# Control Systems/Transforms

## Transforms

There are a number of transforms that we will be discussing throughout
this book, and the reader is assumed to have at least a small prior
knowledge of them. It is not the intention of this book to teach the
topic of transforms to an audience that has had no previous exposure to
them. However, we will include a brief refresher here to refamiliarize
people who maybe cannot remember the topic perfectly. If you do not know
what the **Laplace Transform** or the **Fourier Transform** are yet, it
is highly recommended that you use this page as a simple guide, and look
the information up on other sources. Specifically,
Wikipedia has lots of information on these subjects.

### Transform Basics

A **transform** is a mathematical tool that converts an equation from
one variable (or one set of variables) into a new variable (or a new set
of variables). To do this, the transform must remove all instances of
the first variable, the \"Domain Variable\", and add a new \"Range
Variable\". Integrals are excellent choices for transforms, because the
limits of the definite integral will be substituted into the domain
variable, and all instances of that variable will be removed from the
equation. An integral transform that converts from a domain variable *a*
to a range variable *b* will typically be formatted as such:

$$\mathcal{T}[f(a)] = F(b) = \int_C f(a)g(a,b)da$$

Where the function *f(a)* is the function being transformed, and
*g(a,b)* is known as the **kernel** of the transform. Typically, the
only difference between the various integral transforms is the kernel.

## Laplace Transform

The **Laplace Transform** converts an equation from the time-domain into
the so-called \"S-domain\", or the **Laplace domain**, or even the
\"Complex domain\". These are all different names for the same
mathematical space and they all may be used interchangeably in this book
and in other texts on the subject. The Transform can only be applied
under the following conditions:

1.  The system or signal in question is analog.
2.  The system or signal in question is Linear.
3.  The system or signal in question is Time-Invariant.
4.  The system or signal in question is causal.

The transform is defined as such:

$$\begin{matrix}F(s) = \mathcal{L}[f(t)] = \int_0^\infty f(t) e^{-st}dt\end{matrix}$$

Laplace transform results have been tabulated extensively. More
information on the Laplace transform, including a transform table can be
found in **the
Appendix**.

If we have a linear differential equation in the time domain:

$$\begin{matrix}y(t) = ax(t) + bx'(t) + cx''(t)\end{matrix}$$

With zero initial conditions, we can take the Laplace transform of the
equation as such:

$$\begin{matrix}Y(s) = aX(s) + bsX(s) + cs^2X(s)\end{matrix}$$

And separating, we get:

$$\begin{matrix}Y(s) = X(s)[a + bs + cs^2]\end{matrix}$$

### Inverse Laplace Transform

The **inverse Laplace Transform** is defined as such: {{-}}


$$\begin{matrix}f(t) 
        = \mathcal{L}^{-1} \left\{F(s)\right\}
        = {1 \over {2\pi i}}\int_{c-i\infty}^{c+i\infty} e^{st} F(s)\,ds\end{matrix}$$

The inverse transform converts a function from the Laplace domain back
into the time domain.

### Matrices and Vectors

The Laplace Transform can be used on systems of linear equations in an
intuitive way. Let\'s say that we have a system of linear equations:

$$\begin{matrix}y_1(t) = a_1x_1(t)\end{matrix}$$

$$\begin{matrix}y_2(t) = a_2x_2(t)\end{matrix}$$

We can arrange these equations into matrix form, as shown:

$$\begin{bmatrix}y_1(t) \\ y_2(t)\end{bmatrix} = \begin{bmatrix}a_1 & 0 \\ 0 & a_2\end{bmatrix}\begin{bmatrix}x_1(t) \\x_2(t)\end{bmatrix}$$

And write this symbolically as:

$$\mathbf{y}(t) = A\mathbf{x}(t)$$

We can take the Laplace transform of both sides:

$$\mathcal{L}[\mathbf{y}(t)] = \mathbf{Y}(s) = \mathcal{L}[A\mathbf{x}(t)] = A\mathcal{L}[\mathbf{x}(t)] = A\mathbf{X}(s)$$

Which is the same as taking the transform of each individual equation in
the system of equations.

### Example: RL Circuit

Here, we are going to show a common example of a first-order system, an
**RL Circuit**. In an inductor, the relationship between the current,
*I*, and the voltage, *V*, in the time domain is expressed as a
derivative:

$$V(t) = L\frac{dI(t)}{dt}$$

Where L is a special quantity called the \"Inductance\" that is a
property of inductors.

{RI(t) + L \\frac{dI(t)}{dt}}V\_{in}(t)`</math>`{=html}

This is a very complicated equation, and will be difficult to solve
unless we employ the Laplace transform:

$$V_{out}(s) = \frac{Ls}{R + Ls}V_{in}(s)$$

We can divide top and bottom by L, and move V~in~ to the other side:

$$\frac{V_{out}}{V_{in}} = \frac{s}{\frac{R}{L} + s}$$

And using a simple table look-up, we can solve this for the time-domain
relationship between the circuit input and the circuit output:

$$\frac{V_{out}}{V_{in}} = \frac{d}{dt}e^{\left(\frac{-Rt}{L}\right)}u(t)$$}}

### Partial Fraction Expansion

Laplace transform pairs are extensively tabulated, but frequently we
have transfer functions and other equations that do not have a tabulated
inverse transform. If our equation is a fraction, we can often utilize
**Partial Fraction Expansion** (PFE) to create a set of simpler terms
that will have readily available inverse transforms. This section is
going to give a brief reminder about PFE, for those who have already
learned the topic. This refresher will be in the form of several
examples of the process, as it relates to the Laplace Transform. People
who are unfamiliar with PFE are encouraged to read more about it in
**Calculus**.

### Example: Second-Order System

### Example: Fourth-Order System

\\to \\frac{t\^{n}}{n!}e\^{-\\alpha t} \\cdot u(t) `</math>`{=html}

We can plug in our values for *A*, *B*, *C*, and *D* into our expansion,
and try to convert it into the form above.

$$F(s)=\frac{A}{s}+\frac{B}{(s+10)^3}+\frac{C}{(s+10)^2}+\frac{D}{s+10}$$

$$F(s)=A\frac{1}{s}+B\frac{1}{(s+10)^3}+C\frac{1}{(s+10)^2}+D\frac{1}{s+10}$$

$$F(s)=1\frac{1}{s}+26\frac{1}{(s+10)^3}+69\frac{1}{(s+10)^2}-1\frac{1}{s+10}$$

$$f(t)=u(t)+13t^2e^{-10t}+69te^{-10t}-e^{-10t}$$}}

### Example: Complex Roots

### Example: Sixth-Order System

### Final Value Theorem

The **Final Value Theorem** allows us to determine the value of the time
domain equation, as the time approaches infinity, from the S domain
equation. In Control Engineering, the Final Value Theorem is used most
frequently to determine the steady-state value of a system. The real
part of the poles of the function must be \<0.

$$\lim_{t \to \infty}x(t) = \lim_{s \to 0} s X(s)$$

From our chapter on system metrics, you may recognize the value of the
system at time infinity as the steady-state time of the system. The
difference between the steady state value and the expected output value
we remember as being the steady-state error of the system. Using the
Final Value Theorem, we can find the steady-state value and the
steady-state error of the system in the Complex S domain.

### Example: Final Value Theorem

### Initial Value Theorem

Akin to the final value theorem, the **Initial Value Theorem** allows us
to determine the initial value of the system (the value at time zero)
from the S-Domain Equation. The initial value theorem is used most
frequently to determine the starting conditions, or the \"initial
conditions\" of a system.

$$x(0) = \lim_{s \to \infty} s X(s)$$

### Common Transforms

We will now show you the transforms of the three functions we have
already learned about: The unit step, the unit ramp, and the unit
parabola. The transform of the unit step function is given by:

$$\mathcal{L}[u(t)] = \frac{1}{s}$$

And since the unit ramp is the integral of the unit step, we can
multiply the above result times *1/s* to get the transform of the unit
ramp:

$$\mathcal{L}[r(t)] = \frac{1}{s^2}$$

Again, we can multiply by *1/s* to get the transform of the unit
parabola:

$$\mathcal{L}[p(t)] = \frac{1}{s^3}$$

## Fourier Transform

The **Fourier Transform** is very similar to the Laplace transform. The
fourier transform uses the assumption that any finite time-domain signal
can be broken into an infinite sum of sinusoidal (sine and cosine waves)
signals. Under this assumption, the Fourier Transform converts a
time-domain signal into its frequency-domain representation, as a
function of the radial frequency, ω, The Fourier Transform is defined as
such:

$$F(j\omega) = \mathcal{F}[f(t)] = \int_0^\infty f(t) e^{-j\omega t} dt$$

We can now show that the Fourier Transform is equivalent to the Laplace
transform, when the following condition is true:

$$\begin{matrix}s = j\omega\end{matrix}$$

Because the Laplace and Fourier Transforms are so closely related, it
does not make much sense to use both transforms for all problems. This
book, therefore, will concentrate on the Laplace transform for nearly
all subjects, except those problems that deal directly with frequency
values. For frequency problems, it makes life much easier to use the
Fourier Transform representation.

Like the Laplace Transform, the Fourier Transform has been extensively
tabulated. Properties of the Fourier transform, in addition to a table
of common transforms is available in **the
Appendix**.

### Inverse Fourier Transform

The **inverse Fourier Transform** is defined as follows: {{-}}


$$f(t) 
        = \mathcal{F}^{-1}\left\{F(j\omega)\right\}     
        = \frac{1}{2\pi}\int_{-\infty}^\infty F(j\omega) e^{j\omega t} d\omega$$

This transform is nearly identical to the Fourier Transform.

## Complex Plane

![](S_Plane.svg "S_Plane.svg"){width="200"}

Using the above equivalence, we can show that the Laplace transform is
always equal to the Fourier Transform, if the variable *s* is an
imaginary number. However, the Laplace transform is different if *s* is
a real or a complex variable. As such, we generally define *s* to have
both a real part and an imaginary part, as such:

$$\begin{matrix}s = \sigma + j\omega\end{matrix}$$

And we can show that *s = j*ω if σ*= 0*.

Since the variable *s* can be broken down into 2 independent values, it
is frequently of some value to graph the variable *s* on its own special
\"S-plane\". The S-plane graphs the variable σ on the horizontal axis,
and the value of *j*ω on the vertical axis. This axis arrangement is
shown at right.

{{-}}

## Euler\'s Formula

There is an important result from calculus that is known as **Euler\'s
Formula**, or \"Euler\'s Relation\". This important formula relates the
important values of *e*, *j*, π, 1 and 0:

$$\begin{matrix}e^{j\pi} + 1 = 0\end{matrix}$$

However, this result is derived from the following equation, setting ω
to π:

$$\begin{matrix}e^{j\omega} = \cos(\omega) + j\sin(\omega)\end{matrix}$$

This formula will be used extensively in some of the chapters of this
book, so it is important to become familiar with it now.

## MATLAB

The MATLAB symbolic toolbox contains functions to compute the Laplace
and Fourier transforms automatically. The function **laplace**, and the
function **fourier** can be used to calculate the Laplace and Fourier
transforms of the input functions, respectively. For instance, the code:

`t = sym('t');`\
`fx = 30*t^2 + 20*t;`\
`laplace(fx)`

produces the output:

`ans =`\
\
`60/s^3+20/s^2`

We will discuss these functions more in The
Appendix.

## Further reading

-   Digital Signal Processing/Continuous-Time Fourier
    Transform
-   Signals and Systems/Aperiodic
    Signals
-   Circuit Theory/Laplace
    Transform

