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

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Process Control and Instrumentation
In a closed loop system, the process to be controlled is an integrating process with transfer function l/2s. The controller proposed to be used in an integral controller with transfer function 1/T₁s. When a step change in set point is applied to such a closed loop system, the controlled variable will exhibit

Underdamped response
Unstable response
Overdamped response
Undamped response

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