Chapter 3, Problem 3.11E

Interpretation Introduction

(a)

Interpretation:

Time functions in the given dynamic model of the process where $u\left(t\right)$ is $aS\left(t\right)$ are to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$   ....... (1)

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Laplace transform of higher order derivatives is given by:

$\mathcal{L}\left(\frac{d^{n}f}{d{t}^n}\right)=s^{n}F\left(s\right)-s^{n-1}f\left(0\right)-s^{n-2}{f}^{\left(1\right)}\left(0\right)-\cdots -s{f}^{\left(n-2\right)}\left(0\right)-f^{\left(n-1\right)}\left(0\right)$   ....... (2)

Interpretation Introduction

(b)

Interpretation:

Time functions in the given dynamic model of the process where $u\left(t\right)$ is $b{e}^{-t/\tau }$ are to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$   ....... (1)

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Laplace transform of higher order derivatives is given by:

$\mathcal{L}\left(\frac{d^{n}f}{d{t}^n}\right)=s^{n}F\left(s\right)-s^{n-1}f\left(0\right)-s^{n-2}{f}^{\left(1\right)}\left(0\right)-\cdots -s{f}^{\left(n-2\right)}\left(0\right)-f^{\left(n-1\right)}\left(0\right)$   ....... (2)

Interpretation Introduction

(c)

Interpretation:

Time functions in the given dynamic model of the process where $u\left(t\right)$ is $c{e}^{-t/\tau }$ are to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$   ....... (1)

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Laplace transform of higher order derivatives is given by:

$\mathcal{L}\left(\frac{d^{n}f}{d{t}^n}\right)=s^{n}F\left(s\right)-s^{n-1}f\left(0\right)-s^{n-2}{f}^{\left(1\right)}\left(0\right)-\cdots -s{f}^{\left(n-2\right)}\left(0\right)-f^{\left(n-1\right)}\left(0\right)$   ....... (2)

Interpretation Introduction

(d)

Interpretation:

Time functions in the given dynamic model of the process where $u\left(t\right)$ is $d\sin \omega t$ are to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$   ....... (1)

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Laplace transform of higher order derivatives is given by:

$\mathcal{L}\left(\frac{d^{n}f}{d{t}^n}\right)=s^{n}F\left(s\right)-s^{n-1}f\left(0\right)-s^{n-2}{f}^{\left(1\right)}\left(0\right)-\cdots -s{f}^{\left(n-2\right)}\left(0\right)-f^{\left(n-1\right)}\left(0\right)$   ....... (2)