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FIGURE 3-l Location of typical roots of characteristic equation.
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of the characteristic equation and the roots of the denominator of the transformed forcing function. Consider Fig. 3.1, a drawing of the complex plane, in which several typical roots are located and labeled with their coordinates. lhble 3.1 gives the form of the terms in the equation for x(f), corresponding to these roots. Note that all constants, a 1, ~2, . . . , bl, b2, . . . , are taken as positive. The constants Cl and C2 am arbitrary and can be determined by the partial-fraction expansion techniques. As discussed above, this determination is often not necessary for our work. If any of these roots am repeated, the term given in Table 3.1 is multiplied by a power series in t, K1 + K2t + Kg2 + ..a + K/--l where r is the number of repetitions of the root and the constants Kl, K2, . . . , K, can be evaluated by partial-fraction expansion. It is thus evident that the imaginary axis divides the root locations into distinct areas, with regard to the behavior of the corresponding terms in x(t) as t becomes large. Terms corresponding to roots to the left of the imaginary axis vanish exponentially in time, while those corresponding to roots to the right of
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TABLE 3.1 Roots
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e-@(C1 cos bzt + C2 sin bzt) Cl cos bgt + C2 eQ (C1 cos bqt + C2
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INVERSION BY PARTIAL FRACTIONS
the imaginary axis increase exponentially in time. Terms corresponding to roots at the origin behave as power series in time, a constant being considered as a degenerate power series. Terms corresponding to roots located elsewhere on the imaginary axis oscillate with constant amplitude in time unless they are multiple roots in which case the amplitude of oscillation increases as a power series in time. Much use will be made of this information in later sections of the text.
SUMMARY
The reader now has available the basic tools for the use of Laplace transforms to
solve differential equations. In addition, it is now possible to obtain considerable
information about the qualitative nature of the solution with a minimum of labor.
It should be pointed out that it is always necessary to factor the denominator of x(s) in order to obtain any information about x(t). If this denominator is a polynomial of order three or more, this may be far from a trivial problem.
15 is largely devoted to a solution of this problem within the context of control
applications.
The next chapter is a grouping of several Laplace transform theorems that
will find later application. In addition, a discussion of the impulse function 8(t)
is presented there. Unavoidably, this chapter is rather dry. It may be desirable for
the reader to skip directly to Chap. 5, where our control studies begin. At each
point where a theorem of Chap. 4 is applied, reference to the appropriate section
of Chap. 4 can be made.
PROBLEMS
3.1. Solve the following using Laplace
(a) (b) (c) ~+~+.=1
transforms:
x(0) = x (0) = 0 x(0) = x (0) = 0 x(0) = x (0) = 0
$+%+m $.~.,=I
Sketch the behavior of these solutions on a single graph. What is the effect of the coefficient of dxldt 3.2. Solve the following differential equations by Laplace transforms:
d4x d3x (a) -$ + -j-p- = cos t ( b ) $+s = t2+2t x(0) = x (0) = x (0) = 0 q(O) = 4 q (0) = -2
x (0) = 1
Invert
following
transforms:
@) (s2 + l)(s2 + 4)
THE LAPLACE
TRANSFORM
@I s(s2 - 2s + 5)
(4 3s3 - s2 -3s+2 sqs - I)2 expansion. Do not evaluate coef-
3.4. Expand the following functions by pm+fraction ficients or invert expressions.
(4 X(s) = (s + l)(s2 + 1)2(s + 3) (b) X(s) = s3(s + l)(s + 2)(s + 3)3 Q-3 X(s) = (s + l)(s + 2)(s + 3)(X + 4)
3.5. (a) Invert: x(s) = l/[s(s + 1)(0.5s + l)] (b) Solve: dxldt + 2x = 2, x(O) = 0 3.6. Obtain y(t) for 1 1
(4 Y(S) = , 2 1 :,l+ 5
(b) y(s) = + (cl Y(S) = &
3.7. (a) Invert the following function y(s) = l/(s2 + 1)2
(b) Plot y versus t from 0 to 37r. 3.8. Determine f(t) for f(s) = l/[s2(s + l)].
CHAPTER
FURTHER PROPERTIES OFTRANSFORMS
This chapter is a collection of theorems and results relative to the Laplace transformation. The theorems are selected because of their applicability to problems in control theory. Other theorems and properties of the Laplace transformation are available in standard texts [see Churchill (1972)]. In later chapters, the theorems presented here will be used as needed.
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