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Nonlinear Systems
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
The fundamental difference between autonomous and non-autonomous systems lies in the fact that the state trajectory of an autonomous system is independent of the initial time, while that of a non-autonomous system generally is not. As we will see in the next chapter, this difference requires us to consider the initial time explicitly in defining stability concepts for non-autonomous systems and makes the analysis more difficult than that of autonomous systems. Generally speaking, autonomous systems have relatively simpler properties and their analysis is much easier.
A generalised proportional-derivative type scheme with multiple saturating structure for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with bounded inputs
Published in International Journal of Systems Science, 2023
Arturo Zavala-Río, Griselda I. Zamora-Gómez, Tonametl Sanchez, Fernando Reyes-Cortes
Consider an nth order non-autonomous system where is continuous, is a domain that contains the origin , and , . We denote – or simply whenever convenient or clear from the context – a solution of (6) with initial condition at initial time , and is the set of all the solutions starting at .
Trajectory-tracking in finite-time for robot manipulators with bounded torques via output-feedback
Published in International Journal of Control, 2023
Emmanuel Cruz-Zavala, Emmanuel Nuño, Jaime A. Moreno
Consider a non-autonomous system described by where is the state vector and is a continuous vector field locally bounded uniformly in time. The open connected set is a domain that contains the origin . The solutions of (1) are denoted by with initial condition at initial time and exist in forward time (i.e. ). Assume that is an equilibrium point, i.e. for all .
Ordinary differential equations defined by a trigonometric polynomial field: behaviour of the solutions
Published in Dynamical Systems, 2023
In this article, we study the asymptotic behaviour of solutions for ordinary differential equations (ODE) defined by a trigonometric polynomial field. The idea comes from the scalar case, where in this case H. Poincaré defined the rotation number for circle homeomorphisms [7]. The simple example is a scalar differential equation where is lipschitz, 1-periodic and is the state of the system. There exists a rotation number for which the function is bounded (periodic). We know that any non-autonomous system can be written as an autonomous system. Our result is a generalization of this asymptotic behaviour to any dimension. In this case, λ is a vector and called a rotation vector or rotation set as it is defined in [5]. Under some assumptions of stability [8], [2] proved the existence of the rotation vector. Some biological works use the ODE defined by a trigonometric polynomial field and study the rotation vector components as in [1], [3], [4], [10]. Our contribution to this biological works has two key points, the mathematical proof of existence of the rotation vector and the study of the behaviour of solutions.