hw3_6243_2003

hw3_6243_2003 - u ( t ) = sgn( x 1 ( t ) + . 5 x 2 ( t ) 2...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ² ¯ 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 the form
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 > is small....
View Full Document

This note was uploaded on 11/07/2011 for the course AERO 16.36 taught by Professor Alexandremegretski during the Spring '09 term at MIT.

Page1 / 2

hw3_6243_2003 - u ( t ) = sgn( x 1 ( t ) + . 5 x 2 ( t ) 2...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online