Integral Quadratic Constraints

Integral Quadratic Constraints (IQC) are inequalities used to describe (partially) possible signal combinations within a given dynamical system. IQC offer a framework for abstracting ”challenging” (e.g. non-linear, time-varying, uncertain, or distributed) elements of dynamical system models to aid in rigorous analysis of robust stability and performance (more specifically, to establish L2 gain bounds, passivity, and other system properties which can be expressed, exactly or approximately, in terms of generalized dissipativity). While the technique can be employed to prove general theorems, it is most powerful when used to derive optimization-based algorithms for certification of stability and robustness of specific feedback systems. IQC can be viewed as implicit generalized dissipation inequalities with known quadratic supply rates and unspecified storage functions, not necessarily quadratic or sign definite. Alternatively, they have frequency a domain interpretation as bounds on the degree of harmonic distortion produced by a specific element of the complete model. The past research in nonlinear systems and robust control can be harvested to extract rich IQC descriptions of commonly used components of feedback systems. These IQC can then be re-used in a modular approach to system analysis. The IQC framework is closely related to multiplier based passivity, upper bounding of structured singular values, quadratic relaxations in non-convex optimization (including the sums of squares approach to positivity of multivariable polynomials), and other constructive techniques for handling nonlinearity and uncertainty.