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| موضوع: محاضرة يعنوان Control Systems - Block Diagram Reduction الأحد 09 أكتوبر 2022, 11:30 pm | |
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Cascade Form Parallel Form Feedback Form Moving Blocks Superposition Example Control Systems MEC-C305Block Diagram Reduction1 Block Diagram Reduction Cascade Form Parallel Form Feedback Form Moving Blocks Superposition ExampleComponents of a Block Diagram for a Linear Time Invariant System System components are alternatively called elements of the system. Block diagram has four components: Signals System/ block Summing junction Pick-off/ Take-off pointSubsystems are represented in block diagrams as blocks, each representing a transfer function. In this unit we will consider how to combine the blocks corresponding to individual subsystems so that we can represent a whole system as a single block, and therefore a single transfer function. Here is an example of this reduction: Reduced Form: Block Diagram ReductionFirst we summarize the elements of block diagrams: We now consider the forms in which blocks are typically connected and how these forms can be reduced to single blocks.Cascade Form When multiple subsystems are connected such that the output of one subsystem serves as the input to the next, these subsystems are said to be in cascade form. The algebraic form of the final output clearly shows the equivalent system TF—the product of the cascaded subsystem TF’s.When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other. That is, a subsystem’s output remains the same no matter what the output is connected to. If another subsystem connected to the output modifies that output, we say that it loads the first system. Consider interconnecting the circuits (a) and (b) below:When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other. That is, a subsystem’s output remains the same no matter what the output is connected to. If another subsystem connected to the output modifies that output, we say that it loads the first system. Consider interconnecting the circuits (a) and (b) below: The overall TF is not the product of the individual TF’s!We can prevent loading by inserting an amplifier. This amplifier should have a high input impedance so it does not load its source, and low output impedance so it appears as a pure voltage source to the subsystem it feeds into. If no actual gain is desired then K = 1 and the “amplifier” is referred to as a buffer.Parallel Form Graphs Parallel subsystems have a common input and their outputs are summed together. The equivalent TF is the sum of parallel TF’s (with matched signs at summing junction).Feedback Form Systems with feedback typically have the following form: Noticing the cascade form within the feedforward and feedback paths wecan simplify:We can easily establish the following two facts: E(s) = R(s) ∓ C(s)H(s) C(s) = E(s)G(s) We can now eliminate E(s) to obtain, G(s) Ge(s) = 1± G(s)H(s)Moving Blocks A system’s block diagram may require some modification before the reductions discussed above can be applied. We may need to move blocks either to the left or right of a summing junction:Or we may need to move blocks to the left or right of a pickoff point:Canonical Form of a Feedback Control System The system is said to have negative feedback if the sign at the summing junction is negative and positive feedback if the sign is positive.1. Open loop transfer function 2. Feed Forward Transfer function 3. control ratio 4. feedback ratio 5. error ratio 6. closed loop transfer function 7. characteristic equation 8. closed loop poles and zeros if K=10. H(s)Characteristic EquationUnity Feedback SystemCharacteristic Equation • The control ratio is the closed loop transfer function of the system. • The denominator of closed loop transfer function determines the characteristic equation of the system. • Which is usually determined as: 1 G(s)H(s) = 0Reduction techniques G1 G2 G1G2 1. Combining blocks in cascade 2. Combining blocks in parallel5. Moving a pickoff point ahead of a block 3. Moving a summing point ahead of a block 4. Moving a pickoff point behind a block Reduction techniques6. Eliminating a feedback loop G H 7. Swap with two neighboring summing points Reduction techniques 1 ∓ ????????Block Diagram Transformation Theorems The letter P is used to represent any transfer function, and W, X , Y, Z denote any transformed signals.Transformation Theorems Continue:Transformation Theorems Continue:Reduction of Complicated Block Diagrams:Example 1: Reduce the Following Block Diagram.Example 1 : Continue. However in this example step-4 does not apply. However in this example step-6 does not apply.Example 2: Simplify the Block Diagram.Example 2: Continue.Example 3: Reduce the Block Diagram.Example 3: Continue.Example 4: Reduce the Block Diagram.Example 4: Continue.Example 8: For the system represented by the following block diagram determine: 1. Open loop transfer function 2. Feed Forward Transfer function 3. control ratio 4. feedback ratio 5. error ratio 6. closed loop transfer function 7. characteristic equation 8. closed loop poles and zeros if K=10.First we will reduce the given block diagram to canonical form +Example 9: For the system represented by the following block diagram determine: 1. Open loop transfer function 2. Feed Forward Transfer function 3. control ratio 4. feedback ratio 5. error ratio 6. closed loop transfer function 7. characteristic equation 8. closed loop poles and zeros if K=100.Example-10: Reduce the system to a single transfer function.Example 10: Continue.Example 10: Continue.Example Reduce the following system to a single TF: First we can combine the three summing junctions together...We can now recognize the parallel form in the feedback path: We now have G1 cascaded with a feedback subsystem:Example 2 Reduce the following more complicated block diagram: Steps: Rightmost feedback loop can be reduced Create parallel form by moving G2 leftReduce parallel form involving 1/G2 and unity Push G1to the right past the summing junction to create a parallel form in the feedback pathReduce parallel form on left Recognize cascade form on rightReduce feedback form on leftSuperposition of Multiple InputsExample: Multiple Input System. Determine the output C due to inputs R and U using the Superposition Method.Example: Continue.Example: Continue.Example 15: Multiple-Input System. Determine the output C due to inputs R, U1 and U2 using the Superposition Method.Example-15: Continue.Example 15: Continue.Example 16: Multi-Input Multi-Output System. Determine C1 and C2 due to R1 and R2.Example 16: Continue.Example: Continue. When R1 = 0, When R2 = 0,
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