# Control Systems Unit2 (QB)

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Question Bank Unit 2: Block diagrams and signal flow graphs: Transfer functions Block diagram algebra signal flow graphs state variable formulation excluded Introduction: To understand and control complex systems one must obtain quantitative mathematical models of these systems. It is necessary therefore to analyze the relationships between the system variables and to obtain a mathematical model. Because the systems under consideration are dynamic in nature the descriptive equations are usually differential equations. Furthermore if these equations can be linearized then the Laplace transform can be utilized to simplify the method of solution. • The transfer function of a linear system is defined as “the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable with all initial conditions assumed to be zero”. The transfer function of a system or element represents the relationship describing the dynamics of the system under consideration. s R s C s G input Output function Transfer A transfer function may be defined only for a linear stationary constant parameter system. • The importance of the cause-and-effect relationship of the transfer function is evidenced by the facility to represent the relationship of system variables by diagrammatic means. • A block diagram representation of the system relationships is prevalent in control system engineering. • Block diagram consist of unidirectional operational blocks that represent the transfer function of the variables of interest. Block diagram transformations and reduction techniques are derived by considering the algebra of the diagram variables. • System analysis by the method of block diagram reduction affords a better understanding of the contribution of each component element than is possible to obtain by the manipulation of equations. • The Block diagram provides the analyst with a graphical representation of the interrelationships of controlled and input variables. • Block diagrams are adequate for the representation of the interrelationships of controlled and input variables. However for a system with reasonably complex interrelationships the block diagram reduction procedure is cumbersome and often difficult to complete. • The transition from a block diagram representation to a directed line segment representation is easy to accomplish by reconsidering the systems of the previous block diagram method. • The advantage of the line path method called the signal-flow graph method is the availability of a flow graph gain formula which provides the relation between system variables without requiring any reduction procedure or manipulation of the flow graph.

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• Signal –flow graph is a diagram consisting of nodes that are connected by several directed branches and is a graphical representation of a set of linear relations. Properties of Signal Flow Graphs: • A branch indicates the functional dependence of one signal on another. A signal passes through only in the direction specified by the arrow of the branch. • A node adds the signals of all incoming braches and transmits this sum to all outgoing branches • A mixed node which has both incoming and outgoing branches may be treated as an output node sink by adding an outgoing branch of unity transmittance. • For a given system a signal flow graph is not unique. Many different signal flow graphs can be drawn for a given system by writing the system equations differently. Signals flow graphs are particularly useful for feedback control systems because feedback theory is primary concerned with the flow and processing of signals in systems. • Mason’s Gain Formula for Signal Flow Graphs: In many practical cases we wish to determine the relationship between an input variable and an output variable of the signal flow graph. The transmittance between an input node and an output node is the overall gain or overall transmittance between these two nodes. Mason’s gain formula which is applicable to the overall gain is given by ∆ ∆ ∑ K K K P s R s C s T gain overall Where path forward k the of ce transmit or Gain P th K tan − − − − − − − + − + − ∆ 1 min det loops g nontouchin three of ns combinatio possible the all of products gain of Sum loops g nontouchin two of ns combinatio possible the all of products gain of Sum gains loop individual different the all of Sum graph the of ant er − − − − − − + − + − ∆ ∑ ∑ ∑ f f e d e d c b c b a a L L L L L L 1 loops g nontouchin two of ns combinatio possible all of products gain of Sum L L c b c b ∑ . K K P the of cofactor the ∆ path forward K the touching not loops by formed but as same th ∆ In summary the approach to dynamic system problems can be listed as follows: 1. Define the system and its components. 2. Formulate the mathematical model and list the necessary assumptions. 3. Write the differential equations describing the model. 4. Solve the equations for the desired output variables. 5. Examine the solutions and the assumptions. 6. If necessary reanalyze or redesign the system. gain loop individual all of Sum L a a ∑

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1. Define transfer function and what its properties are. 05 2. Obtain the transfer function for the block diagram shown in figure using block diagram reduction method. 07 3. For the block diagram given in figure obtain over all transfer function using Mason’s gain formula. 08 4. Illustrate how to perform the following in connection with block diagram reduction rules: i shifting a take-off point after a summing point. ii shifting a take-off point before a summing point. 04 5. The performance equations of a controlled system are given by the following set of linear algebraic equations: i draw the block diagram. ii Find the overall transfer function . using block diagram reduction technique 1 − 3 . . 2 1 − 1 . 4 . 3 2 1 . 2 − 2 . . 4 2 . 3 . C 3 . 4 08 6. Draw the corresponding signal flow graph for the given block diagram is shown in figure and obtain the overall transfer function by Mason’s gain formula. 08

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7. Obtain the transfer function for the block diagram shown in figure using block diagram reduction technique. 10 8. Obtain the closed loop transfer function for the signal flow graph of a system shown in figure by use of Mason’s gain formula. 10 9. Using block diagram reduction technique find the overall transform function of the system represented by the block diagram shown in figure. 10 10. Using Mason’s gain formula. Obtain 8 1 for the signal flow graph shown in figure. 10 11. Find using Mason’s gain formula for the signal flow graph shown in the figurea. 08 12. Derive an expression for the closed loop transfer function of a negative feedback system. 04 Fig. a

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13. Find the overall transfer function using block diagram reduction technique for the system shown in figure a. 08 14. Explain the block diagram rule regarding: i combining block in cascade… ii Moving a summing point after a block iii Moving a take off point beyond a block. 06 15. Reduce using block diagram reduction technique the circuit shown in figure and hence find its transfer function . 08 16. Reduce using block diagram reduction technique the circuit shown in figure and hence find its transfer function . 08 17. For the network shown in figure construct the signal flow graph. 08

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18. For the signal flow graphs shown in figure determine the transfer function using mason’s gain formulae. 08

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19. For the signal flow graphs shown in figure determine the transfer function using mason’s gain formulae. 08

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20. For the electrical network shown in figure draw signal flow graph and hence obtain transfer function of V os/Vs 08 21. For the electrical network shown in figure draw signal flow graph and hence obtain transfer function of V os/Vs 08 Mr. Y.L.ChandraShekar Professor Dept. of EC Engg MIT Mysore.