By Galal Rabie
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Additional resources for Automatic Control for Mechanical Engineers
Conventionally, the symbol G(s) is used for the transfer function, but if the element appears in the feedback path, the symbol H(s) is used. The linear, single-input single-output, system can be described by a single linear differential equation of zero initial conditions. This differential equation (DE), can be transformed into Laplace domain by replacing the term (d/dt) by (s). 2) For real systems, the order of P(s) should not exceed that of Q(s); n m . 1 Consider the linear system, of input x(t) and output y(t), described by the following differential equation of zero initial conditions.
17) 1 s(s 2) The inverse Laplace transform of this expression can be obtained after performing a partial fraction expansion as follows. e. Repeated real roots of the polynomial of the denominator). , n 1 s a 1 ds i! 10 Find the partial fractions of Y(s) Y( s ) s4 (s 1)(s 2) 2 (s 1)Y(s) 3 k 3 lim ( s 2 ) 2 Y ( s ) 2 s 2 (s 1)(s 2) 2 k3 k1 k 2 s 1 s 2 ( s 2) 2 k 1 lim s 1 s4 (s 1) (s 4) d s 4 d (s 2) 2 Y(s) lim lim k 2 lim 3 s 2 ds (s 1) 2 s 2 ds s 1 s 2 Y( s ) 3 3 2 s 1 s 2 ( s 2) 2 Or, calculate k1 and k3 as shown above then calculate k2 as follows.
Output node (Sink): is a node, which has only ingoing branches X5. Forward path: is a path from input node to output node along which no node is encountered more than once. Feedback loop: is a path, which originates and terminates at the same node, along which no node is encountered more than once. Path gain: is the product of all gains (transfer functions) along the forward path. 1. M1 t 12 t 23 t 34 t 45 M2 t12 t 24 t 45 M3 t12 t 25 Loop gain: is the product of the gains of branches forming the loop.