Kalman–Yakubovich–Popov lemma

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

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]

References

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.