## ASSIGNMENT:2 question3

POLES AND ZEROS The transfer function provides a basis for determining important system response characteristics without solving the complete differential equation. A transfer function is defined as a ratio of two polynomials: H(S)= N(S)/D(S) Where N(s) and D(s) are simple polynomials These can be written as: N(s)=(s − z1)(s − z2) . . . (s − zm−1)(s − zm) D(s)=(s − p1)(s − p2) . . . (s − pn−1)(s − pn) , Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s. Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s. All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must be either purely real, or appear in complex conjugate pairs. Poles and Zeros of a transfer function are the frequencies for which the value of the transfer function becomes infinity or zero respectively.As s approaches a zero, the numerator of the transfer function (and therefore the transfer function itself) approaches the value 0. When s approaches a pole, the denominator of the transfer function approaches zero, and the value of the transfer function approaches infinity.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the simplest sense, can be designed simply by assigning specific values to the poles and zeros of the system. A system is characterized by its poles and zeros .Because the transfer function completely represents a system differential equation, its poles and zeros effectively define the system response. In particular the system poles directly define the components in the homogeneous response. The locations of the poles, and the values of the real and imaginary parts of the pole helps to determine the response of the system. The stability of a linear system may be determined directly from its transfer function. An nth order linear system is asymptotically stable only if all of the components in the homogeneous response from a finite set of initial conditions decay to zero as time increases.In order for a linear system to be stable, all of its poles must have negative real parts. Reference: http://en.wikibooks.org/wiki/Control_Systems/Poles_and_Zeros