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d u = --sems dt v = f(t)dt I0
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Since f(t) must satisfy the requirements for possession of a transform, it can be shown that the first term on the right, when evaluated at the upper limit of 03, vanishes because of the factor eesr. Furthermore, the lower limit clearly vanishes, and hence, there is no contribution from the first term. The second term may be recognized as sL{&, f (t)dt}, and the theorem follows immediately.
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Example
4.6. Solve the following equation for x(t):
x(t)dt - I
Taking the Laplace theorem
transform of both sides, and making use of the previous sx(s) - 3 = q - L s2
Solving for x(s), x(s) = 3s2 - 1 39 - 1 s(s2 - 1) = s(s -I- l)(s - 1)
This may be expanded into partial fractions according to the usual procedure to give 1 1 x(s) = A + -+s s+1 s - l Hence, x (t) = 1 + e-j + 62 The reader should verify that this function satisfies the original equation.
PROBLEMS
4.1. If a forcing function f(t) has the Laplace transform
f(s) = ; + e-s i2e
graph the function f(t). 4.2. Solve the following equation for y(t):
- e-3s
G(t) I0 y(M7 = dt
Y(O) = 1
4.3. Express the function given in Fig. P4.3 in the t-domain and the s-domain.
FIGURE P4-3
FURTHER
PROPERTIES
TRANSFORMS
4.4. Sketch the following functions: f(t) = u(t) - 2u(t - 1) + u(t - 3) f(t) = 3tu(t) - 3u(t - 1) - u(t - 2) 4.5. The function f(t) has the Laplace transform f(s) = (1 - 2eAS + ee2 j/s2 Obtain the function f(t) and graph f(t). 4.6. Determine f(t) at t = 1.5 and at t = 3 for the following function: f(t) = OSu(t) - OSu(t - 1) + (r - 3)u(r - 2)
THELAPLACETRANSFORM
Example 4.6. Solve the following equation for x(t):
x(t)dt - t
x(0) = 3 Taking the Laplace theorem transform of both sides, and making use of the previous SX(S) - 3 = +) - L s2 Solving for x(s), x(s) = 39 - 1 3s2 - 1 s(s2 - 1) = s(s + I)(s - 1) This may be expanded into partial fractions according to the usual procedure to give 1 1 x(s) = f + -+s+l S--l Hence, i(t) = 1 + eet + et The reader should verify that this function satisfies the original equation.
PROBLEMS
4.1. If a forcing function f(t) has the Laplace transform
f(s) = f + ems ;2ep2s em3
graph the function f(t). 4.2. Solve the following equation for y(t):
y(T)dT = 5 y(0) = 1
4.3. Express the function given in Fig. P4.3 in the t-domain and the s-domain.
FIGURE P4-3
FURTHER PROPERTIES OF TRANSFORMS
Sketch
following
functions: f ( t ) = u(t) - 2u(t - 1) + u(t - 3) f(t) = 3tu(t) - 3u(t - 1) - u(t - 2)
4.5. The function f(t) has the Laplace transform f(s) = (1 - 2emS + e-2s)/s2 Obtain the function f(t) and graph f(t). 4.6. Determine f(t) at t = 1.5 and at t = 3 for the following function: f(t) = 054(t) - OSu(t - 1) + (t - 3)u(t - 2)
PART
LINEAR OPEN-LOOP SYSTEMS
CHAPTER 5 RESPONSEOF FIRST-ORDER SYSTEMS
Before discussing a complete control system, it is necessary to become familiar with the responses of some of the simple, basic systems that often are the building blocks of a control system. This chapter and the three that follow describe in detail the behavior of several basic systems and show that a great variety of physical systems can be represented by a combination of these basic systems. Some of the terms and conventions that have become well established in the field of automatic control will also be introduced. By the end of this part of the book, systems for which a transient must be calculated will be of high-order and require calculations that are time-consuming if done by hand. The reader should start now using Chap. 34 to see how the digital computer can be used to simulate the dynamics of control systems.
TRANSFER
FUNCTION
MERCURY THERMOMETER. We shall develop the transfer function for a Jirst-
order system by considering the unsteady-state behavior of an ordinary mercury-
in-glass thermometer. A cross-sectional view of the bulb is shown in Fig. 5 !l . Consider the thermometer to be located in a flowing stream of fluid for which the temperature x varies with time. Our problem is to calculate the response or the time variation of the thermometer reading y for a particular change in x.*
*In order that the result of the analysis of the thermometer be general and therefore applicable to other first-order systems, the symbols x and y have been selected to represent surrounding temperature and thermometer reading, respectively.
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