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Section Wrap-Up
Published in Niket S. Kaisare, Computational Techniques for Process Simulation and Analysis Using MATLAB®, 2017
A phase plane is a plane with the two state variables plotted as two axes. Each curve on the plot represents a trajectory that the system takes starting at various locations in the plane. A collection of such curves that defines the dynamic response of the system constitutes the phase portrait. Figure 5.5 shows the phase portrait for the linear bioreactor example. Circles denote the various initial conditions in the 2D space. Starting at any point, the system responds rapidly along the first eigenvector v1, whereas the response along the second eigenvector v2 is more sluggish. This gives rise to the phase portrait as seen in Figure 5.5. Eigenvalues and eigenvectors can be used to qualitatively analyze the dynamic response of a system, which will be discussed in this section.
Primer on Spike Processing and Excitability
Published in Paul R. Prucnal, Bhavin J. Shastri, Malvin Carl Teich, Neuromorphic Photonics, 2017
Paul R. Prucnal, Bhavin J. Shastri, Malvin Carl Teich
For a given parameter set Ω, the dynamics of a system governed by Eq. (2.6) can be visualized by analyzing the values of F(X; Ω) for all X ∈ AX. These can be geometrically represented on a phaseplane. If the dynamical system is prepared at an initial condition X0, it is possible to trace the associated trajectory of the system’s state on the phase plane. These trajectories may diverge to the boundary of AX, or end up in a fixed point of the system, where dXdt=0.3 They can also lead back to the original point X0 after an excursion and start over again repeatedly in a periodic orbit. A phase portrait can be constructed with the trajectories represented as lines with arrows, and fixed points with dots. The phase portrait is a useful tool because it reveals the topologicalstructure of the dynamical system, indicating the presence and locations of orbits and fixed points, defining its qualitative behavior at different regions. Examples of phase portraits are shown in Figs. 2.5, 2.6 and 2.7.
Analysis Methods
Published in William S. Levine, Control System Fundamentals, 2019
which is a special case of Equation 19.39. A variety of procedures has been proposed for sketching state [phase] plane trajectories for Equations 19.39 and 19.41. A complete plot showing trajectory motions throughout the entire state (phase) plane is known as a state (phase) portrait. Knowledge of these methods, despite the improvements in computation since they were originally proposed, can be particularly helpful for obtaining an appreciation of the system behavior. When simulation studies are undertaken, phase plane graphs are easily obtained and they are often more helpful for understanding the system behavior than displays of the variables x1 and x2 against time.
Vehicle control synthesis using phase portraits of planar dynamics
Published in Vehicle System Dynamics, 2019
Carrie G. Bobier-Tiu, Craig E. Beal, John C. Kegelman, Rami Y. Hindiyeh, J. Christian Gerdes
A phase portrait illustrates the dynamics of a system graphically by plotting the state derivatives and resulting trajectories as a function of the state for a fixed input. For example, given two states, x and y, and their dynamic equations, the x–y phase portrait depicts the magnitude and direction of the state derivatives and evaluated at a grid of x–y values. A phase portrait of a nonlinear system immediately reveals the location and type of equilibria in the system and the regions of attraction, or stable regions, about those equilibria.
A cooperative control strategy for yaw rate and sideslip angle control combining torque vectoring with rear wheel steering
Published in Vehicle System Dynamics, 2022
A phase portrait graphically illustrates the dynamics of a system by plotting the state derivatives and resulting trajectories as a function of the state for fixed inputs. Phase portrait analysis has proved to be a very effective tool for analysing the dynamics of a nonlinear system as it immediately reveals the location and type of equilibria in the system and the region of attraction, or stable regions, about those equilibria. [18,29,31] showed that the fundamental planar instabilities critical to control problems such as ESP or stabilising a drifting vehicle arise from the yaw and sideslip dynamics.
An experimental study on the motion of buoyant particles in the free-surface vortex flow
Published in Journal of Hydraulic Research, 2021
Alex Duinmeijer, Francois Clemens
A phase portrait depicts the trajectories of a dynamical system. Each set of initial conditions is represented by a vector presenting the direction and magnitude of the systems change. Consequently, the portrait shows for a large ensemble of initial conditions the dynamic behaviour of the system. As portraits based on experiments are very time consuming and difficult to establish numerous combinations of initial conditions with slightly different values, the portraits presented here are constructed with the numerical model. The applied ensemble of initial conditions consists of the radial position r0 from 0.02 to 0.30 m (step size 0.005 m), the radial velocity Ur,0 from −1.0 to 1.0 m s−1 (step size 0.1 m s−1) and two rotations: Ωp,0 = 0 and Ωp,0 = 1.5 rad s−1. So, the portrait is made by an ensemble of 1056 different initial conditions. Figure 5a, b shows the phase portrait for sphere 86 (Ø38 mm) and Fig. 5c, d for sphere 101 (Ø38 mm). All figures includes several “trajectories” with the initial conditions illustrated by a blue dot. The portraits obviously show that, depending on the initial conditions, both spheres can move to a stable limit cycle. In the experiments, this cycle is the stable orbit around the air core at an approximately constant height (e.g. Fig. 4b). This cycle attracts neighbouring trajectories and is defined as an attractor. On the other hand, the sphere can move in an unstable limit cycle with increasing radius and thus move outward of the centre. This cycle repels trajectories and is thus a repellor. In conclusion: the behaviour of the system is strongly influenced by the particle initial conditions and, indeed, the system shows chaotic behaviour that implies a limit to the system’s forecast horizon. However, not all features seen in the experiments are reproduced by the model, which was to be expected given the simplifications made in the model.