Stretched exponential function: Difference between revisions

Revision as of 05:24, 24 January 2014

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 by 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+PA=-qq^{T}\,

Pb-c={\sqrt {\gamma }}q\,

exist if and only if

\gamma +2Re[c^{T}(j\omega I-A)^{-1}b]\geq 0

Moreover, the set $\{x:x^{T}Px=0\}$ 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 Kalman, who brought together results by Vladimir Andreevich Yakubovich and Vasile Mihai Popov.

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+The '''Kalman–Yakubovich–Popov lemma''' is a result in [[system analysis]] and [[control theory]] which states: Given a number <math>\gamma > 0</math>, two n-vectors b, c and an n by n [[Hurwitz matrix]] A, if the pair <math>(A,b)</math> is completely [[controllability|controllable]], then a symmetric matrix P and a vector q satisfying
+:<math>A^T P + P A = -q q^T\,</math>
+:<math> P b-c = \sqrt{\gamma}q\,</math>
+exist if and only if
+:<math>
+\gamma+2 Re[c^T (j\omega I-A)^{-1}b]\ge 0
+</math>
+Moreover, the set <math>\{x: x^T P x = 0\}</math> is the unobservable subspace for the pair <math>(A,b)</math>.
+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 Kalman|Kalman]], who brought together results by [[Vladimir Andreevich Yakubovich]] and [[Vasile M. Popov|Vasile Mihai Popov]].
+{{DEFAULTSORT:Kalman-Yakubovich-Popov Lemma}}
+[[Category:Lemmas]]
+[[Category:Stability theory]]