1 / 49

Chapter 2

Chapter 2. Phase Plane Analysis. Basic Idea. To generate motion trajectories corresponding to various initial conditions in the phase plane. To examine the qualitative features of the trajectories. In such a way, information concerning

urban
Download Presentation

Chapter 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2 Phase Plane Analysis

  2. Basic Idea • To generate motion trajectories • corresponding to various initial conditions in • the phase plane. • To examine the qualitative features of the • trajectories. • In such a way,information concerning • stability and other motion patterns of the • system can be obtained.

  3. Feature 1. A graphical method: to visualize what goes on in a nonlinear system without solving the nonlinear equations analytically. 2. Applying to strong nonlinearities and to “hard” nonlinearities. 3. Limitation: limited for second-order (or first –order) dynamic system; however, some practical control systems can be approximated as second-order systems.

  4. 2.1Concepts of Phase Plane Analysis

  5. Phase Portrait of a Mass-spring System Mass-spring system , Solution: Equation of the trajectories:

  6. Figure 2.1:A mass-spring system and its phase portrait

  7. System response corresponding to various initial conditions is directly displayed on the phase plane. • The system trajectories neither converge to the origin nor diverge to infinity. • They simply circle around the origin, indicating the marginal nature of the system’s stability.

  8. In state space form : • with and General Description of Second-Order System

  9. Phase Plane Method The phase plane method is developed for the dynamics (2.3), and the phase plane is defined as the plane having x and as coordinates.

  10. Equilibrium Point and Singular Points • Equilibrium point is defined as a point where the system states can stay forever, this implies that = 0, and using (2.3) • A singular point is an equilibrium point in the phase plane.

  11. Example 2.2: A nonlinear second-order system The system has two singular points: (0, 0) and (-3, 0). - The trajectories move towards the point (0,0) while moving away from the point (-3,0).

  12. Why Called Singular Point - The slope of the phase trajectories: - The value of the slope is 0/0 ,however, at singular points i. e., the slope is indeterminate.

  13. Example 2.3: A first-order system Consider the system There are three singular points, defined by , namely, x = 0, －2, and 2.

  14. Figure 2.3: Phase trajectory of a first-order system

  15. The arrows in the phase portrait denote the • direction of motion. • The equilibrium point x = 0 is stable, while the • other two are unstable.

  16. 2.1.3 Symmetry in Phase Plane Portraits • Symmetry about the x1 axis: f (x1 ,,x2 ) = f (x1 ,,－ x2 ) • Symmetry about the x2axis: f (x1 ,,x2 ) = － f (－ x1 , x2 ) • Symmetry about the and axis: f (x1 ,,x2 ) = f (x1 ,,－ x2 ) = － f (－ x1 , x2 ) only one quarter of it has to be explicitly considered.

  17. Symmetry about the origin: f (x1 ,,x2 ) = －f (－x1 ,,－ x2 )

  18. 2.2 Constructing Phase Portraits I. Analytical method II. The method of isoclines

  19. I. Analytical method Example 2.4: Mass-spring system Since can be rewritten as Integration of this equation yields ( Note: and )

  20. Figure 2.1:A mass-spring system and its phase portrait

  21. Example 2.5: A satellite control system Figure 2.5:Satellite control using on-off thrusters If u = -U, then

  22. - Mathematical model of the satellite is u : torque provided by thrusters : satellite angle. Figure 2.4: satellite control system

  23. - Control law - The thrusters push in the counterclockwise direction if θ is positive, and vice versa.

  24. If u =U, then • Phase trajectories : • where is a constant determined by the initial condition.

  25. Example 2.5: A satellite control system Figure 2.5:Satellite control using on-off thrusters If u = -U, then

  26. Starting from a nonzero initial angle, the satellite will oscillate in periodic motions. • The systems is marginally stable.

  27. II. The Method of Isoclines • An isoclines is defined to be the locus of the points with a given tangent slope. • Two-step phase portrait : 1st step: obtain a field of directions of tangents to the trajectories. 2nd step: phase plane trajectories are formed from the field of directions.

  28. Example: Mass-Spring System • Slope of the trajectories: • Isoclines equation for a slope i.e., a straight line. - Short line segments with slope α.

  29. Figure 2.7: Isoclines for the mass-spring system - Differentα different isoclines short line segments generate a field of tangent directions to trajectories

  30. Example 2.6: The Van der pol Equation - Isoclines of slope :

  31. Points on the curve • same slope α. • Different  different isoclines • Short line segments on the isoclines generate a • field of tangent directions the phase portraits

  32. Figure 2.8:Phase portrait of the Van der Pol equation - Limit cycle The trajectories starting from both outside and inside converge to this curve.

  33. 2.4 Phase Plane Analysis of Linear Systems

  34. Consider the second-order linear system is the general solution Where the eigenvalues and are the solutions of the characteristic equation

  35. There is only one singular point (assuming ), namely the origin. 1. and are both real and have the same sign (positive or negative) 2. and are both real and have opposite signs (saddle point) 3. and are complex conjugate with non-zero real parts 4. and are complex conjugates with real parts equal to zero (center point)

  36. Figure 2.9：Phase-Portraits of linear systems

  37. 2.5 Phase Plane Analysis of Nonlinear Systems Nonlinear systems can display much more complicated patterns in the phase plane, such as multiple equilibrium points and limit cycles

  38. Multiple equilibrium point : • If the singular point of interest is not at the origin : defining the difference between the original state and the singular point as a new set of state variables shift the singular point to the origin.

  39. Using Taylor expansion, Equations (2.1a) and (2.1b) can be rewritten as Where g1 and g2 contain higher order terms. -The higher order terms can be neglected, the nonlinear system trajectories essentially satisfy the linearized equation

  40. The phase portrait of a nonlinear system around a singular point can be obtained like the phase portrait of a linear system.

  41. In the phase plane, a limit cycle is defined as an isolated closed curve. 1.Stable Limit Cycles：all trajectories in the vicinity of the limit cycle converge to it as 2.Unstable Limit Cycles：all trajectories in the vicinity of the limit cycle diverge from it as

  42. 3. Semi-Stable Limit Cycles：some of the trajectories in the vicinity converge to it, while the other diverge from it as t ∞

More Related