Direct Substitution For Stability Limit

The direct-substitution method is a simple and useful method for finding the values of parameters m the characteristic equation that put the system just at the limit of stability. 

We know the system is stable if all the roots of the characteristic equation are in the LHP and unstable if any of the roots are in the RHP. Therefore the imaginary axis represents the stability boundary. On the imaginary axis s is equal to some pure imaginary number s = ioi. 

The technique consists of substituting iai for s in the characteristic equation and solving for the values of co and other parameters (e.g., controller gain) that satisfy the resulting equations. The method is best understood by looking at the example below. 

Example 10.4. Consider again the three-CSTR system. We have already developed its doscdloop characteristic equation with a proportional controller [Equation (10.22)]. 

Equating the real and imaginary parts of the left- and right-hand sides of the above equation gives two equations  

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