Root Locus: Evans Construction

The root locus is the set of all points in the =s=-plane that a closed-loop pole can occupy as a single real parameter - almost always the loop gain K - sweeps from 0 to +\infty. It is, in the language of dynamical systems, a parametric plot of the spectrum of the closed-loop map against one scalar knob. Walter R. Evans drew it by hand in 1948[fn:: Evans, W. R., "Graphical Analysis of Control Systems," AIEE Transactions, 67 (1948), pp. 547–551.], in an era when "plotting" meant a slide rule, graph paper, and a willingness to re-evaluate a polynomial a hundred times. It survives for the same reason Bode and Nyquist survive: it gives geometric intuition about how the closed loop deforms under a parameter change, and that intuition is cheaper than recomputing.[fn:: A computer can render the locus in milliseconds, but the human who has internalized the construction rules can predict its shape - number of branches, asymptote directions, breakaway point - before the plot finishes rendering. The rules are a lossy compression of the geometry.]

The Equation the Locus Satisfies

Consider a unity-feedback loop with forward-path plant G(s) and gain K \geq 0, so the closed-loop transfer function is

The closed-loop poles are the roots of the characteristic equation

Write G(s) as a ratio of polynomials in the open-loop form

where z_j are the open-loop zeros and p_i the open-loop poles. Substituting into (1) and clearing the denominator,

which is a polynomial whose coefficients are affine in K. The root locus is the locus of roots of (2) as K runs over [0, +\infty). Equivalently, a point s_0 is on the locus iff it satisfies the phase condition

because (1) demands =K G(s_0) = -1=, a negative real number, and K>0 contributes no phase. The magnitude condition

then fixes K once s_0 is known. Equation (3) is the seed from which all of Evans' rules grow: every rule is a geometric specialization of "the angles of the vectors from the open-loop poles and zeros to s_0 must sum to an odd multiple of \pi."

Evans' Construction Rules

The following are the standard rules, presented as tools for sketching the locus by hand. Each is a consequence of (3) or of (2); the derivations are omitted here but are collected in any controls text worth its salt[fn:: See, e.g., Feedback Control of Dynamic Systems (Franklin, Powell, Emami-Naeini) or Modern Control Engineering (Ogata). The rules are not magic; they are the polynomial-coefficient identities of (2) rewritten as angle equalities.]. Notation: n = number of open-loop poles, m = number of open-loop zeros.

Real-axis segments

A point on the real axis is on the locus iff the number of real open-loop poles and zeros to its right is odd. Complex-conjugate poles/zeros contribute angles that cancel in pairs, so they do not affect the count. This is the single most useful rule: it lets you strike the real axis immediately.

Branch count and start/end

The locus has exactly max(n, m) branches. Branches start at the open-loop poles (when =K=0=) and end at the open-loop zeros (when =K=\infty=). The n - m excess branches go to infinity along asymptotes.

Asymptote angles and centroid

The n - m asymptotes radiate from a common centroid on the real axis

at angles

So for =n - m = 1= the lone asymptote is at \pi (negative real axis); for =n - m = 2= the asymptotes are at \pm\pi/2 (vertical); for =n - m = 3= at \pi/3, \pi, 5\pi/3; for =n - m = 4= at =\pm\pi/4, \pm 3\pi/4.

Breakaway and break-in points

On the real axis, two branches can meet and leave the axis (a breakaway, where poles go complex) or return to it (a break-in, where complex poles become real). At such a point s_b,

Candidates are the real roots of this polynomial that lie on a locus segment. Not every root is a genuine breakaway - some lie on segments the locus never visits - so one must cross-check against the real-axis rule.

Departure and arrival angles

At a complex open-loop pole p_k, branches leave at an angle of departure

and at a complex zero z_k, branches arrive at an angle of arrival with the signs of the pole/zero contributions interchanged. These are just (3) evaluated infinitesimally close to p_k or z_k, treating the nearby contribution as the unknown.

Imaginary-axis crossing

If a branch crosses the j\omega axis, that crossing marks the gain K at which the closed loop goes marginally stable. Substitute =s = j\omega= into (2), separate real and imaginary parts, and solve for \omega and K. This is the Routh–Hurwitz criterion in disguise

• see Stability: Routh-Hurwitz.

Worked Example: A Third-Order System

Take the open-loop plant

Open-loop poles at 0, -1, -2; no finite zeros, so =n = 3=, =m = 0=, and =n - m = 3= branches all leave to infinity.

Step 1 - Real-axis segments

Counting real poles to the right: (-2, -1) has one (the pole at -1)... wait - to the right of a point in (-\infty, -2) lie two poles (-2, -1, 0 → three); of a point in (-2, -1) lie the pole at -1 and 0 (two); of a point in (-1, 0) lies only 0 (one). So the on-locus real-axis segments are (-\infty, -2) and (-1, 0). Odd count → on locus.

Step 2 - Asymptotes

Centroid

Angles \pi/3, \pi, 5\pi/3: three rays from (-1, 0) at 60 deg, 180 deg, 300 deg. The 180 deg ray runs along the negative real axis; the other two head into the right half-plane at \pm 60 deg - a warning sign that this plant will go unstable for large K.

Step 3 - Breakaway on (-1, 0)

Solve =d/ds [1 / (s (s+1) (s+2))] = 0=, i.e. =3 s^2 + 6 s + 2 = 0=, giving =s = (-3 \pm sqrt(3)) / 3 \approx -0.423, -1.577=. Only s \approx -0.423 lies on the locus segment (-1, 0); -1.577 lies on (-2, -1) which is not a locus segment, so it is rejected. Breakaway at s_b \approx -0.423.

Step 4 - Imaginary-axis crossing

With =s = j\omega= in =s^3 + 3 s^2 + 2 s + K = 0=:

The =\omega = 0= root gives =K = 0= (the trivial start). The non-trivial crossing is at =\omega = \pm sqrt(2) \approx \pm 1.414=, =K = 3 omega^2 = 6=. So the loop is stable for 0 < K < 6 and unstable beyond - consistent with the \pm 60 deg asymptotes.

Sketch

Three branches leave 0, -1, -2. One branch runs from 0 leftward to the breakaway at -0.423, where it meets the branch from -1 and the pair departs into the complex plane at \pm 90 deg (locally). They curve rightward, cross the j\omega axis at \pm 1.414 j when =K = 6=, and continue into the right half-plane along the \pm 60 deg asymptotes. The third branch runs from -2 leftward to -\infty along the real axis.

Gain for a target damping ratio

Suppose we want =\zeta = 0.5=, i.e. closed-loop poles on the line =/_ s = \pm arccos(0.5) = \pm 60 deg= from the origin (measured from the negative real axis). Intersect this ray with the locus and read off the magnitude condition. On the ray =s = -sigma (1 + j tan 60)=, substitute into (2) and solve numerically; the crossing gives s \approx -0.34 + 0.59 j and K \approx 1.0. At that gain the dominant (slowest) pair has =\zeta = 0.5=, predicting a step overshoot

and a 2% settling time

This is the canonical use of root locus for transient design: pick a ray for the desired damping, intersect with the locus, read K.

Design for Transient Specifications

The =s=-plane encodes transient behavior geometrically - a fact inherited from the Laplace picture (see Control Theory: Signals & Laplace): constant \zeta lines are rays from the origin, constant \omega_n are circles, constant settling time t_s \approx 4/(\zeta \omega_n) are vertical lines =\sigma = -4/t_s=, and constant natural frequency lines are radial. Root locus design is therefore a matter of shaping the locus - by adding poles and zeros via a compensator D(s) so that some branch passes through the region of the =s=-plane that the spec demands.

A lead compensator =D(s) = (s + z_d) / (s + p_d)= with p_d > z_d pulls the locus leftward, improving both \zeta and t_s at the price of added high-frequency gain. A lag compensator p_d < z_d improves steady-state error without disturbing the dominant poles much. This is the bridge to PID Control: Theory & Tuning - a PID is roughly a PI (lag near origin) plus a PD (lead at high frequency) - and the tuning rules can be read as instructions for where to place the compensator's pole and zero so the locus threads the spec region.

Connection to the =s=-plane Geometry

The root locus lives entirely in the =s=-plane, which is the same plane introduced in the Laplace note as the domain of the transform =s = \sigma + j\omega=. Everything we care about statically - poles, zeros, regions of convergence - is already there; root locus adds dynamics in the parameter K. The Laplace note tells you where the poles are now; root locus tells you where they go as you turn the knob. Stability (the left half-plane) is one constraint; damping (the \zeta cone) is another; the locus is the curve through this constrained landscape along which the closed-loop poles are allowed to travel. Routh–Hurwitz (see Stability: Routh-Hurwitz) answers only the binary "does any branch cross into the right half-plane, and at what gain?"; root locus answers the richer question "where is every branch, for every gain?"

When to Use Root Locus vs. Bode / Nyquist

The three classical graphical tools carve the design problem along different axes:

Root locus is the right tool when the design spec is cast in time domain (overshoot, settling time, damping) and the free parameter is a scalar gain. Bode and Nyquist are the right tools when the spec is cast in frequency domain (bandwidth, margins, disturbance rejection) or when one needs to argue robustness against plant uncertainty - because margins are properties of the open-loop frequency response, not of the pole locations per se. See Frequency Response: Bode & Nyquist for the dual view.

When It Breaks

Root locus is not a universal tool, and it is worth being precise about its domain of competence.

• SISO only. The locus is defined for a single loop, single input,

single output. MIMO generalizations exist (root loci of the characteristic polynomial of a state-feedback matrix) but lose the clean angle interpretation that makes hand-sketching possible.

• Single parameter. Classical root locus varies K. Varying two

parameters (e.g. K and a compensator pole) produces a family of loci, not a single curve; the "generalized root locus" varying an arbitrary polynomial coefficient exists[fn:: The extension to arbitrary parameter variation is sometimes called the root contour. It is covered briefly in Ogata but rarely taught, because the geometric rules do not carry over cleanly and the payoff is low - at that point you reach for a numerical eigenvalue sweep.] but is rarely taught because the payoff over direct computation is small.

• Intractable for high-order plants. For n much beyond 4 or 5,

the breakaway equation becomes a polynomial of high degree, the asymptote centroid is easy but the branch shapes are not, and one is reduced to numerical plotting anyway. The rules degrade gracefully - real-axis segments and asymptotes remain cheap - but the hand-sketch loses its value.

• No frequency-domain picture. Root locus tells you where the poles

are, not what the loop looks like as a function of frequency. It will not, by itself, tell you the phase margin, the gain crossover frequency, or the sensitivity peak. For those, you must convert pole/zero locations into a Bode plot - which is exactly the bridge to Frequency Response: Bode & Nyquist.

• Assumes the loop gain is the design knob. If the real design

freedom is a compensator structure, not just a scalar, root locus is a diagnostic, not a synthesis tool. One sketches the locus for each candidate compensator and checks whether it threads the spec region - useful, but not a direct design procedure.

Meta-observation

Evans invented root locus in 1948 as a graphical tool for the slide-rule era, when evaluating a cubic or quartic a hundred times to trace a curve was a genuinely expensive prospect. The rules are a set of shortcuts - polynomial-coefficient identities rewritten as angle equalities - that let an engineer predict the shape of the locus without solving the characteristic equation numerically. A computer can now plot the locus instantly, and one might think the rules are obsolete. They are not, for the same reason that Bode's asymptote construction is not obsolete: the rules are a compressed description of the geometry, and the human who has internalized them can predict the shape - number of branches, asymptote directions, where the breakaway sits, whether the locus ever enters the right half-plane - before the machine finishes rendering. That prediction is a sanity check on the machine and, more importantly, a way of forming intuition that transfers to systems the machine has not yet been asked to plot. The locus is a picture; the rules are the vocabulary for describing it without drawing it.

Related

• Optimal Control - pole placement becomes LQR
• Robust Control - root locus under parameter uncertainty
• Control Theory: Signals & Laplace
• Transfer Functions & Block Diagrams
• Stability: Routh-Hurwitz
• Frequency Response: Bode & Nyquist
• PID Control: Theory & Tuning