The theorem is a key concept in probability theory because it implies that probabilistic and. Lyapunov stability theorem for nonlinear descriptor systems in 16, 28, a suf. Markov and lyapunov on conditions for the applicability of the central limit theorem of probability theory. R be a continuously di erentiable function such that v0 0 and vx 0 0 for some x 0 with arbitrarily small kx 0k. Verified stability analysis using the lyapunov matrix.
Pdf a lyapunovbased smallgain theorem for infinite. Lyapunov functions and feedback in nonlinear control 3 several classical theorems dealing with the uncontrolled case, is due to clarke, ledyaev and stern 9. A smooth converse lyapunov theorem for robust stability. Lyapunovs theorem in probability theory is a theorem that establishes very general sufficient conditions for the convergence of the distributions of sums of independent random variables to the normal distribution. Under these conditions, the solution x of the lyapunov equation is positive semidefinite. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. In the field of crns, one important example is the pseudo. Pdf classical converse theorems in lyapunovs second method. Linear quadratic lyapunov theory stanford university. However, it is a remarkable fact that the converse of theorem 4. Lyapunov stability analysis with solved examples electrical. 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. Boyd linear quadratic lyapunov theory lyapunov equations we assume a 2 rn n, p pt 2 rn n. Pdf lyapunovs second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems.
The possible function definiteness is introduced which forms the building block of lyapunovs direct method. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by. 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 and related equations are named after the russian mathematician aleksandr lyapunov. The goal is to help the reader to understand the most. Scribd is the worlds largest social reading and publishing site. A theorem on average liapunov functions springerlink. Proof suppose trajectory xt does not converge to zero. Lyapunov stability theory research papers academia. Report by electronic transactions on numerical analysis. Lyapunov function partial differential equations for. In this work we prove a converse lyapunov theorem for uniformly asymptotically stable nonlinear switched systems and we allow the systems to depend explicitly on the time t, that is, we work the nonautonomous case out. In the delay free case, if two solutions of this linear differential equation cross, then the two.
Lyapunovrazumikhin and lyapunovkrasovskii theorems for. Lyapunov functions and lyapunovs stability theorem. Notations and definitions by xt we denote the transposition of a vector x2rn. Theorem a matrix a is hurwitz if and only if for any q qt 0 there is p pt 0 that satis. Click download or read online button to textbookofdemoivrestheorem book pdf for free now. The wonder of it all is that the plan of salvation is set before. Estabilidad por lyapunov sistemas dinamicos lineales. Iss lyapunovrazumikhin type theorem and subsection 3. Verified stability analysis using the lyapunov matrix equation. If the curved panel is stable at this mach number, then the free stream mach number is increased. Lyapunov exponents for continuoustime dynamical systems t. A theorem on average liapunov functions for dynamical systems is generalized.
Lyapunov stability theorem for nonlinear descriptor systems. We can expect with probability 1 that two randomly chosen initial conditions will diverge exponentially at a rate given by the largest lyapunov exponent 6, 15. 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. Lecture 12 basic lyapunov theory stanford university. The lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. A practical method for calculating largest lyapunov exponents. Lyapunovrazumikhin and lyapunovkrasovskii theorems for interconnected iss timedelay systems sergey dashkovskiy and lars naujok abstractwe consider an arbitrary number of interconnected nonlinear systems with timedelays and investigate them in view of inputtostate stability iss. A longtime numerical calculation of the leading lyapunov exponent requires rescaling the distance in order to keep the nearby trajectory separation within the linearized. Lecture linear quadratic lyapunov theory the lyapunov equation lyapunov stability conditions the lyapunov operator and integral evaluating quadratic integrals analysis of are discretetime results linearization theorem 1. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. If the derivative of along the trajectories of the system is a negative semidefinite then. In probability theory, the central limit theorem clt establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution informally a bell curve even if the original variables themselves are not normally distributed. Theorem3 supposse that lis a lyapunov functional on oand xt is an orbit lying in a closed bounded set k. The strength of lyapunovs second method as encapsulated in theorems 1.
Let the origin be an isolated critical point for the autonomous systems 4, and let x. The precise statement of lyapunovs theorem is as follows. Then the system is strongly asymptotically stable if and only if there exists a pair of c. A function v satisfying the hypotheses of this theorem is called a lyapunov function for the. Defining stability using lyapunov functions energy functions duration. Sources and studies in the history of mathematics and.
As an illustration the result is used to establish a rather strong coexistence criterion for an ecological system. Finally, we have the following theorem, which follows readily from the last lemma and an argument similar to the proof of theorem 4. Overview of lyapunov stability theory lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. This is a generalization of lyapunovs convexity theorem, which is a particular case of this statement for p x. Lyapunov exponents for the threegeneration leslie model with parameters 0.
Liapunovs second method is a generalization to the theory of nonlinear systems of odes of two basic. This is a paper in a series of investigations into the use of. Helmholtz free energy function, proposed by horn and jackson 18. If rvx fx is negative semide nite, then x e is stable. Lyapunov exponents oberwolfach, 1990 lecture notes in mathematics, 1486. Discontinuity of lyapunov exponents ergodic theory and.
A lyapunov function for an autonomous dynamical 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 the widely spread local. Assuming that each subsystem is exponentially inputtostate stable. Lyapunov in 1901 and was the final step in research of p. Computers and internet mathematics algorithms research technology application usage liapunov functions mathematical research stability stability physics. If the matrix x were specified explicitly, then we could estimate the. It is globally asymptotically stable if the conditions for asymptotic stability hold. Lyapunov functionsbasic idea seek an aggregate summarizing function that continually decreases toward a minimum for mechanical systemsenergy of a free mechanical system with friction always decreases unless the system is at rest, equilibrium. To show that any initial condition can be steered towards zero in the presence of a.
Tutorial on lyapunovs stability utsa college of engineering. A converse lyapunov theorem for discretetime systems with disturbances. Lyapunov functions and feedback in nonlinear control. The second liapunov theorem gives a criterion for instability. Then, in chapter 9, we turn our attention to the contrasting man. The main idea of the lyapunovkrasovskii theorem is that it. Lyapunov function and relate it to these v arious stabilit y notions.
We shall henceforth assume that the matrix a is stable, that q qr i o, and that a, q is a controllable pair. In fact, the range of a nonatomic vector measure is a zonoid the closed and convex set that is the limit of a convergent sequence of zonotopes. The following theorem is a direct application of the standard discretetime lyapunov stability theorem for general dynamical systems to the dynamical system 4. Wang in 35 to prove a converse lyapunov theorem on asymptotically stable nonlinear, autonomous switched systems in 38. As the author, i send this ebook in pdf format to you free of charge.
Theorem lyapunov let x e 0 be an equilibrium point for the system dxtdt fxt. Inria computation of lyapunov functions and stability of interconnected systems. Theorem4 suppose that lis a lyapunov functional on oand that p. As it so often goes with easy ideas, it turns out that lyapunov exponents are not natural for study of dynamics, and we would have passed them. Download textbookofdemoivrestheorem ebook pdf or read online books in pdf, epub, and mobi format. In section 4 an example is given to illustrate the results. We start with the lyapunov theorem of the work by lewis. We show that the lyapunov equation of that theorem can lead to. Lyapunov s stability analysis technique is very common and dominant. View lyapunov stability theory research papers on academia. Since considering the kolob theorem my mind has not rested. Zak school of electrical and computer engineering ece 680 fall 20. 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. Department of mathematics indian institute of science bangalore 560 012, india december 9, 2003 abstract in this article, di.
Pdf a lyapunovbased smallgain theorem for infinite networks. This paper presents a smallgain theorem for networks composed of a countably infinite number of finitedimensional subsystems. The main deficiency, which severely limits its utilization, in reality, is the complication linked with the development of the lyapunov function which is needed by the technique. Let f have compact convex nonempty values and closed graph. Lyapunovs stability analysis technique is very common and dominant. Lyapunov equations arise in several areas of control, including stability theory and the study of the rms behavior of systems. 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. Unlike lyapunov exponents, which measure local instability in terms of geometric. The lyapunov characteristic exponents and their computation. Then the origin is a globally asymptotically stable equilibrium of x0 fx. Yao 3 for autonomous systems, when wx in the above theorem is only positive semidefinite, asymptotic stability may still be obtained by applying the following simplified version of lasalles theorem. Finally section 5 concludes this paper with a short summary. Defining stability using lyapunov functions energy.
Lyapunov exponents for continuoustime dynamical systems. Noncoercive lyapunov functions for infinitedimensional systems. 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 theory is used to make conclusions about trajectories of a system. Nonmonotonic lyapunov functions for stability of nonlinear. A practical method for calculating largest lyapunov. In the theory of vector measures, lyapunovs theorem states that the range of a finitedimensional vector measure is closed and convex. The proposed smallgain criterion for the stability analysis of the network the spectral radius of the gain operator is less than one can b e. Whereas there is no general technique for constructing lyapunov functions for odes, in many specific cases the construction of lyapunov functions is known. We can expect with probability 1 that two randomly chosen initial conditions will diverge exponentially at a.