compensates for model uncertainty. Suppose we analyze a feedback system based on a linear model of dynamics, and the true model is nonlinear. If the linear model captures essential aspects of the dynamics sufficiently, the true nonlinear feedback system will often have the same qualitative response characteristics as the modeled linear system. As Vinnicombe (2001, p. xvii) emphasized: “One of the key aims of using feedback is to minimize the effects of lack of knowledge about a system which is to be controlled.” Second, the fundamental principles of control systems apply to both linear and nonlinear dynamics. The comprehensive theory for linear systems provides insight into nonlinear systems. For example, strong feedback signals often help to minimize error but can create instabilities. Controllers can be added at different points in a system to filter signals or modulate inputs that drive processes. Primary goals of analysis and design often emphasize stability, disturbance rejection, regulation or tracking. Certain tradeoffs inevitably arise. Integral control smooths response toward a long-term target. Derivative control improves the speed of response by using a simple prediction. Third, the main tools for the analysis and design of nonlinear systems typically extend the tools developed for linear systems. For example, nonlinear systems are approximately linear near a particular operating point. One can study the linear approximation around that point, and then switch to an alternative linear approximation as the system moves to another operating domain. By piecing together the different linear approximations in neighboring domains, one develops a sequence of linear systems that together capture much of the nonlinear characteristics. Other tools of nonlinear analysis typically leverage the deep insights and methods of linear analysis (Slotine and Li 1991; Isidori 1995; Khalil 2002). This chapter presents a few brief illustrations of nonlinear control systems.
View Paper: https://www.semanticscholar.org/paper/41fd9a85736bb889da0bdf5297ddfaa6548cfaec
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
Citations in the corpus
- Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption (2022, background): “Conversely, for a further simplification of the parabolic-elliptic formulation, in spatially radial contexts it is known (see [Win18a]) that the value 2 n is critical, in the sense that the boundedness of solutions is guaranteed for any n ≥ 1 and 0 < k < 2 n , whereas for k > 2 n blow-up phenomena m”