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# Lect_34 - Nonlinear Systems and Control Lecture 34 Robust...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 34 Robust Stabilization Lyapunov Redesign & Backstepping – p. 1/ ? ? Lyapunov Redesign (Min-max control) ˙ x = f ( x ) + G ( x )[ u + δ ( t,x,u )] , x ∈ R n , u ∈ R p Nominal Model: ˙ x = f ( x ) + G ( x ) u Stabilizing Control: u = ψ ( x ) ∂V ∂x [ f ( x ) + G ( x ) ψ ( x )] ≤ − W ( x ) , ∀ x ∈ D, W is p.d. u = ψ ( x ) + v bardbl δ ( t,x,ψ ( x ) + v ) bardbl ≤ ρ ( x ) + κ bardbl v bardbl , ≤ κ < 1 ˙ x = f ( x ) + G ( x ) ψ ( x ) + G ( x )[ v + δ ( t,x,ψ ( x ) + v )] ˙ V = ∂V ∂x ( f + Gψ ) + ∂V ∂x G ( v + δ ) – p. 2/ ? ? w T = ∂V ∂x G ˙ V ≤ − W ( x ) + w T v + w T δ w T v + w T δ ≤ w T v + bardbl w bardbl bardbl δ bardbl ≤ w T v + bardbl w bardbl [ ρ ( x )+ κ bardbl v bardbl ] v = − η ( x ) w bardbl w bardbl parenleftbigg w bardbl w bardbl = sgn( w ) for p = 1 parenrightbigg w T v + w T δ ≤ − η bardbl w bardbl + ρ bardbl w bardbl + κ η bardbl w bardbl = − η (1 − κ ) bardbl w bardbl + ρ bardbl w bardbl η ( x ) ≥ ρ ( x ) (1 − κ ) ⇒ w T v + w T δ ≤ ⇒ ˙ V ≤ − W ( x ) – p. 3/ ? ? v = − η ( x ) w bardbl w bardbl , if η ( x ) bardbl w bardbl ≥ ε − η 2 ( x ) w ε , if η ( x ) bardbl w bardbl < ε η ( x ) bardbl w bardbl ≥ ε ⇒ ˙ V ≤ − W ( x ) For η ( x ) bardbl w bardbl < ε ˙ V ≤ − W ( x ) + w T bracketleftbigg −...
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Lect_34 - Nonlinear Systems and Control Lecture 34 Robust...

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