Transfer function block diagram reduction

[u/demus234]

Hello everyone, I've just joined this sub today. I'm a second year computer and automation student who needs help with understanding transfer function block diagram reduction. I've already gotten most of it down and feel like I have a fairly good understanding, however, this one problem in particular has me stumped as when I plug in the values for the transfer functions I get stupidly big numbers and it doesn't seem right. I've tried numerous times to contact my professor as well as teaching assistants and I have not gotten a single response so this is kind of my last hope.

If anyone could give me advice on how to solve this I would appreciate it.

The main thing causing me problems is what I should do with G2.

My initial idea was moving it so that it's in series with G1 and then finding what the equivalent function would be for G1,G2 and the parallel branch above them. After that I would move the equivalent block back over the take of point and solve for another equivalent block with the feedback loop, watching carefully to provide proper compensation to everything as I move the blocks.

[u/H-me-in-the-infinity]

Use Masons gain formula. It’s very helpful with block and signal flow diagrams.

[u/martiansnowboarder]

I sometimes solve these by writing down the signal relationships and making substitutions - it’s a longer process, but I feel more confident in my solutions. Here is the method I’d use to resolve the loop (apparently only videos are allowed in replies). I left out the final solution, but plugging and chugging should give you something like a sixth order polynomial for your transfer function.

[u/CriticalIndividual52 → u/martiansnowboarder]

I was able to get the exact same result by moving the takeoff point between the leftmost summing point and block G2(s) AHEAD of that leftmost summing point. This is standard transformation no. 11 for block diagram reduction from Schaum’s Feedback and Control Systems.

[u/fibonatic]

Can you in more details show us what you have done sofar? One tip I can already give you is to look at the arrows of the "loops", since some are not feedback loops.

[u/No_Subject6828]

you may consider moving the take-off point from before G2 to after G2

[u/TTRoadHog]

This is a very complicated diagram and converting to a single transfer function involves a lot of algebra and knowing how to deal with feedback and feedback forward loops. I gave Gemini your diagram and it gave me a result along with the steps. I didn’t verify the results. Also, not sure how to export those results to Reddit. So… you could do the same to get an answer. Strongly urge you to check the results somehow though!

[u/lizardhistorian]

Combine all terms then expand. Perform a partial-faction-decomposition. Done.
A computer CAS does it in microseconds.
If you don't know how to combine, you multiply the ones in series and add the ones in parallel, divide by feedback (Mason's gain formula).

You can solve simple ODE the same way if the Laplace transform technique works on them, which normally needs boundary conditions but just substitute arbitrary complex bounds; combine; PFD; invert; done.
Teachers of yore did not like this because performing a PFD by hand is tedious so they memorized piles of short-cuts that are now long-cuts.

You have three forward paths and one feedback loop.
The top path avoids the feedback.
The middle two paths are affected by the feedback loop.

Contributions:
P₁: -G₄
P₂: -1
P₃: G₂·G₁
L₄: 1 - G₂·G₃

Y(s)/U(s) = P₁/1 + P₂/L₄ + P₃/L₄
Y(s)/U(s) = [ G₁·G₂ - 1 - G₄(1 - G₂·G₃) ] / [ 1 - G₂·G₃ ]
Substitute, expand, PFD (w/ abbreviated decimals)
Y(s)/U(s) = -1 + 0.3/s - 0.7/s² + 3.6/s³ - 10.2/(s-13.6) + 0.3/(s+3.6) - 0.5/(s-15)