Kalman–Yakubovich–Popov lemma

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The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair is completely controllable, then a symmetric matrix P and a vector Q satisfying

exist if and only if

Moreover, the set is the unobservable subspace for the pair .

The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain.

It was derived in 1962 by Rudolf E. Kalman,[1] who brought together results by Vladimir Andreevich Yakubovich and Vasile Mihai Popov.

Multivariable Kalman–Yakubovich–Popov lemma[edit]

Given with for all and controllable, the following are equivalent:

  1. for all
  2. there exists a matrix such that and

The corresponding equivalence for strict inequalities holds even if is not controllable. [2]


  1. ^ Kalman, Rudolf E. (1963). "Lyapunov functions for the problem of Lur'e in automatic control" (PDF). Proceedings of the National Academy of Sciences. 49 (2): 201–205. doi:10.1073/pnas.49.2.201. 
  2. ^ "Anders Rantzer" (1996). "On the Kalman–Yakubovich–Popov lemma". Systems & Control Letters. 28 (1): 7–10. doi:10.1016/0167-6911(95)00063-1.