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Problem 1: (a) State and prove the final value theorem starting directly from the definition of the Laplace transform. (You may find it useful to prove first the Laplace transform of the derivative and then to use the result to develop your proof. See Section 2.3 in Linear Control Systems – A Neo-Classical Approach)

(b) Using the final value theorem, predict the steady-state value of the step response for the following systems:

i. 1

ii. 1

iii. 1

c Check your result by computing the step response using the inverse Laplace transform for all three cases.

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00:49

In Exercises $1-6,$ use the results of this section to calculate the specified Laplace transforms and state the corresponding inverse transforms. What restrictions on the transform variable $s$ apply? Assume $a$ and $b$ are real constants, $n$ is a nonnegative integer, and $i$ is the imaginary unit $\left(i^{2}=-1\right)$$$L\left\{e^{a t} \cos (b t)\right\}$$

07:15

Show that the Laplace transform is linear, that is, show thatL{af(t) + bg(t)}(s) = aL{f(t)}(s) + bL{g(t)}(s), (3) for s >max{a1, a2}, where a, b are real numbers and f, g are two functionswhose Laplace transform exists for s > a1 and s > a2,respectively. Apply the Laplace transform method, property (3) andL[H(t – c)f(t – c)](s) = e^(-cs)L[f(t)](s), where H(t) = 0, t < 0,1, t â‰¥ 0, to solve the initial value problem y”(t) + 2y'(t) +y(t) = -|t – 1|, y(0) = 0, y'(0) = 0.

01:13

Use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that $a$ and $k$ are positive constants. $\mathscr{L}\left\{e^{\eta}\right\} \quad_{s-1} ; \mathscr{L}\left\{e^{a r}\right\}$

02:08

Use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that $a$ and $k$ are positive constants.$$\mathscr{L}\left\{e^{t}\right\}=\frac{1}{s-1} ; \quad \mathscr{L}\left\{e^{a t}\right\}$$

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