hw7_6243_2003

hw7_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 71 Problem 7.1 A stable linear system with a relay feedback excitation is modeled by x(t) = Ax(t) + B sgn(Cx(t)), ˙ (7.1) where A is a Hurwitz matrix, B is a column matrix, C is a row matrix, and sgn(y ) denotes the sign nonlinearity � y > 0, 1, 0, y = 0, sgn(y ) = � −1, y < 0. For T > 0, a 2T -periodic solution x = x(t) of (7.1) is called a regular unimodal limit cycle if C x(t) = −Cx(t + T ) > 0 for all t → (0, T ), and C Ax(0) > |CB |. (a) Derive a necessary and sufficient condition of exponential local stability of the reg­ ular unimodal limit cycle (assuming it exists and A, B , C, T are given). (b) Use the result from (a) to find an example of system (7.1) with a Hurwitz matrix A and an unstable regular unimodal limit cycle. Problem 7.2 A linear system controlled by modulation of its coefficients is modeled by x(t) = (A + B u(t))x(t), ˙ where A, B are fixed n-by-n matrices, and u(t) → R is a scalar control. 1 Posted November 5, 2003. Due date November 12, 2003 (7.2) 2 (a) Is it possible for the system to be controllable over the set of all non-zero vectors x → Rn , x = 0, when n � 3? In other words, is it possible to find matrices A, B with ¯ ¯ ∞ n > 2 such that for every non-zero x0 , x there exist T > 0 and a bounded function ¯ ¯1 u : [0, T ] ≥� R such that the solution of (7.2) with x(0) = x0 satisfies x(T ) = x1 ? ¯ ¯ (b) Is it possible for the system to be full state feedback linearizable in a neigborhood of some point x0 → Rn for some n > 2? ¯ Problem 7.3 A nonlinear ODE control model with control input u and controlled output y is defined by equations x1 ˙ x2 ˙ x3 ˙ y = = = = x2 + x2 , 3 (1 − 2x3 )u + a sin(x1 ) − x2 + x3 − x2 , 3 u, x1 , where a is a real parameter. (a) Output feedback linearize the system over a largest subset X0 of R3 . (b) Design a (dynamical) feedback controller with inputs x(t), r(t), where r = r(t) is the reference input, such that for every bounded r = r(t) the system state x(t) stays bounded as t � �, and y (t) � r(t) as t � � whenever r = r(t) is constant. (c) Find all values of a → R for which the open loop system is full state feedback linearizable. (d) Try to design a dynamical feedback controller with inputs y (t), r(t) which achieves the objectives from (b). Test your design by a computer simulation. ...
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