Root locus of K/[s(s+1)(s²+4s+5)], critical damping

The root locus is the path of the closed-loop poles in the s-plane as $K$ varies from 0 to $\infty$. The characteristic equation is $1 + G(s) = 0$, i.e.

$$s(s+1)(s^2+4s+5) + K = 0 \;\Rightarrow\; s^4 + 5s^3 + 9s^2 + 5s + K = 0$$

Step 1: Poles, zeros and branches

• Open-loop poles: $s = 0,\ -1,\ -2 \pm j1$ ($n = 4$); zeros: none ($m = 0$).
• Number of branches = 4; all four go to infinity.

Step 2: Real-axis locus

A real-axis point is on the locus if the number of real poles + zeros to its right is odd. Only the segment between 0 and −1 qualifies.

Step 3: Asymptotes

$$\begin{aligned} \sigma_A &= \frac{\sum p - \sum z}{n-m} = \frac{0-1-2-2}{4} = -1.25 \\ \theta_A &= \frac{(2q+1)180^\circ}{4} = \pm 45^\circ,\ \pm 135^\circ \end{aligned}$$

Step 4: Breakaway point

$K = -(s^4+5s^3+9s^2+5s)$, so

$$\frac{dK}{ds} = -(4s^3+15s^2+18s+5) = 0$$

Roots: $s = -0.393$ and $s = -1.679 \pm j0.603$. Only $s = -0.393$ lies on the locus (between 0 and −1). Gain there:

$$K = 0.393 \times 0.607 \times |(-0.393)^2 + 4(-0.393) + 5| = 0.855$$

Step 5: Angle of departure from $-2+j1$

Angles from the other poles to $-2+j1$: from $0$: $153.43^\circ$; from $-1$: $135^\circ$; from $-2-j1$: $90^\circ$.

$$\phi_d = 180^\circ - (153.43^\circ + 135^\circ + 90^\circ) = -198.43^\circ \equiv 161.57^\circ$$

By symmetry, the angle from $-2-j1$ is $-161.57^\circ$.

Step 6: Imaginary-axis crossing (Routh array)

Marginal stability: $40 - 5K = 0 \Rightarrow K = 8$. Auxiliary equation: $8s^2 + 8 = 0 \Rightarrow s = \pm j1$. The locus crosses the $j\omega$ axis at $\pm j1$ rad/s when $K = 8$.

Sketch

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             .....x                .X..
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---------------------------x....*...x-------------
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             .....x                .X..
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                 -2       -1        0
x pole   * breakaway (-0.393)
X jw crossing (+-j1, K = 8)

Branches from 0 and −1 meet at −0.393, break away at ±90°, and bend toward the ±45° asymptotes, crossing at ±j1 (K = 8). Branches from $-2\pm j1$ leave at $\pm 161.6^\circ$ and approach the ±135° asymptotes.

Gain for critical damping

Critical damping occurs when the dominant closed-loop poles are real and equal, i.e. at the breakaway point $s = -0.393$. The other two poles are then at $-2.107 \pm j1.046$.

Answer: Critical damping at K ≈ 0.855 (double pole at s = −0.393). The system is stable for $0 < K < 8$.