Absolute Stability, Sector Criteria & Describing Functions

Many nonlinear feedback loops can be written in Lur'e form: a linear dynamic block $G(s)$ in feedback with a static nonlinearity $y=\phi(u)$. Saturation, dead zones, relay elements, and quantizers fit this pattern under stated memoryless assumptions. The separation enables two very different questions:

1. Can stability be guaranteed for every $\phi$ in a sector class?
2. For one known $\phi$, can a first-harmonic approximation predict a possible limit cycle?

Sector criteria provide sufficient guarantees for a class. The describing-function intersection supplies a candidate amplitude and frequency that nonlinear analysis must verify.

Sector bounds #

A memoryless scalar nonlinearity lies in sector $[\alpha,\beta]$ when

$$\big(\phi(u)-\alpha u\big)\big(\beta u-\phi(u)\big)\ge0$$

for all relevant $u$. Geometrically, its graph lies between lines of slopes $\alpha$ and $\beta$. For $[0,k]$, the output has the same sign as input and gain no larger than $k$.

A local sector over $|u|\le u_{max}$ gives only a regional result unless trajectories are independently shown to remain there. Hysteresis and dynamic nonlinearities are not memoryless graphs and require extended models.

Absolute stability #

Absolute stability asks whether the feedback interconnection is stable for every nonlinearity in the specified sector, usually under assumptions about $G$ such as stability, properness, and well-posedness. The circle criterion converts a sector condition into a geometric exclusion/encirclement condition on the Nyquist plot after an appropriate loop transformation. The exact circle and inside/outside condition depend on feedback sign, sector endpoints, and open-loop pole count.

Because sign mistakes are easy, derive the transformed loop from the stated equations instead of copying a memorized circle. A satisfied criterion is sufficient; failure of the criterion does not prove instability.

Modern conic-sector formulations express trajectory constraints as quadratic integral inequalities. Passivity is a special sector property. This extends the idea to MIMO systems and computational tests, but still requires the stated factorization and minimum-phase conditions for frequency-domain equivalence.

Popov criterion #

For a common negative-feedback Lur'e configuration with stable $G(s)$ and a memoryless nonlinearity in sector $[0,k]$, a Popov sufficient condition can be written: find $q\ge0$ such that

$$\mathop{\mathrm{Re}}\left[(1+j\omega q)G(j\omega)\right]>-\frac{1}{k}$$

for all $\omega\ge0$, with the theorem's regularity assumptions satisfied. Geometrically, the Popov plot must remain on the permitted side of a line through $-1/k$.

The multiplier $1+qs$ incorporates a derivative-weighted energy argument and can certify cases that a simpler circle test misses. It is not a tuning knob in the physical loop unless explicitly implemented; it is part of the proof.

Describing function #

For sinusoidal input

$$u(t)=A\sin\omega t,$$

pass it through $\phi$ and retain only the fundamental Fourier component. The complex ratio of that fundamental to the input is the describing function $N(A)$ (or $N(A,\omega)$ for dynamic/non-memoryless elements).

For an ideal symmetric relay with output $\pm M$,

$$N(A)=\frac{4M}{\pi A}.$$

For a symmetric saturation, $N(A)$ equals the small-signal gain below the saturation amplitude and decreases with amplitude above it. Hysteresis produces an imaginary component because the output fundamental is phase shifted.

The harmonic-balance condition is

$$1+G(j\omega)N(A)=0,$$

or

$$G(j\omega)=-\frac{1}{N(A)}.$$

An intersection between the Nyquist locus of $G$ and the amplitude-parametrized $-1/N(A)$ locus yields candidate frequency $\omega$ and amplitude $A$.

What the approximation discards #

The linear block filters the nonlinear output's higher harmonics, but they do not vanish. The approximation is most credible when $G$ strongly attenuates harmonics $3\omega,5\omega,\ldots$ relative to the fundamental and the nonlinearity is dominated by one periodic mode.

It can fail through:

• weak harmonic attenuation;
• multiple nonlinearities or multiple candidate intersections;
• subharmonic, quasiperiodic, or chaotic behavior;
• bias/asymmetry producing a DC component;
• memory, rate dependence, or switching delay;
• sampling and quantization;
• predicted amplitude crossing a different operating regime.

An intersection neither proves a periodic solution exists nor proves its orbital stability. A local slope test sometimes suggests stability, but simulation, a Poincaré map, harmonic balance with more terms, or rigorous nonlinear analysis must close the claim.

Worked relay reasoning #

For the ideal relay, harmonic balance becomes

$$G(j\omega)=-\frac{\pi A}{4M}.$$

The right side lies on the negative real axis and moves left as $A$ grows. Therefore candidate frequencies occur where the linear plant Nyquist plot crosses the negative real axis. If the crossing value is $G(j\omega_*)=-g_*$ with $g_*>0$, the predicted amplitude is

$$A=\frac{4Mg_*}{\pi}.$$

Check that this amplitude is consistent with the relay model and that higher harmonics are attenuated. Then simulate the exact relay loop from multiple initial conditions and estimate the cycle with a Poincaré section.

Choosing the method #

Absolute-stability criteria are conservative guarantees; describing functions are informative approximations. Confusing their evidentiary strength is the central failure mode of this topic.