1.2 Pole Zero Analysis

  • The dynamic output \(y(t)\) of the system for changes in the input \(x(t)\) can be written in a polynomial form as: \[a_ny^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_1y^{(1)} + a_oy = b_mx^{(m)} + b_{m-1}x^{(m-1)} + \cdots + b_1x^{(1)} + b_ox\] where \(y^{(n)}\) is the \(n^{\text{th}}\) derivative of \(y\). With \(n\ge m\), and with zero initial conditions, i.e., \(y^{(n)}=y^{(n-1)}=y=0\) at \(t=0\), we can write the above differential equation using Laplace transform as below: \[\frac{Y(s)}{X(s)} = \frac{b_m s^m + b_{m-1} s^{m-1} + \cdots + b_1 s + b_o}{a_n s^n + a_{n-1} s^{n-1} + \cdots + a_1 s + a_o} = G(s) = \frac{Q(s)}{P(s)}\] This ratio \(\dfrac{Y(s)}{X(s)}=G(s)\) is called as the transfer function of the system.

    The above equation in pole-zero form is given as \[G(s) = \frac{Q(s)}{P(s)} = K\frac{(s-z_1)(s-z_2)\cdots (s-z_m)}{(s-p_1)(s-p_2)\cdots (s-p_n)}\] where \(p\) are poles and \(z\) are zeros of the transfer function. The zeros affect only the coefficients of the solution \(y(t)\), but not the time dependent functions. Therefore in qualitative discussions, the focus is only on the poles.

  • ‘Poles’ are values of \(s\) that make the denominator of transfer function zero. Similarly, ‘zeros’ are values of \(s\) that make the numerator of transfer function zero.

  • Pole is defined as the frequency at which the transfer function of the system becomes infinite. Zero is defined as the frequency at which the transfer function of the system becomes 0.

  • The poles and zeros are properties of the transfer function, and therefore of the differential equation describing the input-output system dynamics. Together with the gain constant \(K\) they completely characterize the differential equation, and provide a complete description of the system.

  • For a physically realizable system, the order of the numerator of transfer function must be less than or equal to that of the denominator. e.g.: \[\begin{aligned} {3} G_1(s) &= \frac{(s+1)(s+2)(s+3)}{(s+4)(s+5)} \qquad& \text{(Physically unrealizable)} \\ G_1(s) &= \frac{(s+1)(s+2)}{(s+4)(s+5)} & \text{(Physically realizable)}\end{aligned}\]