# lec11 - ME6402, Nonlinear Control Systems Outline...

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1 Outline ± Motivation stems ± Describing Function Method Fundamentals (DFM) ± Applications of DFM ± DFM of Nonlinear Systems ME6402, Nonlinear Control Sys ± Examples Motivation ± Describing function method is an extension of the frequency response method for the frequency response method for approximate analysis of nonlinear systems ± It is a powerful design and analysis tool because of its graphical representation and use of physical insight ± It is different from input-output stability as it is used for predicting limit cycles rather than absolute stability ± It can deal with ‘hard’ nonlinearities

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2 Problem Statement ± Estimate existence of a limit cycles and its characteristics such as amplitude stems its characteristics, such as and frequency for a single feedback loop with a nonlinear element: LTI r=0 u y ME6402, Nonlinear Control Sys System NL - Frequency Response (FR) Principle LTI System t A u ω = sin ( ) ϕ + ω = t M y ss sin ), ( , ) ( ω = ϕ ω = j G j G A M G : LTI System TF NL Element t A x ω = sin () ϕ + ω t M w sin ? ? ?, = ϕ = A M
3 Alternative Representation of FR t ω t j e AG y ω ω stems G(j ω ) j Ae u = ss j = ) ( ME6402, Nonlinear Control Sys Assumption on Nonlinearity ± Single Nonlinear Component ± Time-Invariant ± Only the 1st frequency response component is significant ± Odd nonlinearity (I.e., f(-x)=-f(x))

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4 Basic Input-Output Relationship NL Element f(x) stems t A t x ω = sin ) ( ) sin ( ) ( t a f t w i ω = w(t) is periodic of period 2 π / ω Fourier series of w(t) : ω +
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## lec11 - ME6402, Nonlinear Control Systems Outline...

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