Process Control and Instrumentation
The Laplace transform of exp(at), where a > 0, is defined only for the Laplace parameter, s > a since

The Laplace transform integral of exp(at) has initial values only for s > a
The function exp(at) is piece-wise continuous only for s > a
The Laplace transform of integral of exp(at) has finite values only for s > a
The function is exponential

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Process Control and Instrumentation
Suppose that the gain, time constant and dead time of a process with the following transfer function:Gc(s) = 10 exp( - 0.1s)/(0.5s + 1) are known with a possible error of ± 20% of their values. The largest permissible gain Kc of a proportional controller needs to be calculated below taking the values of process gain, time constant and dead time as

8, 0.6, 0.12
12, 0.4, 0.08
12, 0.6, 0.12
8, 0.6, 0.08

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Process Control and Instrumentation
For a feed back control system to be stable, the

Poles of the closed loop transfer function should lie in the left half of the complex plane
Roots of the characteristic equation should be real
Bode plots of the corresponding open loop transfer function should monotoni-cally decrease
Poles of the closed loop transfer function should lie in the right half of the complex plane

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