In simply word, B must be the combinations of all eigen vectors of A, that is, B can not belong to any sub eigen vector combinations of A for fully control of all eigen values of A.
Does enforcing balance have any particular value, or is it just "nice" because it isn't biased towards either control or observability? I imagine it might benefit numerical stability, for the same reason that the rectangle that maximizes the area for a given perimeter is a square. But if I know ahead of time that I'm going to be kicking some controls harder or more frequently than others, and paying closer attention to some subspace of the observable output... Ah and in that case I should rewrite my Wo and Wc to incorporate those biases, at which point balanced reduction does make the most sense! So I should really be thinking of these systems as already "closed" or balanced in the sense that observation and controllability are equally important, I guess modulo the case when I have 0 control, which is probably best handled by working with pseudo-inverses aka retractions of the (possibly trivial) invertible subspace.
Data science is usually involving solutions to inverse problems using some sort of regualarization that projects various elements into the compliment of the null space of the operator. One sentence explanation that most people probably wont get.
As a wargamer I am thinking that there might be historical. sociological, and exosmic application to this as well. For example one might have a databases of combat losses in WW2 battles verses the forces involved plus the terrain and such and the derive simplified models that can predict those outcomes, these being the rules for a wargame. This is essentially what wargame designers do but they traditional do not use machine learning to do that but by hand. I would be interesting to see just how well (or poorly) such machine derived models could work.
Method 1) (- x= 6) equation is given Multiplying both sides by (-1) -1*-x= -1*6 Then x= -6 or Method 2) Let the equation be (- x= 6) If we multiply both sides with "MINUS" sign -(- x)= -(6) Then x= -6. Which one method is correct or both methods are correct Please help 🙏🙏
The physical systems usually take t=0 as the starting point so the 'one sided', and the stable systems by definition have to be stable so 'weighted' in order to introduce the negative gain along the timeline. Is that a correct way of generalizing the practicality of the Laplace transform in control systems theory? By the way, Thank you so much for this brilliant lecture Sir!
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