# Control Systems/System Delays

## Delays

A system can be built with an inherent **delay**. Delays are units that
cause a time-shift in the input signal, but that don\'t affect the
signal characteristics. An **ideal delay** is a delay system that
doesn\'t affect the signal characteristics at all, and that delays the
signal for an exact amount of time. Some delays, like processing delays
or transmission delays, are unintentional. Other delays however, such as
synchronization delays, are an integral part of a system. This chapter
will talk about how delays are utilized and represented in the Laplace
Domain. Once we represent a delay in the Laplace domain, it is an easy
matter, through change of variables, to express delays in other domains.

### Ideal Delays

An ideal delay causes the input function to be shifted forward in time
by a certain specified amount of time. Systems with an ideal delay cause
the system output to be delayed by a finite, predetermined amount of
time.

![](Ideal_Delay.svg "Ideal_Delay.svg"){width="400"}

## Time Shifts

Let\'s say that we have a function in time that is time-shifted by a
certain constant time period *T*. For convenience, we will denote this
function as *x(t - T)*. Now, we can show that the Laplace transform of
*x(t - T)* is the following:

$$\mathcal{L}\{x(t - T)\} \Leftrightarrow e^{-sT}X(s)$$

What this demonstrates is that time-shifts in the time-domain become
exponentials in the complex Laplace domain.

### Shifts in the Z-Domain

Since we know the following general relationship between the Z Transform
and the Star Transform:

$$z \Leftrightarrow e^{sT}$$

We can show what a time shift in a discrete time domain becomes in the Z
domain:

$$x((n-n_s)\cdot T)\equiv x[n - n_s] \Leftrightarrow z^{-n_s}X(z)$$

## Delays and Stability

A time-shift in the time domain becomes an exponential increase in the
Laplace domain. This would seem to show that a time shift can have an
effect on the stability of a system, and occasionally can cause a system
to become unstable. We define a new parameter called the **time margin**
as the amount of time that we can shift an input function before the
system becomes unstable. If the system can survive any arbitrary time
shift without going unstable, we say that the time margin of the system
is infinite.

## Delay Margin

When speaking of sinusoidal signals, it doesn\'t make sense to talk
about \"time shifts\", so instead we talk about \"phase shifts\".
Therefore, it is also common to refer to the time margin as the **phase
margin** of the system. The phase margin denotes the amount of phase
shift that we can apply to the system input before the system goes
unstable.

We denote the phase margin for a system with a lowercase Greek letter φ
(phi). Phase margin is defined as such for a second-order system:

$$\phi_m = \tan^{-1} \left[ \frac{2 \zeta}{(\sqrt{4 \zeta^4 + 1} - 2\zeta^2)^{1/2}}\right]$$

Oftentimes, the phase margin is approximated by the following
relationship:

$$\phi_m \approx 100\zeta$$

The Greek letter zeta (ζ) is a quantity called the **damping ratio**,
and we discuss this quantity in more detail in the next chapter.

## Transform-Domain Delays

The ordinary Z-Transform does not account for a system which experiences
an arbitrary time delay, or a processing delay. The Z-Transform can,
however, be modified to account for an arbitrary delay. This new version
of the Z-transform is frequently called the **Modified Z-Transform**,
although in some literature (notably in Wikipedia), it is known as the
**Advanced Z-Transform**.

### Delayed Star Transform

To demonstrate the concept of an ideal delay, we will show how the star
transform responds to a time-shifted input with a specified delay of
time *T*. The function $$X^*(s, \Delta)$$ is the delayed star transform
with a delay parameter Δ. The delayed star transform is defined in terms
of the star transform as such:

$$X^*(s, \Delta)
        = \mathcal{L}^* \left\{ x(t - \Delta) \right\}
        = X(s)e^{-\Delta T s}$$

As we can see, in the star transform, a time-delayed signal is
multiplied by a decaying exponential value in the transform domain.

### Delayed Z-Transform

Since we know that the Star Transform is related to the Z Transform
through the following change of variables:

$$z = e^{-sT}$$

We can interpret the above result to show how the Z Transform responds
to a delay:

$$\mathcal{Z}(x[t - T]) = X(z)z^{-T}$$

This result is expected.

Now that we know how the Z transform responds to time shifts, it is
often useful to generalize this behavior into a form known as the
**Delayed Z-Transform**. The Delayed Z-Transform is a function of two
variables, *z* and Δ, and is defined as such:

$$X(z, \Delta)
        = \mathcal{Z} \left\{ x(t - \Delta) \right\}
        = \mathcal{Z} \left\{ X(s)e^{-\Delta T s} \right\}$$

And finally:

$$\mathcal{Z}(x[n], \Delta) = X(z, \Delta) = \sum_{n=-\infty}^\infty x[n - \Delta]z^{-n}$$

## Modified Z-Transform

The Delayed Z-Transform has some uses, but mathematicians and engineers
have decided that a more useful version of the transform was needed. The
new version of the Z-Transform, which is similar to the Delayed
Z-transform with a change of variables, is known as the **Modified
Z-Transform**. The Modified Z-Transform is defined in terms of the
delayed Z transform as follows:

$$X(z, m) 
        = X(z, \Delta)\big|_{\Delta \to 1 - m} 
        = \mathcal{Z} \left\{ X(s)e^{-\Delta T s} \right\}\big|_{\Delta \to 1 - m}$$

And it is defined explicitly:

$$X(z, m) = \mathcal{Z}(x[n], m) = \sum_{n = -\infty}^{\infty} x[n + m - 1]z^{-n}$$

