problem6_sol

# problem6_sol - EE221A Problem Set 6 Solutions Fall 2011...

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Unformatted text preview: EE221A Problem Set 6 Solutions - Fall 2011 Problem 1. Linear systems. a) Call this dynamical system L = ( U , Σ , Y ,s,r ) , where U = R n i , Σ = R n , Y = R n o . So clearly U , Σ , Y are all linear spaces over the same field ( R ). We also have the response map ρ ( t,t ,x ,u ) = y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) and the state transition function s ( t,t ,x ,u ) = x ( t ) = Φ( t,t ) x + ˆ t t Φ( t,τ ) B ( τ ) u ( τ ) dτ We need to check the linearity of the response map; we have that, ∀ t ≥ t ,t ∈ R + : ρ ( t,t α 1 x 1 + α 2 x 2 ,α 1 u 1 + α 2 u 2 ) = C ( t ) Φ( t,t )( α 1 x 1 + α 2 x 2 ) + ˆ t t Φ( t,τ ) B ( τ )( α 1 u 1 ( τ ) + α 2 u 2 ( τ )) dτ + D ( t )( α 1 u 1 ( τ ) + α 2 u 2 ( τ )) = α 1 C ( t )Φ( t,t ) x 1 + ˆ t t Φ( t,τ ) B ( τ ) u 1 ( τ ) dτ + D ( t ) u 1 ( t ) + α 2 C ( t )Φ( t,t ) x 1 + ˆ t t Φ( t,τ ) B ( τ ) u 2 ( τ ) dτ + D ( t ) u 2 ( t ) = α 1 ρ ( t,t ,x 1 ,u 1 ) + α 2 ρ ( t,t ,x 2 ,u 2 ) b) Using the definition of time-invariance for dynamical systems, check: ρ ( t 1 + τ,t + τ,x ,T τ u ) = Cx ( t 1 + τ ) + Du (( t 1 + τ )- τ ) = C e A ( t 1 + τ- ( t + τ )) x + ˆ t 1 + τ t + τ e A ( t 1 + τ- σ ) Bu ( σ- τ ) dσ + Du ( t 1 ) = Ce A ( t 1- t ) x + ˆ t 1 t e A ( t 1- s ) Bu ( s ) ds + Du ( t 1 ) = ρ ( t 1 ,t ,x ,u ) Problem 2. A linear time-invariant system....
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problem6_sol - EE221A Problem Set 6 Solutions Fall 2011...

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