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1 © Copyright F.L. Lewis 1999 All rights reserved EE 4314 - Control Systems LECTURE 6 Updated: Sunday, February 14, 1999 BLOCK DIAGRAM AND MASON'S FORMULA A linear time-invariant (LTI) system can be represented in many ways, including: differential equation state variable form transfer function impulse response block diagram or flow graph Each description can be converted to the others. In this lecture we shall see how to represent systems in terms of block diagrams, and how to determine the transfer function of a block diagram system using Mason's Formula. SYSTEM INTERCONNECTIONS Systems can be interconnected in a variety of ways, including the following. Series Interconnection The overall transfer function in ) ( ) ( ) ( s U s H s Y = is given by the product ) ( ) ( ) ( 2 1 s H s H s H = . H 1 (s) H 2 (s) u(t) y(t)

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2 Parallel Interconnection The overall transfer function in ) ( ) ( ) ( s U s H s Y = is given by the sum ) ( ) ( ) ( 2 1 s H s H s H + = . Feedback Interconnection The overall transfer function in ) ( ) ( ) ( s U s H s Y = is given by ) ( ) ( 1 ) ( ) ( 2 1 1 s H s H s H s H + = . We shall see later how to derive this using Mason's Formula. Example 1- Transfer Function of Feedback Configuration An unstable plant has transfer function 1 1 ) ( 1 - = s s H which has one pole in the right-half plane at s=1 . To stabilize the plant, one may use a feedback configuration with compensator 5 ) 2 ( 10 ) ( 2 + + = s s s H . H 1 (s) H 2 (s) u(t) y(t) H 1 (s) H 2 (s) u(t) y(t)
3 The closed-loop transfer function is ) 83 . 12 )( 17 . 1 ( 5 15 14 5 20 10 ) 5 )( 1 ( 5 5 ) 2 ( 10 1 1 1 1 1 ) ( ) ( 1 ) ( ) ( 2 2 1 1 + + + = + + + = + + + - + = + + - + - = + = s s s s s s s s s s s s s s s H s H s H s H This is stable with two left-half plane poles. The compensator has stabilized the plant. Superposition for Linear Systems For LTI systems superposition holds, so that the effects of different inputs can simply be added together. The formula above can be applied several times and mixed together to derive the transfer function in many systems. Example 2- Superposition In this system, u(t) might represent the control input while d(t) is a disturbance input. The system can be viewed as two systems, one driven by input u(t) and one driven by input d(t) . Setting to zero u(t) one can determine that ) ( ) ( ) ( 2 s D s H s Y = . Setting to zero d(t), one can consider the transfer from u(t) to y(t) as two parallel branches of systems in series. Thus, the formulae above yield ) ( )] ( ) ( ) ( ) ( [ ) ( 4 3 2 1 s U s H s H s H s H s Y + = . Since the system is LTI, the overall output is the sum (superposition) of the effects of the two inputs ) ( ) ( ) ( )] ( ) ( ) ( ) ( [ ) ( 2 4 3 2 1 s D s H s U s H s H s H s H s Y + + = .

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Masonì˜ ì´ë“ê³µì‹

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