# Analysis of the temporal response of a system - Análisis de la respuesta temporal de un sistema

## Deterministic test signals

### System response to a step type input

An input signal of the step type ( en: Step response ) allows to know the response of the system to abrupt changes in its input. Likewise, it gives us an idea of ​​the establishment time of the signal, that is, how long it takes for the system to reach its steady state. Another characteristic of this signal is that, as a result of the discontinuity of the jump, it contains a frequency spectrum in a wide band, which makes it equivalent to applying a large number of sinusoidal signals with a large frequency range to the system. Mathematically, this signal is expressed as:${\displaystyle r(t)=A\cdot u(t)}$. Where${\displaystyle u(t)}$: unit step; ${\displaystyle A}$: constant

In the figure below, the step begins at time t = 1 (not at t = 0), ${\displaystyle A=3}$ Y ${\displaystyle u(t)={\begin{cases}0,&{\mbox{si }}t<1\\1,&{\mbox{si }}t\geq 1\end{cases}}}$

### System response to a ramp-type input

This signal allows us to know what is the response of the system to input signals that change linearly with time. Mathematically it is represented as:${\displaystyle r(t)=A\cdot t\cdot u(t)}$. Where${\displaystyle t}$:weather; ${\displaystyle A}$: constant

### System response to a unit impulse input

The response of the system to an input of the unit impulse type allows to have an idea about the intrinsic behavior of the system. The mathematical representation of the unit impulse function, or Dirac's Delta is:${\displaystyle u(t)={\begin{cases}0,&{\mbox{si }}t<0\\\infty ,&{\mbox{si }}t=0\\0,&{\mbox{si }}t>0\end{cases}}}$

A system is represented mathematically through its transfer function . In the laplace plane the mathematical expression that represents it is:${\displaystyle G(s)={\frac {Y(s)}{U(s)}}}$; where${\displaystyle Y(s):}$ system exit and${\displaystyle U(s):}$ system input . The Laplace transform of the unit impulse is unity; that is to say,${\displaystyle U(s)=1}$ Therefore, the output signal has as a Laplace transform the transfer function of the process ${\displaystyle G(s)=Y(s).}$ From this, it follows that the impulse response and the transfer function contain the same information.

## Temporal response

### First order system without delay

A first order system can be modeled by the following ordinary differential equation.

${\displaystyle {\dot {y}}(t)+a_{0}y(t)=b_{0}u(t)}$

To calculate its transfer function we apply the laplace transform and consider the initial null condition

${\displaystyle L\left\{{\dot {y}}(t)+a_{0}y(t)\right\}=L\left\{b_{0}u(t)\right\}}$

Once the transform has been applied to each of the terms of the differential equation we have:

${\displaystyle sY(s)-y(0)+a_{0}Y(s)=b_{0}U(s)}$

Factoring and writing as a transfer function.

${\displaystyle Y(s)[s+a_{0}]=b_{0}U(s)}$
${\displaystyle F(s)={\frac {Y(s)}{U(s)}}={\frac {b_{0}}{s+a_{0}}}}$

The transfer function can also be written as follows

${\displaystyle F(s)={\frac {k}{\tau s+1}}}$

${\displaystyle k={\frac {b_{0}}{a_{0}}}}$, ${\displaystyle \tau ={\frac {1}{a_{0}}}}$ The constant ${\displaystyle k}$is the steady state gain, which gives us the value that the response of the system takes for a time tending to infinity. The constant${\displaystyle \tau }$is the time constant, which will indicate the time in which the system has 63.21% of the value in steady state. It can be seen that this type of system has a pole that is${\displaystyle =-a_{0}}$

### First order system with delay

The easiest way to analyze the temporal response of a system, even though it seems paradoxical, is through its transfer function, which for a first-order system with a delay is the following:

${\displaystyle G(s)={\frac {ke^{-Ls}}{\tau s+1}}}$

Where L is the time it takes for the system to react from the moment the step is applied.

### Second order system

A second order system has the following equation as a transfer function:

${\displaystyle {\frac {k\omega _{n}^{2}}{s^{2}+2\xi \omega _{n}s+\omega _{n}^{2}}}}$

Where: ${\displaystyle \omega _{n}:}$ natural frequency of oscillation,${\displaystyle \xi :}$damping coefficient and ${\displaystyle k:}$ steady state gain.

The steady state gain corresponds to the constant value that the system takes for a very long time. It can be calculated through the final theorem of the limit of the transfer function ${\displaystyle F(s)}$.

${\displaystyle y_{ee}=\lim _{t\to +\infty }y(t)=\lim _{s\to 0}sF(s)=F(0)}$

The response of the system depends on the roots of the denominator (poles of the system). For a second order system the poles are expressed as:

${\displaystyle s_{1,2}=-\xi \omega _{n}+\omega _{n}{\sqrt {1-\xi ^{2}}}}$

Depending on the value ${\displaystyle \xi }$, second order systems present different behaviors.

As seen in the figure when ${\displaystyle \xi =0}$(blue color curve) the oscillations will continue indefinitely. For values ​​greater than${\displaystyle \xi }$ a faster decay of the oscillations is obtained, but with a slower rise of the response (The curve in green has a value ${\displaystyle \xi =0.1}$, while for the red ${\displaystyle \xi =0.5}$. In the case where${\displaystyle \xi =1}$, the system becomes critically damped to the point that the oscillations disappear (See pink curve).

### Specifications in the time domain of second order systems

In control theory, the characterization of the temporal response of a system is usually done using what are known as system specifications. The most used specifications are:

• Delay time. It is the time required for the response to reach 50% of the final value.
• Rise time. It is the time required for the answer to go from 10 to 90% of the final value. It can also be defined as the step time from 5% to 95% or from 0% to 100%.
• Peak time: It is the time that passes until the first overshoot peak is reached.
• Overshoot: It is the maximum peak value per unit. It is usually expressed as a percentage.
• Establishment time: It is the time necessary for the response of the system to be within a percentage (about 5%, although it varies according to the author) of the final value.