The root locus

The Routh criterion tells the designer where a loop stops being stable. It does not show where the closed-loop poles are on the way, and lesson 4 showed that their exact location decides the step response. In 1948 Walter Evans, working on aircraft guidance, published a graphical method that shows exactly that: the root locus, the path traced in the s-plane by the closed-loop poles as one gain varies from zero to infinity. A sketch of the locus, made in minutes with a few rules, shows at a glance whether a specification can be met by adjusting gain and, if not, what kind of compensator would help. This lesson develops the rules and applies them to the PT-1 pan servo, with a result that explains why Amara's proportional designs could never settle fast enough.

The angle and magnitude conditions

Write the loop gain as KG(s), where K ≥ 0 is the gain being varied and G(s) = N(s)/D(s) contains everything else. The closed-loop poles are the roots of 1 + KG(s) = 0, or KG(s) = −1. A complex number equals −1 when its magnitude is one and its angle is an odd multiple of 180°. Because K is real and positive, these split into two conditions. The angle condition says a point s is on the locus if and only if the angle of G(s) is 180° plus a multiple of 360°. The magnitude condition then gives the gain that puts a closed-loop pole there.

The angle condition does not involve K at all. That is the key insight: the shape of the locus depends only on the open-loop poles and zeros, and the gain merely says where on the locus the closed-loop poles sit. Each term in the angle condition can be drawn as a vector from a pole or zero to the trial point, so the test is geometric. A designer with a protractor, or a few lines of code, can decide whether any point is on the locus and, if not, by how many degrees it misses.

Rules for sketching

Evans's rules follow from the two conditions and from the behaviour of the polynomial D(s) + KN(s) at the extremes of K. With n open-loop poles and m zeros, the main rules are these.

The real-axis rule is the angle condition in disguise. A complex pair of poles or zeros contributes equal and opposite angles to a real-axis point and so cancels, and real poles or zeros to the left contribute 0°, so only those to the right matter, each contributing 180°. The asymptote rule describes the locus far from all the poles and zeros, where it looks as if n − m poles were concentrated at the centroid. For n − m = 1 the single asymptote is the negative real axis; for 2 it is a vertical line through the centroid; for 3 it is three lines at ±60° and 180°.

The pan servo under proportional control

Amara applies the rules to the PT-1 pan axis with the full plant of lesson 2, KG(s) = 16 667K/(s(s + 21.2)(s + 1979)). There are three poles and no zeros, so three branches, all going to infinity.

The real-axis segments are from 0 to −21.2, where there is one pole to the right, and to the left of −1979, where there are three. The three asymptotes leave the centroid (0 − 21.2 − 1979)/3 = −666.7 at angles of +60°, −60° and 180°. The branches starting at 0 and −21.2 move toward each other, meet at a breakaway point and leave the real axis as a complex pair; the branch from −1979 moves left along the real axis forever. The breakaway point satisfies dK/ds = 0, where K = −s(s + 21.2)(s + 1979)/16 667. The complex pair heads up and down, bending slowly to the right toward the ±60° asymptotes, and crosses the imaginary axis at the frequency and gain lesson 5 found.

import math, cmath
def roots(p, iters=400):
    a = [c / p[0] for c in p]; n = len(a) - 1
    z = [(0.4 + 0.9j) ** k * 100 for k in range(n)]
    f = lambda x: sum(c * x ** (n - i) for i, c in enumerate(a))
    for _ in range(iters):
        z = [zi - f(zi) / math.prod(zi - zj for j, zj in enumerate(z) if j != i) for i, zi in enumerate(z)]
    return sorted(z, key=lambda r: (round(r.real, 6), r.imag))
# open loop K * 16667 / (s (s + 21.2)(s + 1979)): den = s^3 + 2000.2 s^2 + 41975 s
den = [1.0, 2000.2, 21.2 * 1979.0, 0.0]; g = 16667.0
poles = [0.0, -21.2, -1979.0]
# asymptotes: n - m = 3 branches to infinity
centroid = sum(poles) / 3
print(f"asymptote centroid {centroid:.1f}, angles 60, 180, 300 deg")
# breakaway: dK/ds = 0 with K = -den(s)/g  ->  3 s^2 + 4000.4 s + 41975 = 0
a, b, c = 3.0, 2 * 2000.2, 21.2 * 1979.0
sb = (-b + math.sqrt(b * b - 4 * a * c)) / (2 * a)
Kb = -(sb**3 + 2000.2 * sb**2 + 21.2 * 1979.0 * sb) / g
print(f"breakaway at s = {sb:.2f}, gain {Kb:.2f} V/rad")
# imaginary-axis crossing from the Routh condition
print(f"j-axis crossing at {math.sqrt(21.2 * 1979.0):.0f} rad/s, gain {2000.2 * 21.2 * 1979.0 / g:.0f} V/rad")
# walk along the locus and report damping and settling estimates
print("   K     dominant poles        zeta   4/(zeta wn)   third pole")
for K in (5.0, 13.3, 20.0, 36.8, 100.0, 300.0, 1000.0, 4000.0):
    r = roots([1.0, 2000.2, 21.2 * 1979.0, g * K])
    dom = max(r, key=lambda x: x.real); third = min(r, key=lambda x: x.real)
    wn = abs(dom); zeta = -dom.real / wn
    print(f"{K:6.1f}  {dom.real:8.2f} ± j{abs(dom.imag):6.2f}   {zeta:5.2f}   {4/(zeta*wn)*1000:7.0f} ms   {third.real:8.1f}")
# angle condition check at a trial point: the point -20 + j26.6 is not on the locus
s0 = complex(-20, 26.6)
ang = -sum(math.degrees(cmath.phase(s0 - p)) for p in poles)
print(f"angle of G at -20 + j26.6: {ang:.1f} deg (a locus point needs -180); shortfall {(-180 - ang):.1f} deg")

asymptote centroid -666.7, angles 60, 180, 300 deg
breakaway at s = -10.57, gain 13.27 V/rad
j-axis crossing at 205 rad/s, gain 5035 V/rad
   K     dominant poles        zeta   4/(zeta wn)   third pole
   5.0     -2.22 ± j  0.00    1.00      1801 ms    -1979.0
  13.3    -10.57 ± j  0.50    1.00       378 ms    -1979.1
  20.0    -10.56 ± j  7.55    0.81       379 ms    -1979.1
  36.8    -10.52 ± j 14.11    0.60       380 ms    -1979.2
 100.0    -10.39 ± j 27.10    0.36       385 ms    -1979.4
 300.0     -9.96 ± j 49.25    0.20       402 ms    -1980.3
1000.0     -8.46 ± j 91.28    0.09       473 ms    -1983.3
4000.0     -2.14 ± j182.75    0.01      1867 ms    -1995.9
angle of G at -20 + j26.6: -215.1 deg (a locus point needs -180); shortfall 35.1 deg

The breakaway point is at −10.57, reached when K = 13.3 V/rad. From there the dominant pair rises almost straight up, its real part creeping from −10.6 to −10.4 as K rises to 100 and to −8.5 at 1000. Damping falls steadily: ζ = 0.6 at K = 36.8, 0.36 at 100, 0.09 at 1000. The settling-time estimate 4/(ζω_n) = 4/|real part| barely changes from 380 ms until the pair begins to bend toward the imaginary axis, and the electrical pole meanwhile wanders from −1979 to −1996, never close enough to matter.

The conclusion is the most useful thing a root locus can tell a designer: no value of proportional gain meets the specification. The settling requirement needs the dominant poles to the left of −16, and the dominant branches never get there. Raising the gain only adds oscillation. This is not a matter of tuning; it is a matter of where the open-loop poles are. The branches leave the real axis halfway between the poles at 0 and −21.2, and with no zeros nearby to pull them left, they stay there.

Testing a desired pole location

Lesson 3's cascade design put the closed-loop poles at −20 ± j26.6, inside the specification region. The angle condition shows how far that point is from the proportional locus. The angle of G at s₀ = −20 + j26.6 is −(126.9° + 87.4° + 0.8°) = −215.1°, the three terms being the angles of the vectors from the poles at 0, −21.2 and −1979. A point on the locus needs −180°, so the point misses by 35.1°. A compensator that adds 35.1° of phase at s₀ would put the locus through it. That is exactly what the lead compensator of lesson 8 does, and the number 35.1° is its design target.

The rate loop of lesson 3 reached the point a different way. Feeding back speed changed the plant the outer loop sees, moving its pole from −21 to −40, and with the poles at 0 and −40 the breakaway point moves to −20. The proportional locus for that modified plant rises vertically through −20, and K_p = 133 V/rad places the pair at ±j26.6. Root locus thinking explains the cascade too: the designer chose K_d to move the open-loop pole, then chose K_p to slide along the new locus.

Other uses of the locus

The locus need not be drawn against a controller gain. Any parameter that enters the characteristic equation linearly can be isolated: rewrite the equation as 1 + αG′(s) = 0 and draw the locus of the new G′ against α. Daniel uses this to check the effect of the camera head's inertia. The PT-1 will be sold with two housings, the standard one and a heated version with a larger window that adds 40 % to the load inertia, and the locus against inertia shows the cascade's poles moving toward the imaginary axis by a few degrees, comfortably inside the specification region. A locus against a sensor time constant or a delay approximation answers similar questions.

The locus also extends to negative K, the complementary root locus, which describes positive feedback or a plant whose gain changes sign. It appears when a sensor is wired with the wrong polarity, a classic commissioning error. With the encoder's A and B channels swapped, the PT-1's feedback becomes positive, the complementary locus puts one closed-loop pole in the right half plane at any gain, and the camera spins away until it reaches a limit. Leo's commissioning checklist starts by jogging the motor open loop and confirming that the encoder counts in the right direction.

Software and judgement

Computers draw loci accurately in milliseconds, and the code above simply finds the roots at many gains. The value of the sketching rules is not in drawing but in reasoning. A designer who knows that branches bend toward zeros, that asymptotes depend on n − m and that poles near the origin dominate the response can look at a plant and say, before any computation, that proportional control will be too slow, that a zero is needed to pull the locus left, and roughly where to put it. That reasoning is the subject of lesson 8.