To elaborate my question, the considered MIMO system is a stirred tank reactor which has two inputs and outputs. The transfer function is as follows:
G=[tf(1,[0.7 1]) tf(5,[0.3 1]);tf(1,[0.5 1]) tf(2,[0.4 1])];
I'm looking for finding the bandwidth of this system. I searched in some references and I found two theorems.
First, In [1], it's mentioned that the closed-loop bandwidth is the frequency where |S(jw)| first crosses 0.707 (-3 db) from below which S is the sensitivity function. I tried to form S(jw) by S=inv(I+KG) but after I tried to plot the bode diagram it had multi channels and I didn't find a unique answer.
Second, in [2], it's said that the closed-loop bandwidth is the frequency where upper bound of sigma(S) crosses -6 db. I tried to plot this too but the upper bound didn't cross -6db, however, the lower bound crossed it at w=10 rad/s.
I really appreciate if you can help me to find a solution for this problem.
Thanks in advance.
Refs:
[1] Skogestad, Sigurd, and Ian Postlethwaite. Multivariable feedback control: analysis and design. Vol. 2. New York: Wiley, 2007.
[2] Albertos, Pedro, and Sala Antonio. Multivariable control systems: an engineering approach. Springer Science & Business Media, 2006.
I think it would be easier if you converted your G into a single closed loop transfer function and then plot the bode plot using MATLAB, which I think you are doing.
Then the you can find the bandwidth as the frequency where the magnitude plot crosses the -3dB mark.
Thank you for your reply dear Harsh, would you mind telling me how i can convert my multi channel G into a single closed loop transfer function to gain a single unique bode plot using MATLAB?
I tried to plot the determinant of the sensitivity function ( inv(I+GK) ) but it's ascending and doesn't cross the -3db.
Any reply would be appreciated.
Dear Hirad,
remarks:
-The Bandwidth word used this way can insinuated a designation other than that intended by the author of the question.
- G (s) is called the transfer matrix and not transfer fontion because it is in the MIMO case.
- The theorems mentioned correspond to the case SISO and for MIMO system can not be applied.
The technique used in this case may be Nyquist arrays and Gershgorin bands.
Best regards
Dear Mohamad-Mourad,
Thanks for your reply, would you tell me a practical way to find bandwidth of a MIMO system by employing Nyquist arrays and gershgorin bands?
Any answer would be appreciated.
I am new to multi-channel systems myself. Thanks for the resources Mr. Lafifi
Thank you so much dear Mohamad-Mourad, There resources are useful but I guess the nyquist array method or gershgorin bands which show the diagonal dominance of the system or high interaction in channels of the system. But my issue is finding a practical way of finding the bandwidth of a MIMO system (even if it's extended from the concepts of the SISO systems). I'm going to use this bandwidth in a particular pre-compensation form which Skosgetad in his book calls it "Approximate Decoupling at bandwidth frequency". Although these resources are great but I didn't find my answer in them. Thanks again.
Thank you so much for your reply dear Alexander,
Something baffles me, Skogestad in his book [1] says that the prerequisite to employ the method of approximate decoupling at w0 (which is a method to design a pre-compensator to make the system diagonally dominant) is the bandwidth frequency to consider it as w0. So by your answer, there are 4 bandwidth frequency and it's the designer option to choose which?
[1] Skogestad, Sigurd, and Ian Postlethwaite. Multivariable feedback control: analysis and design. Vol. 2. New York: Wiley, 2007.
Hirad - Please look at H-infinity control design methods. The bandwidth of the MIMO controller can be seen by generating the Bode plot for the largest singular value of the MIMO controlled system.
Let us look at a simple multivariable control system with only 2 loops (the simulation example in [HY05a]).
Assume that each loop has no interaction,
the phase margin of loop #1 is \phi_1,
the phase margin of loop #2 is \phi_2,
both \phi_1 and \phi_2 should be positive in order to make the whole system stable. That is, \phi_1>0 and \phi_2>0.
By taking into account the interaction between loop #1 and loop #2, the real-world phase margins for both loops are {\phi_1}^' and {\phi_2}^' instead.
Direct Nyquist Array (DNA) can be used to analyze the interaction of different loops over a system.
According to DNA stability theorem, {\phi_1}^'0 and {\phi_2}^'>0.
As suggested by Mohamed-Mourad Lafifi, "The technique used in this case may be Nyquist arrays and Gershgorin bands."
Quote Hirad Assimi's comment "would you tell me a practical way to find bandwidth of a MIMO system by employing Nyquist arrays and gershgorin bands?"
The following two papers apply Direct Nyquist Array and Gershgorin circle to measure the stability margin of multivariable system.
[HY05a] Y. Hong and O.W.W. Yang, "Adaptive Multiloop PI Rate-based Controller Design for a MIMO IP Router Based on Phase Margin," Proceedings of IEEE Globecom, St. Louis, MO, U.S.A, November 2005, pp. 1070-1074.
https://www.researchgate.net/publication/4213069_Adaptive_multiloop_PI_rate-based_controller_design_for_a_MIMO_IP_router_based_on_phase_margin
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1577801
[HY05b] Y. Hong and O.W.W. Yang, "Self-Tuning Multiloop PI Rate Controller for an MIMO AQM Router With Interval Gain Margin Assignment," Proceedings of IEEE HPSR, Hong Kong, China, May 2005, pp. 401-405.
https://www.researchgate.net/publication/4171680_Self-tuning_multiloop_PI_rate_controller_for_an_MIMO_AQM_router_with_interval_gain_margin_assignment
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1503263
The above two papers have been reviewed by the master thesis "Analysis of Stability and Stabilization for Second-Order Vector Differential Systems" written by Wen-Yan Huang.
https://hdl.handle.net/11296/4g6447
Discussion on control systems engineering
"What are trends in control theory and its applications in physical systems (from a research point of view)?"
https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2
Conference Paper Self-tuning multiloop PI rate controller for an MIMO AQM rou...
Conference Paper Adaptive multiloop PI rate-based controller design for a MIM...
Dear Hirad,
Since you tackled this problem 3 years ago, I wonder if you would share the final definition of bandwidth which was consistent with your problem.
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