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Good time of the day. In my research (I'm an electrical engineer) I'm doing a kind of quaternion analysis for a control system design. So, in my particular case, I just have a "vector part" of a voltage quaternion associated with the grid voltage waveform. And I want to find grid voltage derivative in the quaternion basis. I decided to perform partial finite-difference for each quaternion part like $q_1$, $q_2$, $q_3$ (I don't have a real part in this quaternion $q_0$). So, is it legal to call it quaternion time derivative or can you suggest a more strict way of a numeric quaternion differentiation?

$Q'=\left([q_1(k+1)-q_1(k)]/T_s,[q_2(k+1)-q_2(k)]/T_s,[q_3(k+1)-q_3(k)]/T_s\right)$, where $k$ is a sample instant and $T_s$ is a sample time.

zipirovich
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Иван Александров
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  • It isn't really a derivative until some kind of limit is taken. Your formula just gives a difference of two quaternians, assuming the "real parts" are the same. You could argue it approximates the derivative providing each $q_j$ happens to be relatively large, so that a change of one unit was relatively small. – coffeemath Nov 27 '21 at 05:08
  • Thanks a lot! As I understand you right I might call it approximated derivative of quaternion for numeric differentiation purposes? Obviously, each part of the quaternion is a time-dependent real-valued function. – Иван Александров Nov 27 '21 at 08:01
  • Well I might use "approximate" rather than "approximated" there. And to me "for numeic differentiated purposes" is redundant, since saying it is the approximate derivative already implies it is numeric and is for differention purposes. – coffeemath Nov 27 '21 at 11:00
  • You can do it this way though $[q(k+1)-q(k-1)]/(2T_s)$ is a better approximation. This is not a "quaternion derivative", but just the usual derivative of a quaternion-valued function (the former phrase often means that the argument is of quaternion type). – fedja Nov 28 '21 at 14:20
  • @fedja Thank you for your answer! I think that the word quaternion is a bit redundant for this problem. But my advisor told me that it is a quaternion function and I think that it is a set of three independent time-valued functions. I have a misconception about quaternions and their application. Can I call a set of three-phase voltages a voltage quaternion or something... In my case, it is more similar to 'vector' – Иван Александров Nov 29 '21 at 15:29
  • Using the word "quaternion" assumes four coordinates, not three, and is most appropriate if you use the quaternion multiplication in computations. Otherwise it is, indeed, just a vector-valued function. In general you are free to choose any name that is not outright misleading but I would certainly not call a quaternion something with only 3 components. – fedja Nov 29 '21 at 15:45
  • @fedja Oh, currents and voltages of a three-phase electrical system are called quaternions which have only vector part, but their combinations like power quaternion (which is obviously quaternion product of three-phase voltage and current of the grid may be or any AC load) has scalar part of its quaternion (four dimension) which represents usefull power flow in the grid and vector part of the power quaternion represent a useless part of the power flow. – Иван Александров Nov 29 '21 at 15:59
  • @fedja But anyway. why to use this type of algebra? The results of the quaternionic multiplication are just equal to scalar and vector multiplication of three-phase vectors and it really is a scalar and vector multiplication but called by a fancy word isnt it? – Иван Александров Nov 29 '21 at 15:59
  • It is just a convenient formalism in the electrodynamics. Consider it "an established historical practice", if you want. It is an interesting algebra by itself, but I'm not sure whether the purely algebraic aspects of it have any relation to the physics (though the answer is probably "yes": the modern physics resembles abstract algebra more and more :-)). So, if something is 4D and has those multiplication rules, then calling it "quaternions" is completely appropriate and conveys the whole structure in one medium length word. – fedja Nov 29 '21 at 16:09
  • Quaternions with zero real part and nonzero vector part are known as "pure quaternions". – Mauricio Cele Lopez Belon Nov 30 '21 at 09:12

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