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by: Miss Damien Crooks


Miss Damien Crooks
GPA 3.52

Alan Desrochers

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About this Document

Alan Desrochers
Class Notes
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This 13 page Class Notes was uploaded by Miss Damien Crooks on Monday October 19, 2015. The Class Notes belongs to ECSE 6400 at Rensselaer Polytechnic Institute taught by Alan Desrochers in Fall. Since its upload, it has received 20 views. For similar materials see /class/224770/ecse-6400-rensselaer-polytechnic-institute in ELECTRICAL AND COMPUTER ENGINEERING at Rensselaer Polytechnic Institute.





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Date Created: 10/19/15
22 Lyapunov Stability Topics Be able to Ref Sections 9 79 9 915 9 I 7 assess stability in the sense of Lyapunov using the direct method of Lyapunov determine asymptotic stability using Lyapunov s 2nd method generate Lyapunov functions from Vx explain and use Lyapunov s instability theorem nd Lyapunov functions for Lienard39s equation design feedback controllers using Lyapunov methods Fa 2001 Direct Method of Lyapunov Theorem If Vxt is PD andVx t is NSD then the origin is stable isL Example Show that the following system is stable isL 39l in I W il Lyapunov39s 2nd method Theorem If VXt is PD and vow is ND then the origin is asymptotically stable Theorem If VXt is PD and VX t is ND and VX t gt 00 as I Xt e 00 then the origin is globally asymptotically stable Example Show that the following system is globally asymptotically stable C 3L I R 5 example continued So farVXt 2 x2 1 R looks NSD 1f x2 1 0 and x10 at 0 Show that when x2 I 0 it is not possible for x10 to be nonzero Note x10 9e 0 are not equilibrium points Show global asymptotic stability Example Derive conditions for stability isL 3510 0 1 3610 x20 a b x2t Strategy Guess a Vxt which is NSD then integrate to get Vxt and see if it is PD if it isn t guess again Lyapunov s Instability Theorem If there exists Vxtwhich is ND and VXt which is not PSD nor PD then the system is not stable and xt is not bounded Note Vxtwhich is PD and Vxt which is PD or PSD also shows instability Example Investigate 56t xt xt 0 for instability Solution 0 1 m L 1xrgt Try Vxt XtT Qxt then Finding Lyapunov Functions If the system can be put in the form of Lienard s equation 550 f 960 950 5 xt 0 then a Lyapunov function can always be constructed Assumptions 0 f 0 is an even function of xt and f 2 0 g0 0 and is monotone increasing De ne the following state variables Where are the equilibrium points for these state variable equations Lyapunov functions for Lienard39s equation 2 x10 x2 I T 890615 VX With this definition we can find VX Example Investigate the stability of 561 36t x3t 0 Solution Example Investigate the stability of Van der Pol s equation 550 ex2t 13ampt xt 0 Solution The state variable equatiogis are x 0 x10 x2r e 3 x1t x2 x1 Locate the equilibrium points Example Nonlinear Systems Investigate the stability properties of 9610 x1t fx3t 9521 f 3631 3630 Z 1x1t Yx2t Rfx3t where y R gt 0 Assume f is a first and third quadrant nonlinearity Solution Since f is a function of x3t try 3531 1 1 VXt Elefa Eszfa g fx3dx3 Feedback Controller Design Via Lyapunov Theory Example Consider the nonlinear system below x10 2x13t x2 I x1 t 3x2 2 ut Use full state feedback to stabilize the system Use Lyapunov theory to find a range on the controller gains k1 and k2 that ensures the closed loop system is stable 0A14


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