ELEN90055 · Control Systems
Stability analysis: Routh, root-locus, Nyquist
This chapter of ELEN90055 Control Systems at the University of Melbourne tackles the central question of feedback design: are every one of the closed-loop poles in the open left-half plane, so the loop is stable? It develops three complementary tests — the Routh-Hurwitz array (stability and parameter ranges straight from the characteristic polynomial), the root locus (how the poles move as one gain varies), and the Nyquist criterion built on Cauchy's argument principle Z = N + P (counting unstable poles from a single frequency-response plot).
It sits in Part III of the subject and underpins the short-answer stability question and the larger feedback-design question, where the three tools are used as cross-checks of one another.
What this chapter covers
- 01Apply the necessary coefficient test to reject an unstable polynomial at a glance
- 02Build a Routh-Hurwitz array and read stability from the first-column signs
- 03Use the cubic shortcut a2 a1 > a3 a0 for third-order characteristic polynomials
- 04Find the range of a gain K that keeps a feedback loop stable, and the marginal gain
- 05Read the imaginary-axis crossing frequency from the Routh auxiliary polynomial
- 06State the root-locus magnitude condition K = 1/|F(z)| and phase condition ∠F(z) = (2l+1)π
- 07Sketch a root locus: branches, real-axis segments, and asymptote centroid and angles
- 08Apply the Nyquist criterion Z = N + P by counting encirclements of the -1 point
- 09Compute a gain margin in dB and predict instability at high gain
- 10Cross-check a stability verdict across Routh, root-locus and Nyquist
Find the stable gain range with Routh-Hurwitz
- +1Characteristic polynomial. Closed-loop poles solve 1 + K·G(s) = 0, i.e. s(s+2)(s+4) + K = 0, which expands to s³ + 6s² + 8s + K = 0.
- +1Necessary screen. For stability every coefficient must be positive; the coefficients are 1, 6, 8 and K, so we need K > 0 (no term is missing).
- +2Routh array. Rows: s³ row [1, 8]; s² row [6, K]; s¹ row [(6×8 − 1×K)/6, 0] = [(48 − K)/6, 0]; s&sup0; row [K]. The first column is 1, 6, (48 − K)/6, K.
- +1First-column rule. Stability needs no sign change in the first column, so (48 − K)/6 > 0 and K > 0, giving 0 < K < 48.
- +1Marginal gain and crossing frequency. At K = 48 the s¹ entry is zero, signalling imaginary-axis poles; the auxiliary polynomial from the s² row is 6s² + 48 = 0, so s² = −8 and s = ±j√8 ≈ ±j2.83 rad/s.
- +1Verdict and cross-check. Stable for 0 < K < 48; marginal (sustained oscillation) at K = 48; unstable for K > 48. The cubic shortcut a2 a1 > a3 a0 gives 6×8 > K, i.e. K < 48, and factoring at K = 48 gives (s + 6)(s² + 8) with roots −6 and ±j2.83 — both checks agree.
Key terms
- Characteristic polynomial
- The polynomial whose roots are the closed-loop poles; for a unity-feedback loop with open loop A0(s) it is the numerator of 1 + A0(s). The loop is stable exactly when all its roots have real part less than zero.
- Routh-Hurwitz criterion
- A necessary-and-sufficient stability test: build the Routh array from the characteristic polynomial's coefficients; all roots lie in the left-half plane if and only if the first-column entries all share the sign of the leading coefficient. Each first-column sign change counts one right-half-plane root.
- Necessary coefficient test
- A quick screen: if all roots have negative real part then every coefficient of the (monic) characteristic polynomial is strictly positive. Any missing or non-positive coefficient proves instability. It is necessary but not sufficient for order three or higher.
- Root locus
- The set of closed-loop pole locations traced in the s-plane as a single gain K sweeps from 0 to infinity, found from 1 + K·F(s) = 0. Branches start at the poles of F and end at its zeros or at infinity.
- Magnitude and phase conditions
- A point z lies on the root locus when the phase condition ∠F(z) = (2l+1)π holds (this fixes the shape); the gain that places a pole there is then K = 1/|F(z)| from the magnitude condition.
- Asymptote centroid and angles
- The n - m root-locus branches going to infinity approach straight asymptotes meeting the real axis at the centroid σ = (sum of poles - sum of zeros)/(n - m), leaving at angles (2k-1)π/(n - m).
- Nyquist criterion (Z = N + P)
- From Cauchy's argument principle, the number of unstable closed-loop poles is Z = N + P, where N is the number of clockwise encirclements of the point -1 + j0 by the plot of A0(jω) and P is the number of open-loop right-half-plane poles. The loop is stable when Z = 0.
- Gain margin
- The factor by which the loop gain can rise before instability, read at the frequency where the phase is -180 degrees: Mg = 20·log10(1/|A0(jω_180)|) in dB. A larger gain margin means more room before the Nyquist plot reaches -1.
Stability analysis: Routh, root-locus, Nyquist FAQ
When should I use Routh, root locus, or Nyquist?
Use Routh-Hurwitz when you have a polynomial characteristic equation and want a yes/no stability answer or a parameter range — it is purely algebraic and needs no plot. Use the root locus when you want to see how the closed-loop poles move as one gain varies, and to choose a gain for a target damping or speed. Use Nyquist when the loop contains something Routh cannot handle as a polynomial, such as a pure time delay e^(-sT), or when the plant is open-loop unstable (non-zero P), because Nyquist counts unstable poles as Z = N + P and also hands you the gain and phase margins directly.
Why do we count encirclements of -1 rather than the origin in the Nyquist test?
Cauchy's argument principle is applied to 1 + A0(s), whose zeros are the closed-loop poles. Encirclements of the origin by 1 + A0 are the same as encirclements of the point -1 by A0 itself, because subtracting 1 just shifts the origin. Plotting A0(jω) and watching the -1 point is more convenient than plotting 1 + A0 and watching the origin, so that is the standard convention; the count N feeds straight into Z = N + P.
Can AI help me with stability analysis (Routh, root-locus, Nyquist) in ELEN90055?
Yes, as a study aid. An AI tutor such as Sia can explain the Routh array construction step by step, walk you through the root-locus sketching rules and the Nyquist Z = N + P count, and check your algebra on a parameter range or a gain margin. Use it to learn the method and the sign conventions and to practise; it does not sit your open-book exam for you or guarantee a mark, and you should confirm every rule and formula against the current unit notes on Canvas.
Studying with AI? Sia — free AI electrical engineering tutor works through ELEN90055 step by step.
Exam move
Treat stability as three views of one question — do every closed-loop pole sit in the open left-half plane? First, make Routh automatic: for a cubic, memorise the shortcut a2 a1 > a3 a0, and always run the necessary coefficient screen before building an array. Second, drill the root-locus rules until you can sketch from memory: branches start at poles and end at zeros or infinity, real-axis points lie to the left of an odd number of poles and zeros, and the asymptotes meet at the centroid σ = (sum poles - sum zeros)/(n - m) at angles (2k-1)π/(n - m). Third, practise Nyquist as a counting exercise: identify P, sketch the A0(jω) plot, count clockwise encirclements of -1 to get N, then state Z = N + P and the verdict with its reason. The most valuable habit is cross-checking: a marginal gain found by Routh should match the imaginary-axis crossing on the root locus and the -1 crossing on the Nyquist plot. The exam is 3 hours, open book, worth 70% with a hurdle, and you answer four questions summing to 50 marks — budget about 3.6 minutes per mark. Because it is open book, marks reward method shown with correct sign conventions and units, not memorised results, so lay out each array, angle sum or encirclement count explicitly. Confirm the exact date of the next (Semester 1, ~June 2027) sitting on Canvas.
