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hw8_6243_2003

# hw8_6243_2003 - Massachusetts Institute of Technology...

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± ± Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 8 1 Problem 8.1 Autonomous system equations have the form y ( t ) (8.1) y ¨( t ) = y y ˙( ( t t ) ) Q y ˙( t ) , where y is the scalar output, and Q = Q is a given symmetric 2-by-2 matrix with real coeﬃcients. (a) Find all Q for which there exists a C bijection : R 2 ≥� R 2 , matrices A, C , and a C function : R ≥� R 2 such that z = ( y, y ˙) satisfies the ODE z ˙( t ) = Az ( t ) + ( y ( t )) , y ( t ) = Cz ( t ) whenever y ( · ) satisfies (8.1). (b) For those Q found in (a), construct C functions F = F Q : R 2 × R ≥� R 2 and H = H Q : R 2 ≥� R such that H Q ( ( t )) y ˙( t ) 0 as t � → whenever y : [0 , ) ≥� R is a solution of (8.1), and ˙( t ) = F Q ( ( t ) , y ( t )) . 1 Posted November 19, 2003. Due date November 26, 2003

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2 Problem 8.2 A linear constrol system x ˙ 1 ( t ) = x 2 ( t ) + w 1 ( t ) , x ˙ 2 ( t ) = x 1 ( t ) x 2 ( t ) + u + w 2 ( t ) is equipped with the nonlinear sensor y ( t ) = x 1 ( t ) + sin( x 2 ( t )) + w 3 ( t ) , where w i ( · ) represent plant disturbances and sensor noise satisfying a uniform bound | w i ( t ) | � d . Design an observer of the form ˙( t ) = F ( ( t ) , y ( t ) , u ( t )) and constants d 0 > 0 and C > 0 such that | ( t ) x ( t ) | Cd t 0 whenever (0) = x (0) and d < d 0 . (Try to make d 0 as large as possible, and C as small
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