Stability analysis method for fuzzy control systems dedicated. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. Ship stability for masters and mates ship stability for masters and mates fifth editioncaptain d. Optimizationbasedclearanceofflightcontrollaws varga free ebook download as pdf file. Power system stability and control third edition by leonard l. The stability and instability of steady states duration. Python control library documentation read the docs.
Section 3 gives a stability theorem for nonlinear systems with ts flcs and an algorithm for the design of a stable fuzzy control system. Optimizationbasedclearanceofflightcontrollaws varga control. Overview of lyapunov stability theory lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. Takagisugeno fuzzy control systems controlling nonlinear processes. I will assume knowledge of my autumn 2002 model theory lecture notes 1. This process is experimental and the keywords may be updated as the learning algorithm improves. The notion of exponential stability guarantees a minimal rate of decay, i.
Asymptotic stability of linear systems an lti system is asymptotically stable, meaning, the equilibrium state at the origin is asymptotically stable, if and only if the eigenvalues of a have negative real parts for lti systems asymptotic stability is equivalent with convergence stability condition automatically satisfied. This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of stability continue to be built. Lyapunovbased approach introduces advanced tools for stability analysis of nonlinear systems. Anand pillay september 29, 2003 1 introduction and preliminaries the aim of this course and these notes is to present an exposition of the basics of stability theory, stable group theory, and geometric stability theory. View lyapunov stability theory research papers on academia. The problem of the stability of the equilibrium position or the steady motion of many dynamical systems reduces to the problem of the stability of the trivial solution of the system of equations differentiation with respect to time is denoted by a dot 1. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local. Jun 05, 2017 stability analysis, state space 3d visualization duration. View lyapunov stability ppts online, safely and virus free. The beginning of what is known today as the lyapunov stability theory can be traced back to the original publication of alexander mikhailovich lyapunovs doctoral thesis on the general problem of the stability of motion, which he defended at the university of moscow in 1892. Open loop we know that the stability of linear timeinvariant lti dynamical systems can be determined by examining the system poles. These stable region free parameters allow the optimization of the. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at.
The lyapunov first and second methods are investigated and the stability analysis of fractional differential systems is highlighted. An equilibrium is stable if for any neighbourhood n of the equilibrium there is a neighbourhood n0 contained in nsuch that all. Pdf power system stability and control by prabha kundur book free download. Lyapunov s second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a dynamical system or systems of differential equations. Rninto rnand there is at least one equilibrium point x, that is f x 0. Power system stability and control by prabha kundur. Stability is a classical issue in dynamical system theory.
El2620 nonlinear control exercises and homework henning schmidt, karl henrik johansson, krister jacobsson, bo wahlberg. But perhaps the most important stability concept is that of stability in the sense of lyapunov or simply lyapunov stability. Lyapunov stability the stability of solutions to odes was. In recent years, considerable attention has been paid to control systems. Lyapunov stability analysis of large scale power systems by vijay vittal. Power system stability and control third edition by leonard l grigsby. Hence, stability is correlated with the idea of visiting a particular compact set k. R a system trajectory is completely contained within n if and only if vt 0. An equilibrium is stable if for any neighbourhood n of the equilibrium there is a neighbourhood n0 contained in nsuch that all solutions starting in n0 remain in n. Purchase stability by liapunovs direct method with applications by joseph l salle and solomon lefschetz, volume 4 1st edition. Lyapunov matrix equation in system stability and control. A wikipedia page gives a general idea about the lyapunov stability.
Yao 1 lyapunov stability stability of equilibrium points 1. R n is an equilibrium point of the system if fxe 0 xe is an equilibrium point xt xe is a trajectory suppose xe is an equilibrium point system is globally asymptotically stable g. We now consider using lyapunovs direct method for testing the stability of a linear system. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. It is basically concerned with the determination of the upper limit of. Please feel free to send me comments on the contents as well as the bulletted notes. Lyapunov stability is named after aleksandr lyapunov, a russian mathematician who published his book the general problem of stability of motion in 1892. A particular feature which makes the framework attractive is that it appeals to all our intuition regarding energy conserving systems. Lyapunov stability analysis free download as powerpoint presentation.
Lyapunov stability of the solution of an equation given on is lyapunov stability of the point relative to the family of mappings. Alternatively, to use setuptools, first download the source and unpack it. An autonomous system of odes is one that has the form y0 fy. Stability in the sense of lyapunov university of notre dame. Taking into account the importance of stability and index one, some investigations have considered them simultaneously. Portable document format, jpeg joint photographic experts group, png portable network graphics and metapost, while. Stability analysis method for fuzzy control systems. Lyapunov stability is named after aleksandr mikhailovich lyapunov, a russian mathematician who defended the thesis the general problem of stability of motion at kharkov university in 1892. One of the most widely adopted stability concepts is lyapunov stability, which plays important roles in system and control theory and in the analysis of engineering systems. Mar 15, 2001 lyapunovs second theorem is a standard tool for stability analysis of ordinary differential equations.
This content was uploaded by our users and we assume good faith they have the permission to share this book. Stability analysis of nonlinear systems using lyapunov theory. A free powerpoint ppt presentation displayed as a flash slide show on id. Lyapunov stability analysis of large scale power systems. It is p ossible to ha v e stabilit y in ly apuno without ha ving asymptotic stabilit y, in whic h case w e refer to the equilibrium p oin t as mar ginal ly stable. Pdf power system stability and control by prabha kundur book. Ppt fundamentals of lyapunov theory powerpoint presentation. Burton northwest research institute 732 caroline st. Lyapunovs second theorem is a standard tool for stability analysis of ordinary differential equations. Purchase lyapunov matrix equation in system stability and control, volume 195 1st edition. Lyapunov stability analysis stability theory systems. One the major stability technique for non linear dynamic system is called lyapunov.
Lyapunov stability free download as powerpoint presentation. The derivative of v along the state trajectories of x is given by. We recommend those of you that are not familiar with l a tex to see 1 for an introduction. Stability in the sense of lyapunov stability is one of the most important properties characterizing a systems qualitative behavior. A detailed post on the lyapunov stability criteria will be uploaded soon. Lyapunov stability is a very mild requirement on equilibrium points. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and. We extend the wellknown artsteinsontag theorem to derive the necessary and sufficient conditions for the inputtostate stabilization of stochastic control systems. From existence of a scalar function satisfying certain inequalities it follows that almost all trajectories of the system tend to zero. Stability of non linear system depends on both initial value and its input unlike liner system.
Various types of stability are there for solution of differential equation of the dynamic systems. Here we introduce a theorem which can be viewed as a dual to lyapunovs result. Lecture 12 basic lyapunov theory stanford university. We will demonstrate that c is a necessary and sufficient condition for a and b. Lyapunovs second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a dynamical system or systems of differential equations. Stability analysis, state space 3d visualization duration.
This method of testing stability relies on the state model of a system. Stability behaviour of nonlinear systems need not be always global unlike linear systems. November 15, 2009 1 1 lyapunov theory of stability introduction. The possible function definiteness is introduced which forms the building block of lyapunovs direct method. Stability by liapunovs direct method with applications by. We then analyze and apply lyapunovs direct method to prove these stability properties, and develop a nonlinear 3axis attitude pointing control law using lyapunov theory. There are a number of stability concepts used in the study of dynamical systems. Third edition by leonard l grigsby is available for free download in pdf format. Lyapunov function asymptotic stability distillation column balance volume quadratic lyapunov function these keywords were added by machine and not by the authors. Lyapunov stability of a point relative to the family of mappings is equivalent to the continuity at this point of the mapping of a neighbourhood of this point into the set of functions defined by the formula, equipped with the topology of uniform convergence on. The idea of lyapunov stability can be extended to infinitedimensional manifolds, where it is known as structural stability, which concerns the behavior of different but nearby solutions to differential equations.
As it is for deterministic systems, lyapunov theory is a powerful tool for qualitative analysis of stochastic differential equations. Apr 25, 2015 4 lyapunov stability a state of an autonomous system is called an equilibrium state, if starting at that state the system will not move from it in the absence of the forcing input. This formalizes the problematic concept by integrating it into a. We then analyze and apply lyapunov s direct method to prove these stability properties, and develop a nonlinear 3axis attitude pointing control law using lyapunov theory. Numerous reports have been published on this matter, with particular emphasis on the application of lyapunovs second method 5, 6. Lyapunov and nonlyapunov stability of linear discrete time delay systems 1149 analysis of nonlinear singular and implicit dynamic systems in terms of the generic qualitative and quantitative concepts, which contain technical and practical stability types as special cases, have been introduced and studied in bajic 1988, 1992. In using lypunov s direct method for stability analysis, it is sometimes possible to manipulate into the form 3. Stability of linear system is independent of initial conditions. A dual to lyapunovs stability theorem sciencedirect. An illustrative example presented in section 4 shows that good control system performance can be obtained by applying the suggested algorithm. Ship stability for masters and mates pdf free download epdf. Asymptotic stability of linear systems an lti system is asymptotically stable, meaning, the equilibrium state at the origin is asymptotically stable, if and only if the eigenvalues of a have negative real parts for lti systems asymptotic stability is equivalent with convergence stability. Lyapunov stability of the solution of a differential equation given on is lyapunov stability of the point relative to the family of mappings, where is the cauchy operator of this equation.
In particular, it does not require that trajectories starting close to the origin tend to the origin asymptotically. This is a paper in a series of investigations into the use of. Lyapunov characterization for the stability of stochastic. Lecture notes stability theory math 414 spring 2003. We say that y0 is a critical point or equilibrium point of the system, if it is a constant solution of the system, namely if fy0 0. Pdf power system stability and control by prabha kundur. Lyapunovlike characterization for the problem of inputtostate stability in the probability of nonautonomous stochastic control systems is established. Lyapunov stability theory research papers academia.
Physics videos by eugene khutoryansky 54,222 views. Introduction lyapunov stability examplesconclusions lyapunov stability let v. Lap lambert academic publishing february 23, 2017 language. The intuitive picture is that of a scalar outputfunction, often thought. Lyapunov stability of a point relative to a mapping is defined as lyapunov.
381 618 466 615 1380 55 1006 249 222 1124 60 441 1479 650 554 131 1578 406 224 1168 416 1288 1354 1499 1240 94 555 490 533 879 1246 136 1195