Update Content - 2024-12-13

content/zettels/system_identification.md

@@ -8,39 +8,174 @@ Tags
 : [Modal Analysis]({{< relref "modal_analysis.md" >}})
 
 
-## SISO Identification {#siso-identification}
+## System Identification {#system-identification}
 
 
-### Problem Description {#problem-description}
+### Problem Definition {#problem-definition}
 
-<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" >}}
+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 2: </span>Simpler Block diagram of the SISO system identification" >}}
+{{< 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" >}}
 
-<div class="note">
+The goal of system identification is to compute the transfer function \\(G\\) from known excitation signal \\(u\\) and from a measure of \\(y\_m\\).
 
-If the open-loop system is unstable, first design a simple controller that stabilizes the system and then identify the closed-loop system.
+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  |
 
-### Design of the Excitation Signal {#design-of-the-excitation-signal}
+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.
 
 
-#### Introduction {#introduction}
+### 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
+-   Random noise, Periodic signals (PRBS)
+-   Multi-Sine
 
 
-#### Random noise with specific ASD {#random-noise-with-specific-asd}
+### Random noise with specific ASD {#random-noise-with-specific-asd}
 
 The ASD of the measured output is:
 
@@ -82,7 +217,7 @@ u = lsim(G_u, u_norm, t);
 ```
 
 
-#### Choose Sampling Frequency and Duration of Excitation {#choose-sampling-frequency-and-duration-of-excitation}
+### Choose Sampling Frequency and Duration of Excitation {#choose-sampling-frequency-and-duration-of-excitation}
 
 <div class="important">
 
@@ -134,39 +269,102 @@ T\_{\text{exc}} = \frac{10}{1} = 10\\,s
 </div>
 
 
-### Computation of the Frequency Response Function {#computation-of-the-frequency-response-function}
+### Multi-Sine {#multi-sine}
 
 
-#### Windowing Function {#windowing-function}
-
-
-#### Example {#example}
-
-`tfestimate`
+### `generatemultisine` - Matlab Function {#generatemultisine-matlab-function}
 
 ```matlab
-[G, f] = tfestimate(u, y, win, [], [], 1/Ts);
-```
+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
 
-### Verification of the Identification Quality {#verification-of-the-identification-quality}
+        mag = (args.asd*2*sqrt(Fs)/sqrt(Ns)).^2;
+    end
 
-`mscohere`
+    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
 
-```matlab
-[coh, f] = mscohere(u, y, win, [], [], 1/Ts);
+    %% 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}
 
--   (<a href="#citeproc_bib_item_1">Pintelon and Schoukens 2012</a>)
--   (<a href="#citeproc_bib_item_2">Schoukens, Pintelon, and Rolain 2012</a>)
+-   <pintelon12_system_ident>
+-   <schoukens12_master>
 
 
 ## 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>
+<./biblio/references.bib>