+
Pintelon, Rik, and Johan Schoukens. 2012.
System Identification : a Frequency Domain Approach. Hoboken, N.J. Piscataway, NJ: Wiley IEEE Press. doi:
10.1002/9781118287422.
+
-
Lee, Eun Joong, Youngok Park, Chul Sung Kim, and Taejoon Kouh. 2010. “Detection Sensitivity of the Optical Beam Deflection Method Characterized with the Optical Spot Size on the Detector.”
Current Applied Physics 10 (3): 834–37. doi:
10.1016/j.cap.2009.10.003.
-
Li, Qing, Shaoxiong Xu, Jiawei Yu, Lingjie Yan, and Yongmei Huang. 2019. “An Improved Method for the Position Detection of a Quadrant Detector for Free Space Optical Communication.”
Sensors 19 (1): 175. doi:
10.3390/s19010175.
-
Manojlović, Lazo M. 2011. “Quadrant Photodetector Sensitivity.”
Applied Optics 50 (20). Optical Society of America: 3461–69.
-
Wu, Jiabin, Yunshan Chen, Shijie Gao, Yimang Li, and Zhiyong Wu. 2015. “Improved Measurement Accuracy of Spot Position on an Ingaas Quadrant Detector.”
Applied Optics 54 (27). Optical Society of America: 8049–54.
+
Azaryan, N. S., J. A. Budagov, M. V. Lyablin, A. A. Pluzhnikov, B. Di Girolamo, J.-Ch. Gayde, and D. Mergelkuhl. 2019. “Position-Sensitive Photoreceivers: Sensitivity and Detectable Range of Displacements of a Focused Single-Mode Laser Beam.”
Physics of Particles and Nuclei Letters 16 (4): 354–76. doi:
10.1134/s1547477119040058.
+
Lee, Eun Joong, Youngok Park, Chul Sung Kim, and Taejoon Kouh. 2010. “Detection Sensitivity of the Optical Beam Deflection Method Characterized with the Optical Spot Size on the Detector.”
Current Applied Physics 10 (3): 834–37. doi:
10.1016/j.cap.2009.10.003.
+
Li, Qing, Shaoxiong Xu, Jiawei Yu, Lingjie Yan, and Yongmei Huang. 2019. “An Improved Method for the Position Detection of a Quadrant Detector for Free Space Optical Communication.”
Sensors 19 (1): 175. doi:
10.3390/s19010175.
+
Manojlović, Lazo M. 2011. “Quadrant Photodetector Sensitivity.”
Applied Optics 50 (20). Optical Society of America: 3461–69.
+
Ng, T.W., H.Y. Tan, and S.L. Foo. 2007. “Small Gaussian Laser Beam Diameter Measurement Using a Quadrant Photodiode.”
Optics &Amp; Laser Technology 39 (5): 1098–1100. doi:
10.1016/j.optlastec.2006.06.001.
+
Wu, Jiabin, Yunshan Chen, Shijie Gao, Yimang Li, and Zhiyong Wu. 2015. “Improved Measurement Accuracy of Spot Position on an Ingaas Quadrant Detector.”
Applied Optics 54 (27). Optical Society of America: 8049–54.
+
+If the open-loop system is unstable, first design a simple controller that stabilizes the system and then identify the closed-loop system.
+
+
+
+
+### 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}
+
+
+
+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}
+
+
+
+
+
+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}
+
+
+
+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}
+
+
+
+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}
+
+
+
+
+### 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}
+
+- (
+
Pintelon, Rik, and Johan Schoukens. 2012.
System Identification : a Frequency Domain Approach. Hoboken, N.J. Piscataway, NJ: Wiley IEEE Press. doi:
10.1002/9781118287422.
+
Schoukens, Johan, Rik Pintelon, and Yves Rolain. 2012.
Mastering System Identification in 100 Exercises. John Wiley & Sons.
+
Elias, Said, and Vasant Matsagar. 2017. “Research Developments in Vibration Control of Structures Using Passive Tuned Mass Dampers.”
Annual Reviews in Control 44 (nil): 129–56. doi:
10.1016/j.arcontrol.2017.09.015.
+