Root locus of a sampled system (T = 0.5 s)

The root locus is drawn in the z-plane for the open-loop pulse transfer function with gain K in the forward path.

Open-loop pulse transfer function

With $T = 0.5$ s and $e^{-0.5} = 0.6065$:

$$\begin{aligned} G(z) &= K(1-z^{-1})\,\mathcal{Z}\left[\frac{1}{s^2(s+1)}\right] = K\,\frac{(T-1+e^{-T})z + (1-e^{-T}-Te^{-T})}{(z-1)(z-e^{-T})}\\ &= \frac{K(0.1065z + 0.0902)}{(z-1)(z-0.6065)} = \frac{0.1065K(z+0.8467)}{(z-1)(z-0.6065)} \end{aligned}$$

• Open-loop poles: $z = 1$, $z = 0.6065$
• Open-loop zero: $z = -0.8467$ (the other branch ends at $-\infty$)

Construction

1. Real-axis locus: between 0.6065 and 1, and to the left of −0.8467.
2. Breakaway / break-in points from $dK/dz = 0$, where $K = -\dfrac{(z-1)(z-0.6065)}{0.1065(z+0.8467)}$: $$z^2 + 1.6935z - 1.9667 = 0 \;\Rightarrow\; z = 0.7915\ \ (K = 0.221),\quad z = -2.4850\ \ (K = 61.7)$$
3. Complex part: for one real zero and two real poles the complex part is a circle centred at the zero: centre $= -0.8467$, radius $= \sqrt{(1+0.8467)(0.6065+0.8467)} = 1.638$.
4. Crossing of the unit circle: characteristic equation $$z^2 + (0.1065K - 1.6065)z + (0.6065 + 0.0902K) = 0$$ For complex roots $|z|^2 = 0.6065 + 0.0902K = 1$, so $K_c = 4.36$. The roots are then $z = 0.571 \pm j0.821$ (angle 55.2°, $\omega = 1.93$ rad/s).

Sketch

                         Im
                          |
              .  -  -  -  | #  K = 4.36: z = 0.57 + j0.82 (on unit circle)
          .               |     .
        .                 |       .
 <======*=========o-------+----x==*==x----> Re
      -2.48     -0.85     0  0.61 0.79 1
    break-in    zero              breakaway
        .                 |       .
          .               |     .
              .  -  -  -  | #  K = 4.36: z = 0.57 - j0.82
                          |
  x = open-loop pole, o = open-loop zero, * = break point,
  # = unit-circle crossing, == real-axis locus
  Complex branches: circle, centre -0.85, radius 1.64

Result

• For $0 < K < 0.221$: two real closed-loop poles between 0.6065 and 1 (overdamped).
• For $0.221 < K < 61.7$: complex poles on the circle. They lie inside the unit circle only while $K < 4.36$.
• The system is stable for $0 < K < 4.36$; at $K = 4.36$ it oscillates at about 1.93 rad/s.
• The input $R(s) = 1/s$ does not change the locus; it only decides the step response.