Kalman–Yakubovich–Popov lemma

The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number ${\displaystyle \gamma >0}$, two n-vectors B, C and an n x n Hurwitz matrix A, if the pair ${\displaystyle (A,B)}$ is completely controllable, then a symmetric matrix P and a vector Q satisfying

${\displaystyle A^{T}P+PA=-QQ^{T}}$

${\displaystyle PB-C={\sqrt {\gamma }}Q}$

exist if and only if

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

Moreover, the set ${\displaystyle \{x:x^{T}Px=0\}}$ is the unobservable subspace for the pair ${\displaystyle (C,A)}$.

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.

The Kalman–Popov–Yakubovich lemma which was first formulated and proved in 1962 by Vladimir Andreevich Yakubovich^[1] where it was stated that for the strict frequency inequality. The case of nonstrict frequency inequality was published in 1963 by Rudolf E. Kálmán.^[2] In that paper the relation to solvability of the Lur’e equations was also established. Both papers considered scalar-input systems. The constraint on the control dimensionality was removed in 1964 by Gantmakher and Yakubovich^[3] and independently by Vasile Mihai Popov.^[4] Extensive reviews of the topic can be found in ^[5] and in Chapter 3 of.^[6]

Multivariable Kalman–Yakubovich–Popov lemma

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

1. for all ${\displaystyle \omega \in \mathbb {R} \cup \{\infty \}}$

   ${\displaystyle \left[{\begin{matrix}(j\omega I-A)^{-1}B\\I\end{matrix}}\right]^{*}M\left[{\begin{matrix}(j\omega I-A)^{-1}B\\I\end{matrix}}\right]\leq 0}$

2. there exists a matrix ${\displaystyle P\in \mathbb {R} ^{n\times n}}$ such that ${\displaystyle P=P^{T}}$ and

   ${\displaystyle M+\left[{\begin{matrix}A^{T}P+PA&PB\\B^{T}P&0\end{matrix}}\right]\leq 0.}$

The corresponding equivalence for strict inequalities holds even if ${\displaystyle (A,B)}$ is not controllable. ^[7]