Moving Poles For Desirable Locations

Moving Poles For Desirable Locations - Chapter 4 How the...

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Chapter 4 How the Laplace transform greatly simplifies system representation and manipulation 4.1 Laplace transform techniques Many useful techniques depend on the Laplace transform. The Laplace transform of a function f (t) is denoted sometime s by £ { f (t) } and sometimes by F (s). The inverse Laplace transform of F(s) is denoted sometimes by £-l{F(s)} and sometimes by f(t). Figure 4.1 makes the relation clear; s is a complex variable whose role is defined by eqn. 4.1. 4.2 Definition of the Laplace transform By definition ~ {f(t)} = exp(-st)f(t) dt Examples (1) (4.]) Let f(t) : a constant k, and let R(s) denote the real part of the complex number s fo (k) = exp(-st)kdt = -1/s exp(-st) o = 0 - (-k/s) = k/s provided that R(s) is positive (for otherwise the integral does not exist). -~ F(s)=~{fit)} Figure 4.1 The Laplace transform operation
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30 Control theory (2) Let f(t) : exp(at) £ {exp(at)} = exp(-st) exp(at) dt 1 s)t o 1 (a - s) exp(a - s + a This will be true provided that R(s) > a. The chore of calculating Laplace transforms of particular time functions and the converse problem - calculating the time function, by inverse Laplace transformation, corresponding with a particular Laplace transform - can be avoided by the use of software packages or tables of transform pairs. Small tables are to be found as appen- dices in many introductory control textbooks. A larger set of tables can be found in McCollum and Brown (1965) and a very comprehensive set in Prudnikov et al. (1992). 4A Convergence of the integral that defines the Laplace transform It is quite typical, as in the last example, for the integral that defines the Laplace transform to be finite (and hence defined), only for restricted values of s. However, there seems to be a tacit agreement in the teaching of control theory to avoid any discussion of the distracting question: what is the significance of the region of convergence of the integral that defines the Laplace transform ? For example, let a = 2 in the transform 1 / ( s + a ) that we have just derived. Then it is clear that the transform is only defined and valid in the shaded region in Figure 4.2 where the real part of s is strictly greater than 2. However, later in this chapter, we shall see that, for this transform, the value of s for which s + a = 0 is highly significant i:/ y:i/ /[ /2t%: ,Z-2 yZ // Figure 4.2 The transform 1/(s + 2) is only defined in the shaded region, yet the point s -- (-2, O) is the one of interest and the transform is universally used at that point without further question
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The Laplace transform 31 (i.el thepoint s = (-2,0)). We, in common with the whole control fraternity, blithely use the transform at the point s = (-2, 0) where it is undefined. Notice also that the re~ion in which the integral converges may be empty. For example, the function exp(t ~) has no Laplace transform for this reason. 4B Problems with 0- and 0 + (i) Anyone who has used Laplace transforms to solve differential equations will be used to obtaining solutions such as y(t) = y(O) exp('t) where by y(O) is meant y (0 +) which has to be calculated independently. One is expected to know y(O+), but y(O +) is really part of the solution that is to be determined. Clearly
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Moving Poles For Desirable Locations - Chapter 4 How the...

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