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Unformatted text preview: The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic Control—Fall 2009 Solutions to Problem Set Three C. Caramanis October 5, 2009. 1. • We use the notation  A  to denote the determinant of a square matrix A . The rule for determinant of the product of matrices gives  T 1 AT  =  T  A  T 1  . Also TT 1 = I gives  T 1  =  T  1 , therefore,  T 1 AT  =  T  A  T 1  =  T  A  T  1 =  A  . Denoting the characteristic polynomials of A and T 1 AT by p A ( λ ) and p T 1 AT ( λ ) gives p A ( λ ) =  λI A  =  T  1  λI A  T  =  T 1  λI A  T  =  T 1 ( λI A ) T  =  λT 1 T T 1 AT  =  λI T 1 AT  = p T 1 AT ( λ ) Since the characteristic polynomials of A and T 1 AT are same, they have the same eigenvalues as eigenvalues are roots of the characteristic polynomial. • For a square matrix A , its eigenvalue λ and corresponding eigenvector v satisfy Av = λv . Since TT 1 = I , we have ATT 1 v = λv . Multiplying both sides by T 1 from the left gives T 1 ATT 1 v = λ ( T 1 v ) ⇒ ( T 1 AT )( T 1 v ) = λ ( T 1 v ) ⇒ ( T 1 AT ) w = λw , where w , T 1 v . Hence, w = T 1 v is the eigenvector corresponding to the eigenvalue λ for the matrix T 1 AT . 2. (a) We assume α 6 = 0 or else the utility of feedback would be eliminated. Substituting u = αx in the differential equation and solving for ˙ x = 0 gives 1 1 + x 1 αx = 0 ⇒ x x + 1 + 1 α = 0 ⇒ x = 0 , 1 1 α Hence, the equilibrium points are x e = 0 , 1 1 α . (b) V ( x ) = 1 2 x 2 is positive definite about x e = 0 as V (0) = 0 and V ( x ) > 0 for x 6 = 0. ˙ V ( x ) = x ˙ x = αx 2 x + 1 + 1 α x + 1 ! For stability about x e = 0, ˙ V ( x ) must be locally negative semidefinite about x e = 0. As ˙ V (0) = 0, we only require ˙ V ( x ) ≤ 0 for x 6 = 0. Since x 2 > 0 for x 6 = 0, we have ˙ V ( x ) = αx 2 x + 1 + 1 α x + 1 ! ≤ ⇒ x + 1 + 1 α x + 1 ≥ 0 if α > ⇒ x + 1 + 1 α x + 1 ≤ 0 if α < 1 Consider the case α > 0; the range of x satisfying the inequality can be found to be x ≥  1 or x ≤  1 1 α i.e. (∞ , 1 1 α ] ∪ [ 1 , ∞ ). This range always includes a neighborhood around...
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This note was uploaded on 10/26/2009 for the course EE 362K taught by Professor Friedrich during the Fall '08 term at University of Texas.
 Fall '08
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