Ieee transactions on automatic control 1 nonlinear control. Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. Inputtostate stability iss applies lyapunov notions to systems with inputs. 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.
In extending the technique of lyapunov functions to control systems, a number of new issues arise. Multiple lyapunov functions see 9, 22, 38 arise as an alternative to the search of common lyapunov functions. Lyapunov matrix equation in system stability and control pdf. Later, in the classical works of massera, barbashin and krasovskii, and kurzweil, this su. Lyapunov function from a set of demonstrations, 3 proposing an optimal control problem based on the learned lyapunov function to ensure stability of nonlinear autonomous ds, and 4 extending the classical clf control scheme and present it in the context of learning robot discrete motions from demonstrations. Control lyapunov functions have been shown to be very. Stability analysis of nonlinear systems using lyapunov theory. Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. Common polynomial lyapunov functions for linear switched systems. Lyapunov stability theory an overview sciencedirect topics. 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.
The most important type is that concerning the stability of solutions near to a point of equilibrium. Robust asymptotic stabilization of hybrid systems using. The paper proposes a controltheoretic framework for veri. The ordinary lyapunov function is used to test whether a dynamical system is stable more. Here again, multiple quadratic lyapunov functions such as those introduced in 7, 12, 14, 25 hold special interest because checking for their existence also amounts to solving a set of linear matrix inequalities. I am struggling to see how differentiation of the lyapunov function is carried out. The possible function definiteness is introduced which forms the building block of lyapunovs direct method.
Failure of a lyapunov function candidate to satisfy the conditions for stability or asymptotic stability does not mean that the equilibrium point is not stable or asymptotically stable. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. Systems with inputs we now consider systems with inputs, i. In the control and systems theory developed for linear repetitive processes, the stability along the pass property. Detecting new e ective families of lyapunov functions can be seen as a serious advance.
We present a survey of the results that we shall need in the sequel, with no proofs. Lyapunov theory is used to make conclusions about trajectories of a system x. Kellett and teel, on the robustness of klstability for difference inclusions. Pdf liapunov functions and stability in control theory. Overview of lyapunov stability theory lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. Lyapunov theory lyapunov theory is used to make conclusions about trajectories of a system x. Neural networks, fuzzy logic, genetic algorithm etc. Liapunov functions and stability in control theory springerlink.
Theorems on implicit lyapunov functions ilf for finitetime and fixedtime. Lyapunov functions for nonlinear discretetime systems. Liapunov functions and stability in control theory request pdf. As a result, the existence of the weak control lyapunov functions can be specified by the investigation of property of manifolds. 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. They provide sufficient conditions for the stability of equilibria or more. Multiple lyapunov functions and other analysis tools for. Lyapunov stability theory research papers academia. However, it is a remarkable fact that the converse of theorem 4. Up to 2nd order or maxi 3rd order system graphical method differential geometry feedback linearization lyapunov theory intelligent techniques.
The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. Ieee transactions on automatic control 1 nonlinear. A wikipedia page gives a general idea about the lyapunov stability. 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. A relaxation of lyapunovs direct method has been proposed elsewhere that allows for an algorithmic construction of lyapunov functions to prove stability of. Liapunov functions and stability in control theory. Sevilla lyapunov krasovskii functionals for the study of stability and stabilisation of timedelay systems with application to networked control systems2 15.
View lyapunov stability theory research papers on academia. One of the most widely adopted stability concepts is lyapunov stability, which plays important roles in sys tem and control theory and in the analysis of engineering systems. We also discuss the case where the switched systems are indexed by an arbitrary compact set. In recent years, the problem of stability and stabilizability of switched systems has attracted increasing attentions see for instance 1, 4, 5, 7, 10, 12, 16, and still many questions remain unsolved. While the results therein involving iss lyapunov functions can certainly be used for design, constructive design tools that guarantee robustness of asymptotic stability to large disturbances are not yet available. 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. If the lyapunov function satis es the conditions of asymptotic stability over a domain. Contents 1 introduction to the lyapunov krasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ. Sevilla lyapunovkrasovskii functionals for the study of stability and stabilisation of timedelay systems with application to networked control systems2 15. A detailed post on the lyapunov stability criteria will be uploaded soon. In control theory, a controllyapunov function is a lyapunov function v x \ displaystyle vx vx for a system with control inputs. Finally, we look at alternate feedback control laws and closed loop dynamics. 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. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. From this structure, we can construct a finite number of iterations to define weakcontrollyapunov functions. Lyapunov function and relate it to these v arious stabilit y notions. These tools will be used in the next section to analyze the stability properties of a robot controller. The idea is that if one can select a suitable lyapunov function and force it to decrease along the trajectories of the system, the resulting system will converge to its equilibrium. Afterward, we present results on asymptotic stability in hybrid systems, with an emphasis on robustness.
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. Initially, we focus on lyapunov functions as the primary stability analysis tool and show how lyapunov functions are used in hybrid control design. The use of vector lyapunov functions in control theory is not frequent. 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. Techniques of nonlinear control systems analysis and design phase plane analysis. Following posts gives a very basic example to hel user use the lyapunov function in matlab.
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. Describing functions optimization theory variational optimization, dynamic. Lyapunov stability theory provides a means of stabilizing unstable nonlinear systems using feedback control. One the major stability technique for non linear dynamic system is called lyapunov. Fixedtime stability looks promising if a controller. 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. Liapunov functions and stability in control theory andrea. 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 game the adversary picks a region in the state space of radius.
It only means that such stability property cannot be. Lyapunov functions, in many cases it is almost impossible to get ones hands onto one such lyapunov function. Contents 1 introduction to the lyapunovkrasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ. As a result, the existence of the weakcontrollyapunov functions can be specified by the investigation of property of manifolds. Common polynomial lyapunov functions for linear switched. In this paper, we address the problem of existence of common polynomial lyapunov functions for linear switched systems. However, it seems reasonable to think that the flexibility shown by vector lyapunov functions in stability theory can be utilized to our advantage for feedback control design in complex systems.
Stability notions and lyapunov functions for sliding. A particular focus is on the problem of the existence of liapunov functions converse liapunov theorems and their regularity, whose. An optimization theory perspective a boris polyak a. This makes it insufficient to simply engineer v or. The control lyapunov function is used to test whether a system is feedback stabilizable, that is whether for any state x there exists a control. In simple terms, if the solutions that start out near an equilibrium point. Vector lyapunov function based stability of a class of. 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.
The idea is to transfer lyapunov functions and the associated computational techniques from control systems. What links here related changes upload file special pages permanent link page information. Lyapunov functions and stability in control theory request pdf. Is there any standard procedure to find out the lyapunov fucnion. Thus, lyapunov functions allow to determine the stability or instability of a system. Lecture 12 basic lyapunov theory stanford university. Nonlinear systems, iterative modeling and control design 1. Ifac papersonline 501 2017 7456a7461 7457 lyapunov functions. Lyapunovkrasovskii functionals for the study of stability. Lyapunov functions and feedback in nonlinear control. Matrix inequalities in system and control theory, volume 15 of siam. This book presents a modern and selfcontained treatment of the liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. 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.
In addition, this method allows to study the stability of equilibrium points of nonrough systems, for example, in the case when the equilibrium point is a center. Stability is a classical issue in dynamical system theory. Example of stability problem we consider the system x0 y x3. More formally, suppose we are given an autonomous dynamical system. From this structure, we can construct a finite number of iterations to define weak control lyapunov functions. As perturbations are incorporated in dynamical systems, robust stability becomes impor tant in control theory with. A particular focus is on the problem of the existence of liapunov functions converse. Kalman brings lyapunov theory to the eld of automatic control kalman and bertram \control system analysis and design via the second method of lyapunov. Stability notions and lyapunov functions for sliding mode control. I will assume knowledge of my autumn 2002 model theory lecture notes 1. Computation of lyapunov functions and stability of. The advantage of this method is that we do not need to know the actual solution xt. The intuitive picture is that of a scalar output function, often thought.
Finally, we present tools for stability analysis based on limited events in hybrid systems. Lyapunov functions are used for stability analysis and control lyapunov functions clf are used in the design of stabilizing feedback controllers. Lecture notes in control and information sciences 267. 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 may be discussed by the theory of aleksandr lyapunov. It only means that such stability property cannot be established by using this lyapunov function. Stability analysis and stabilization of nonlinear systems are two of the most important and extensively studied problems in control theory. Learning control lyapunov function to ensure stability of. 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. Abstractthe paper proposes a controltheoretic framework for veri.
A particular focus is on the problem of the existence of liapunov functions converse liapunov theorems and their regularity, whose interest is especially motivated by applications. A relaxation of lyapunov s direct method has been proposed elsewhere that allows for an algorithmic construction of lyapunov functions to prove stability of equilibria in nonlinear systems, but. Lecture notes stability theory math 414 spring 2003. 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 idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex op. One difficulty with explicitly constructing such examples stems from non uniqueness of lyapunov functions. Lyapunov stability analysis of switching controllers in presence of. Request pdf on jan 1, 2005, andrea bacciotti and others published lyapunov functions and stability in control theory find, read and cite all the research.
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