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 \gamma > 0, two n-vectors B, C and an n x n Hurwitz matrix A, if the pair (A,B) is completely controllable, then a symmetric matrix P and a vector Q satisfying

A^T P + P A = -Q Q^T\,
 P B-C = \sqrt{\gamma}Q\,

exist if and only if

\gamma+2 Re[C^T (j\omega I-A)^{-1}B]\ge 0

Moreover, the set \{x: x^T P x = 0\} is the unobservable subspace for the pair (A,B).

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 A \in \R^{n \times n}, B \in \R^{n \times m}, M = M^T \in \R^{(n+m) \times (n+m)} with \det(j\omega I - A) \ne 0 for all \omega \in \R and (A, B) controllable, the following are equivalent:

  1. for all \omega \in \R \cup \{\infty\}
     \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right]^*   M   \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right] \le 0
  2. there exists a matrix P \in \R^{n \times n} such that P = P^T and
    M + \left[\begin{matrix} A^T P + PA & PB \\ B^T P & 0 \end{matrix}\right] \le 0.

The corresponding equivalence for strict inequalities holds even if (A, B) is not controllable. [2]


  1. ^ Kalman, Rudolf E. (1963). "Lyapunov functions for the problem of Lur'e in automatic control". 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.