A function of a vector and of time which is positivedefinite and has a negativedefinite derivative with respect to time for nonzero vectors, is identically zero for the zero vector, and approaches infinity as the norm of the vector approaches infinity. A general indicator of the presence of chaos in a dynamical system is the largest lyapunov exponent. Belur,department of electrical engineering,iit bombay. It is customary to normalize storage and lyapunov functions so that their minimum equals zero, which yields c 2 and. Ecuaciones diferenciales con maxima jose antonio vallejo fcuaslp. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof.
Estabilidad segun lyapunov y estabilidad asintotica. Liapunov o lyapunov en ruso puede hacer referencia a. Pdf a general indicator of the presence of chaos in a dynamical system is the largest lyapunov exponent. The method of lyapunov functions plays a central role in the study of the controllability and stabilizability of control. Linear quadratic lyapunov theory stanford university. Lyapunov equations arise in several areas of control, including stability theory and the study of the rms behavior of systems. Linear quadratic lyapunov theory lyapunov equations we assume a 2 rn n, p pt 2 rn n. Lectures on lyapunov exponents the theory of lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Use v x xtpx as a lyapunov function candidate for x. 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. In control theory, the discrete lyapunov equation is of the form. The following code solves the system of the odes and also plots the output 3d orbit.
Constructions of lyapunov functions for classics sis, sir. This quantity provides a measure of the mean exponential rate of divergence of nearby orbits. The lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. Liapunov function article about liapunov function by the. Lecture 12 basic lyapunov theory stanford university. In this paper, we show how to implement the socalled variational method for calculating the largest lyapunov exponent in mathematica. We revisit the canonical continuoustime and discretetime matrix algebraic and ma. Integrated wearable and selfcarrying active upper limb orthosis. Calculation lyapunov exponents for ode file exchange.
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. Lyapunov design shuzhi ge department of electrical and computer engineering, the national university of singapore, singapore keywords. Finite time stabilization of an uncertain chain of integrators by. W e shall assume that equilibrium p o i n t of in terest is at the origin, since if x 6 0, a simple translation can alw ys b e applied to obtain an equiv alen t system with the equilibrium at 0. 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. If a is stable, lyapunov equation has a unique solution p, for any q qt.
Finding lyapunov functions mit opencourseware free. Lyapunov functions and feedback in nonlinear control. We study a new model describing the transmission of influenza virus with disease resistance in human. Finding lyapunov functions1 this lecture gives an introduction into basic methods for. Lyapunov equations arise in several areas of control, including stability theory and. Taking into account that you give specific initial conditions, then the solution of the system will correspond to a threedimensional orbit. Finding lyapunov functions free online course materials. Mathematical analysis shows that dynamics of the spread is determined by the basic reproduction number r 0. It defines a lyapunov function on the homogeneous space s1. Lyapunov functions article about lyapunov functions by the. Estabilizacion del pendulo invertido sobre dos ruedas.
Control lyapunov function, lyapunov design, model reference adaptive control, adaptive control, backstepping design. Stability analysis of an influenza virus model with disease. Jan 16, 2001 lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. In the theory of ordinary differential equations odes, lyapunov functions are scalar functions. Lyapunov theory is used to make conclusions about trajectories of a system x. Theorem a matrix a is hurwitz if and only if for any q qt 0 there is p pt 0 that satis. A lyapunov function for an autonomous dynamical system. Capitulo iii estabilidad segun liapunov 1 definiciones fing. A practical method for calculating lyapunov exponents from. Article information, pdf download for integrated wearable and. This and related equations are named after the russian mathematician aleksandr lyapunov. Suppose there is a continuous stabilizing state feedback control u. S1s1 this trick generalizes to many homogeneous spaces geometry and symmetries in coordination control alain sarlette ph. We present a new method for calculating the largest lyapunov exponent from an experimental time series.
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