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barcode reader integration with asp.net dk = dt in Software
dk = dt USS Code 128 Recognizer In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Create Code 128 Code Set C In None Using Barcode printer for Software Control to generate, create Code 128B image in Software applications. Y 2579 +1
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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 IslAl= 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 313 Apical motion of secondorder 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 phasediagram 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 firstorder differential equation. Hence, the substitution X = VX yields XdV = lwv=(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 314 Phase plane corresponding to motion of Fig. 3 1.3.
476 NO NLINEAR
CONTROL
FIGURE 315 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. 287289.) 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 springmassdamper system. No matter where the system starts, it spirals to the condition XO = Xc = 0, the steadystate 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.

