hw8_6243_2003

hw8_6243_2003 - Massachusetts Institute of Technology...

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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 81 Problem 8.1 Autonomous system equations have the form � � � y (t ) y (t) y (t) = ¨ Q , y t) ˙( y t) ˙( (8.1) 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 � : R2 ≥� R2 , matrices A, C , and a C � function � : R ≥� R2 such that z = � (y, y satisfies the ODE ˙) z t) = Az (t) + �(y (t)), y (t) = C z (t) ˙( whenever y (·) satisfies (8.1). (b) For those Q found in (a), construct C � functions F = FQ : R2 × R ≥� R2 and H = HQ : R2 ≥� R such that HQ (� (t)) − y t) � 0 as t � → whenever ˙( y : [0, →) ≥� R is a solution of (8.1), and � t) = FQ (� (t), y (t)). ˙( 1 Posted November 19, 2003. Due date November 26, 2003 2 Problem 8.2 A linear constrol system � x1 (t) = ˙ x2 (t) + w1 (t), x2 (t) = −x1 (t) − x2 (t) + u + w2 (t) ˙ is equipped with the nonlinear sensor y (t) = x1 (t) + sin(x2 (t)) + w3 (t), where wi (·) represent plant disturbances and sensor noise satisfying a uniform bound |wi (t)| � d. Design an observer of the form � t) = F (� (t), y (t), u(t)) ˙( and constants d0 > 0 and C > 0 such that |� (t) − x(t)| � C d � t ∀ 0 whenever � (0) = x(0) and d < d0 . (Try to make d0 as large as possible, and C as small as possible.) Problem 8.3 Is it true or false that the set � = �F = {P } of positive definite quadratic forms VP (x) = ¯ � � x P x, where P = P > 0, which are valid control Lyapunov function for a given ODE ¯¯ model x(t) = F (x(t), u(t)), ˙ in the sense that x¯ inf x� P F (¯, u) � −|x|2 � x ≤ Rn , ¯ ¯ uR ¯ is linearly connected for all continuously differentiable functions F : Rn × R ≥� Rn ? (Remember that a set � of matrices is called linearly connected if for every two matrices P0 , P1 ≤ � there exists a continuous function p : [0, 1] ≥� � such that p(0) = P0 and p(1) = P1 . In particular, the empty set is linearly connected.) ...
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