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 , 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 bardbl x bardbl = (1 ) c 3 bardbl x bardbl 2 c 3 bardbl x bardbl 2 + c 4 bardbl x bardbl < < 1 (1 ) c 3 bardbl x bardbl 2 , bardbl x bardbl c 4 / ( c 3 ) def = p. 3/1 8 Apply Theorem 4.18 bardbl x ( t ) bardbl 1 2 ( 1 ( r )) bardbl x ( t ) bardbl r radicalbigg c 1 c 2 < 1 2 ( 1 ( r )) c 4 c 3 < r radicalbigg c 1 c 2 < c 3 c 4 radicalbigg c 1 c 2 r b = 1 1 ( 2 ( )) b = radicalbigg c 2 c 1 b = c 4 c 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 + x 3 2 + d ( t ) , | d ( t ) | , t V ( x ) = x T Px...
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Lect_19 - Nonlinear Systems and Control Lecture # 19...

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