# Lect_19 - Nonlinear Systems and Control Lecture 19...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability – p. 1/1 8 Perturbed Systems: Nonvanishing Perturbation Nominal System: ˙ x = f ( x ) , f (0) = 0 Perturbed System: ˙ x = f ( x ) + g ( t,x ) , g ( t, 0) negationslash = 0 Case 1: The origin of ˙ x = f ( x ) is exponentially stable c 1 bardbl x bardbl 2 ≤ V ( x ) ≤ c 2 bardbl x bardbl 2 ∂V ∂x f ( x ) ≤ − c 3 bardbl x bardbl 2 , vextenddouble vextenddouble vextenddouble vextenddouble ∂V ∂x vextenddouble vextenddouble vextenddouble vextenddouble ≤ c 4 bardbl x bardbl ∀ x ∈ B r = {bardbl x bardbl ≤ r } – p. 2/1 8 Use V ( x ) to investigate ultimate boundedness of the perturbed system ˙ V ( t,x ) = ∂V ∂x f ( x ) + ∂V ∂x g ( t,x ) Assume bardbl g ( t,x ) bardbl ≤ δ, ∀ t ≥ , x ∈ B r ˙ V ( t,x ) ≤ − c 3 bardbl x bardbl 2 + vextenddouble vextenddouble vextenddouble ∂V ∂x vextenddouble vextenddouble vextenddouble bardbl g ( t,x ) bardbl ≤ − c 3 bardbl x bardbl 2 + c 4 δ bardbl x bardbl = − (1 − θ ) c 3 bardbl x bardbl 2 − θc 3 bardbl x bardbl 2 + c 4 δ bardbl x bardbl < θ < 1 ≤ − (1 − θ ) c 3 bardbl x bardbl 2 , ∀ bardbl x bardbl ≥ δc 4 / ( θc 3 ) def = μ – p. 3/1 8 Apply Theorem 4.18 bardbl x ( t ) bardbl ≤ α − 1 2 ( α 1 ( r )) ⇔ bardbl x ( t ) bardbl ≤ r radicalbigg c 1 c 2 μ < α − 1 2 ( α 1 ( r )) ⇔ δc 4 θc 3 < r radicalbigg c 1 c 2 ⇔ δ < c 3 c 4 radicalbigg c 1 c 2 θr b = α − 1 1 ( α 2 ( μ )) ⇔ b = μ radicalbigg c 2 c 1 ⇔ b = δc 4 θc 3 radicalbigg c 2 c 1 For all bardbl x ( t ) bardbl ≤ r radicalbig c 1 /c 2 , the solutions of the perturbed system are ultimately bounded by b – p. 4/1 8 Example ˙ x 1 = x 2 , ˙ x 2 = − 4 x 1 − 2 x 2 + βx 3 2 + d ( t ) β ≥ , | d ( t ) | ≤ δ, ∀ t ≥ V ( x ) = x T Px...
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## This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Lect_19 - Nonlinear Systems and Control Lecture 19...

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