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A pictorial representation of the functions performed by each component and of the flow of signals. Block diagram. A pictorial representation of the functions performed by each component and of the flow of signals.Shifting of take-off point ahead of the block
If we need to shift the take-off point ahead of the block, then we must keep ‘p’ as it is. So, even after shifting p must be X(s) and for this, we have to add a block with gain which is reciprocal of the gain of the originally present block.
- Rule 1 − Check for the blocks connected in series and simplify.
- Rule 2 − Check for the blocks connected in parallel and simplify.
- Rule 3 − Check for the blocks connected in feedback loop and simplify.
- Rule 1: Blocks in cascade/series.
- Rule 2: Blocks in parallel.
- Rule 3: Feedback loop elimination.
- Rule 4: Associative law for summing.
- Rule 5: Shifting a summing point before a block.
- Rule 6: Shifting a summing point after a block.
- Rule 7: Shifting take-off point before a block.
Table of Contents
Block Diagram Reduction Rules – Part 1
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Spoiler: This post is long but fun! As we all know, control systems consist of mathematical models. The transfer function is the mathematical representation of every physical system. To show the function performed by each component, we usually use block diagrams. Therefore, in order to analyze complex control systems (and thus complex block diagrams), it is highly desirable to reduce block diagrams to simpler terms (using block diagram reduction techniques). Simply put, a block diagram is a visual representation of the entire system. In this way, it represents the relationship between the input and output of the entire system.
Block diagram definitions:
1. Block Diagram: This gives the graphical relationship between the outputs and inputs of the system. 2. Output: defined as the product of input and gain, namely H. output = gain x input. 3. Summation point: The point where multiple signals can be added or subtracted. 4. Starting point: From this point the output can be fed back to the input, so this point is used for feedback. 5. Forward path: This path represents the direction in which the signal flows from the input se to the output se in the system. 6. Feedback path: This path represents the direction in which the signal in the system flows from the output se to the input se. The first step is to get a block diagram of the physical system (can be an amplifier, potentiometer, whatever). Having obtained the block diagram, we now need to obtain the transfer function. For analysis, we need to reduce the (complex) block diagram to a single block. This method of reducing a complex block diagram into a single block representing the transfer function of the entire control process is called “block diagram reduction technique”.
Basic Closed loop Transfer Function (most important formula) !!
Before moving on to finding transfer functions for complex block diagrams, let’s look at the basic closed-loop transfer function. Conser a simple closed loop system, R(s) = Laplace of input signal C(s) = Laplace of output signal E(s) = Laplace of error signal B(s) = Feedback signal The Laplace of G(s) = forward transfer function H(s ) = feedback transfer function when there is negative feedback in the system, when there is positive feedback in the system. Also for the open-loop transfer function, since there is no feedback, the transfer function can be given as: Open-loop transfer function = G(s).H(s)
Block Diagram Reduction Rules
We must follow certain rules to convert any complex control system simplified to a more simple form. Here are the most commonly used block diagram reduction rules (each discussed in detail) Rule 1: Cascaded/Concatenated Blocks. Rule 2: Parallel blocks. Rule 3: Eliminate feedback loops. Rule 4: The associativity of sums. Rule 5: Move the summing point to the front of the block. Rule 6: Move a summation point after a block. Rule 7: Start offset before block. Rule 8: Start offset after one block. Rule 9: Move the summing point to the front of the block. Rule 10: Move a summation point after a block. For simplicity, we will write G(s) as “G” and H(s) as “H”. Now, in the next section, we will examine each of the above rules in detail and fully understand their application.
Shortcut Rules For Reducing Any Complex Block Arrangement !!
Rule 1: Blocks In Series Any finite specific number of blocks arranged in series can be combined together by multiplication as shown below: The above blocks shown can be combined together and replaced with single block as Output C(s) = G1 x G2 x R(s) If there is a take-off point or summing point between the blocks, the blocks cannot be sa to be in cascade/series.(the take-off/summing point has to be shifted before or after the block using another rule) Rule 2: Blocks In Parallel When the blocks are connected in parallel combination, they get added algebraically (consering the sign of the signal) this can be combined as(refer both the diagrams) The above blocks can be replaced with a single block as C(s) = R(s)G1 + R(s)G2 – R(s)G3 C(s) = R(s) (G1 + G2 – G3) If any summing point/take-off point is present in between the blocks, then that has to be shifted first.(in a parallel arrangement, the direction of signal flow must be in the same direction through all the blocks) Rule 3: Elimination of feedback Loop We can use Closed loop transfer function to eliminate the feedback loop present.(Always remember for applying this method the direction of flow of signals should be in opposite direction, otherwise, if they are in the same direction, then we need to apply parallel reduction technique discussed above) Now conser the application of the above three rules together and refer to the block diagram above. Rule 4: Associative Law For Summing Point This can be better explained by taking below diagram Y= R(s) – B1 C(s) = y – B2 = R(s) – B1 – B2 This law is applicable only to summing points that are connected directly to each other. Note: If there is a block present between two summing points(and hence they are not connected directly) then this rule can’t be applied. Rule 5: Shifting of a Summing Point before a block When we shift the summing point before a block, we need to do the transformation in order to achieve the same result. Please refer to the diagram below : C(s) = GR(s) + X After shifting the summing point, we will get C(s) = [R +(X/G) ] G = GR + X which is same as output in the first case. Hence to shift a summing point before a block, we need o to add another block of transfer function ‘1/G’ before the summing point as shown in figure. Rule 6: Shifting of the Summing Point after a block When we generally shift the summing point after any block, we required to do the transformation to attain the same (required) result. Please refer the below diagram . C(s) = (R + X)G After shifting the summing point, we will get C(s) = (R +X) G = GR + XG which is same as output in the first case. Hence to shift a summing point before a block, we need to add another block having the same transfer function at the summing point as shown in fig Rule 7: Shifting of Take-off point after a block Here we want to shift the take – off point after a block, as shown in the diagram Here we have X = R and C = RG (initially) In order to achieve this, we need to add a block of transfer function ‘1/G’ in series with signal taking off from that point. Rule 8 : Shifting of Take-off point before a block Here we want to shift the take – off point before a block, as shown in the diagram Here we have X = R and C = RG (initially) In order to achieve this, we need to add a block of transfer function ‘G’ in series with X signal taking off from that point. Rule 9 : Shifting a Take-off point after a Summing Point can be transformed to (refer both the diagrams) Before shifting take-off point, initially, we have: C(s)=R±Y and Z = R ± Y (initially) Hence if we want to shift a take-off point after a summing point, one more summing point needs to be added in series with take-off point. Rule 10: Shifting a take-off point before a summing point Suppose if we want to shift take-off point before a summing point, then initially we have C(s)=R±Y and Z = R ± Y (initially) this can be transformed to (refer both the diagrams) In order to satisfy this condition, we need to add a summing point in series with the take-off point. Note: 1. Rules 9 and 10 are critical rules, their usage should be avoed as much as possible(i.e if there is no other options present then only we should use them). 2. Always try to shift take-off points towards right and summing points towards left If there are multiple inputs present, then we have to use the superposition theorem(which means we have to conser one input at a time and find out the output consering all other inputs to be zero and finally taking the summation of all the output thus obtained).
Advantages Of Block Reduction Technique !!
Though there are many other methods present to solve any complex block arrangements (seen in the later posts) , this block reduction rules often come handy due to the following reasons : 1. With these blocks reducing shortcuts, you can easily break a complex LTI (linear time-invariant) control system into simple blocks and then reduce it to get the required output easily . 2. You can indivually visualize the various components of any given complicated control system easily , which can be helpful while analyzing any issue or to study separate parts of any LTI system . Finally, we reach the end of this wonderful(time-consuming :-p) post. Block reduction technique is one of the many (other methods discussed in later posts) methods available to deal with complex block diagrams in the Control System. Hope you really liked this post , let me know your thoughts in the comments section. Stay tuned for the next post in this series (with solved examples for block diagram reduction related to this topic). Read More : 1. Complete analysis of Bode Plot 2. A to Z about the Root Locus in Control System 3. Complete overview of the Polar Plot 4. Details about the Nyquist Plot report this ad
What are the various rules for block reduction technique explain any seven of them?
- Rule 1: Blocks in cascade/series.
- Rule 2: Blocks in parallel.
- Rule 3: Feedback loop elimination.
- Rule 4: Associative law for summing.
- Rule 5: Shifting a summing point before a block.
- Rule 6: Shifting a summing point after a block.
- Rule 7: Shifting take-off point before a block.
What is meant by block diagram reduction technique?
A pictorial representation of the functions performed by each component and of the flow of signals. Block diagram. A pictorial representation of the functions performed by each component and of the flow of signals.
What is the rule to move a take-off point after a block in block diagram reduction?
Shifting of take-off point ahead of the block
If we need to shift the take-off point ahead of the block, then we must keep ‘p’ as it is. So, even after shifting p must be X(s) and for this, we have to add a block with gain which is reciprocal of the gain of the originally present block.
Why is block diagram Reduction important?
The use of block diagrams to illustrate a cause-and-effect relationship is prevalent in control. We use operational blocks to represent transfer functions and lines for unidirectional information transmission. It is a nice way to visualize the interrelationships of various components.
What is the basis for framing the rules of block diagram reduction technique?
5. What is the basis for framing the rules of block diagram reduction technique? The rules for block diagram reduction technique are framed such that any modification made on the diagram does not alter the input output relation.
What are the advantages and disadvantages of block diagram reduction technique?
- Advantages of Block Diagram Representation.
- • Very simple to construct block diagram for a complicated system.
- • Function of individual element can be visualized.
- • Individual & Overall performance can be studied.
- • Over all transfer function can be calculated easily.
- Disadvantages of Block Diagram Representation.
What are the types of control system?
- Open loop control systems (non-feedback control systems)
- Closed loop control systems (feedback control systems)
What is the simplest form of block diagram?
The simplest form of the block diagram is the single block, with one input and one output, as shown in Fig. 2-1.
How do you block G1 and G2 connected in parallel are combined?
Parallel Connection
In the following figure, two blocks having transfer functions G1(s) and G2(s) are connected in parallel. The outputs of these two blocks are connected to the summing point. Compare this equation with the standard form of the output equation, Y(s)=G(s)X(s).
What is stability in control system?
What is Stability? A system is said to be stable, if its output is under control. Otherwise, it is said to be unstable. A stable system produces a bounded output for a given bounded input. The following figure shows the response of a stable system.
What is gain in control system?
Gain is a proportional value that shows the relationship between the magnitude of the input to the magnitude of the output signal at steady state. Many systems contain a method by which the gain can be altered, providing more or less “power” to the system.
What are the basic components of block diagram?
The basic elements of a block diagram are a block, the summing point and the take-off point.
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