Process Control and Instrumentation
A typical example of a physical system with under damped characteristic is a

Spring loaded diaphragm valve
U-tube manometer
Thermocouple kept immersed in a liquid filled thermowell
CSTR with first order reaction

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Process Control and Instrumentation
Routh test

Can not determine as to how many roots of the characteristics equation have positive real roots
Cannot be used to test the stability of a control system containing transportation lag
Criterion is not applicable to systems with polynomial characteristic equation
Criterion provides information about the actual location of roots

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