Remember: Stability

A stable system is one for which the output response is bounded for all bounded inputs..

All real poles are –ve.. For imaginary poles all real parts are –ve..

+ 2öü)nS + S2. The general form of the characteristic equation of a 2nd order system.

52 —öOn — O —1. abstract. — öü.)n + On —1. . Both roots are real and negative.

Both roots are equal, real and negative. abstract.

Roots are complex conjugates with negative real parts.

Both roots are complex with no real parts. System is unstable.

Of the above cases, the underdamped response occurs most frequently in control systems. Hence a number of terms are used to describe the underdamped response quantitatively..

Decay ratio. T he ratio of the sizes of successive peaks and is given by C/A in Figure above ..

T he time required for the response to come within  5 percent of its ultimate value and remain there..

Substituting each S in the equation above by j  we obtain.

I —j2ö + 2ö. + 26 + 26 26. Ø = tan¯l 26 = —tan-.