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# HW1 - M such that A = M-1 ¯ AM Obtain the ±ollowing ±or...

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ECE 515 Assignment # 1 Issued: January 18 Due: January 28, 2010 Reading Assignment: Lecture Notes Ch. 1 and 2. See also Brogan , § 3.1–3.4, 5.1–5.8, or Chen § 2.1–2.8, 3.3–3.5. Problems: 1 . Based on Sections 1.1 and 1.2 of the lecture notes : You are given a nonlinear input- output system which satisfies the nonlinear differential equation: ¨ y ( t ) = y 2 ( u y )+2 ˙ u . (a) Obtain a two-dimensional nonlinear state-space representation with output y , input u , and states x 1 = y and x 2 = ˙ y 2 u . (b) Obtain an all-integrator block diagram system description consistent with (a). The ‘blocks’ will include nonlinearities, such as y^2 (powers) or y.u (products). (c) Linearize this system of equations around its equilibrium output trajectory when u ( · ) 0, and write it in state space form. (d) Find the transfer function for the linear system obtained in (c). 2 . Let A be an n × n matrix, and suppose that the infinite sum exists U = I + A + A 2 + A 3 + · · · where I denotes the identity matrix. Verify that U is the inverse of the matrix I A 3 . Two square matrices A and ¯ A are called similar if there is an invertible matrix
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Unformatted text preview: M such that A = M-1 ¯ AM Obtain the ±ollowing ±or two similar matrices A and ¯ A . (i) Show that A m is similar to ¯ A m ±or any m ≥ 1, where the superscript “ m ” denotes matrix product, A 1 = A, A m = A ( A m-1 ) , m ≥ 1 . (ii) Show that v is an eigenvector ±or A i± and only i± Mv is an eigenvector ±or ¯ A . (iii) Suppose that ¯ A is diagonal ( ¯ A ij = 0 i± i n = j ). Suppose moreover that | ¯ A ii | < 1 ±or each i . Conclude that I − A admits an inverse by combining Prob. 2 and Prob. 3 (i). 4 . Which o± the ±ollowing sets are linearly independent? (a) 1 3 2 , 1 , 1 in ( R 3 , R ). (b) 1 3 2 , 1 , 1 , 1 √ 2 π , in ( R 3 , R ). (c) 1 3 2 , 1 , 1 , 1 √ 2 j π , in ( C 3 , C )....
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