Liapunovs second method is a generalization to the theory of nonlinear systems of odes of two basic physical principles. 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. The total energy of a conservative physical system is constant during the evolution of the system. Pdf liapunov functions and stability in control theory.
On merging constraint and optimal controllyapunov functions. Lyapunov stability theory was come out of lyapunov, a russian mathematician in 1892, and came from his doctoral dissertation. Assume next that is a spherical region dfx jkx nxk r 0g d. 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.
This article is devoted to a brief description of the basic stability theory, criteria, and methodologies of lyapunov, as well as a few related important stability concepts, for nonlinear dynamical systems. Detecting new e ective families of lyapunov functions can be seen as a serious advance. Lyapunov function and relate it to these v arious stabilit y notions. In control theory, a control lyapunov function is a lyapunov function for a system with control inputs. Pdf in this paper a class of sirs epidemic dynamical models with nonlinear incidence rate \\beta fsgi\, vaccination in susceptible and.
Stability of continuous systems stability of linear systems. However, a subset of trajectories may still be asymptotically periodic or quasiperiodic in a chaotic system. 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. More recently, clarke 22 showed how to solve the problem of merging two semiconcave continuous. Lyapunov functions and feedback in nonlinear control.
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 lyapunov functions and other analysis tools for. Control lyapunov functions for adaptive nonlinear stabilization. More recently, clarke 22 showed how to solve the problem of merging two. Since the q i are orthogonal that is, you can think of them as a change of basis, you can assume that the cocycle is given by uppertriangular matrices. Rninto rnand there is at least one equilibrium point x, that is f x 0. Theorem3 supposse that lis a lyapunov functional on oand xt is an orbit lying in a closed bounded set k.
The story begins in about 1890, when lyapunov published his seminal work introducing what we now call lyapunov theory. If it is positive, bounded ows will generally be chaotic. Theorem4 suppose that lis a lyapunov functional on oand that p. Lecture 17 perronfrobenius theory stanford university. Pdf given two control lyapunov functions clfs, a merging is a new clf whose gradient is a positive combination of the gradients of the two parents. 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. 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. The lyapunov and lasalle theorems university of michigan. Lyapunov stability game the adversary picks a region in the state space of radius.
Computation of lyapunov functions and stability of. First of all, the lyapunov stability theory is understood through the picture. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Although there is no general systematic method to find a lyapunov function, research over the years has shown how to choose lyapunov function candidates for. Solutions to assignment 5 university of nebraskalincoln. A longtime numerical calculation of the leading lyapunov exponent requires rescaling the distance in order to keep the nearby trajectory separation within the linearized. The main challenge is finding the lyapunov function. System is deterministic, the irregular behavior is due to. Lyapunov exponents for the threegeneration leslie model with parameters 0. If the pair meets the merge criteria, the two merges are merged into one. Introduction control design must quite often compromise among performance, robustness, and constraints, and lyapunov theory offers suitable tools in this. Lyapunov function is sufficient condition for stability. A elements of lyapunov theory 279 and condition c generalizes the wellknown property of mechanical systems that the energy of a free mechanical system never increases and with friction it always decreases.
It follows then that the cocycle is an x q nr r 1 r. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. 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 notes on nonlinear systems and control spring semester 2018 eth zurich peter al hokayem and eduardo gallestey abb switzerland, ltd. Lyapunov theory for zeno stability andrew lamperski and aaron d. Inputtostate stability iss applies lyapunov notions to systems with inputs. Nonmonotonic lyapunov functions for stability of nonlinear. The lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. Evaluating eigenvalues is necessary and sufficient for stability.
Finding, for a given supply rate, a valid storage function or at least proving that one exists is a major challenge in constructive analysis of nonlinear systems. To introduce the concepts, consider the simple scalar equation y0tayt. Linear quadratic lyapunov theory stanford university. I will assume knowledge of my autumn 2002 model theory lecture notes 1. The lyapunov function method is applied to study the stability of various differential equations and systems. Select multiple pdf files and merge them in seconds. Study the stability of the origin for all b 0 by using the lyapunov function given by the energy of the system, i.
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. Moreover, the notion of lyapunov function has played a predominant role in the investigation of liveness properties. Introduction to dynamic systems network mathematics. Lecture 17 perronfrobenius theory positive and nonnegative matrices and vectors perronfrobenius theorems markov chains economic growth population dynamics maxmin and minmax characterization power control linear lyapunov functions metzler matrices 171. Pdf controlsharing and merging control lyapunov functions. 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. Pdf applications of lyapunov methods in stability researchgate. Lyapunov s stability theory100 years on 277 central limit theorem and other deep investigations in areas of mechanics and mathematical analysis. Below, we restrict ourselves to the autonomous systems. Index termscomposite control lyapunov functions, stabilizability of linear differential inclusions. Here we present a simple method for checking the validity of a. Pdf a new lyapunov function for sirs epidemic models.
We can solve for this exponent, asymptotically, by. Lyapunov s direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. The method is a generalization of the idea that if there is some measure of energy in a system, then. Computing lyapunov exponents 3 in order to get the next qand r. This behavior commonly arises in mechanical systems undergoing impacts and optimal. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and. Ames abstractzeno behavior is a dynamic phenomenon unique to hybrid systems in which an in. Finally, we look at alternate feedback control laws and closed loop dynamics. Gosselin abstract for humanrobot cooperation in the context of humanaugmentation tasks, the stability of the control model is of great concern due to the risk for the human safety represented by a powerful robot. In the three intervals where there seem to be multiple lyapunov exponents, the lowest two lyapunov exponents are very negative and do not appear in the plot. This and related equations are named after the russian mathematician aleksandr lyapunov. Lecture 12 basic lyapunov theory stanford university. We will demonstrate that c is a necessary and sufficient condition for a and b. The most com mon approach is based on considering a linearly parameterized subset of storage function.
On common quadratic lyapunov functions for stable discrete. Lyapunov function share a common control 18, proposition. Lecture notes stability theory math 414 spring 2003. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunov s second method for stability are important to stability theory of dynamical systems and control theory. Calculating the entire lyapunov spectra of the lorenz. Lyapunov stability theory research papers academia. A state of a conservative physical system is stable only if its potential enery has a local minimum at that state.
A lyapunov function is a scalar function defined on the phase space, which can be used to prove the stability of an equilibrium point. A be a discretetime stable lti system, and p pt 0 be a solution of the discretetime lyapunov equation atpa. Lyapunovs direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. Our next purpose is to show that by combining the improvement in one step and the. 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. 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. 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. Fundamentals of lyapunov theory the objective of this chapter is to present lyapunov stability theorem and illustrate its use in the analysis and the design of nonlinear systems. Some of these will be described in later sections of this paper.
The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. Tutorial on lyapunovs stability utsa college of engineering. The objective of this paper is to refocus the discussion on safety by introducing control barrier functions that play a role equivalent to lyapunov functions in the study of liveness properties. View lyapunov stability theory research papers on academia. This allows you to estimate the lyapunov exponent of a scalar map by only knowing the. We introduce multiple lyapunov functions as a tool for analyzing lyapunov stability and use iterated function systems ifs theory as a tool for lagrange stability. Until now, the theory of lyapunov stability is still the main theoretical basis of almost all systemcontroller design chen, 1984. Boyd linear quadratic lyapunov theory lyapunov equations we assume a 2 rn n, p pt 2 rn n.
To successfully merge two clfs, the controlsharing condition is crucial. Lyapunov functions are the centerpiece of the lyapunov stability theory for dynamical systems in general. Pdf on feb 9, 1993, aristide halanay and others published applications of lyapunov methods in stability find, read and cite all the research you need on. This discoverywent against the intuition drawn from the classical theory of furstenberg. The following lemma relates the discretetime lyapunov equation 1 to the continuoustime lyapunov equation 6, and introduces a notation that we shall use to state our results. Controlsharing and merging control lyapunov functions. Lyapunov theory is very versatile and applies to a wide range of mathematical models. November 15, 2009 1 1 lyapunov theory of stability introduction. Lyapunov stability theory an overview sciencedirect topics. Example of stability problem we consider the system x0 y x3. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at least c1 righthand side and the origin is an equilibrium. The system in 1 is autonomous time invariant if f fx.