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(31.2)
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It is usually convenient in phase-plane analysis to write the variables in terms of deviation about the final condition. In this case, the spring will ultimately come to rest at Y = 1. Hence we define X = Y - 1 i=jJ Then, Eq. (31.2) becomes
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dX -= x dt di -= dt
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(31.3)
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These are now viewed as two simultaneous, first-order differential equations in the variables X and X. To solve E&s. (3 1.3), we may use the methods presented in Chaps. 28 and 29. For this purpose, let Xr = X and X2 = X. Eqs. (31.3) may be written in the form
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whereX=[i:]
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(31.4)
i $1
Equation (31.4) is in the standard form of a matrix differential equation [Eq. (28.7)]. Notice that the term bu of Eq. (28.7) is not present because no
NONLINEAR CONTROL
forcing term is present in Eqs. (3 1.3): Equation (3 1.4) may be solved by use of Eq. (29.7): X(t) = eA X(0) (29.7) where eAr = L- ((~1 - A)- } (29.13)
Following the usual steps required to solve these equations gives the result Xl = X = Clesl + C2eS2 X2 = X = slCle lr + s2C2es2t where Ci =
c2 s2xo - xo s2 - Sl = xo - SlXO s2 - Sl
(31.5)
and Xc and Xa are the initial conditions; thus Xa = X(O) and Xa = X(O). The terms st and s2 are the roots of the characteristic equation Isl-Al= Expanding this equation gives T2S2 + 25rs + 1 = 0 This quadratic equation has two roots: s12 = -!c* m 7 If we take s2 as the root with the positive sign s2= -r+m 7 the constants take the form 0 (31.6)
2Jm 2Jn
(SZXO
- Xo)
(31.7)
(io - Gfo)
Equations (31.5) and (31.7) together give X(t) and X(t) for all possible initial conditions Xa and Xa. For a given set of initial conditions, we compute C 1 and C2 from (3 1.7)) and then each value of t in Eq. (3 1.5) yields a pair of values for X and X. These may be plotted as a point on an XX diagram (i.e., a phase plane). the locus of these points as t varies from zero to infinity @ll be a curve in the XX plane. As an example, consider the case Xc = - 1, Xa = 0, C < 1. The solution is already known to us in the form of X versus t (Chap. 8) and is replotted in Fig. 3 1.3 for convenience, together with a plot of X
EXAMPLES
OF NONLINEAR SYSTEMS
b=1
dX TW
FIGURE 31-3 Apical motion of second-order system.
versus t. If these curves are replotted as X versus X, with t as a parameter, the result is as shown in Fig. 31.4. The reader should carefully compare Figs. 3 1.3 and 3 1.4 to be satisfied that they ate indeed equivalent. The relationship between the two may be expressed by the statement that Fig. 3 1.3 is a parametric representation of Fig. 31.4. Having only the curve X versus t of Fig. 3 1.3, one can construct Fig. 31.4. To explore the phase-diagram concept further, note that division of the second of Eqs. (31.3) by the first yields
di - x - 2&i -= dX 72i
(31.8)
in which the variable t has been eliminated. Equation (3 1.8) may be recognized as a homogeneous first-order differential equation. Hence, the substitution X = VX yields XdV = -l-w-v=-(l + 25 V + V) TW
an equation which is separable in X and V. This can then be easily solved for V in terms of X. Finally, replacing V = X/X gives the solution for X versus X, or
at origin
FIGURE 31-4 Phase plane corresponding to motion of Fig. 3 1.3.
476 NO NLINEAR
CONTROL
FIGURE 31-5 Interpolation on the phase plane.
the equation for the curve of Fig. 31.4. The algebraic details of this rather tedious process are omitted. (See Graham and McRuer, 1961, pp. 287-289.) The point of the discussion is to emphasize further the equivalence between the description of the motion as X versus t or X versus X. A convenient feature of the phase diagram is that several motions, corresponding to different initial conditions, can be readily plotted on the same diagram. Thus, if we add to Fig. 31.4 a curve for the motion under the initial condition Xc = 1, Xa = 0, we obtain Fig. 31.5. This new trajectory represents the motion of the system after it is stretched 2 units and released from rest. (This follows from the definition X = Y - 1.) Furthermore, we have also interpolated in Fig. 31.5 to obtain the motion corresponding to Xc = 0, Xc = 1. As we shall see later, this interpolation is justified. Hence, it is evident that the phase diagram gives us the big picture of the motion of the underdamped spring-mass-damper system. No matter where the system starts, it spirals to the condition XO = Xc = 0, the steady-state position. This spiral motion in the phase plane corresponds to the oscillatory nature of the X versus t curve of Fig. 31.3. Before beginning a more detailed study of the mechanics of phase analysis, it may be worthwhile to see how situations amenable to such analysis arise naturally in the physical world.
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