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lec15 - 15 TRANSFER FUNCTIONS STABILITY The reader is...

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15 TRANSFER FUNCTIONS & STABILITY The reader is referred to Laplace Transforms in the section MATH FACTS for preliminary material on the Laplace transform. Partial fractions are presented here, in the context of control systems, as the fundamental link between pole locations and stability. 15.1 Partial Fractions Solving linear time-invariant systems by the Laplace Transform method will generally create a signal containing the (factored) form Y ( s ) = K ( s + z 1 )( s + z 2 ) · · · ( s + z m ) . (190) ( s + p 1 )( s + p 2 ) · · · ( s + p n ) Although for the moment we are discussing the signal Y ( s ), later we will see that dynamic systems are described in the same format: in that case we call the impulse response G ( s ) a transfer function. A system transfer function is identical to its impulse response, since L ( ζ ( t )) = 1. (Continued on next page)
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74 15 TRANSFER FUNCTIONS & STABILITY The constants z i are called the zeros of the transfer function or signal, and p i are the poles. Viewed in the complex plane, it is clear that the magnitude of Y ( s ) will go to zero at the zeros, and to infinity at the poles. Partial fraction expansions alter the form of Y ( s ) so that the simple transform pairs can be used to find the time-domain output signals. We must have m < n ; if this is not the case, then we have to divide
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