e483h6_07 - Simon Fraser University School of Engineering...

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Unformatted text preview: Simon Fraser University School of Engineering Science ENSC 483 - Modern Control Systems Spring 2007Assignment 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 x , with P being a nonsingular transfor- mation into an equivalent system with: 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 c = 0 0 0 0 1 given that 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 re-define the state variables of this system as x = 1 1 1 1 1 2 3 4 2 1 2 2 2 3 2 4 3 1 3 2 3 3 3 4 x where i s are the eigenvalues of the system. Find the equivalent system described in the following form x = A x + Bu y = C x (b) Calculate the transfer function of the original system. (c) Calculate the transfer function of the transformed system. Hint: You may find MATLABs 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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e483h6_07 - Simon Fraser University School of Engineering...

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