# recN - Sam Burden Linear Systems EE 221 Recitation#N —...

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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 = ˙ ξ =- A ( t ) * ξ- B ( t ) * μ η = B * ( t ) ξ + D ( t ) * μ . Let Φ , Ψ be the state transition matrices for R , e R . Lemma Ψ( t,τ ) = Φ( τ,t ) * . Proof. By the Pairing Lemma with u ≡ μ ≡ 0, h ξ ( t ) ,x ( t ) i = h ξ ( τ ) ,x ( τ ) i . But h ξ ( t ) ,x ( t ) i = h ξ ( t ) , Φ( t,τ ) x ( τ ) i = h Φ * ( t,τ ) ξ ( t ) ,x ( τ ) i . Therefore ∀ x ( τ ) : h ξ ( τ )- Φ * ( t,τ ) ξ ( t ) ,x ( τ ) i = 0, whence ξ ( τ ) = Φ * ( t,τ ) ξ ( t ) = Ψ( τ,t ) ξ ( 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. ξ unobservable for e R on [ t ,t 1 ] ⇐⇒ e L o ξ = 0 ⇐⇒ B ( t ) * Ψ( t,t ) ξ = 0 on [ t ,t 1...
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recN - Sam Burden Linear Systems EE 221 Recitation#N —...

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