I have a Twin Rotor MIMO system represented by a 2x2 transfer matrix.
I have identified 3 different models for the same using different ways. Let's say the three different transfer matrix are given by G1(s), G2(s) and G3(s). The singular value plots for the same are attached herewith. I would like to know what would be the most significant observations from the plot. I understand that singular value decomposition gives an idea about the input and output directions and also whether the system is ill-conditioned or not but how do I interpret the results from these three plots in a relative manner.
NB:- The plot has been obtained in matlab by using the command sigma(G(s)) .
Please help.
When one singular value is (near) zero, then it means that the singular vector for that singular value is a solution for the system output in the absence of forcing at its input (or, if there is energy loss, that only relatively weak forcing is required to get a significantly large output).
As I see it, then, knowing little about your particular system, if you drive the system at that resonant frequency, the output will be dominated by an amplifying or run-away version of the singular vector. It is the mode of behavior with no (or little) forcing.
The amplitude (energy) of the output will tend to grow uncontrollably, perhaps very quickly, until the system is driven out of its regime of intended (designed) operation into saturation or into hard mechanical limits, possibly suffering electrical or mechanical damage. (Look up the "death wobble" that can attack a motorcycle and some cars as an illustration of a disastrous resonance in link:
https://www.youtube.com/watch?v=ObdequWUJ3U)
One place to look for a theoretical treatment of this is the admittedly abstract theory of singular values, and in the existence and uniqueness of linear systems of equations, which come to life in applications like the one you have, and for linear systems of equations in as they are applied in many places in engineering and physics more generally. It is worth revisiting the abstract theory with practical applications in mind.
If the system has high energy loss, then the system may remain tame, but then the resonant frequency will be a pole in the complex frequency plane -- as in the theory of functions of a complex variable -- with an imaginary part to the resonant frequency that gives exponential decay in the time response of the output. You can find this complex pole by searching for it somewhat off the real frequency axis in the complex frequency plane rather than on the real frequency axis as you are doing.
Your third plot -- the plot with the very narrow, sharp notches in it -- suggests that there may be at least two free modes in your system, at the frequencies of those notches.
If you probe with high resolution into those narrow notches in the smallest singular value, then you may find that the smallest singular values in those notches are in fact very deep minima, dropping down to machine precision (to zero in effect).
These narrow notches would easily be missed in the plot if your step size in frequency is too large. Perhaps the other graphs have them as well and they were missed. And perhaps there are more than two in the third graph.
You might do some very high-resolution searches to confirm that you have not missed some (assuming that such modes would be of interest).
I have used SVs to find modes in large systems, with a mode identified when the smallest SV drops to zero. The advantage of SV is that you can also monitor the second smallest SV in case it too drops to zero at of very near the frequency where the smallest SV drops to zero, indicating that you have a double mode or a close more pair. Finding and probing deep into the SV minima was problematic because they were bounded by relatively narrow notches, much as you may have them in the third graph.
Thank you for the analysis, Sir. As you have said that the third plot indicates that the system contains at least 2 free modes, can you kindly point out what effects will that have on the controllability of the system? Will it be difficult to control the third system compared to first and the third one ?
When one singular value is (near) zero, then it means that the singular vector for that singular value is a solution for the system output in the absence of forcing at its input (or, if there is energy loss, that only relatively weak forcing is required to get a significantly large output).
As I see it, then, knowing little about your particular system, if you drive the system at that resonant frequency, the output will be dominated by an amplifying or run-away version of the singular vector. It is the mode of behavior with no (or little) forcing.
The amplitude (energy) of the output will tend to grow uncontrollably, perhaps very quickly, until the system is driven out of its regime of intended (designed) operation into saturation or into hard mechanical limits, possibly suffering electrical or mechanical damage. (Look up the "death wobble" that can attack a motorcycle and some cars as an illustration of a disastrous resonance in link:
https://www.youtube.com/watch?v=ObdequWUJ3U)
One place to look for a theoretical treatment of this is the admittedly abstract theory of singular values, and in the existence and uniqueness of linear systems of equations, which come to life in applications like the one you have, and for linear systems of equations in as they are applied in many places in engineering and physics more generally. It is worth revisiting the abstract theory with practical applications in mind.
If the system has high energy loss, then the system may remain tame, but then the resonant frequency will be a pole in the complex frequency plane -- as in the theory of functions of a complex variable -- with an imaginary part to the resonant frequency that gives exponential decay in the time response of the output. You can find this complex pole by searching for it somewhat off the real frequency axis in the complex frequency plane rather than on the real frequency axis as you are doing.
Hi Biswajit,
Thank you once again for having asked me to join this discussion, and sorry once again for this .delayed reply. In the meantime, I read Dr Kessel's replies with interest, which amounted to learning in their own right.
Now for my own two-penny bit ! Let me go through each frequency response with a pin !!
G1(s) :
G2(s) :
G3(s) :
If my interpretations detailed above are correct (touchwood !!), the G3(s) should have serious controllability and/or observability issues due to #4, that is, the pole-zero cancellation !
The other two transfer functions should turn out to be perfectly controllable and observable, subject solely to accuracy of numerical computation. They may have other issues with the gain/phase margins, inadequate damping at selected frequencies, etc. But controllability and observability aspects should not pose associated challenges.
Hope all that helped !!
With best wishes.
-Sanjay
Dear, Sir Sanjoy, please don't be sorry about that. I am rather overjoyed that you have found the time out of your busy schedule to help me out of my ignorance.
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