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I have a good basic understanding on classical control theory. Now I am studying modern control theory.I encountered the following equations as seen below.

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Online I have only been able to find representation of differential equations in the form on these equations.

Im more interested in the background (or derivation) of these equations themselves or atleast some background on from where these equations came about, I dont want to accept them as is.

If my question too rudimentary, please know im just starting out in this subject. Thanks!

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    \$\begingroup\$ The source of the idea is actually the easiest part. Just like matrix form is a compact way of writing out systems of linear equations, so the state space model is a compact way of writing out Nth order differential equations as a bunch of 1st order equations in matrix form, instead. Plus, there are many alternative approaches. Creativity can apply. So I'm not sure if there's some bright line definition that can be just handed to you on a silver platter. Most of it just comes from sitting down and looking at the machinery over and over until you feel comfortable with it. \$\endgroup\$ – jonk Aug 18 at 1:10
  • \$\begingroup\$ But have you wikied? "State-space representation - Wikipedia": en.wikipedia.org/wiki/State-space_representation. \$\endgroup\$ – tlfong01 Aug 18 at 1:21
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    \$\begingroup\$ Buy derivation, do you mean modelling (from physics principles to differential equations) ? If so, it varies from case to case. For mechanical systems, it usually starts from Newton's laws. For electrical systems, it starts from kcl, kvl, Faraday law, Maxwell equations, etc. Similarly for biological, chemical systems there will be laws from respective fields. First few chapters of most basic control system text books cover this in detail. \$\endgroup\$ – AJN Aug 18 at 3:00
  • \$\begingroup\$ Ah, yes, there are many state space models. Actually I googled a couple of state models on different areas, before I recommended the wiki state space model more specific to (control) engineering. \$\endgroup\$ – tlfong01 Aug 18 at 3:30
  • \$\begingroup\$ Do you understand transfer functions? Because you can easily get from a transfer function to the state space model \$\endgroup\$ – Pangus Aug 18 at 14:00

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