+++
title = "System Identification"
author = ["Dehaeze Thomas"]
draft = false
+++

Tags
: [Modal Analysis]({{< relref "modal_analysis.md" >}})


## SISO Identification {#siso-identification}


### Problem Description {#problem-description}

<a id="figure--fig:siso-identification-schematic"></a>

{{< figure src="/ox-hugo/siso_identification_schematic.png" caption="<span class=\"figure-number\">Figure 1: </span>Block diagram of the SISO system identification" >}}

<a id="figure--fig:siso-identification-schematic-simplier"></a>

{{< figure src="/ox-hugo/siso_identification_schematic_simplier.png" caption="<span class=\"figure-number\">Figure 2: </span>Simpler Block diagram of the SISO system identification" >}}

<div class="note">

If the open-loop system is unstable, first design a simple controller that stabilizes the system and then identify the closed-loop system.

</div>


### Design of the Excitation Signal {#design-of-the-excitation-signal}


#### Introduction {#introduction}

There are several choices for excitation signals:

-   Impulse, Steps
-   Sweep Sinus
-   Random noise, Periodic signals


#### Random noise with specific ASD {#random-noise-with-specific-asd}

The ASD of the measured output is:

\begin{equation}
\Gamma\_{y\_m}(\omega) = \Gamma\_d(\omega) + \Gamma\_u(\omega) \cdot |G(j\omega)|
\end{equation}

And we want the effect of the excitation signal to be much higher than the effect of the exogenous signals (measurement noise, input noise, disturbances).

\begin{equation}
\Gamma\_u(\omega) \gg \Gamma\_d(\omega) \cdot |G(j\omega)|^{-1}
\end{equation}

Note that \\(\Gamma\_d(\omega)\\) can be estimated by measuring the system output in the absence of any excitation signal.
The plant magnitude \\(|G(j\omega)|\\) can be roughly estimated from a first identification with bad coherence.

In order to design a random excitation signal with specific spectral characteristics, first a signal with an ASD equal to one is generated (i.e. white noise with unity ASD):

```matlab
Ts = 1e-4; % Sampling Time [s]
t = 0:Ts:10; % Time Vector [s]

%% Signal with an ASD equal to one
u_norm = sqrt(1/2/Ts)*randn(length(t), 1);
```

Then, a transfer function whose magnitude \\(|G\_u(j\omega)|\\) has the same shape as the wanted excitation ASD \\(\Gamma\_u(\omega)\\) is designed:

```matlab
%% Transfer function representing the wanted ASD
G_u = tf([1], [1/2/pi/100 1]);
```

Finally `lsim` is used to compute the shaped excitation signal.

```matlab
%% Shape the ASD of the excitation signal
u = lsim(G_u, u_norm, t);
```


#### Choose Sampling Frequency and Duration of Excitation {#choose-sampling-frequency-and-duration-of-excitation}

<div class="important">

The sampling frequency \\(F\_s\\) will determine the maximum frequency \\(F\_{\text{max}}\\) that can be estimated (see Nyquist theorem):

\begin{equation}
F\_{\text{max}} = \frac{1}{2} F\_s
\end{equation}

</div>

<div class="important">

The duration of excitation \\(T\_{\text{exc}}\\) will determine the minimum frequency \\(F\_{\text{min}}\\) that can be estimated:

\begin{equation}
F\_{\text{min}} = \frac{1}{T\_{\text{exc}}}
\end{equation}

It will also corresponds to the frequency resolution \\(\Delta f\\):

\begin{equation}
\Delta f = \frac{1}{T\_{\text{exc}}}
\end{equation}

</div>

In order to increase the estimation quality, averaging can be use with a longer excitation duration.
A factor 10 is usually good enough, therefore the excitation time can be taken as:

\begin{equation}
T\_{\text{exc}} \approx \frac{10}{F\_{\text{min}}}
\end{equation}

<div class="exampl">

Therefore, if the system has to be identified from 1Hz up to 500Hz, the sampling frequency should be:

\begin{equation}
F\_s = 2 F\_{\text{max}} = 1\\,\text{kHz}
\end{equation}

Then, the excitation duration should be (10 averaging):

\begin{equation}
T\_{\text{exc}} = \frac{10}{1} = 10\\,s
\end{equation}

</div>


### Computation of the Frequency Response Function {#computation-of-the-frequency-response-function}


#### Windowing Function {#windowing-function}


#### Example {#example}

`tfestimate`

```matlab
[G, f] = tfestimate(u, y, win, [], [], 1/Ts);
```


### Verification of the Identification Quality {#verification-of-the-identification-quality}

`mscohere`

```matlab
[coh, f] = mscohere(u, y, win, [], [], 1/Ts);
```


## Reference Books {#reference-books}

-   (<a href="#citeproc_bib_item_1">Pintelon and Schoukens 2012</a>)
-   (<a href="#citeproc_bib_item_2">Schoukens, Pintelon, and Rolain 2012</a>)


## Bibliography {#bibliography}

<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Pintelon, Rik, and Johan Schoukens. 2012. <i>System Identification : a Frequency Domain Approach</i>. Hoboken, N.J. Piscataway, NJ: Wiley IEEE Press. doi:<a href="https://doi.org/10.1002/9781118287422">10.1002/9781118287422</a>.</div>
  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Schoukens, Johan, Rik Pintelon, and Yves Rolain. 2012. <i>Mastering System Identification in 100 Exercises</i>. John Wiley &#38; Sons.</div>
</div>