Linearization and lyapunov functions are used to deal with nonlinear stability problems. Learning control lyapunov function to ensure stability of. Notes pdf file on hrushovskis stable group theory and approximate subgroups. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and.
Lyapunov theory lyapunov theory is used to make conclusions about trajectories of a system x. Lyapunov matrix equation in system stability and control pdf educationtempe158. View lyapunov stability theory research papers on academia. A free powerpoint ppt presentation displayed as a flash slide show on id. Sep 16, 2014 basic concepts for the lyapunov stability are introduced. Stability analysis of nonlinear systems using lyapunov theory. In using lypunov s direct method for stability analysis, it is sometimes possible to manipulate into the form 3. The stability of such systems is a function of both the dynamics and the interconnection topology.
In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. Various types of stability are there for solution of differential equation of the dynamic systems. A detailed post on the lyapunov stability criteria will be uploaded soon. The interest in this class of systems has been motivated traditionally by sampleddata systems in which a process is sampled periodically and then controlled via a computer. A wikipedia page gives a general idea about the lyapunov stability. Stability analysis ferenc szidarovszky university of arizona a. Lyapunov stability is named after aleksandr lyapunov, a russian mathematician who published his book the general problem of stability of motion in 1892. Stability theory by liapunovs direct method nicolas rouche. The context is such that there should be no risk of serious am. Analysis, stability and control outline eecs 222 spring 2007 linear vs. The research presented in this thesis considers the stability analysis and control of discretetime systems with delay. Lyapunov stability of the solution of an equation given on is lyapunov stability of the point relative to the family of mappings.
Pdf documents can contain links and buttons, form fields, audio, video, and business logic. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. Ppt fundamentals of lyapunov theory powerpoint presentation. There is a strong form of stability which demands an exponential rate of convergence. This is a paper in a series of investigations into the use of. Simple nonlinear models planar dynamical systems chapter 2 of textbook. Lecture notes stability theory math 414 spring 2003. This behavior commonly arises in mechanical systems undergoing impacts and optimal. Vector lyapunov functions have been used for a long time in stability theory for nonlinear systems see 3,6,7,11,12,14,15,18,19,20,21,27 and the references in 7. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i.
He was inspired by the following theorem from mechanics lagrange in 1800. Ferrari trecate dipartimento di ingegneria industriale e dellinformazione universit a degli studi di pavia advanced automation and control ferrari trecate dis nonlinear systems advanced autom. 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. Fundamentals of lyapunov theory autonomous systems lyapunov s pioneering work on stability is based on finding a energylike function with special features. Lyapunov s second or direct method provides tools for studying asymp totic stability properties of an equilibrium point of a dynamical system or systems of dif ferential equations. One the major stability technique for non linear dynamic system is called lyapunov. Chetaev was so significant that many mathematicians, physicists and engineers consider him lyapunovs direct successor and the nextinline scientific descendant in the creation and development of the mathematical theory of stability. The intuitive picture is that of a scalar outputfunction, often thought. Lyapunov functions and feedback in nonlinear control. Lecture 12 basic lyapunov theory stanford university. A mechanical system composed by two rigid bodies subject to the gravitational force is at a stable equilibrium if the total energy is. The most important type is that concerning the stability of solutions near to a point of equilibrium.
Stability theory for gaussian rough differential equations. This may be discussed by the theory of aleksandr lyapunov. Lyapunov theory for zeno stability andrew lamperski and aaron d. Stability theory plays a significant role in engineer ing systems. 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. In simple terms, if the solutions that start out near an equilibrium point stay near forever. Feb 29, 2020 eigenvalues or poles of the system the stability characteristic of the system can be achieved. Next study how lyapunov theory looks like when applied to linear systems. Eigenvalues or poles of the system the stability characteristic of the system can be achieved. Example of stability problem we consider the system x0 y x3. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Lyapunov stability theory excerpted from\a mathematical. At the basis of it lie the concepts of lyapunov stability and asymptotic stability see asymptoticallystable solution, introduced by lyapunov, lyapunovs theorem on stability in the first approximation on which lyapunovs first method for investigating stability is based and lyapunovs. We present a survey of the results that we shall need in the sequel, with no proofs.
The lyapunov concept of stability from the standpoint of. Stability is a classical issue in dynamical system theory. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. Vector lyapunov functions were first introduced by bellman in 3 and have been acknowledged to be a more flexible tool for proving stability than the usual single. 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.
Stability theory and its variants the library at msri. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunovs second method for stability are important to stability theory of dynamical systems and control theory. Lecture 14 lyapunov theory with inputs and outputs systems with inputs and outputs reachability bounding bounds on rms gain boundedreal lemma feedback synthesis via controllyapunov functions 141. Exponential stability, rate of convergence the equilibrium point. For any given control system, it is for any given control system, it is required to have a stable system since an unsta ble. Stability of dynamical systems on a graph mohammad pirani, thilan costa and shreyas sundaram abstractwe study the stability of largescale discretetime dynamical systems that are composed of interconnected subsystems. 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. The theory of lyapunov function is nice and easy to learn, but nding a good lyapunov function can often be a big scienti c problem. Multiple equilibria, limit cycles, complex dynamics.
Conditions are obtained for the stability of linear equations with constant, periodic, and general variable coefficients. At the basis of it lie the concepts of lyapunov stability and asymptotic stability see asymptoticallystable solution, introduced by lyapunov, lyapunov s theorem on stability in the first approximation on which lyapunov s first method for investigating stability is based and lyapunov s. Eventually this theory was seen as having a \geometric meaning. Lyapunov stability rizki adi nugroho 1410501075 rizki nugroho. 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. In the theory of ordinary differential equations odes, lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ode.
Burton northwest research institute 732 caroline st. Lyapunov stability theory with some applications paperback february 23, 2017 by ronak saeed author see all formats and editions hide other formats and editions. The lyapunov concept of stability from the standpoint of poincares approach. Lecture 5 inverse of matrices section 22 and 23 njitronbrown. They can be signed electronically, and you can easily view pdf files.
November 15, 2009 1 1 lyapunov theory of stability introduction. When every point has this property, and when additionally the origin has the familiar local stability property known as lyapunov stability, it is said in the literature to be gac. This theory still dominates modern notions of stability, and provides the foundation upon which alternative notions of stability continue to be built. Lyapunov stability theory encyclopedia of mathematics. Detecting new e ective families of lyapunov functions can be seen as a serious advance. These tools will be used in the next section to analyze the stability properties of a robot controller. We propose a quantitative direct method of proving the stability result for gaussian. 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. R a system trajectory is completely contained within n if and only if vt 0. Rninto rnand there is at least one equilibrium point x, that is f x 0.
Classical stochastic stability theory current developments further open problems lasalletype theorems stochastic stabilization numerical stability scalar linear sde dxt. This means, on the one hand, that various structural results have a geometric avour. Systems with inputs we now consider systems with inputs, i. Ames abstractzeno behavior is a dynamic phenomenon unique to hybrid systems in which an in.
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of. Basic concepts for the lyapunov stability are introduced. Stability of linear system is independent of initial conditions. This formalizes the problematic concept by integrating it into a welldeveloped mathematical theory. 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. Lyapunov stability the stability of solutions to odes was. Stability of non linear system depends on both initial value and its input unlike liner system. Nonlinear systems lyapunov stability theory part 2. Stability behaviour of nonlinear systems need not be always global unlike linear systems. Control lyapunov function clf control scheme 20, 21, 22 is one of these techniques that is designed based on the following intuitive idea. Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. I will assume knowledge of my autumn 2002 model theory lecture notes 1. 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. Stability analysis and control of discretetime systems.