Best to do some searches about this but here's a good site:

http://www.engr.udayton.edu/faculty/rkashani/mee527/nyq_stab/nyq_stability.htm
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The Nyquist criterion is used to analyze a system's stability using frequency response methods. The RouthHurwitzCriterion ( http://www-me.mit.edu/Lectures/RLocus/2.12-rationale.html) examines the characteristic polynomial. 

A linear system is stable if and only if all  
the poles of the system TransferFunction have negative real parts. ControllabilityAndObservability for a system can be determined from its TransferFunction.
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CategoryMath