feedback cannot be maintained. If loop 2 is reduced, the signal between G

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1 ME 3600 Control Systems Block Diagram eduction Examples Example :.C. Dorf and.h. Bishop, Modern Control Systems, th Ed., Pearson Prentice-Hall, 008. Given: The system shown in the block diagram has one input signal, s, () and one output signal, () s. Find: Using block diagram reduction, find the transfer function () s. Example.7 from Dorf and Bishop Solution: First, it should be noted that there are many solution paths that can be taken to reduce complex systems using block diagram reduction techniques. The system shown here is not very complex, and yet, there are at least half a dozen approaches that can be taken. To encourage the reader to consider different solution paths, multiple approaches are presented here. As shown, the system has three feedback loops. Loops and are nested in loop 3, but loops and are overlapping. To determine if two loops are overlapping, consider collapsing one of the two loops. If any signals required by the second loop would be removed by collapsing of the first, then the two loops are overlapping. In this system, if loop is reduced, the summing block between G and G 3 is eliminated, so the H feedback cannot be maintained. If loop is reduced, the signal between G 3 and G is eliminated, so the H feedback cannot be maintained. Approach : In this first approach, loop is nested within loop by expanding loop as shown in the diagram below. This is accomplished by moving the pick-off point of loop to be behind G. This requires that feedback transfer function H be divided by G. Loop is now nested within loop which is nested within loop 3. Hence, reduction of loop, followed by reduction of loop, and finally the reduction of loop 3 will produce the system transfer function. Kamman Control Systems Block Diagram Examples page: /8

2 Loop transfer function: G G3G GH H G G 3 (positive feedback) Transfer functions G 3 and G are combined, then the positive feedback loop is collapsed. Now, substitute this result into the original diagram and then reduce the modified loop. Loop transfer function: G GH 3G HG3G G G G H H G G G 3 3 3G H G G H G G 3 3 Now, substitute this result into the diagram and reduce the modified loop 3 to find the system transfer function. Approach : G3G G HG3G H3 G3G s GH G H G G H G G H G G G G 3G H3 HG3G H3 In this approach, loop is nested within loop by contracting loop as shown in the diagram. This is accomplished by moving the pick-off point of loop ahead of G. This requires that feedback transfer function H be multiplied by G As in the first approach, loop is now nested within loop which is nested within loop 3. Kamman Control Systems Block Diagram Examples page: /8

3 Loop transfer function: G G3 GH H G G 3 (positive feedback) Now, substitute this result into the original diagram and then reduce the modified loop. Loop transfer function: 3 G H G3G 3 GH H G G H G G 3 H HG3G 3 3 Substituting this result into the diagram and reducing loop 3 gives the system transfer function. G3G G HG3G H3 G3G s GH G H G G H G G H G G G G 3G H3 HG3G H3 Approach 3: Loop can also be nested within loop by expanding loop as shown in the diagram below. This is accomplished by moving the summing block of loop ahead of G. This requires that feedback transfer function H be divided by G. Now, loop is nested within loop which is nested within loop Kamman Control Systems Block Diagram Examples page: 3/8

4 Loop transfer function: G 3 GH H G G 3 Now, substitute this result into the original diagram and then reduce the modified loop. Loop transfer function: G GH 3G H3 H G 3 G H G G 3 3G H G G H G G 3 3 (positive feedback) Substituting this result into the diagram and reducing loop 3 gives the system transfer function. G3G G H3 HG3G G3G s GH G H G G H G G H G G G G 3G H3 H3 HG3G Example : Benjamin C. Kuo, Automatic Control Systems, 7 th Ed., Prentice-Hall, 995. Given: The system shown in the block diagram has two input signals, s () and Ns, () and one output signal, () s Find: Using block diagram reduction, find the transfer functions () s N and () s. Problem 3-7 from Kuo Kamman Control Systems Block Diagram Examples page: /8

5 Solution: When a system has more than one input signal, transfer functions are found by setting all but one of the input signals to zero. So, in this case, if both input signals are non-zero, then the output can be written as ( s) s N( s) s ( s) N It is understood (but not always written) that ( ) 0 finding N s N s when finding s, and Ns ( ) 0 when s. So, each transfer function contains information about a single input, only. : Approach Setting s ( ) 0 in the diagram above gives the diagram on the right. The diagram shows the output () s is fed back through the transfer functions G and H, and those two signals are combined at a summing block before passing through G. To show this more clearly, the system is redrawn with the input signal on the left and the output signal on the right. Note that in the second diagram the signals are added before passing them through G, and then the combined signal is negated at the next summing block. This is equivalent to the sign changes made in the original diagram. To simplify this diagram, note in the feedback loop that () s is passed through both G and H, and then the two signals are added together. This process is the same as passing () s through the transfer function G H. The system transfer function can now be calculated as follows. Kamman Control Systems Block Diagram Examples page: 5/8

6 G s N GH G G H G G HG Note here that the transfer functions G and G are forward-path transfer functions for the input s, () but they are feedback-path transfer functions for the input Ns. () N s : Approach In the solution above, Ns () was kept at the same summing block as in the original diagram. As an alternative, consider moving Ns () to the middle summing block. To do this, Ns () must be divided by G as shown in the next diagram. Also, the signals for the H and G paths have been added together before negating the sum of the two signals at the next summing block. As noted above, the transfer functions in the parallel paths can be summed. G s N G GH G G G G H G G HG (as before) Note that Ns () could also be moved to the first summing block. In this case, it would have to be divided by the product. s : Approach Setting Ns ( ) 0 in the original diagram gives the diagram on the right. To get started, the input signal s () needs to be isolated to a single summing block. Kamman Control Systems Block Diagram Examples page: 6/8

7 Moving the output of G 3 to the summing block ahead of G requires the new signal be divided by G. Isolating the input diagram for s () gives the diagram on the right. 3 3 Input transfer function: G G G (parallel paths) G G Substituting this result into the original diagram gives the diagram on the right. Now, the two feedback loops can be reduced, and that result can be cascaded with the input transfer function. G G Inner feedback loop transfer function: GH HG Outer feedback loop transfer function: G G HG GH G HG G G HG s G G G 3 s G G G G G3 HG G G HG G G : Approach In this second approach, consider moving the outer feedback loop return to the second summing block as shown in the diagram. This requires the signal to be multiplied by G. In the diagram, the two feedback signals are added first and then negated at the next summing block. (negative unity feedback) Note here that the signal entering G (which ultimately gets fed back through the two feedback loops) is the same in this diagram as it is in the original. Hence, the diagrams are equivalent. Kamman Control Systems Block Diagram Examples page: 7/8

8 Combining the parallel paths into single transfer functions, the diagram can be redrawn as shown here. The system transfer function can now be calculated by reducing the closed loop and cascading the result with G G3. s G G Example 3: G G G G G H G HG G G Given: The system shown in the block diagram has one input signal, s, () and one output signal, () s. Find: Using block diagram reduction, find the transfer function () s. 3 3 (as before) Solution: It is tempting with this system to simply separate the two feedback loops, but this approach leads to incorrect results. As the diagram indicates the two feedback loops are overlapping. If the upper loop is collapsed, the second summing block is removed and the lower loop cannot be completed, and if the lower loop is collapsed the pick-off point for the upper loop is removed and the upper loop cannot be completed. When two feedback loops overlap, one of the loops should be expanded to nest the other loop within it. In this example, the pick-off point for H could be moved behind G, or the output of H could be moved ahead of G. The latter approach is taken here. Inner loop transfer function: G G GH G H G GH s GH H G G G GH G H G H Kamman Control Systems Block Diagram Examples page: 8/8