module 8 p3 - Nonlinear & Adaptive Control Module...

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1 Nonlinear & Adaptive Control Module 8-Part III NONLINEAR DESIGN TOOLS
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2 Topics 1. Backstepping. 2. Passivity-Based Designs Material in Chapter 14 of Khalil and in attached notes.
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3 Why Other Design Tools? 1. Feedback-Linearization may not work! 2. Feedback-Linearization may cancel out “good” nonlinearities. 3. Feedback-Linearization may not be robust. 4. We need a larger set of tools for the large class of nonlinear systems. I will not discuss the sliding-control method, nor the Lyapunov redesign approach.
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4 Backstepping 1 Let us start with Consider the system: ( ) ( ) where [ , ] , . Let , : be smooth on a domain TT n n n Integrator Backstepping. fg u uf g D D ηη η ζ ηζ + =+ = ∈∈ i i \\ \ \ containing the origin, and let (0) 0. Design a state-feedback control law to stabilize the origin. f = Problem : () + 1 s 1 s u i
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5 Cascade Stabilization Assume that, considering as an input to the subsystem, we are able to find ( ) with (0)=0 to stabilize the subsystem, i.e. to make the origin of ( ) ( ) ( ) asymptotically stable. Sup fg ζη ζφ η φ ηη = =+ i i i () ()() pose also that we have a smooth, positive-definite Lyapunov function ( ) such that: [ ( )] [ ( ) ( ) ( )] ( ), with ( ) positive-definite. Then, V V LV f g W D W + [ ( ) ( ) ( )] ( )[ ( )] g u ζ + = i i
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6 Cascade System Transformation (1/4) ζ η () g 1 s 1 s u i + (.) f f g u i + + (.) (.) (.) fg φ + φη
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7 Cascade System Transformation (2/4) Let the error variable be ( ), to obtain: [() ()() ] () Now, we ( ) through the integrator and obtain: [() ()] as an input to the integrator. Finally, we let z fg g z zu ζ φη ηη η φ = =+ + =−
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module 8 p3 - Nonlinear & Adaptive Control Module...

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