Control theory: Laplace versus state space representation

Mårten

Okey, since I got no reply to my previous post above, I guess that means everything there was correct...?

Then to the following question: One of the benefits with Laplace transforms, is that you can just mulitply boxes in a row to get the transfer function for all the boxes together. In general, it's pretty simple to handle different configurations with boxes in series and in parallel, when doing transfer functions.

Is it possible as well to put up a state space representation when you have several boxes in series and so on, and how is that done then? Maybe that's a drawback to state space representations, that it's not done so easily as with transfer functions?

rbj

Marten,

i just now saw this thread was left hanging. i just received a few infractions from Evo (from a totally different thread) and i need some time to think about what you were asking in post #17. there is a straight-forward way to go from state-space representation to transfer function. it's in the text books but it's pretty easy to derive. of course, there is loss of information (since many different state-space representations can turn into a single transfer-function representation). it's maybe not as straight-forward, but you can combine the states of one box and the states of another box (in series or parallel) into a single box where the number of states is the sum of the two and where the states are defined exactly as they were before.

Mårten

I'll be happy for any comments!

The reason I'm so eager to stick to state space representations, is that I would like to be able to handle, and to fully understand, systems with multiple inputs/outputs and with arbitrary initial values, and still have pretty simple calculations (for instance, arbitrary initial values is not good for transfer functions cause then simplicity disappears; multiple inputs/outputs only possible to handle with state space representations).

Okey. I'll try to do this myself here to see what happens. Imagine two boxes G_1 and G_2 after each other in the following way

u --> [itex]G_1[/itex] --> v --> [itex]G_2[/itex] --> y

where [itex]G_1: a_1v'' + b_1v' + c_1v = u[/itex] and [itex]G_2: a_2y'' + b_2y' + c_2y = v[/itex], i.e. G_1 has u and v as input/output and G_2 has v and y as input/output. For simplicity, [itex]a_1 = a_2 = 1[/itex]. The state space representation for this then becomes (as far as I can see)

[tex]
\left(
\begin{array}{c}
x'_1 = v' \\
x'_2 = v'' \\
x'_3 = y' \\
x'_4 = y'' \\
\end{array}
\right)

=

\left(
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
-c_1 & -b_1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & -c_2 & -b_2 \\
\end{array}
\right)

\left(
\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
\end{array}
\right)

+

\left(
\begin{array}{c}
0 \\
u \\
0 \\
0 \\
\end{array}
\right).
[/tex]

Okey, it seems to work! The output from G_1 becomes the input to G_2 through the number 1 in the A-matrix' lower left corner (element [itex]a_{41}[/itex], sort of the "chain element" which connects the two boxes).

Now comes the question of how to solve this, ehhh... Using eigenvalue techniques or [itex]e^{At}[/itex]-matrices, and so forth, maybe leading to unsolvable nth degree equations (n>4). Or simply using a computer...

Proton Soup

yeah, it's been a good 12 or 13 years since i modelled state space systems, which is why i haven't tried to help. but yeah, computing the discrete state-space matrix from the continuous time transfer function is done with a computer. something like Matlab or Scilab, maybe Octave, should have the functions available. Matlab i know for sure does, because that's what i used to use.

Mårten

Of course, MATLAB would do the trick! I should know that.

Okey, I guess I won't get out any more in this particular area. Thanks for all help! Don't hesitate to comment further, what I've said above.

I've now started a new thread on the topic of frequency response,
see here.

Proton Soup

well, it requires computing a matrix exponential, doesn't it? iirc, there's really no good way to do that by hand.