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

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


## System Identification {#system-identification}


### Problem Definition {#problem-definition}

The goal of a system identification is to extract a model (usually a LTI transfer function) from experimental data.
The system is represented in Figure <fig:siso_identification_schematic_simplier> with one input \\(u\\) and one output \\(y\_m\\) affected by some disturbances and noise \\(d\\).

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

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

The goal of system identification is to compute the transfer function \\(G\\) from known excitation signal \\(u\\) and from a measure of \\(y\_m\\).

There are different ways of obtaining a model \\(G\\) of the system summarized in Table <tab:system_identification_model_types>.
For mechatronics systems, _frequency domain identification_ is the norm.

<a id="table--tab:system-identification-model-types"></a>
<div class="table-caption">
  <span class="table-number"><a href="#table--tab:system-identification-model-types">Table 1</a>:</span>
  Different ways to obtain a model
</div>

|                                     | **Advantages**                                          | **Disadvantages**                               |
|-------------------------------------|---------------------------------------------------------|-------------------------------------------------|
| Physical Modeling                   | Understanding dynamics, no measurements                 | Low accuracy, time consuming                    |
| Time domain identification          | Parametric models, small data sets, non-linearity, MIMO | order/structure selection can be time consuming |
| **Frequency domain identification** | easy, intuitive, user-defined model fitting             | fitting MIMO parametric model can be difficult  |

As an example, a second order plant will be used:

```matlab
G = 1/(1 + 0.1*s/(2*pi*40) + s^2/(2*pi*40)^2); % Plant model
```

The sampling time of the recorded digital signal is 1ms.


### Open-loop identification {#open-loop-identification}

In open-loop identification (Figure <fig:siso_identification_schematic_simplier_open_loop>), a test signal \\(u\\) is used to _excite_ the system in the frequency range of interest.
The signal \\(u\\) can typically be a swept sine, noise or multi-sine.

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

{{< figure src="/ox-hugo/siso_identification_schematic_simplier_open_loop.png" caption="<span class=\"figure-number\">Figure 2: </span>Open Loop signal identification" >}}

The plant estimate \\(\hat{G}(j\omega)\\) is computed as follows:
\\[ \hat{G}(j\omega) = \frac{\Phi\_{yu}(\omega)}{\Phi\_{u}(\omega)} \\]
with \\(\phi\_{u}(\omega)\\) is the spectrum of \\(u\\) and \\(\Phi\_{yu}(\omega)\\) is the cross-spectrum of \\(u\\) and \\(y\\).

In Matlab, the transfer function can be extracted from the data using:

```matlab
Nfft = floor(1/Ts); % Size of the window [-]
win = hanning(Nfft); % Hanning window
Noverlap = floor(Nfft/2); % Overlap of 50%

% Identified Transfer Functions
[Gm, f] = tfestimate(data.du, data.y, win, Noverlap, Nfft, 1/Ts);
```

Then, the bode plot of the obtained transfer function is compared against the plant model including a 1.5 sample time delay (Figure <fig:system_identification_ol_comp_plant>).

<a id="figure--fig:system-identification-ol-comp-plant"></a>

{{< figure src="/ox-hugo/system_identification_ol_comp_plant.png" caption="<span class=\"figure-number\">Figure 3: </span>Comparison of the plant transfer function with the identified FRF in Open-Loop" >}}


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

In order to assess the quality of the obtained FRF, the _coherence_ can be computed using the `mscohere` Matlab function.

```matlab
[coh, f] = mscohere(data.du, data.y, win, Noverlap, Nfft, 1/Ts);
```

The result for the example is shown in Figure <fig:system_identification_ol_coh>.
At high frequency, the measurement noise dominates and the coherence is poor.

<a id="figure--fig:system-identification-ol-coh"></a>

{{< figure src="/ox-hugo/system_identification_ol_coh.png" caption="<span class=\"figure-number\">Figure 4: </span>Comparison of the plant transfer function with the identified FRF in Open-Loop" >}}


### Closed-Loop identification {#closed-loop-identification}

If the open-loop system is unstable, a first simple controller needs to be designed to stabilizes the system.
Then, the plant can be identified from closed-loop system identification (Figure <fig:siso_identification_schematic_simplier_closed_loop>).

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

{{< figure src="/ox-hugo/siso_identification_schematic_simplier_closed_loop.png" caption="<span class=\"figure-number\">Figure 5: </span>Closed Loop signal identification" >}}

To perform plant identification in closed-loop:

-   Chose \\(d\_u\\) (sweep sine, multi-sine, noise, ...)
-   Measure \\(u\\) and \\(y\\)
-   Compute the process sensitivity:
    \\[ GS(j\omega) = \frac{\Phi\_{yd\_u}(\omega)}{\Phi\_{d\_u}(\omega)} \\]
-   Compute the Sensitivity:
    \\[ S(j\omega) = \frac{\Phi\_{ud\_u}(\omega)}{\Phi\_{d\_u}(\omega)} \\]
-   Then the plan estimate \\(\hat{G}(j\omega)\\) can be computed:
    \\[ \hat{G}(j\omega) = \frac{GS(j\omega)}{S(j\omega)} \\]

In Matlab, this can be done with the following code:

```matlab
Nfft = floor(1/Ts); % Size of the window [-]
win = hanning(Nfft); % Hanning window
Noverlap = floor(Nfft/2); % Overlap of 50%

[S,  f] = tfestimate(data.du, data.u, win, Noverlap, Nfft, 1/Ts);
[GS, ~] = tfestimate(data.du, data.y, win, Noverlap, Nfft, 1/Ts);

Gm = GS ./ S;
K = (1 ./ S - 1) ./ Gm;
T = 1 - S;
```


### Multi-Input Multi-Output Plant {#multi-input-multi-output-plant}

This can be generalized to a MIMO plant (Figure <fig:siso_identification_schematic_simplier_closed_loop>).

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

{{< figure src="/ox-hugo/mimo_identification_schematic_simplier_closed_loop.png" caption="<span class=\"figure-number\">Figure 6: </span>Closed Loop signal identification (MIMO case)" >}}

Suppose a plan with \\(m\\) inputs and \\(n\\) outputs.
\\(\bm{G}\\) is therefore a \\(n \times m\\) plant, and the controller \\(\bm{K}\\) an \\(m \times n\\) system.
Input sensitivity is an \\(m \times m\\) system:
\\[ \bm{S}\_i = (\bm{I}\_{m} + \bm{K}\bm{G})^{-1} \\]
And process sensitivity is an \\(m \times m\\) system:
\\[ \bm{GS} = \bm{G} (I\_m + KG)^{-1} \\]

And the \\(n \times m\\) plant can be computed using:
\\[ GS \cdot S^{-1} = \bm{G} (I\_m + KG)^{-1} (\bm{I}\_{m} + \bm{K}\bm{G}) = G \\]

To estimate the full plant, \\(m\\) separate identification needs to be performed (one for each input):

-   Chose the excitation signal for the \\(i^{th}\\) input: \\(d\_{ui}\\) (sweep sine, multi-sine, noise, ...)
-   Measure \\(u\_i\\) and \\(\bm{y}\\)
-   Compute the \\(i^{th}\\) column of the process sensitivity (\\(j\\) from \\(1\\) to \\(n\\)):
    \\[ GS\_{ij}(j\omega) = \frac{\Phi\_{y\_jd\_{ui}}(\omega)}{\Phi\_{d\_{ui}}(\omega)} \\]
-   Compute the \\(i^{th}\\) column of the input sensitivity (\\(j\\) from \\(1\\) to \\(n\\)):
    \\[ S\_{ij}(j\omega) = \frac{\Phi\_{u\_jd\_{ui}}(\omega)}{\Phi\_{d\_{ui}}(\omega)} \\]

When the complete \\(GS\\) and \\(S\\) matrices are identified, the plan estimate \\(\hat{\bm{G}}(j\omega)\\) can be computed:
\\[ \hat{\bm{G}}(j\omega) = \bm{GS}(j\omega) \bm{S\_i}^{-1}(j\omega) \\]


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

<span class="org-target" id="org-target--sec-system-identification-excitation-signal"></span>


### Introduction {#introduction}

There are several choices for excitation signals:

-   Impulse, Steps
-   Sweep Sinus
-   Random noise, Periodic signals (PRBS)
-   Multi-Sine


### 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>


### Multi-Sine {#multi-sine}


### `generatemultisine` - Matlab Function {#generatemultisine-matlab-function}

```matlab
function y = generate_multisine(Fs, Ns, args)
%% Input parsing
    arguments
        Fs % Sampling frequency of the generated test signal
        Ns % Length of the generated test signal
        args.asd   double {mustBeNumeric, mustBeNonnegative} = 0
        args.type  char   {mustBeMember(args.type,{'schroeder', 'normal'})} = 'schroeder'
    end

    %% Find amplitude response
    if args.asd == 0
        % If magnitude response is zero, set default to "unitary ASD"
        mag = (ones(1, Ns)*2*sqrt(Fs)/sqrt(Ns)).^2;
        mag(round(Ns/2+1):end) = 0;
    else
        if length(args.asd) ~= Ns
            error('ASD must be of length Ns');
        end

        mag = (args.asd*2*sqrt(Fs)/sqrt(Ns)).^2;
    end

    if any(mag(round(Ns/2+1):end))
        warning('Non-zero magnitude values present outside of frequency limits.');
        mag(round(Ns/2+1):end) = 0;
    end

    %% Find phase response
    switch args.type
      case 'schroeder'
        phase = schroederPhases(Ns, mag);
      case 'normal'
        phase = randn(1, Ns);
      otherwise
        error('type must be "Schroeder" or "normal"');
    end

    %% Generate multisine signal
    % Frequency domain representation
    Y = sqrt(mag/2).*exp(sqrt(-1).*phase);

    % IFFT to time domain
    y = ifft(forceFFTSymmetry(Y))*(Ns/2);

    %% Find zero crossing closest to zero
    % Heuristic method of finding minimum gradient zero crossing
    ySign = y>0;
    zeroInds = find((ySign(2:end) ~= ySign(1:end-1)));

    % Find the index with the smallest gradient around the zero crossing.
    zeroGrad = zeros(1,length(zeroInds));
    for nn=1:length(zeroInds)
        zeroGrad(nn) = abs(y(zeroInds(nn)) - y(zeroInds(nn)+1));
    end
    [~, minInd] = min(zeroGrad);

    yWrapped = [y(zeroInds(minInd):end) y(1:zeroInds(minInd)-1)];
    y = yWrapped;
end

%% Generate phases as defined in "Synthesis of Low-Peak-Factor Signals and
% Binary Sequences With Low Autocorrelation" by M. R. Schroeder
function phase = schroederPhases(Ns, mag)
    rel_mag = mag./sum(mag); % Normalize magnitude for Schroeder's algorithm
    sum(rel_mag)
    phase = zeros(1, Ns);
    for nn=2:floor(Ns/2+1)
        ll=1:(nn-1);
        phase(nn) = -2*pi*sum((nn-ll).*rel_mag(ll));
    end
end

%% forceFFTSymmetry  A function to force conjugate symmetry on an FFT such that when an
% IFFT is performed the result is a real signal.
% The function has been written to replace MATLAB's ifft(X,'symmetric'), as this function
% is not compatible with MATLAB Coder.
function Y = forceFFTSymmetry(X)
    Y = X;
    XStartFlipped = fliplr(X(2:floor(end/2)));
    Y(ceil(end/2)+2:end) = real(XStartFlipped) - sqrt(complex(-1))*imag(XStartFlipped);
end
```


## Reference Books {#reference-books}

-   <pintelon12_system_ident>
-   <schoukens12_master>


## Bibliography {#bibliography}

<./biblio/references.bib>