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equation
s2+6s+5=0
so that
has a critical point such as that of Fig. 32.2~. Here the trajectories enter the critical point directly, without oscillation. This type of critical point is called a node. For comparison, a typical focus is sketched in Fig. 32.2b. In fact, other types of behavior may be exhibited by critical points of a second-order system, depending on the nature of the roots of the characteristic equation. These are summarized for linear systems in Table 32.1 and sketched in Fig. 32.2. The distinction between
METHODS OF PHASE-PLANE ANALYSIS
FIGURE 32-2 Second-order critical points: (a) stable node, (b) stable focus, (c) unstable focus, (d) unstable node, (e) saddle point.
stable and unstable nodes or foci is made to indicate that the trajectories move toward the stable type of critical point and away from the unstable point. The saddle point arises when the roots of the characteristic equation are real and have opposite sign. In this case there are only two trajectories that enter the critical point, and after entering, the trajectories may leave the critical point (permanently)
TABLE 32.1
Classification
critical
points
Pertinent values of C = = = = = 0 0 0 0 0 i l O<{<l -l<l<O 5-c - 1 All Nature of roots Real Complex Complex Real Real Sign of roots Both Real parts both Real parts both + Both + One +, one -
WPe of
critical point Stable node Stable focus Unstable focus Unstable node Saddle point
Characteristic equation 7212 722 7292 $2 .2s2 + + + + + 2{rs 2575 2lrs 257s 24.7s + + + + 1 1 1 1 1
NONLINEAR CONTROL
on either of two other trajectories. No other trajectory can enter the critical point, although some approach it very closely. This categorization of critical points according to the particular linear system is often of value in the analysis of nonlinear systems. The reason for this is that, in a sufficiently small vicinity of a critical point, a nonlinear system behaves approximately linearly. Thus, the system of Eq. (31.10) for the pendulum is nonlinear. It has two physically distinguishable steady states, corresponding to the pendulum pointing up or down. The nonlinear term sin 8 may be linearized around each steady state. Near the steady state at 8 = 0,
and near the steady state at 8 = r, a Taylor series yields sin8 = -(e - 7~) Therefore, near 8 = 0, Eqs. (31.10) are closely approximated by the linear equations de -= e _ dt (32.4) dl) -= -028 - Dl, dt andnear = n,by
dx . -=x
dt (32.5) di - = 0,2x - Di dt where x = 8 - rr. These linearized versions of Eqs. (3 1.10) can be easily solved to determine the nature of the linear upproximutions to the critical points. Thus, the characteristic equation for Eqs. (32.4) is s*+DS+w* n = 0 while that for Eqs. (32.5) is s*+DS-w; = 0 (32.7) As shown in Table 32.1, Eq. (32.6) yields a stable critical point, which may be a node or focus depending on the degree of damping. (Note that, as the damping is increased, the behavior changes from focus to node, or from oscillatory to nonoscillatory.) On the other hand, Eq. (32.7) indicates a saddle point for the motion near 8 = TT. These conclusions apply strictly only to the linearized phase equations, Eqs. (32.4) and (32.5). To compare them with the behavior of the true system of Eqs. (3 1. lo), the actual phase diagram is sketched for a lightly damped case in Fig. 32.3. For simplicity, this diagram is extended beyond the range 0 9 8 9 2~ even though this is the only region of physical significance. Actually, the section for 0 5 8 5 2n should be cut out and rolled into a cylinder so that the lines (32.6)
METHODS
OF PHASE-PLANE ANALYSIS
corresponding to 8 = 0 and 8 = 27r coincide. This phase cylinder would more realistically represent the motion of the pendulum. As seen from Fig. 32.3, the point at 8 = 7~ is, indeed, a saddle point and the point 0 = 0 (or 27r) is a stable focus. If the system were more heavily damped, this latter point would be a stable node. A greater understanding of the saddle point may now be obtained by analyzing the 8 = T point in terms of what we know to be the physical behavior of the pendulum at this point. That is, the point may be approached from either of two directions. When the pendulum is at the point, an infinitesimal disturbance will cause it to fall in either of two directions. Other trajectories narrowly miss this point, indicating that just the right initial velocity must be imparted to the pendulum at a given initial point to cause it to stop in the 8 = v position. In summary, it can be concluded that in this case the linearized equations give valuable, accurate information about the behavior of the nonlinear system in the vicinity of the critical points. Because the linearized equations are more easily solved, it is always desirable to be able to relate the behavior of the actual system to the behavior of the linearized solutions in the vicinity of the operating point. In fact, in our previous work on control systems, we have assumed for nonlinear systems that design of a stable control system based on the linearized equations was adequate to ensure stable operation of the actual system. The basis for this assumption is given by the following theorem of Liapunov (see Letov, 1961). Let the nonlinear equations of a motion be linearized by expansion in deviation variables around a particular critical point. If the linearized solution for the deviation variables is stable, the actual motion will be stable in some vicinity of the critical point. If the. linearized solution is neutrally stable (i.e., its characteristic equation has roots on the imaginary axis), no statement can be made about the actual motion. If the linearized solution is unstable, then the actual motion will be unstable. It is necessary to define what is meant by stability and instability of the actual nonlinear motion in the vicinity of the critical point. Although stability in nonlinear systems is a complex subject, for our purposes it will suffice to state that a stable nonlinear motion in the vicinity of a critical point is one for which all phase-plane trajectories in this vicinity travel toward and end at the critical point. An unstable motion is one for which trajectories move away from the critical point. This would mean that, while theoretically the state of the system may remain at the critical point indefinitely, any slight disturbance causes the unstable system
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