Classical Stability Criteria: Routh–Hurwitz, Mikhailov & Nyquist

For a finite-dimensional continuous-time LTI realization, asymptotic internal stability requires every internal mode to lie in the open left half-plane. For a minimal proper transfer function, BIBO stability is equivalent to all poles lying there. Hidden unstable cancellations break the equivalence, so establish which stability claim is being made before applying a polynomial or frequency test.

Routh–Hurwitz, Mikhailov, and Nyquist do not create three definitions of stability. They infer the same right-half-plane pole count from different available data.

The characteristic equation #

For negative feedback with loop transfer $L(s)=N_L(s)/D_L(s)$,

$$1+L(s)=0$$

gives characteristic polynomial

$$\Delta(s)=D_L(s)+N_L(s).$$

Do not use a cancelled external transfer function if the cancellation hides internal dynamics. Include delays separately: a true $e^{-sT}$ makes the characteristic equation transcendental and invalidates a finite polynomial Routh table unless an approximation is explicitly introduced.

Routh–Hurwitz criterion #

For

$$\Delta(s)=a_ns^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0,$$

construct the Routh array. The first two rows contain alternating coefficients; subsequent elements are determinant-like combinations of the two rows above. Assuming no zero special case, the number of sign changes in the first column equals the number of roots in the open right half-plane.

All first-column entries with the same sign imply no right-half-plane roots. Positive coefficients alone are necessary for a Hurwitz polynomial with positive leading coefficient, but not sufficient above second order.

Worked cubic #

Let

$$\Delta(s)=s^3+3s^2+2s+K.$$

The Routh first column is

$$1,\quad 3,\quad \frac{6-K}{3},\quad K.$$

Strict stability therefore requires

$$K>0,\qquad 6-K>0,$$

so $0<K<6$. At $K=6$, an auxiliary polynomial from the $s^2$ row gives $3s^2+6=0$, hence an imaginary pair $s=\pm j\sqrt2$: the boundary is marginal, not asymptotically stable.

Special rows #

• A zero first element with a nonzero row is handled by a positive $\varepsilon$ limit; interpret the sign as $\varepsilon\to0^+$.
• An entire zero row indicates roots symmetric about the origin. Form the auxiliary polynomial from the row above, differentiate it, and replace the zero row.

These are structural cases, not calculator tricks. They identify imaginary-axis or symmetric root patterns.

Hurwitz determinants #

The equivalent Hurwitz criterion forms a matrix from polynomial coefficients and requires leading principal determinants to have the correct sign. It is compact for symbolic inequalities and parameter regions, while the Routh table is often easier for counting roots and locating a crossing boundary. Both assume a real polynomial with nonzero leading coefficient.

Mikhailov criterion #

Evaluate the characteristic polynomial on the imaginary axis:

$$\Delta(j\omega),\qquad 0\le\omega<\infty.$$

For a real $n$th-order Hurwitz polynomial with positive coefficients and no imaginary-axis roots, the Mikhailov curve begins on the positive real axis and its continuous argument increases by $n\pi/2$ as frequency runs from zero to infinity, traversing the quadrants in order without passing through the origin.

The method is an argument-principle test on the characteristic polynomial. It is useful for geometric understanding and hand checks, but direct numerical roots or Nyquist analysis are usually more convenient in modern workflows. A coarse frequency grid can miss an origin crossing; unwrap phase continuously and bracket close approaches.

Nyquist criterion #

Nyquist works directly with loop transfer $L(s)$ and can handle open-loop unstable poles and exact delays. Map a closed contour enclosing the open right half-plane through $L(s)$. The closed-loop characteristic zeros are zeros of $1+L(s)$, so the critical point is $-1$.

Using the convention that the right-half-plane contour is traversed clockwise and $N$ counts clockwise encirclements of $-1$,

$$N=Z-P,$$

where $P$ is the number of open-loop poles of $L$ in the right half-plane and $Z$ is the number of closed-loop right-half-plane poles. Closed-loop stability requires $Z=0$, hence $N=-P$: the mapped curve must make $P$ counterclockwise encirclements. With the alternative convention that counterclockwise encirclements are positive, the formula is $N=P-Z$. State the convention; do not memorize a sign without the contour orientation.

Poles on the imaginary axis require indentations in the contour and careful limiting arcs. A plot of only positive frequencies is complete only when conjugate symmetry and the closing arcs are correctly supplied.

Gain and phase margins #

For a simple loop with one relevant crossover and no unusual open-loop instability:

• phase margin is the additional phase lag needed at gain crossover to reach $-180^\circ$;
• gain margin is the factor by which gain can change at phase crossover before reaching unit magnitude.

Multiple crossings produce multiple candidate margins. A positive scalar margin does not prove robust performance, and traditional margins can be misleading for loops with delay, resonances, nonminimum-phase dynamics, multiple loops, or near pole-zero cancellations. Also inspect Nyquist distance to $-1$, the sensitivity peak $\|S\|_\infty$, uncertainty models, and internal stability.

Parameter-space stability #

Routh entries can be inequalities in controller or plant parameters. Solving them partitions parameter space into stable and unstable regions. Boundaries occur where a first-column entry vanishes, a root reaches the origin, or an imaginary pair crosses. Confirm each boundary by substituting $s=j\omega$ and solving real/imaginary parts, then test one interior point per region.

The complete geometric construction—including zero-root, imaginary-pair and degree-loss boundaries, singular frequencies, discrete-time unit-circle mapping, and a worked two-parameter proof—is developed in Parameter-Space Stability & D-Decomposition.

A reliable stability proof capsule #

1. State the exact plant, controller, loop sign, delays, and operating point.
2. State whether the claim is external/BIBO, internal, asymptotic, marginal, or robust stability.
3. Count unstable open-loop poles before drawing Nyquist conclusions.
4. Apply one primary test with all special cases resolved.
5. Cross-check closed-loop poles or a state-space realization numerically.
6. Test parameter uncertainty, delay, and hidden cancellations.
7. Verify nonlinear constraints and sampling separately.

Nominal stability is a necessary gate. It is not a complete performance or safety certificate.