EE_302_2010_HW8_soln - EE302 HW8 Q1. Consider the following...

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EE302 HW8 Solution Q1. Consider the following system: ... y + 4¨ y + 5 ˙ y + 2 y = 5 ˙ u + u. Obtain controllable canonical, observable canonical, and (if possible) diagonal canonical state space representations. Sol’n. The transfer function of the system is Y ( s ) U ( s ) = 5 s + 1 s 3 + 4 s 2 + 5 s + 2 from which we can readily write the controllable canonical representation ˙ x = 0 1 0 0 0 1 2 5 4 x + 0 0 1 u y = [1 5 0] x. The observable canonical representation is ˙ x = 0 0 2 1 0 5 0 1 4 x + 1 5 0 u y = [0 0 1] x. To be able to write a diagonal canonical representation the denominator polynomial of the trans- fer function need have distinct roots. Note that the denominator polynomial can be factored as D ( s ) = ( s + 1) 2 ( s + 2). Due to the double root at s = 1 diagonal canonical representation is not possible for this system. Q2. Consider the following system: ˙ x = [ 1 1 7 7 ] x + [ 1 2 ] u y = [1 1] x. Obtain the transfer function Y ( s ) /U ( s ) and find its poles and zero(s). Obtain controllable canonical and observable canonical state space representations. Sol’n. We can write Y ( s ) U ( s ) = C ( sI A ) - 1 B = [1 1] [ s + 1 1 7 s + 7 ] - 1 [ 1 2 ] = [1 1] { 1 s ( s + 8) [ s + 7 1 7 s + 1 ]}[ 1 2 ] = 3 s + 18 s 2 + 8 s .
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The poles are { 0 , 8 } and the zero is {− 6 } . The controllable canonical representation is ˙ x = [ 0 1 0 8 ] x + [ 0 1 ] u y = [18 3] x. The observable canonical representation is ˙ x = [ 0 0 1 8 ] x + [ 18 3 ] u y = [0 1] x. Q3. Consider the following system: ˙ x = Ax + Bu y = Cx where A = [ 1 1 7 7 ] (a) Find the eigenvalues of A . (b) Find (right) eigenvectors r 1 , r 2 R 2 × 1 of A . (c) Is system controllable for B = r 1 ? How about for B = r 2 ? (d) Find (left) eigenvectors 1 , ℓ 2 R 2 × 1 of A . (Vector is said to be a left eigenvector with corresponding eigenvalue λ if it satisfies T A = λℓ T .) (e) Is system observable for C = T 1 ? How about for C = T 2 ? (f) Let V R 2 × 2 be defined as V = [ r 1 r 2 ]. Compute matrix Λ defined as Λ = V - 1 AV . (g) Compute matrix exponential e At = L - 1 { ( sI A ) - 1 } . Compute matrix exponential e Λ t . Show that
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EE_302_2010_HW8_soln - EE302 HW8 Q1. Consider the following...

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