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Unformatted text preview: Topic #21 16.30/31 Feedback Control Systems Systems with Nonlinear Functions Describing Function Analysis Fall 2010 16.30/31 212 NL Example Another classic example Van Der Pol equation 1 : x + ( x 2 1) x + x = 0 which can be written as linear system G ( s ) = s 2 s + 1 in negative feedback with a nonlinear function f ( x,x ) = x 2 x f ( x, x ) G ( s ) x ( t ) q ( t ) x ( t ) Would expect to see different behaviors from the system depending on the value of 321 1 2 3 44321 1 2 3 4 x x =1 =2 =0.2 Of particular concern is the existence of a limit cycle response Sustained oscillation for a nonlinear system, of the type above 1 Slotine and Li, page 158 November 23, 2010 Fall 2010 16.30/31 213 In this case the signal x ( t ) would be of the form of an oscillation x ( t ) = A sin( t ) so that x ( t ) = A cos( t ) Note that A and are not known, and may not actually exist. Given the form of x ( t ) , we have that q ( t ) = x 2 x = A 2 sin 2 ( t ) A cos( t ) A 3 = 4 (cos( t ) cos(3 t )) Thus the output of the nonlinearity (input of the linear part) contains the third harmonic of the input Key point: since the system G ( s ) is low pass, expect that this third harmonic will be suciently attenuated by the linear sys tem that we can approximate A 3 q ( t ) = x 2 x cos( t ) 4 A 2 d = [ A sin( t )] 4 dt Note that we can now create an effective transfer function of this nonlinearity by defining that: A 2 j q = N ( A, )( x ) N ( A, ) = 4 which approximates the effect of the nonlinearity as a frequency re sponse function. N ( A, ) G ( s ) x ( t ) q ( t ) x ( t ) November 23, 2010 Fall 2010 16.30/31 214 What are the implications of adding this nonlinearity into the feedback loop? Can approximately answer that question by looking at the stability of G ( s ) in feedback with N . x = A sin( t ) = G ( j ) q = G ( j ) N ( A, )( x ) which is equivalent to: (1 + G ( j ) N ( A, )) x = 0 that we can rewrite as: A 2 ( j ) 1 + = 0 4 ( j ) 2 ( j ) + 1 which is only true if A = 2 and = 1 These results suggest that we could get sustained oscillations in this case (i.e. a limit cycle) of amplitude 2 and frequency 1. This is consistent with the response seen in the plots independent of we get sustained oscillations in which the x ( t ) value settles down to an amplitude of 2. Note that does impact the response and changes the shape/fea tures in the response. Approach (called Describing Functions ) is generalizable.......
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This note was uploaded on 02/03/2012 for the course AERO 16.30 taught by Professor Ericferon during the Fall '04 term at MIT.
 Fall '04
 EricFeron

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