Stability & Routh–Hurwitz

A stable system returns to equilibrium after a disturbance. Learn BIBO stability, pole locations, the Routh–Hurwitz criterion, marginal stability and critical gain, and relative stability measures.

Stability is the first requirement of any control system: a controller that makes the output oscillate ever more wildly is worse than no controller at all. Feedback can stabilise unstable plants — and destabilise stable ones if the gain is too high.

For linear systems, stability is decided entirely by where the closed-loop poles lie in the s-plane, and the Routh–Hurwitz test tells you without solving for them.

An analogy

A marble in a bowl is stable (it settles back), a marble on an upturned bowl is unstable (it rolls away), and a marble on a flat table is marginally stable (it stays wherever it's pushed).

What stability means

BIBO stability: every bounded input gives a bounded output. For linear time-invariant systems this is equivalent to all poles having negative real parts. Poles on the imaginary axis give sustained oscillation (marginal); any pole in the right half-plane gives exponential growth.

Routh–Hurwitz

Build the Routh array from the characteristic polynomial's coefficients. The system is stable if all first-column entries are positive; each sign change indicates one right-half-plane pole. A row of zeros signals poles symmetric about the origin (often on the jω axis), found from the auxiliary polynomial.

Critical gain

With a gain K in the loop, Routh gives the range of K for stability. At the boundary gain the system oscillates at a frequency given by the auxiliary equation. That critical gain and period are exactly what Ziegler–Nichols tuning uses.

Relative stability

Being stable isn't enough — a system with poles barely in the left half-plane rings for ages and is fragile to model errors. Gain margin, phase margin, damping ratio of the dominant poles and the distance of poles from the jω axis quantify how stable it is.

Beyond linear LTI

Delays, saturation, backlash and parameter variation all affect stability. Lyapunov methods prove stability for non-linear systems by finding an energy-like function that always decreases; robust-control methods guarantee stability for a whole family of plants.

Formulas

• Characteristic equation: 1 + G(s)H(s) = 0
• Stability condition: Re(p_i) < 0 for all closed-loop poles
• Routh first-column rule: sign changes in first column = number of RHP poles
• Cubic s³ + a₂s² + a₁s + a₀: stable ⇔ all a_i > 0 and a₂·a₁ > a₀

Worked example: Routh test with a gain

Given: G(s) = K/(s(s + 1)(s + 5)), unity feedback

Find: Range of K for stability and the oscillation frequency at the limit.

• Characteristic: s³ + 6s² + 5s + K = 0
• Routh: s³ [1, 5]; s² [6, K]; s¹ [(30 − K)/6]; s⁰ [K]
• Stable ⇔ K > 0 and 30 − K > 0 → 0 < K < 30
• At K = 30: 6s² + 30 = 0 → s = ±j√5 → ω = 2.24 rad/s (period 2.81 s)

Answer: 0 < K < 30; oscillates at 2.24 rad/s when K = 30

Worked example: Reading poles directly

Given: s² + 2s + 5 = 0

Find: Poles and stability.

• s = −1 ± j2
• Real parts negative → stable, oscillatory decay (ζ = 1/√5 = 0.45)

Answer: Stable, underdamped

Common mistakes

• Wrong: Concluding stability from positive coefficients alone. — Right: Positive coefficients are necessary but not sufficient beyond second order — complete the Routh array.
• Wrong: Treating marginal stability as acceptable. — Right: Sustained oscillation is a failure in practice; design with gain and phase margins.

Applications

• Power grids: Small-signal stability studies find poorly damped inter-area oscillations (0.1–1 Hz) and tune power-system stabilisers.
• Aircraft: Fighter jets are deliberately unstable for agility and are stabilised by fly-by-wire feedback.
• Power converters: Every regulator's loop is checked for stability over line, load and temperature variations.

Key takeaways

• Stable ⇔ all closed-loop poles in the left half-plane.
• Routh–Hurwitz tests stability without solving for roots.
• Critical gain and frequency come from the Routh boundary.
• Design for margins, not merely for stability.

Going further

• Time delay and stability: Delay adds phase lag ωT without changing gain, so it steadily erodes phase margin at higher frequency. The Smith predictor handles long dead times in process control.
• Lyapunov's direct method: If V(x) > 0 and dV/dt < 0 along trajectories, the equilibrium is asymptotically stable. It works for non-linear systems where poles have no meaning.
• Kharitonov's theorem: For polynomials with interval coefficients, stability of four special polynomials guarantees stability of the whole family — a robust-stability shortcut.