Stability and roots of the characteristic equation

The characteristic equation was defined in section 3.6.2 for a second-order system as 

The roots of the characteristic equation given in equation (5.5) were shown in section 3.6.2. to be 

These roots determine the transient response of the system and for a second-order system can be written as 

The oniy difference between the roots given in equation (5.9) and those in equation (5.i0) is the sign of the reai part. If the real part cr is negative then the system is stabie, but if it is positive, the system wiii be unstabie. This iioids true for systems of any order, so in generai it can be stated If any of the roots of the characteristic equation have positive reai parts, then the system wiii be unstabie . 

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