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# lec10 - ME6402 Nonlinear Control Systems Outline Motivation...

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1 Outline ± Motivation stems ± Absolute Stability Formulation ± Nyquist Criterion ± Aizerman’s Conjecture ± Circle Criterion ME6402, Nonlinear Control Sys ± Examples Motivation ± Many nonlinear systems can be represented as a feedback connection of a linear system as a feedback connection of a linear system and nonlinear element ± Input-out stability techniques can deal with common actuator nonlinearities such as saturation, ripple, etc. It also can be useful in nonlinear control ± It also can be useful in nonlinear control design such as adaptive control ± It extends Nyquist criterion to feedback loop with nonlinear elements

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2 Problem Statement ± Consider the feedback system r=0 y u stems LTI System Nonlinear Element - φ (y) ME6402, Nonlinear Control Sys ± Nonlinearity belongs to a sector [k 1 ,k 2 ]: ( ) 2 1 k y y k φ y Special Cases ± Before tackling the general sector nonlinearity we consider the following nonlinearity, we consider the following special cases: ± φ (y)=ky, i.e., k 1 =k 2 = (linear system) ± φ∈ [0, ), i.e., k 1 =0, k 2 >0 is arbitrary
3 Linear Case ± Linear feedback system r u y stems ± Closed-loop system ± State-space: G(s) K - ( ) x BKC A B Ax x = + = u & ME6402, Nonlinear Control Sys ± Transfer function: ± Stability condition: Roots of 1+KG (or eig. of A-BKC) are in the LHP complex plane Cx = y KG G R Y + = 1 Frequency Domain Approach ± The closed-loop stability can be analyzed based on the frequency response of G(s), i.e., Bode or Nyquist diagram ± The Nyquist diagram of G(s) is the graph of Im[G(s)] vs. Re[G(s)] as s varies along a certain complex contou certain complex contour ± Nyquest criterion provides critical information about the closed loop system based on its open-loop Nyquist diagram.

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4 Nyquist Criterion ± Nyquist Path: RHP Semicircle j Im(G(s)) stems G( s ) -1/K Re(G(s)) ME6402, Nonlinear Control Sys ± P=number of RHP poles of G(s) ± Z=number of RHP closed-loop poles ± N=CCW encirclements of –1/K by Nyquist diagram of G(s), then P=N+Z -j How to Costruct a Nyquist Diagram ± In most cases only G(j ω ) for 0 ≤ω < needs to be plotted since ± the graphs of G(-j ω ) and G(j ω ) are mirror images of each other.
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lec10 - ME6402 Nonlinear Control Systems Outline Motivation...

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