Module 6 - ECE 514 Nonlinear & Adaptive Control Module 6...

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1 ECE 514 Nonlinear & Adaptive Control Module 6 ABSOLUTE STABILITY
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2 Topics ± Absolute Stability ± PR & SPR ± KYP Lemma and applications ± Circle & Popov Criteria Material in Chapter 6 of Khalil
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3 The Problem of Absolute Stability G ( s ) Plant + compensator Actuator 0 = v R y u u = ψ + _ Assume that the actuator is linear Ky = ]. , [ for stable is system loop closed e that th assume and max min K K K y min K max K Hurwitz sector
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4 Assume now that the actuator is nonlinear, for instance, a saturator y u y = ) ( ψ [] . , ) , ( if stable asym. be system the Would in time. change may i.e., , ) , ( assume Moreover, max min K K y t y t = This question was posed by M.A. Aizerman in 1940’s. Lure’s Problem
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5 Conjectures,…. . Kalman conjecture: . 0 such that ) , ( all for stable also is it ], , 0 [ all for stable asym. is system loop closed the If * ) , ( * K y t K K y t y ψ BOTH FALSE! A correct answer first was given by A.I. Lurie. The Popov, Kalman, Yakubovich, others contribute to the solution. Sometimes this problem is called the Lurie problem. ]. , 0 [ ) , ( all for stable asym. also is it ], , 0 [ all for stable asym. is system loop closed the If * * K y t K K Aizerman conjecture:
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6 Robustness & Feedback In modern terms, this is a problem of robustness : If we have a stabilizing feedback ( ), what is the class of sector nonlinearities, where asymtotic stability is maintained? Similarl y ψ y, how much destabilizing uncertain terms can be tolerated for an open-loop stable system? and finally, can one design nonlinear feedback to achieve certain performance objectives, while maintaining open-loop stability?
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7 Problem Formulation Plant p p n R y Cx y R u R x Bu Ax x = + = , ± ( , ) : pp ut y R R R ψψ + =− × Controller Assumptions: 1) (A,B,C) controllable & observable 2) A Hurwitz piecewise continuous in t 3) ψ (t,y) and ψ satisfies sector conditions locally Lipschitz in y
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8 Sector Nonlinearities y u y β y α Generalization for 1 (if is decentralized) p ψ > We can rewrite the sector condition º as , 0 ] ) , ( ][ ) , ( [ y y t y y t + R t −∞ ] , [ ] , [ where or b a y 1 For = p ± ± ±² ± ± ±³ ´ condition sector 2 2 ) , ( y y t y y º −∞ + condition sector global ) , ( condition sector local ] , [ b a y R t
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9 Decentralized Feedback Case Consider the decentralized feedback = ) , ( ) , ( ) , ( 2 1 1 p y t y t y t ψ # . , , , with conditions sector the satisfies ) , ( each Assume i i i i i i b a y t β α Define ) , , ( diag ) , , ( diag 1 max 1 min p p K K " " = = and } : { i i i P b y a R y = Γ Then p -dim sector condition is 0 where , 0 ] ) , ( [ ] ) , ( [ min max max min > Γ + ±² ±³ ´ K K y R t y K y t y K y t T symmetric positive definite diagonal matrix
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10 Centralized Feedback Case p p p R y R t y Ly y t y t R L Γ + × , ) , ( satisfies ) , ( such that Assume 2 2 γ ψ Introduce I L K I L K + = = max min Then 0 ) , ( ] ) , ( [ ] ) , ( [ 2 2 2 2 2 max min = y Ly y t y K y t y K y t T where again 0 min max > K K Definition: . or , , 0 ] ) , ( [ ] ) , ( [ and symmetric and p.d. is such that , if ty nonlineari sector a called is : ty
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This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.

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Module 6 - ECE 514 Nonlinear & Adaptive Control Module 6...

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