lyapunoiva stabilnost

7/23/2019 lyapunoiva stabilnost

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Brief introduction

Russian mathematician AM Lyapunov and mechanics founded in 1892years for the analysis of system stability theory. For the control systemstability is a fundamental problem to be studied. !n studies linear time"

invariant systems there are many criteria such as al#ebraic stabilitycriterion $y%uist stability criterion etc. can be used to determine thestability of the system. Lyapunov stability theory can be applied to bothlinear system analysis and nonlinear systems time"invariant systems andtime"varyin# system stability is more #eneral stability analysis. Lyapunovstability theory mainly refers to the Lyapunov second method also &no'nas the Lyapunov direct method. Lyapunov second method can be used atany sta#e of the system the use of this approach can solve the systemdoes not have to directly determine the stability of the e%uation of state.For nonlinear systems and time"varyin# systems state e%uations is oftenvery di(cult therefore Lyapunov second method sho's #reat advanta#es.)orresponds 'ith the second method is the *rst method of Lyapunov also&no'n as the Lyapunov indirect method it is throu#h the study ofnonlinear systems of linear e%uations of state to determine the distributionof ei#envalues of system stability. +he *rst method and the secondmethod of far. !n modern control theory Lyapunov second method is tostudy the stability of the main method of control systems theory as afundamental tool for the problem but also a detailed analysis of thestability of the control system a common method. Limitations of Lyapunovsecond method is to use re%uires considerable e,perience and s&ill andthe conclusions #iven system is stable or unstable only a su(cient

condition- ineective in other 'ays this approach also can solve somenonlinear system stability issues. $o' 'ith the development of computertechnolo#y usin# a di#ital computer can not only *nd the re%uiredLyapunov function but also to determine the system/s stability re#ion.0o'ever for any system you 'ant to *nd a set of commonly usedmethods are still di(cult.

evelopment vervie'

From the late 19th century Lyapunov stability theory has #uided the

research and application about stability. Many scholars follo' Lyapunov

line of research opened up by the second method made some ne'

developments. n the one hand Lyapunov second method is e,tended to

study the stability of the #eneral system. For e,ample in 1934 5. 6.

7ubov 'ill Lyapunov method is used to study the same set of metric space

stability. ubse%uently : Laalle and other abstract systems and various

forms of Lyapunov stability 'ere studied. !n these studies the description

of the system are not limited to dierential or dierence e%uations sports

e%uilibrium state has adopted the same collection that Lyapunov function

is de*ned in a more #eneral sense. 19;4 . Bu,iao ri#ht level of

representation in the collection and mappin# systems on the


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establishment of the Lyapunov second method. At this time Lyapunov

function is no lon#er real number *eld value but in an orderly #rid de*ned

on the half"value. n the other hand the second method is used to study

the Lyapunov lar#e multi"level system the stability of the system. At this

point Lyapunov function is #enerali


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motion D t- ,? t? of the rail line is not beyond the domain G then 'e

say that the ori#in e%uilibrium ,e @ ? is the sense of Lyapunov stability.

J asymptotically stable

!f the ori#in of the e%uilibrium state is stable in the sense of Lyapunov andat time t tends to infinity the disturbed motion D t- ,? t? conver#es to

the e%uilibrium condition ,e @ ? and this process 'ithout departin# from

the G then the system is asymptotically stable e%uilibrium state. From

a practical standpoint asymptotically stable important than stability. !n the

application determine the ma,imum ran#e asymptotic stability is

necessary it can determine the disturbed motion is asymptotically stable

under the premise of the initial perturbation ,? ma,imum allo'able ran#e.

K 'ide ran#e of asymptotically stable

Also &no'n #lobally asymptotically stable 'hen a state space for all non"

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lyapunoiva stabilnost