Update Content - 2024-12-13

content/zettels/system_identification.md

@@ -174,6 +174,8 @@ There are several choices for excitation signals:
 -   Random noise, Periodic signals (PRBS)
 -   Multi-Sine
 
+A good review is given in <&pintelon12_system_ident> (chapter 5).
+
 
 ### Random noise with specific ASD {#random-noise-with-specific-asd}
 
@@ -271,9 +273,71 @@ T\_{\text{exc}} = \frac{10}{1} = 10\\,s
 
 ### Multi-Sine {#multi-sine}
 
+Multi-sine signal excitation has many advantages as compared to random noise:
+
+-   the signal power at each frequency can be precisely chosen
+-   the signal is periodic and therefore necessitate no windowing (therefore increasing the obtained FRF quality)
+
+It can be generated with the following code.
+
+```matlab
+%% Generate multinsine signal
+Fs = 1e3; % Sampling Frequency [Hz]
+Ns = 1e3; % Signal length [-]
+
+f = linspace(0, Fs/2, Ns/2); % Frequency Vector [Hz]
+
+% Define the wanted ASD of the test signal [unit/sqrt(Hz)]
+wanted_asd = 3*ones(1,Ns);
+f_min = 10; % [Hz]
+f_max = 300; % [Hz]
+wanted_asd([1:round(Ns*f_min/Fs)]) = 0;
+wanted_asd([round(Ns*f_max/Fs+1):end]) = 0;
+
+% Generate the multi-sine signal
+u = generate_multisine(Fs, Ns, ...
+                       'asd', wanted_asd, ...
+                       'type', 'schroeder');
+```
+
+The ASD of the generated signal is exactly as expected (Figure <fig:system_identification_multi_sine_asd>)
+
+```matlab
+[pxx, f] = pwelch(u, ones(Ns,1), [], Ns, Fs);
+```
+
+<a id="figure--fig:system-identification-multi-sine-asd"></a>
+
+{{< figure src="/ox-hugo/system_identification_multi_sine_asd.png" caption="<span class=\"figure-number\">Figure 7: </span>Amplitude Spectral Density of the multi-sine signal" >}}
+
+In the time domain, it is shown in Figure <fig:system_identification_multi_sine_time>.
+
+<a id="figure--fig:system-identification-multi-sine-time"></a>
+
+{{< figure src="/ox-hugo/system_identification_multi_sine_time.png" caption="<span class=\"figure-number\">Figure 8: </span>Generated Multi-Sine signal" >}}
+
+Then, the open-loop identification is performed, and the FRF is computed using the following code (not that no window is being used!).
+Only the first period (here of 1s) is discarded to remove transient effects.
+
+```matlab
+% Skip the first period (transient)
+[Gm, f] = tfestimate(data.du(Ns:end), data.y(Ns:end), ones(Ns,1), [], Ns, Fs);
+```
+
+The obtained FRF is shown in Figure <fig:system_identification_multi_sine_frf>.
+The quality of the obtained FRF is only good in the defined range.
+
+<a id="figure--fig:system-identification-multi-sine-frf"></a>
+
+{{< figure src="/ox-hugo/system_identification_multi_sine_frf.png" caption="<span class=\"figure-number\">Figure 9: </span>Obtained FRF using the multi-sine excitation signal" >}}
+
 
 ### `generatemultisine` - Matlab Function {#generatemultisine-matlab-function}
 
+The synthesis of multi-sine with minimal "crest factor" is taken from <&schroeder70_synth_low_peak_factor_signal>.
+
+The Matlab code is adapted from [here](https://bholmesqub.github.io/thesis/chapters/identification-design/multi-sine/).
+
 ```matlab
 function y = generate_multisine(Fs, Ns, args)
 %% Input parsing
@@ -339,7 +403,6 @@ end
 % 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);