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In this chapter, we have introduced the concept of a phase analysis and some of its basic elements. We have seen how physical situations give rise naturally to phase solutions. Furthermore, we have had our first look at true nonlinear behavior. In so doing, we have come to at least one interesting conclusion: a nonlinear motion or control-system response may have more than one steady-state solution. This was true for the chemical reactor and for the pendulum. In contrast, the linear motions and control-system responses we studied in the previous chapters had only one steady-state solution. In the next chapter, we shall discover still more differences which render nonlinear analysis more difficult than linear analysis.
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The advantages of the phase analysis introduced in Chap. 31 can be more fully appreciated after some acquaintance with the tools available for such analysis. To give a detailed exposition of all, or even most, of the aspects of this subject is not intended. Instead, this chapter strives to indicate its flavor and to stimulate further study.
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In 3 1, we considered three examples for which the dynamic response can be described by two state variables. For the cases of the damped oscillator and the pendulum, the state variables were phase variables in which the dependent variable and its derivative (X, X or 8, h) were chosen as the state variables. For the exothermic chemical reactor, the state variables selected were temperature and composition (T, xA); these variables, which arose naturally in the analysis of the chemical reactor, were called physical variables in Chap. 28. In general, an nth-order dynamic system can be described by II state variables. The state variables (x 1, x 2, . . . ,x,) can be located in a coordinate system called phase space. Each value of t, say t t, defines a point in this space: x i(t I), x2(t1), . . . ,x,, (t i). The solution curve is a locus of these points for all values of t. It is called a trajectory and connects successive states of the system. For
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the damped oscillator presented in Chap. 3 1, the coordinate system was a plane with an axis for each state variable; we shall refer to this coordinate system as a phase plane. Figure 3 1.5 is a typical phase-plane representation of a dynamic system. When the physical system is third-order, the coordinate system consists of three axes, one for each state variable. Of course, systems of fourth- or higherorder require treatment in space that is of too many dimensions to be visualized. The graphic aspects of phase-space representation are advantageous primarily in the case of two dimensions (the phase plane) and to a limited extent for three dimensions. The bulk of practical use of phase-space analysis has been made in the two-dimensional autonomous (time invariant) case:
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= fl(Xl,X2)
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dx2 dt
(32.1)
= f2(Xl>X2)
For this reason, we largely confine our attention in the remainder of this study to systems that may be written in the form of Eqs. (32.1). As we have seen, there is no loss in conceptial generality, but we cannot expect the graphical aspects of the material we shall develop to generalize to higher-dimensional phase space. The solution of the system (32.1) may be presented as a family of trajectories in the x 2~ r plane. If we am given the initial conditions
x1(to) X200) = Xl0 = x20
the initial state of the system is the point (~10~20) in the x2x 1 plane and the trajectory may, in principle, be traced from this point. By dividing the second of Eqs. (32.1) by the first, we obtain
f2(XltX2) flbl, x2)
(32.2)
Now dxT/dx 1 is merely the slope of a trajectory, since a trajectory is a plot of x 2 versus xi for the system. Hence, at each point in the phase plane (x i,x2), Eq. (32.2) yields a unique value for the slope of a trajectory through the point, namely, f2(x 1 ,x2)/fr (x r ,x2). This last statement should be amended to exclude any point (XI,X.L) at which fi(xi,x2) and f2(xirx2) are both zero. These important points are called critical points and will be examined in more detail below. Since the slope of the trajectory at a point, say (xi,xz), is by Eq. (32.2) unique, it is clear that trajectories cannot intersect except at a critical point, where the slope is indeterminate.
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