Root Locus

Consider a unity feedback system with reference signal $r(t)$$r(t)$, output signal $y(t)$$y(t)$, plant $P(s)$$P(s)$ and controller $C(s)$$C(s)$.

If we define $L(s)$$L(s)$ such that,

Then, transfer function from $r$$r$ to $y$$y$ becomes:

Thus the system is stable if and only if,

Rules for Plotting Positive Root Locus

The positive root locus is the set of all points $s$$s$ in the complex plane for which $\mathrm{\angle }L(s)=(2l+1)\pi$$\angle L(s) = (2l + 1)\pi$ radians (where $l$$l$ is an integer).

Rule 1.

The $n$$n$ branches begin at the open loop poles (when $K=0$$K = 0$). Of the $n$$n$ branches, $m$$m$ end at the open loop zeros (when $K=\mathrm{\infty }$$K = \infty$).

Rule 2.

The positive root locus contains all points on the real axis that are to the left of an odd number of zeros and poles.

Rule 3.

Of the $n$$n$ branches in root locus, $n-m$$n - m$ branches go to infinity, and asymptotically approach lines coming out of the point $s=\alpha$$s = \alpha$ with angles ${\mathrm{\Phi }}_l$$\Phi_l$, where

for $l=0,1,2,...,n-m-1$$l = 0, 1, 2, ..., n - m - 1$.

Rule 4.

The root locus will have multiple roots at $\overline{s}$$\bar{s}$ if the following are satisfied:

At a particular point $\overline{s}$$\bar{s}$ on the positive root locus,

where $p_1,p_2,...,p_n$$p_1, p_2,...,p_n$ and $z_1,z_2,...,z_m$$z_1, z_2,...,z_m$ are the zeros and poles respectively.

Rules for Plotting Negative Root Locus

The negative root locus is the set of all points $s$$s$ in the complex plane for which $\mathrm{\angle }L(s)=2l\pi$$\angle L(s) = 2l\pi$ radians (where $l$$l$ is an integer).

Rule 1.

The $n$$n$ branches begin at the open loop poles (when $K=0$$K = 0$). Of the $n$$n$ branches, $m$$m$ end at the open loop zeros (when $K=-\mathrm{\infty }$$K = -\infty$).

Rule 2.

The negative root locus contains all points on the real axis that are to the left of an even number of zeros and poles.

Rule 3.

Of the $n$$n$ branches in root locus, $n-m$$n - m$ branches go to infinity, and asymptotically approach lines coming out of the point $s=\alpha$$s = \alpha$ with angles ${\mathrm{\Phi }}_l$$\Phi_l$, where

for $l=0,1,2,...,n-m-1$$l = 0, 1, 2, ..., n - m - 1$.

Rule 4.

The root locus will have multiple roots at $\overline{s}$$\bar{s}$ if the following are satisfied:

At a particular point $\overline{s}$$\bar{s}$ on the negative root locus,

where $p_1,p_2,...,p_n$$p_1, p_2,...,p_n$ and $z_1,z_2,...,z_m$$z_1, z_2,...,z_m$ are the zeros and poles respectively.

NOTE: The root locus is always symmetric about the real axis provided all coefficients of $N(s)$$N(s)$ and $D(s)$$D(s)$ are real.