The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y(t) to the Laplace transform of the input function g(t), when all initial conditions are zero. If a linear system is governed by the differential equation below, Y(s) for this system. use the linearity property of the Laplace transform and Theorem 5 to determine the transfer function H(s) = G(s) y'' (t) + 8y' (t) +9y(t) = g(t). t> 0 Click here to view Theorem 5 H(s) = Let f(t) f'(t)... -1) (t) be continuous on [0,∞) and let f(n) (t) be piecewise continous on [0,00), with all these functions of exponential order a. Then for s> a, the following equation holds true. L {f{")} (s) = s^L{f}(s) — s"~1¹f(0) -sª-²f'(0) --- (-¹) (0)
The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y(t) to the Laplace transform of the input function g(t), when all initial conditions are zero. If a linear system is governed by the differential equation below, Y(s) for this system. use the linearity property of the Laplace transform and Theorem 5 to determine the transfer function H(s) = G(s) y'' (t) + 8y' (t) +9y(t) = g(t). t> 0 Click here to view Theorem 5 H(s) = Let f(t) f'(t)... -1) (t) be continuous on [0,∞) and let f(n) (t) be piecewise continous on [0,00), with all these functions of exponential order a. Then for s> a, the following equation holds true. L {f{")} (s) = s^L{f}(s) — s"~1¹f(0) -sª-²f'(0) --- (-¹) (0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**The Transfer Function of a Linear System**
The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function \( y(t) \) to the Laplace transform of the input function \( g(t) \), given all initial conditions are zero. For a linear system governed by the differential equation below, we use the linearity property of the Laplace transform and Theorem 5 to determine the transfer function \( H(s) = \frac{Y(s)}{G(s)} \) for this system.
**Differential Equation:**
\[ y''(t) + 8y'(t) + 9y(t) = g(t), \quad t > 0 \]
[Link to Theorem 5]
**Theorem Context:**
Let \( f(t), \ldots, f^{(n-1)}(t) \) be continuous on \([0, \infty)\) and let \( f^{(n)}(t) \) be piecewise continuous on \([0, \infty)\), with all these functions of exponential order \(\alpha\). Then for \( s > \alpha \), the following equation holds true:
\[
\mathcal{L}\{f^{(n)}\}(s) = s^n \mathcal{L}\{f(t)\} - s^{n-1} f(0) - s^{n-2} f'(0) - \cdots - f^{(n-1)}(0)
\]
**Diagram Explanation:**
The image includes a formula for \( H(s) \) with an empty box indicating the position for the expression. It is labeled as \( \frac{1}{s} h(p) \) with an arrow pointing to the empty box. This denotes substituting a specific expression into the indicated location after applying the Laplace Transform to solve for the transfer function \( H(s) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1720cd77-2294-45a6-8b70-43447dc02920%2Ff95f1e95-ff6b-4dca-b773-949a266fd939%2Fyhukobd_processed.png&w=3840&q=75)
Transcribed Image Text:**The Transfer Function of a Linear System**
The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function \( y(t) \) to the Laplace transform of the input function \( g(t) \), given all initial conditions are zero. For a linear system governed by the differential equation below, we use the linearity property of the Laplace transform and Theorem 5 to determine the transfer function \( H(s) = \frac{Y(s)}{G(s)} \) for this system.
**Differential Equation:**
\[ y''(t) + 8y'(t) + 9y(t) = g(t), \quad t > 0 \]
[Link to Theorem 5]
**Theorem Context:**
Let \( f(t), \ldots, f^{(n-1)}(t) \) be continuous on \([0, \infty)\) and let \( f^{(n)}(t) \) be piecewise continuous on \([0, \infty)\), with all these functions of exponential order \(\alpha\). Then for \( s > \alpha \), the following equation holds true:
\[
\mathcal{L}\{f^{(n)}\}(s) = s^n \mathcal{L}\{f(t)\} - s^{n-1} f(0) - s^{n-2} f'(0) - \cdots - f^{(n-1)}(0)
\]
**Diagram Explanation:**
The image includes a formula for \( H(s) \) with an empty box indicating the position for the expression. It is labeled as \( \frac{1}{s} h(p) \) with an arrow pointing to the empty box. This denotes substituting a specific expression into the indicated location after applying the Laplace Transform to solve for the transfer function \( H(s) \).
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