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Unformatted text preview: Sam Burden Linear Systems ( EE 221 ) Recitation #N Dec 3, 2010 1 Duality Definition The following system representations are dual : R = x = A ( t ) x + B ( t ) u y = C ( t ) x + D ( t ) u , e R = =- A ( t ) * - B ( t ) * = B * ( t ) + D ( t ) * . Let , be the state transition matrices for R , e R . Lemma ( t, ) = ( ,t ) * . Proof. By the Pairing Lemma with u 0, h ( t ) ,x ( t ) i = h ( ) ,x ( ) i . But h ( t ) ,x ( t ) i = h ( t ) , ( t, ) x ( ) i = h * ( t, ) ( t ) ,x ( ) i . Therefore x ( ) : h ( )- * ( t, ) ( t ) ,x ( ) i = 0, whence ( ) = * ( t, ) ( t ) = ( ,t ) ( t ). Theorem ( Duality ) The subspace of states controllable to zero for R is the orthogonal complement of the subspace of states unobservable for e R : R ( L c ) = N ( e L o ) . Proof. unobservable for e R on [ t ,t 1 ] e L o = 0 B ( t ) * ( t,t ) = 0 on [ t ,t 1...
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