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Parallel Connection Y  G1 X  G 2 X  ( G1  G 2 ) X 5. Series Connection Z  G2 Y Y  G1 X Z  G1G 2 X 6. Loop With loop gain = G1G2H 7. 1 shows a signal flow graph of a system described by five state variables, X1, X 2 , X 3 , X 4 and X 5 . This graph was constructed for a system described by the following equations, where ti is a transfer function. 1 Signal flow graph The transfer function of the system can be obtained from the signal flow graph, in a relatively simple way, by applying Mason's formula.

25) This is a unity gain first order transfer function of time constant T=2/K. 29) v = Applied electric potential difference, V. 7, can be described by the following equations. 31) Assuming zero initial conditions, the application of Laplace transform to Eq. 4 Second Order Element The second order element has a transfer function of the form: G(s)  k . 8 Quarter car (mass-spring-damper) system The quarter car model shown by Fig. 8, is a typical example of the second order element. Assuming ideal linear spring and damper and neglecting the spring inertia, the following equation can be deduced.

21 indicates that the output y(t) is directly proportional to the time integral of the input  e(t) dt . Therefore the shown valve controlled actuator is said to be an integrating element. 3 First Order Element The first order element has a transfer function of the form: G(s)  k 1 / Ts  1 . It is also called simple lag element. The following are some examples of first order elements. 5, is actuated by a low power input signal x(t) and produces a corresponding sufficiently high power output signal y(t).

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Automatic Control for Mechanical Engineers by Galal Rabie

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