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Special Types of Closed-Loop Drug Input Controllers
Published in Robert B. Northrop, Endogenous and Exogenous Regulation and Control of Physiological Systems, 2020
Phase plane descriptions of the performance of regulators and control systems provide a useful, qualitative, and quantitative contrast to the conventional frequency-domain and time-domain means of analysis. Phase plane analysis generally assumes that the plant is second order. On Cartesian coordinates, we plot a parameter such as the system error on the x-axis and its derivative on the y-axis, each point being taken at some time, tk. Thus a phase plane plot, or phase “portrait,” parametrically eliminates time. Phase plane plots can be made for (1) initial conditions on (e, e˙), (2) an impulse input to the system, or (3) a step input to the system. Phase plane plots are useful for describing the behavior of nonlinear parameter-switching or discontinuous control systems and time-optimal controllers where a controller switching rule is devised that will take the system from one set of states to another in minimum time, under certain constraints. The path a system takes in going from one set of states to another in the phase plane is called a trajectory.
Ordinary Differential Equations Part I
Published in Edward B. Magrab, Advanced Engineering Mathematics with Mathematica®, 2020
in which the independent variable t does not appear explicitly in F and G. It is assumed that F and G are continuously differentiable in a region R in the (x,y)-plane, which is called the phase plane. Any curve in the phase plane is called a trajectory. At t = t0, the system satisfies the initial conditions x(t0)=x0y(t0)=y0
Advanced Control Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
For any second-order system, there are two states x1 and x2. A plane formed by these states is called the phase plane. Given some initial conditions of the system dynamics, one can plot the variation of the two states in this phase plane. This plot is known as the phase motion trajectory of the system for the given initial condition. Different initial conditions result in different motion trajectories forming system phase portraits. Phase plane analysis is a graphical method that uses the information from the phase portrait of the system to analyze the dynamics of the system in question. It enables one to visualize the dynamics of the system without having to solve nonlinear differential equations analytically. One of the major advantages is that it can handle all types of nonlinearities, i.e., smooth, strong, and hard nonlinearities. On the other hand, however, it has the limitation that it cannot handle higher-order systems because of the computational complexity as well as the complexity associated with graphical presentation of higher-order systems. The only way phase plane methods can be applied to higher-order systems is by approximation of these systems to second-order equivalents. This section discuses the fundamentals of phase plane analysis, presenting the theoretical basics of the method that eventually lays a strong groundwork for understanding the stability of systems. The methods for plotting and interpretation of the system phase plots are presented.
Robust limit cycle control in a class of nonlinear discrete-time systems
Published in International Journal of Systems Science, 2018
Ali Reza Hakimi, Tahereh Binazadeh
Limit cycle is an isolated periodic orbit in the phase plane which is an important phenomenon in nonlinear dynamical systems (Haddad & Chellaboina, 2008; Khalil & Grizzle, 2002). If the trajectories beginning near the limit cycle converge to it, then the limit cycle is attractive. This is a very rich dynamical behaviour which has the numerous applications in engineering systems like walking and running (Geyer, Seyfarth, & Blickhan, 2006; Laszlo, van de Panne, & Fiume, 1996; McGeer, 1990), power converters (Benmiloud & Benalia, 2016; Oviedo, Vazquez, & Femat, 2017), satellite altitude control (Clark, 1970), aero elastic problems (Bialy, Chakraborty, Cekic, & Dixon, 2016; Khalid & Akhtar, 2017), boiling-water reactors (Farawila & Pruitt, 2006), diffusively coupled models (Shafi, Arcak, Jovanović, & Packard, 2013), hybrid systems (Benmiloud, Benalia, Djemai, & Defoort, 2017; Flieller, Riedinger, & Louis, 2006) and port Hamiltonian systems (Aguilar-Ibañez, Mendoza-Mendoza, Martinez, Jesus Rubio, & Suarez-Castanon, 2015b).
Imaging phase plane models
Published in International Journal of Mathematical Education in Science and Technology, 2023
Richard F. Melka, Hashim A. Yousif
The phase plane is a graphical representation of the solution of a set of differential equations. It is a standard topic in a first subject on non-linear differential equations; (Perko, 1991; Simmons & Krantz, 2007) are examples. Phase plane analysis leads to the identification of trajectories, the phase portrait, critical points, saddle points, stable and unstable solutions. Phase plane analysis is applied in various disciplines, such as physics, engineering and biology (Arnold, 1989; Edelstein-Keshat, 2005). A brief discussion of the phase plane for oscillatory systems is provided in (Hale & Kocak, 1991), the phase portrait of a mechanical system is investigated in (Al Quran, 2007).