Integral Quadratic Constraints
IQCs certify stability of uncertain and nonlinear feedback loops through frequency-domain quadratic bounds on the troublesome component's signals.
A unifying framework
An integral quadratic constraint describes a system component by a frequency-domain inequality: the integral over frequency of a quadratic form in the input and output spectra, weighted by a multiplier Pi, is nonnegative. Many components, saturations, time delays, sector nonlinearities, uncertain gains, satisfy IQCs. The framework of Megretski and Rantzer turns diverse stability problems into one template.
The stability theorem
If the nominal system is stable and there exists a valid multiplier Pi such that a single frequency-domain inequality holds combining the nominal transfer function with Pi, then the feedback interconnection is stable. Searching over admissible multipliers is a convex LMI problem once the multiplier is restricted to a finite basis, so the test is computable.
Why it beats small gain
The small-gain theorem is the special case where the multiplier is a scaled identity. Richer multipliers, such as Zames-Falb multipliers for monotone nonlinearities or Popov multipliers, encode phase and slope information the small-gain test throws away, giving far less conservative results. The same machinery recovers the circle and Popov criteria as special cases.
- Component described by a quadratic frequency-domain bound
- One LMI over multipliers certifies loop stability
- Generalizes small-gain, circle, and Popov criteria
- Handles delays, saturation, sector and slope-bounded nonlinearities
IQC analysis is primarily a verification tool: given a controller and a menu of uncertainties and nonlinearities, it produces a rigorous stability certificate or reveals the margin. Synthesis is harder, but IQC-based robustness checks routinely validate controllers designed by other means.
For a design-stage loop containing actuator saturation and an uncertain delay, IQCs let one certificate cover both effects at once, which piecewise arguments cannot do cleanly. The method is a cornerstone of modern robustness analysis for nonlinear and uncertain systems.