Root locus criteria; poles 0, −1±j2

Criteria of root locus

Every point $s$ on the root locus satisfies the characteristic equation $1 + G(s)H(s) = 0$, i.e. $G(s)H(s) = -1$. This gives two conditions:

1. Angle criterion: $\angle G(s)H(s) = \pm(2q+1)180^\circ$, $q = 0, 1, 2, \dots$ The sum of angles from the zeros minus the sum of angles from the poles must be an odd multiple of $180^\circ$. A point is on the locus only if it meets this condition.
2. Magnitude criterion: $|G(s)H(s)| = 1$. This gives the value of $K$ at a point already known to be on the locus:

$$K = \frac{\prod |s - p_i|}{\prod |s - z_j|}$$

Open-loop transfer function

Poles at $0$ and $-1 \pm j2$, no zeros:

$$G(s) = \frac{K}{s\left[(s+1)^2 + 4\right]} = \frac{K}{s(s^2+2s+5)}$$

Characteristic equation: $s^3 + 2s^2 + 5s + K = 0$.

Sketching steps

1. Branches: $n = 3$, $m = 0$, so there are 3 branches and all go to infinity.
2. Real-axis locus: the whole negative real axis, from $0$ to $-\infty$ (one pole to the right).
3. Asymptotes:

$$\sigma_A = \frac{0 - 1 - 1}{3} = -0.667, \qquad \theta_A = 60^\circ,\ 180^\circ,\ 300^\circ$$

4. Breakaway: $dK/ds = -(3s^2 + 4s + 5) = 0 \Rightarrow s = -0.667 \pm j1.106$. These are complex, so there is no breakaway point. The branch from the origin moves straight along the negative real axis to $-\infty$.
5. Angle of departure from $-1 + j2$: from pole $0$: $116.57^\circ$; from $-1-j2$: $90^\circ$.

$$\phi_d = 180^\circ - (116.57^\circ + 90^\circ) = -26.57^\circ$$

At $-1-j2$, it is $+26.57^\circ$. 6. $j\omega$-axis crossing (Routh):

$K_{mar} = 10$; auxiliary equation $2s^2 + 10 = 0 \Rightarrow s = \pm j\sqrt5 = \pm j2.236$.

Sketch

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                           x........X
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        -3                -1        0
x pole (0, -1 +- j2)
X jw crossing (+-j2.24, K = 10)

The complex poles leave at $\mp 26.6^\circ$, bend toward the ±60° asymptotes from $-0.667$, and cross the imaginary axis at $\pm j2.236$.

Answer: $G(s) = K/[s(s^2+2s+5)]$; stable for $0 < K < 10$; at $K = 10$ it oscillates at $\omega = 2.236$ rad/s.