Q5Control System

Given: G(s)H(s) = K/[s(s+3)(s²+2s+2)]. Factoring the quadratic term s²+2s+2=0 gives roots s = -1±j1 (using the quadratic formula: s = [-2±√(4-8)]/2 = -1±j1).

Step 1 — Open-loop poles and zeros: the open-loop poles are at s=0, s=-3, s=-1+j1, and s=-1-j1 (four poles total, n=4); there are no finite open-loop zeros (m=0).

Step 2 — Number of branches and asymptotes: the root locus has n=4 branches (equal to the number of open-loop poles), all of which proceed to infinity as K→∞ (since m=0, i.e., n-m=4 branches go to infinity). These 4 branches approach infinity along 4 asymptotes at angles:

giving asymptote angles of 45°, 135°, 225°, and 315°.

Step 3 — Centroid (intersection of asymptotes on the real axis):

Step 4 — Real-axis segments: the real-axis segment between s=0 and s=-3 lies on the root locus, since to the right of any point in this segment there is exactly one real open-loop pole (at s=0), an odd number, satisfying the standard root-locus real-axis rule (a point on the real axis lies on the locus if the total number of real poles and zeros to its right is odd).

Step 5 — Breakaway point: since two branches start at s=0 and s=-3 (both real poles) and must eventually head off toward the complex asymptotes, a breakaway point exists somewhere on the real-axis segment between 0 and -3, found by solving dK/ds=0 for the characteristic equation K = -s(s+3)(s²+2s+2); differentiating and solving numerically within the valid segment (0,-3) gives a breakaway point at approximately s ≈ -1.1 (found by locating the local maximum of K(s) along this real-axis segment).

Step 6 — Maximum gain K for closed-loop stability: the closed-loop characteristic equation is 1+G(s)H(s)=0, i.e., s(s+3)(s²+2s+2)+K=0. Expanding: s(s+3)=s²+3s; multiplying by (s²+2s+2): (s²+3s)(s²+2s+2) = s⁴+2s³+2s²+3s³+6s²+6s = s⁴+5s³+8s²+6s. Adding K:

Constructing the Routh array:

For stability, all elements of the first column must be positive: K>0, and (40.8-5K)/6.8 > 0, which requires 40.8-5K>0, i.e., K < 40.8/5 = 8.16.

Step 7 — jω-axis crossing point: at K=Kmax=8.16, the s¹ row becomes exactly zero, indicating the root locus crosses the imaginary axis at this gain; the crossing frequency is found from the auxiliary equation formed from the s² row: 6.8s²+K=0, i.e., 6.8s²+8.16=0, giving s²=-8.16/6.8=-1.2, so s=±j√1.2=±j1.095.

Summary: the root locus starts at the four open-loop poles (0, -3, -1±j1), with the two real-axis-originating branches (from 0 and -3) meeting at a breakaway point near s≈-1.1 before departing into the complex plane, and all four branches ultimately proceeding to infinity along asymptotes at ±45° and ±135°, centered at the centroid s=-1.25 on the real axis; the maximum permissible gain for closed-loop stability is K=8.16, beyond which the locus crosses into the right-half plane at the jω-axis crossing frequency of ω≈±1.095 rad/s, causing the closed-loop system to become unstable.

Angle of Departure from the Complex Poles

Since the open-loop poles at s=-1±j1 are complex, the root locus branches originating from them do not simply move along the real axis; instead, they depart at a specific angle governed by the angle-condition of the root locus. The angle of departure from the pole at s=-1+j1 is computed as:

The angle from the pole at s=0 to the point s=-1+j1 is 180°-45°=135° (measuring from the positive real axis to the vector pointing from 0 to -1+j1); the angle from the pole at s=-3 to s=-1+j1 is arctan(1/2)=26.57° (measured above the real axis, since the vector from -3 to -1+j1 has real part +2 and imaginary part +1); and the angle from the conjugate pole at s=-1-j1 to s=-1+j1 is exactly 90° (a purely vertical vector of length 2). Summing these contributing angles: 135°+26.57°+90°=251.57°, so the departure angle is θdep=180°-251.57°=-71.57° (i.e., 288.43°, or equivalently -71.57° measured below the positive real axis direction). By symmetry, the locus branch from the conjugate pole at s=-1-j1 departs at the mirror-image angle of +71.57°. This shows that immediately upon leaving the complex poles, the two locus branches initially move outward and slightly toward the real axis before curving toward the asymptotic directions of ±45°/±135° at large K, illustrating that the asymptotic-angle result derived in Step 2 describes only the locus's ultimate large-K behavior, not its immediate direction of departure from the starting poles.

Cross-Check of Kmax via the Direct jω-Substitution Method

As an alternative, independent method of finding the same critical gain Kmax and crossing frequency (without constructing the full Routh array), the characteristic equation s⁴+5s³+8s²+6s+K=0 can be evaluated directly at s=jω, splitting into real and imaginary parts, both of which must vanish simultaneously at the critical (marginally stable) condition:

Setting the imaginary part to zero: 6ω-5ω³=0, i.e., ω²=6/5=1.2, giving ω=√1.2≈1.095 rad/s, exactly matching the jω-crossing frequency found earlier via the auxiliary equation. Substituting ω²=1.2 into the real part and setting it to zero: (1.2)²-8(1.2)+K=0, i.e., 1.44-9.6+K=0, giving K=8.16, exactly matching Kmax found via the Routh array — this direct jω-substitution method independently confirms both the critical gain and crossing frequency obtained from the Routh-array approach, and is in fact the more fundamental derivation from which the Routh-array shortcut is derived.

Relation to Routh-Hurwitz and Practical Design Significance

This problem illustrates the close relationship between the root-locus method and the Routh-Hurwitz criterion: the root locus graphically traces how the closed-loop poles move continuously as K increases from 0 to ∞, while the Routh-Hurwitz criterion, applied at a specific trial value of K, gives an algebraic yes/no stability verdict for that particular gain — used together (as done here), the Routh array pinpoints the exact critical gain Kmax at which the locus crosses the imaginary axis, information that would otherwise require either a very precise graphical root-locus sketch or full numerical rootfinding to obtain with comparable accuracy. From a design standpoint, this result means that any practical controller gain K must be kept safely below 8.16 to guarantee closed-loop stability, and in practice a designer would typically choose K well below this critical value (incorporating a suitable gain margin, e.g., choosing K such that the resulting dominant closed-loop poles have adequate relative stability and acceptable transient-response damping) rather than operating close to the marginal-stability boundary, since real-world parameter variations or unmodeled dynamics could otherwise easily push the true system past the theoretical Kmax=8.16 limit and into instability.