# Control Systems/Poles and Zeros

## Poles and Zeros

**Poles** and **Zeros** of a transfer function are the frequencies for
which the value of the denominator and numerator of transfer function
becomes infinite and zero respectively. The values of the poles and the
zeros of a system determine whether the system is stable, and how well
the system performs. Control systems, in the most simple sense, can be
designed simply by assigning specific values to the poles and zeros of
the system.

Physically realizable control systems must have a number of poles
greater than the number of zeros. Systems that satisfy this relationship
are called **Proper**. We will elaborate on this below.

## Time-Domain Relationships

Let\'s say that we have a transfer function with 3 poles:

$$H(s) = \frac{a}{(s - l)(s - m)(s - n)}$$

The poles are located at s = **l**, **m**, **n**. Now, we can use
partial fraction expansion to separate out the transfer function:

$$H(s) = \frac{a}{(s - l)(s - m)(s - n)} = \frac{A}{s-l} + \frac{B}{s-m} + \frac{C}{s-n}$$

Using the inverse transform on each of these component fractions
(looking up the transforms in our table), we get the following:

$$h(t) = Ae^{lt}u(t) + Be^{mt}u(t) + Ce^{nt}u(t)$$

But, since s is a complex variable, **l** **m** and **n** can all
potentially be complex numbers, with a real part (σ) and an imaginary
part (jω). If we just look at the first term:

$$Ae^{lt}u(t) = Ae^{(\sigma_l + j\omega_l)t}u(t) = Ae^{\sigma_l t}e^{j\omega_l t}u(t)$$

Using **Euler\'s Equation** on the
imaginary exponent, we get:

$$Ae^{\sigma_l t}[\cos(\omega_l t) + j\sin(\omega_l t)]u(t)$$

If a complex pole is present it is always accompanied by another pole
that is its complex conjugate. The imaginary parts of their time domain
representations thus cancel and we are left with 2 of the same real
parts. Assuming that the complex conjugate pole of the first term is
present, we can take 2 times the real part of this equation and we are
left with our final result:

$$2Ae^{\sigma_l t}\cos(\omega_l t)u(t)$$

We can see from this equation that every pole will have an exponential
part, and a sinusoidal part to its response. We can also go about
constructing some rules:

1.  if σ~l~ = 0, the response of the pole is a perfect sinusoidal (an
    oscillator)
2.  if ω~l~ = 0, the response of the pole is a perfect exponential.
3.  if σ~l~ \< 0, the exponential part of the response will decay
    towards zero.
4.  if σ~l~ \> 0, the exponential part of the response will rise towards
    infinity.

From the last two rules, we can see that all poles of the system must
have negative real parts, and therefore they must all have the form (s +
l) for the system to be stable. We will discuss stability in later
chapters.

## What are Poles and Zeros

Let\'s say we have a transfer function defined as a ratio of two
polynomials:

$$H(s) = {N(s) \over D(s)}$$

Where *N(s)* and *D(s)* are simple polynomials. **Zeros** are the roots
of *N(s)* (the numerator of the transfer function) obtained by setting
*N(s) = 0* and solving for *s*.

**Poles** are the roots of *D(s)* (the denominator of the transfer
function), obtained by setting *D(s) = 0* and solving for *s*. Because
of our restriction above, that a transfer function must not have more
zeros than poles, we can state that the polynomial order of *D(s)* must
be greater than or equal to the polynomial order of *N(s)*.

{{-}}

### Example

## Effects of Poles and Zeros

As *s* approaches a zero, the numerator of the transfer function (and
therefore the transfer function itself) approaches the value 0. When *s*
approaches a pole, the denominator of the transfer function approaches
zero, and the value of the transfer function approaches infinity. An
output value of infinity should raise an alarm bell for people who are
familiar with BIBO stability. We will discuss this later.

As we have seen above, the locations of the poles, and the values of the
real and imaginary parts of the pole determine the response of the
system. Real parts correspond to exponentials, and imaginary parts
correspond to sinusoidal values. Addition of poles to the transfer
function has the effect of pulling the root locus to the right, making
the system less stable. Addition of zeros to the transfer function has
the effect of pulling the root locus to the left, making the system more
stable.

## Second-Order Systems

The canonical form for a second order system is as follows:

$$H(s) = \frac{K\omega^2}{s^2 + 2\zeta\omega s + \omega^2}$$

Where K is the **system gain**, ζ is called the **damping ratio** of the
function, and ω is called the **natural frequency** of the system. ζ and
ω, if exactly known for a second order system, the time responses can be
easily plotted and stability can easily be checked. More information on
second order systems can be found
here.

### Damping Ratio

The **damping ratio** of a second-order system, denoted with the Greek
letter zeta (ζ), is a real number that defines the damping properties of
the system. More damping has the effect of less percent overshoot, and
slower settling time. Damping is the inherent ability of the system to
oppose the oscillatory nature of the system\'s transient response.
Larger values of damping coefficient or damping factor produces
transient responses with lesser oscillatory nature.

### Natural Frequency

The natural frequency is occasionally written with a subscript:

$$\omega \to \omega_n$$

We will omit the subscript when it is clear that we are talking about
the natural frequency, but we will include the subscript when we are
using other values for the variable ω. Also, $\omega~=~\omega_n$ when
$\zeta~=0$.

## Higher-Order Systems

