Control theory: block diagram, problem (detailed below)

In summary, the author simplified the transfer function for the negative feedback loop from step c) to step d). The equivalent transfer function is not similar to what I get. I'll explain you what I did and I will compare the results:First of all let's start in step c). The blocks containing the transfer functions \frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}} and G_{3} are in series, therefore the equivalent transfer function is \frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}. The reduced diagram is the following
  • #1
thegreengineer
54
3
I´m taking a course on control engineering and I have a test next Tuesday so I need to study the basics which are: Laplace transform, simplification of block diagrams, and analysis of transient and steady state responses. Right now I am dealing with the second one.

I know the basic rules of block diagram simplification and the specific case I want to deal in this thread is this one:

NEGATIVE FEEDBACK LOOP
Rule-6.gif

Image taken from: http://www.msubbu.in/sp/ctrl/BD-Rules.htm

In other words, the equivalent transfer function is the product of the feed-forward path divided by one plus the product of the transfer functions that make the loop.

So I get stuck on this and the problem is that the author (the book is MODERN CONTROL ENGINEERING by KATSUHIKO OGATA 5th ed.) skipped steps explaining how to simplify this:

This is the way the author simplified it:
https://scontent-lax3-1.xx.fbcdn.net/v/t1.0-9/14358649_1778386355779234_9131464410329635376_n.jpg?oh=e7cd4c25c4caafab8260b0e61c814aa2&oe=583D6FC2
The author simplified from step c) to step d). The equivalent transfer function is not similar to what I get. I'll explain you what I did and I will compare the results:

First of all let's start in step c). The blocks containing the transfer functions [itex]\frac{G_{1}G_{2}}{1-G_{1}G_{2}H_{1}}[/itex] and [itex]G_{3}[/itex] are in series, therefore the equivalent transfer function is [itex]\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}[/itex]. The reduced diagram is the following:
https://scontent-lax3-1.xx.fbcdn.net/v/t1.0-9/14355645_1778854725732397_2673141930938132618_n.jpg?oh=187ae0673bdb279bd12d64a28207f93c&oe=5874F734
Here's now where my problem starts. As far as I see the loop formed by the second-to right summation point, the transfer function [itex]\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}[/itex], and the transfer function [itex]\frac{H_{2}}{G_{1}}[/itex] are forming the negative feedback loop configuration, when I did it the result was not the same as the author's. Trust me, I actually did it and I would put the proof below but my computer is failing so don't insist please, I'm trying to understand this topic but I keep stuck so I would want someone to tell me what the author exactly did step by step.
 
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  • #2
I mean, how did he actually simplify to that?
 
  • #3
$$Y=\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}X -
\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}\frac{H_{2}}{G_{1}}Y
$$

$$Y=\frac{\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}}{1+\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}\frac{H_{2}}{G_{1}}}X
$$

$$Y=\frac{\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}}{1+\frac{1G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}\frac{H_{2}}{1}}\frac{1-G_{1}G_{2}H_{1}}{1-G_{1}G_{2}H_{1}}X
$$

$$Y=\frac{\frac{G_{1}G_{2}G_{3}}{1}}{1-G_{1}G_{2}H_{1}+\frac{1G_{2}G_{3}}{1}\frac{H_{2}}{1}}X
$$

You probably just mixed up a numerator and denominator somewhere

edit: Earlier I had a '+' instead of '-' in that first equation. Fixed!
 
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  • #4
Poster has been reminded not to post in all capital letters (and post has been fixed)
Please someone answer. I have an exam of this on Tuesday. Please, I'm desperate.

I find this example everywhere and everyone skips the same damn steps. Please, I'm tired of getting stuck and confused. I did this following the definition of negative feedback, and I got the result I am showing in the photo which is completely different, and my teacher will just give us thirty minutes for the exam and I can't waste too much time with long steps.

Please I need help. If I can't even do this simple problem, who thinks I'm going to do harder problems?

https://scontent-lax3-1.xx.fbcdn.net/v/t1.0-9/14368868_1779167525701117_2312698362951297674_n.jpg?oh=166ebb913302a2f873efc60a6c857bb5&oe=58811457
 
Last edited by a moderator:
  • #5
Take a deep breath. This part that you did was unnecessary.

$$Y=\frac{\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}}{1+\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}\frac{H_{2}}{G_{1}}}X=\frac{\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}}{1+\frac{G_{1}G_{2}G_{3}H_{2}}{G_{1}^2-G_{1}G_{2}H_{1}}}X$$

It's easier to simply eliminate the term ##\frac{1}{1-G_{1}G_{2}H_{1}}## from the numerator and denominator like so:

$$Y=\frac{\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}}{1+\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}\frac{H_{2}}{G_{1}}}X=\frac{\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}}{1+\frac{1G_{2}G_{3}}{1-G_{1}G_{2}H_{1}}\frac{H_{2}}{1}}\frac{1-G_{1}G_{2}H_{1}}{1-G_{1}G_{2}H_{1}}X=\frac{G_{1}G_{2}G_{3}}{1-G_{1}G_{2}H_{1}+1G_{2}G_{3}H_{2}}X$$

Does that make it more clear?
 
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What is a block diagram in control theory?

A block diagram in control theory is a graphical representation of the components and their connections in a control system. It is used to illustrate the flow of signals and information within the system, and is an important tool for analyzing and designing control systems.

What is the purpose of a block diagram in control theory?

The purpose of a block diagram in control theory is to provide a visual representation of a control system, which helps in understanding the system's functionality and behavior. It also serves as a communication tool for engineers and scientists working on control systems, as it allows them to easily identify the different components and their connections.

What are the main components of a block diagram in control theory?

The main components of a block diagram in control theory are blocks, arrows, and summing points. Blocks represent the different components of the control system, such as sensors, actuators, and controllers. Arrows represent the flow of signals and information between blocks, while summing points are used to combine multiple signals.

What is the problem addressed by control theory?

The problem addressed by control theory is how to design systems that can achieve a desired behavior or output despite having uncertainties and disturbances. Control theory provides mathematical tools and techniques for analyzing and designing systems that can maintain stability and performance in the presence of these uncertainties and disturbances.

What are the different types of control systems?

The different types of control systems are open-loop, closed-loop, and feedback control systems. In an open-loop system, the output is not used to make adjustments to the input. In a closed-loop system, the output is used to adjust the input and maintain a desired output. Feedback control systems use sensors to continuously monitor the output and make adjustments to the input based on the feedback.

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