# poles and zeros of control system

Does number of poles always equal to number of zeros in a control system? means if number poles is greater than number of zeros then remaining number of zeros lie at "infinity". I found this concept in the book "Linear Control Systems, with Matlab Applications: B S Manke". But there is no explanation.

No, the number of poles does not need to be equal to the number of zeros. One could be greater/lesser than the other. For instance, a differentiator or integrator could have an unequal number of poles and zeros. It is simply the nature of the control system that determines this relation.

The assumption that zeros lie at infinity is done only in the case of root locus plots. These zeros are only introduced to complete the definition of the root locus equation, but do not actually occur when you take a look at the transfer function.

Generally no. Whatever your book said, it's not that there are always as many zeros as poles. For instance, if H is a strictly proper transfer function, then unity feedback on H gives

$$H_c = \frac{H}{1+H} = \frac{N_H}{D_H+N_H},$$

where N_H and D_H are the numerator and denominator of H respectively. Since H is strictly proper, deg D_H > deg N_H, hence deg (D_H + N_H) > deg N_H, so the closed loop transfer function is also strictly proper. This always means that the number of the poles, not including their order, is greater than the number of zeros (not including their multiplicity). As order and multiplicity count toward the shape of the frequency response and timescales of the transient response, the overall system is minimum phase.

If you still protest that the poles may not be distinct, not that one may easily find specific H for which each pole is simple and each zero has multiplicity one. This system will be minimum phase.

There is no such thing as a "zero at infinity"--each root of a polynomial over an algebraically closed field is contained in that field, and in the case of the complex numbers there are at most as many zeros as the order of the polynomial.