KalmanYakubovichPopov

The Kalman–Yakubovich–Popov lemma, often shortened to the KYP lemma, is a foundational result in systems and control theory that connects frequency-domain properties of a linear time-invariant system with a time-domain quadratic form. Named after Rudolf Kalman, Sergei Yakubovich, and Valentin Popov, the lemma provides a bridge between energy-based (dissipative) descriptions and state-space realizations, enabling the use of linear matrix inequalities (LMIs) for analysis and design.

In its standard continuous-time formulation, for a system with state-space realization x_dot = A x + B u, y = C x + D u and transfer function G(s) = C(sI − A)^{-1} B + D, G is positive real (or passive) if and only if there exists a symmetric matrix P ≽ 0 such that:

A^T P + P A ≤ 0

P B = C^T

D^T + D ≽ 0

Variants exist depending on strictness, time domain (continuous vs discrete), and whether the system is strictly positive real. The lemma thus translates a frequency-domain condition into a set of LMIs, making verification and controller synthesis amenable to convex optimization techniques.

Applications span stability analysis of interconnected systems, passive and dissipative control design, H-infinity control, model reduction, and robust control. The KYP lemma has become a standard tool in both theoretical investigations and practical engineering, enabling systematic, computational approaches to classical problems.

Historically, the result appeared in the 1960s through the independent contributions of Kalman, Yakubovich, and Popov, and it has since generated numerous extensions and variants in continuous and discrete time.