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hw3_6243_2003

# hw3_6243_2003 - u t = sgn x 1 t 5 x 2 t 2 sgn x 2 t(Do you...

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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 3 1 Problem 3.1 Find out which of the functions V : R 2 R , 2 (a) V ( x 1 , x 2 ) = x 2 + x 2 ; 1 (b) V ( x 1 , x 2 ) = | x 1 | + | x 2 | ; (c) V ( x 1 , x 2 ) = max | x 1 | , | x 2 | ; are valid Lyapunov functions for the systems (1) x ˙ 1 = x 1 + ( x 1 + x 2 ) 3 , x ˙ 2 = x 2 ( x 1 + x 2 ) 3 ; x 1 = x 2 x 1 ( x 2 2 x 2 = x 1 x 2 ( x 2 2 (2) ˙ + x 2 ), ˙ + x 2 ); 1 1 (3) x ˙ 1 = x 2 | x 1 | , x ˙ 2 = x 1 | x 2 | . Problem 3.2 Show that the following statement is not true. Formulate and prove a correct version: if V : R n ≤� R is a continuously differentiable functional and a : R n ≤� R n is a continuous function such that V x ) 0 ¯ x ) = 1 , (3.1) x ) a x : V then V ( x ( t )) 1 for every solution x : [0 , ) R n of x ˙( t ) = a ( x ( t )) (3.2) with V ( x (0)) 1 . 1 Posted September 24, 2003. Due date October 1, 2003

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2 Problem 3.3 The optimal minimal-time controller for the double integrator system with bounded con- trol x ˙ 1 ( t ) = x 2 ( t ) , | u ( t ) | 1 x ˙ 2 ( t ) = u ( t ) , has
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Unformatted text preview: u ( t ) = sgn( x 1 ( t ) + . 5 x 2 ( t ) 2 sgn( x 2 ( t ))) . (Do you know why ?) (a) Find a Lyapunov function V : R 2 R 2 for the closed loop system, such that V ( x ( t )) is strictly decreasing along all solutions of system equations except the equilibrium solution x ( t ) 0. (b) Find out whether the equilibrium remains asymptotically stable when the same controller is used for the perturbed system x 1 ( t ) = x 2 ( t ) , | u ( t ) | 1 , x 2 ( t ) = x 1 ( t ) + u ( t ) , where &amp;gt; is small....
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• Spring '09
• AlexandreMegretski
• Dynamics, Massachusetts Institute of Technology, Stability theory, Department of Electrical Engineering and Computer Science, DYNAMICS OF NONLINEAR SYSTEMS, 6.243j

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hw3_6243_2003 - u t = sgn x 1 t 5 x 2 t 2 sgn x 2 t(Do you...

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