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Unformatted text preview: Simon Fraser University School of Engineering Science ENSC 483  Modern Control Systems Spring 2007–Assignment 6 1. Transformation into OCF. Consider the SISO system described by ˙ x = Ax + B u y = Cx Prove that the system can be transformed by x = P ˆ x , with P being a nonsingular transfor mation into an equivalent system with: ˆ A = 0 0 ··· 0 0 a 1 0 ··· 0 0 a 1 . . . . . . . . . . . . . . . . . . 0 0 ··· 1 0 a n 2 0 0 ··· 0 1 a n 1 ˆ c = ‡ 0 0 0 ··· 0 1 · given that ρ c cA cA 2 . . . cA n 1 = n Hint: To prove this, follow along the same lines as the proof for transformation into CCF which we did in the class, and take P = ‡ v Av A 2 v ··· A n 1 v · 2. Consider the system ˙ x = Ax + Bu y = Cx where A = 0 1 0 0 0 1 0 0 0 1 6 1 7 1 , B = 0 0 1 0 0 0 0 1 , C = ‡ 1 0 1 0 · (a) Let us redefine the state variables of this system as x = 1 1 1 1 λ 1 λ 2 λ 3 λ 4 λ 2 1 λ 2 2 λ 2 3 λ 2 4 λ 3 1 λ 3 2 λ 3 3 λ 3 4 ˆ x where λ i s are the eigenvalues of the system. Find the equivalent system described in the following form ˙ ˆ x = ˆ A ˆ x + ˆ Bu y = ˆ C ˆ x (b) Calculate the transfer function of the original system. (c) Calculate the transfer function of the transformed system. Hint: You may find MATLAB’s ss2tf useful ! Remark: Note that the matrix A in the above is in companion form, and has distinct eigen values. The transformation matrix that I defined above is a special one called a Vandermonde Matrix , and will do what it did in the above for any size matrix in companion form and with distinct eigenvalues....
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 Spring '07
 MehrdadSaif
 Matrices, Eigenvalue, eigenvector and eigenspace, Modern Control Systems

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