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MIT16_30F10_lec21

MIT16_30F10_lec21 - Topic#21 16.30/31 Feedback Control...

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Topic #21 16.30/31 Feedback Control Systems Systems with Nonlinear Functions Describing Function Analysis
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Fall 2010 16.30/31 21–2 NL Example Another classic example Van Der Pol equation 1 : x ¨ + α ( x 2 1) ˙ x + x = 0 which can be written as linear system α G ( s ) = s 2 αs + 1 in negative feedback with a nonlinear function f ( x, x ˙) = x 2 x ˙ 0 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 α -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 x ˙ x α =1 α =2 α =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
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Fall 2010 16.30/31 21–3 In this case the signal x ( t ) would be of the form of an oscillation x ( t ) = A sin( ωt ) so that x ˙( t ) = cos( ωt ) Note that A and ω 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 ( ωt ) cos( ωt ) A 3 ω = 4 (cos( ωt ) cos(3 ωt )) 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 “sufficiently attenuated” by the linear sys- tem that we can approximate A 3 ω q ( t ) = x 2 x ˙ ≈ − cos( ωt ) 4 A 2 d = [ A sin( ωt )] 4 dt Note that we can now create an effective “transfer function” of this nonlinearity by defining that: A 2 q = N ( A, ω )( x ) N ( A, ω ) = 4 which approximates the effect of the nonlinearity as a frequency re- sponse function. N ( A, ω ) G ( s ) x ( t ) q ( t ) x ( t ) 0 November 23, 2010
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Fall 2010 16.30/31 21–4 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( ωt ) = G ( ) q = G ( ) N ( A, ω )( x ) which is equivalent to: (1 + G ( ) N ( A, ω )) x = 0 that we can rewrite as: A 2 ( ) α 1 + = 0 4 ( ) 2 α ( ) + 1 which is only true if A = 2 and ω = 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 α we get sustained oscillations in which the x ( t ) value settles down to an amplitude of 2. Note that α does impact the response and changes the shape/fea- tures in the response. Approach (called Describing Functions ) is generalizable .... November 23, 2010
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Fall 2010 16.30/31 21–5 Describing Function Analysis Now consider a more general analysis of the describing function ap- proach. In this case consider the input to the nonlinearity to be x ( t ) = A sin ωt . Would expect that the output y = f ( x ) is a complex waveform, which we represent using a Fourier series of the form: y ( t ) = b 0 + ( a n sin nωt + b n cos nωt ) n =1 So it is explicit that the output of the nonlinearity contains multiple harmonics of the ingoing signal.
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