Energy function analysis of a twomachine infinitebus. You could think of this as kinetic energy, and in that case with a spring mass potential kinetic energy right. Below, we restrict ourselves to the autonomous systems \\ \\mathbfx. The proposed approach allows learning a larger set of robot mo tions compared to existing methods that are based on quadratic energy function. The approach generalizes the idea of energy methods, and extends the concept of energy function to a more general lyapunov functions family lff constructed via semide. The first method developed the solution in a series which was then proved convergent within limits. The basis for all what he was proving is entirely on the well known fact that towards the equilibrium position, the total energy in a system is either constant or decreasing. This and related equations are named after the russian mathematician aleksandr lyapunov. Energy functions, an extension of the lyapunov functions, have been practically used in electric power systems for several applications. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. Pdf lyapunov functions are an essential tool in the stability analysis of. Defining stability using lyapunov functions energy. One the major stability technique for non linear dynamic system is called lyapunov.
Lyapunov stability theory an overview sciencedirect topics. Lyapunov s direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. However, we need the lyapunov function to be smooth, so we prove that if the. Krylov subspaces and reducedorder solution methods 207 numerical solution of the lyapunov equation 1. Phys 221a lecture notes lyapunov exponents and their. 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 stability of equilibria of a differential equation.
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. Under these conditions, the solution x of the lyapunov equation is positive semidefinite. Pdf lyapunov energy function based control method for three. Lyapunov theory is used to make conclusions about trajectories of a system x. There are two lyapunov methods for stability analysis. Their concept was introduced by lyapunov when studying the stability of nonstationary solutions of ordinary di. In the main part, the computational methods are presented, ordered by the type of method used to construct a lyapunov function.
Lyapunov s stability theory is the single most powerful method in stability analysis of nonlinear. A lyapunov optimization approach for green cellular. Then, in chapter 9, we turn our attention to the contrasting man. Lyapunov equation an overview sciencedirect topics. Then, by combining a lyapunov function construction method by massera for odes, see, for example, 42 or section 5. The lyapunov characteristic exponents and their computation. So thats what makes this lyapunov stability means you can find such a function with all these properties around the states of interest.
The reason for this is the use of workspace gradients and parameterization invariance. Lyapunov function v can be thought of as generalized energy function for. The first method usually requires the analytical solution of the differential equation. The method has more theoretical importance than practical value and can be. The lyapunov s second method or the method of lyapunov functions is the only universal technique for the investigation of the stability of the dynamical systems of the most general configurations and properties including all possible types of motions nonstationary, stationary, periodic, nonlinearities, dimensionalities, parametric dependence. Signs of lyapunov exponents for various attractors. Lyapunov theory with inputs and outputs 145 reachable set with integral quadratic bounds we consider x.
Pdf lyapunov energy function based control method for. The lyapunov function method is applied to study the stability of various differential equations and systems. Lecture 12 basic lyapunov theory stanford university. See 1 for a more detailed treatment of lyapunov functions and nonlinear stabilit. The approach in based on a postulating a lyapunov function candi date consisting of the total energy stored in the system expressed in terms of momenta.
This idea is illustrated for threedimensional state space in fig. 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. In several applications, all that is needed is the cholesky factor l of the symmetric positive definite solution x of a lyapunov equation. In this study, a lyapunov energy function based control method with output voltage feedback loops is proposed for threephase uninterruptible power supply ups inverters. 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. Lyapunovs methods, the quadratic lyapunov function, the threshold accepting algorithms and illustrative examples, respectively.
Dec 15, 2014 mod01 lec08 extension of lyapunovs theorem. The application of this theory to control is mainly based on the lyapunov stability theorem. Numerical methods for lyapunov equations methods for lyapunov equations this chapter is about numerical methods for a particular type of equation expressed as a matrix equality. To show that a system is stable in the sense of lyapunov, a positive definite function of the system states, which decreases along system trajectories lyapunov function, should be found. Following posts gives a very basic example to hel user use the lyapunov function in matlab. It is based on the lyapunov s second method also called lyapunov s direct method. Keywords stability, lyapunovs methods, threshold accepting algorithms, numerical computing. Lyapunov stability of the solution of an equation given on is lyapunov stability of the point relative to the family of mappings. In this paper, we present an automated way of generating quadratic lyapunov functions for certifying linear conver.
Then the method studies stability by looking at the rate of change of this measure of energy. Construction of lyapunov functions for some fourth order. Lyapunov stability analysis very general method to prove or disprove stability of nonlinear sys tems. Formalizes idea that all systems will tend to a minimum energy state.
Alternatively, the method in 21 shows that a local energy like lyapunov function exists, in general, for stable systems with transfer conductances. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. 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. Still, the pseudohelmholtz free energy function serves as the lyapunov function. The intuitive picture is that of a scalar outputfunction, often thought of as a generalized energy that is bounded below, and decreasing along solutions. Lyapunov direct method is based on energylike functions vx and the analysis of the function t. Lyapunovs second method tells us, roughly speaking, that if we succeed in nding a lyapunov functionan energy like function of the state that decreases along trajectoriesthen we have proven that the dynamical system in question is stable. Lyapunov functionsbasic idea seek an aggregate summarizing function that continually decreases toward a minimum for mechanical systems energy of a free mechanical system with friction always decreases unless the system is at rest, equilibrium. Scribd is the worlds largest social reading and publishing site. Lyapunov methods in power system stability springerlink. Control design based on lyapunov s direct method estimating region of attraction farzaneh abdollahi nonlinear control lecture 4 270. Functions which are globally like energy functions are called positive def inite functions.
Pdf the lyapunov direct method is on the verge of being implemented for assessment of online dynamic security. Lyapunov exponent for the tent map is the same as the lyapunov exponent for the 2xmod 1 map, that is h lnj2j, thus the tent map exhibits chaotic behavior as well. Bochi phenomenon of systems whose lyapunov spectra are generically not sim. 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. Energy functions are a specific form of lyapunov functions that guarantee the system convergence to stable equilibrium points. Lyapunovs stability theory is the single most powerful method in stability analysis of nonlinear systems. Lyapunov energy function based control method for threephase ups inverters with output voltage feedback loops. Furthermore, we assume that there exists a continuously di. Lyapunov direct method is the most effective method for studying nonlinear and timevarying systems and is a basic method for stability analysis and control law desgin. An energybased lyapunov function for physical systems. A similar concept appears in the theory of general state space markov chains, usually.
Lyapunov theory is used to make conclusions about trajectories of a system. Unlike the traditional energy function and its variations, the constructed lyapunov functions. Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. In the theory of ordinary differential equations, lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ode. In the mid 1900s, a series of converse lyapunov theorems were developed which established that any. Since these results are local in character, they can only determine the stability of the equilibrium point and cannot be used to determine the domain of attraction. Lyapunov functions are also basis for many other methods in analysis of dynamical system, like frequency criteria and the method of comparing with other systems. Since our pendulum is a mechanical system, we can use the standard notions of potential and kinetic energy. Lyapunov functions and feedback in nonlinear control.
First, the di erent types of systems, where lyapunov functions are used, are brie y discussed. 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. It 3 oct 2015 1 a lyapunov optimization approach for green cellular networks with hybrid energy supplies yuyi mao, student member, ieee, jun zhang, senior member, ieee, and khaled b. Therefore define the energy quadratic function 1 2 vx xpx t. The lyapunov stability theory is used to describe the stability of a dynamic system fig. It is wellknown that the lyapunov s second method is one of the most useful and effective way to analyze the stability of a power system without solving the differential equations 68. The interest in energy methods was recently revived by athay et al. 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.
For lti systems it is easy to find a lyapunov function. Definition of the lyapunov function a lyapunov function is a scalar function defined on the phase space, which can be used to prove the stability of an equilibrium point. Noltllt numerical solution of the lyapunov equation by. Named after the russian mathematician aleksandr mikhailovich lyapunov, lyapunov functions also called the lyapunovs second method for stability are important to stability. In this paper, we attempt to make a bridge between fast. Early algorithms employed a kronecker product expansion that transformed the lyapunov. Consider the continuoustime ct linear time invariant lti autonomous system x ax x,0 with state x tr. General elemental velocity lyapunov function overview.
Energy is constant and you have a v dot that is zero, thats whats happening. We shall henceforth assume that the matrix a is stable, that q qr i o, and that a, q is a controllable pair. Algorithmic construction of lyapunov functions for power. Stability analysis of power systems by lyapunovs direct method. Lyapunov stability theory, we assume that the ode in consideration has a smooth at least c1 righthand side and the origin is an equilibrium. One can think of the lyapunov function as a generalization of the idea of the energy of a system. Region of attraction when the origin x 0 is asymptotically stable, we are often. Lyapunov matrix equation in system stability and control pdf. The method of lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. Via averaging methods, a nonlinear but autonomous system is derived whose linearized equivalent, supposed to be controllable, will provided a stabilizing feedback law.
The lces are asymptotic measures characterizing the average rate of growth or shrinking of small perturbations to the solutions of a dynamical system. Generalized energies and the lyapunov method springerlink. The second method, which is now referred to as the lyapunov stability criterion or the direct method, makes use of a lyapunov function vx which has an analogy to the potential function of classical dynamics. This chapter is about numerical methods for a particular type of equa tion expressed as a matrix equality. An important means for stability analysis of a mas is to construct a lyapunov.
For nonlinear systems, it turns out to be essential to consider nonsmooth lyapunov functions, even if the underlying control dynamics are themselves smooth. Lyapunov functions family approach to transient stability. Q moreover, if a is hurwitz, then p is the unique solution idea of the proof. The existence of such an energy function thus provides a straightforward certi. The proposed approach yields a larger stability region for a polynomial system than an existing method does. If the matrix x were specified explicitly, then we could estimate the. Energybased lyapunov functions for forced hamiltonian. Facts devices, lyapunov energy functions, lyapunov direct methods, powersystem control.
Stability in the sense of lyapunov a natural way to investigate whether an equilibrium is stable is to see whether it dissipates energy. Lyapunov function and relate it to these various stability notions. In control theory, the discrete lyapunov equation is of the form. Lyapunov s indirect method and linking lyapunov function to lqr duration.
The lyapunov stability theorem can be applied without solving the ode system the theorem provides a su cient condition for stability the theorem does not provide a systematic method for constructing the lyapunov function v of a system. A lyapunovbased approach for the control of biomimetic robotic systems with periodic forcing inputs domenico campolo. The lyapunov equation is the most common problem in the class of problems called matrix equations. Pdf computational methods for lyapunov functions researchgate. It will then proved, by means of lyapunov energy functions, that this law can also be used to bound the original nonlinear nonautonomous system. Letaief, fellow, ieee abstractpowering cellular networks with renewable energy. Lyapunov s method has been applied by many researchers in the past century to investigate the stability of nonlinear systems 1. We assume that x 0 0 which can be accomplished with a change of coordinates. A comprehensive energy function theory for general nonlinear autonomous dynamical systems along with its applications to electric power systems will be summarized in this article. The size of attraction domains of asymptotically lyapunov stable equilibria and the rate systems return to them, for instance, are definable in terms of lyapunov.
The possible function definiteness is introduced which forms the building block of lyapunov s direct method. There are two methods for specific application, and the popular one is the lyapunov second method. A wikipedia page gives a general idea about the lyapunov stability. Numerical approach to lyapunovs stability analysis of.