Kalman–Yakubovich–Popov lemma - Alchetron, the free social encyclopedia

The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number γ > 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 = γ Q

exist if and only if

γ + 2 R e [ C T ( j ω I − A ) − 1 B ] ≥ 0

Moreover, the set { x : x T P x = 0 } is the unobservable subspace for the pair ( 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.

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

Multivariable Kalman–Yakubovich–Popov lemma

Given A ∈ R n × n , B ∈ R n × m , M = M T ∈ R ( n + m ) × ( n + m ) with det ( j ω I − A ) ≠ 0 for all ω ∈ R and ( A , B ) controllable, the following are equivalent:

for all ω ∈ R ∪ { ∞ } [ ( j ω I − A ) − 1 B I ] ∗ M [ ( j ω I − A ) − 1 B I ] ≤ 0
there exists a matrix P ∈ R n × n such that P = P T and M + [ A T P + P A P B B T P 0 ] ≤ 0.

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

References

Kalman–Yakubovich–Popov lemma Wikipedia

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