Lyapunov functions and stability in control theory pdf files

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. What links here related changes upload file special pages permanent link page information. These tools will be used in the next section to analyze the stability properties of a robot controller. 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. Kellett and teel, on the robustness of klstability for difference inclusions. Up to 2nd order or maxi 3rd order system graphical method differential geometry feedback linearization lyapunov theory intelligent techniques. Example of stability problem we consider the system x0 y x3. Liapunov functions and stability in control theory request pdf.

Sevilla lyapunovkrasovskii functionals for the study of stability and stabilisation of timedelay systems with application to networked control systems2 15. The idea is to transfer lyapunov functions and the associated computational techniques from control systems. Liapunov functions and stability in control theory andrea. Afterward, we present results on asymptotic stability in hybrid systems, with an emphasis on robustness. They provide sufficient conditions for the stability of equilibria or more.

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. As a result, the existence of the weak control lyapunov functions can be specified by the investigation of property of manifolds. 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. Stabilization of nonlinear systems using weakcontrol. Systems with inputs we now consider systems with inputs, i. Multiple lyapunov functions and other analysis tools for. 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. 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.

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. 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. 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. Stability is a classical issue in dynamical system theory. A wikipedia page gives a general idea about the lyapunov stability. Neural networks, fuzzy logic, genetic algorithm etc. 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. Liapunov functions and stability in control theory. One difficulty with explicitly constructing such examples stems from non uniqueness of lyapunov functions. The most important type is that concerning the stability of solutions near to a point of equilibrium. The ordinary lyapunov function is used to test whether a dynamical system is stable more. The paper proposes a controltheoretic framework for veri. 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. Stability analysis of nonlinear systems using lyapunov theory.

A particular focus is on the problem of the existence of liapunov functions converse liapunov theorems and their regularity, whose. In the control and systems theory developed for linear repetitive processes, the stability along the pass property. Vector lyapunov function based stability of a class of. Kalman brings lyapunov theory to the eld of automatic control kalman and bertram \control system analysis and design via the second method of lyapunov. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Sevilla lyapunov krasovskii functionals for the study of stability and stabilisation of timedelay systems with application to networked control systems2 15. Detecting new e ective families of lyapunov functions can be seen as a serious advance. 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. Stability notions and lyapunov functions for sliding mode control. 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. Later, in the classical works of massera, barbashin and krasovskii, and kurzweil, this su.

Pdf liapunov functions and stability in control theory. Stability notions and lyapunov functions for sliding. Nonlinear systems, iterative modeling and control design 1. One the major stability technique for non linear dynamic system is called lyapunov. Contents 1 introduction to the lyapunov krasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ. 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. 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. Common polynomial lyapunov functions for linear switched systems. Multiple lyapunov functions see 9, 22, 38 arise as an alternative to the search of common lyapunov functions. 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. Lyapunov stability theory provides a means of stabilizing unstable nonlinear systems using feedback control. 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.

Ifac papersonline 501 2017 7456a7461 7457 lyapunov functions. It only means that such stability property cannot be. Lyapunov functions and feedback in nonlinear control. Finally, we present tools for stability analysis based on limited events in hybrid systems. The use of vector lyapunov functions in control theory is not frequent.

Lyapunov functions, in many cases it is almost impossible to get ones hands onto one such lyapunov function. Lecture notes stability theory math 414 spring 2003. Stability analysis and stabilization of nonlinear systems are two of the most important and extensively studied problems in control theory. This may be discussed by the theory of aleksandr lyapunov. I am struggling to see how differentiation of the lyapunov function is carried out. 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. As perturbations are incorporated in dynamical systems, robust stability becomes impor tant in control theory with. Lyapunov function and relate it to these v arious stabilit y notions. However, it is a remarkable fact that the converse of theorem 4. Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems.

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. 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. A particular focus is on the problem of the existence of liapunov functions converse. Lyapunov functions are used for stability analysis and control lyapunov functions clf are used in the design of stabilizing feedback controllers. Following posts gives a very basic example to hel user use the lyapunov function in matlab. The possible function definiteness is introduced which forms the building block of lyapunovs direct method. A relaxation of lyapunovs direct method has been proposed elsewhere that allows for an algorithmic construction of lyapunov functions to prove stability of. In control theory, a controllyapunov function is a lyapunov function v x \ displaystyle vx vx for a system with control inputs. Computation of lyapunov functions and stability of. It only means that such stability property cannot be established by using this lyapunov function. 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. Contents 1 introduction to the lyapunovkrasovskii functionals 2 objectives 3 ncs model 4 general procedure 5 example of application 6 summary univ. More formally, suppose we are given an autonomous dynamical system.

Describing functions optimization theory variational optimization, dynamic. From this structure, we can construct a finite number of iterations to define weak control lyapunov functions. This book presents a modern and selfcontained treatment of the liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. As a result, the existence of the weakcontrollyapunov functions can be specified by the investigation of property of manifolds. 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. Institute of control sciences, russian academy of sciences. 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. Control lyapunov functions have been shown to be very.

If the lyapunov function satis es the conditions of asymptotic stability over a domain. 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. I will assume knowledge of my autumn 2002 model theory lecture notes 1. Initially, we focus on lyapunov functions as the primary stability analysis tool and show how lyapunov functions are used in hybrid control design. An optimization theory perspective a boris polyak a. Lyapunov stability theory an overview sciencedirect topics. A detailed post on the lyapunov stability criteria will be uploaded soon.

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. Thus, lyapunov functions allow to determine the stability or instability of a system. From this structure, we can construct a finite number of iterations to define weakcontrollyapunov functions. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. Inputtostate stability iss applies lyapunov notions to systems with inputs. 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. Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. Lyapunov functions and stability in control theory request pdf. Lyapunov stability game the adversary picks a region in the state space of radius.

Ieee transactions on automatic control 1 nonlinear. View lyapunov stability theory research papers on academia. The advantage of this method is that we do not need to know the actual solution xt. In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. In simple terms, if the solutions that start out near an equilibrium point. 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. Lecture notes in control and information sciences 267. In this paper, we address the problem of existence of common polynomial lyapunov functions for linear switched systems.

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. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. Abstractthe paper proposes a controltheoretic framework for veri. Lyapunov functions for nonlinear discretetime systems. Lyapunov matrix equation in system stability and control pdf. The idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex op. Liapunov functions and stability in control theory springerlink. Lecture 12 basic lyapunov theory stanford university. Lyapunovkrasovskii functionals for the study of stability. Robust asymptotic stabilization of hybrid systems using.

This makes it insufficient to simply engineer v or. 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. Lyapunov stability analysis of switching controllers in presence of. Lyapunov theory is used to make conclusions about trajectories of a system x. 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. Is there any standard procedure to find out the lyapunov fucnion. Ieee transactions on automatic control 1 nonlinear control.

Theorems on implicit lyapunov functions ilf for finitetime and fixedtime. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. Finally, we look at alternate feedback control laws and closed loop dynamics. Common polynomial lyapunov functions for linear switched. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. The intuitive picture is that of a scalar output function, often thought. Techniques of nonlinear control systems analysis and design phase plane analysis.

We present a survey of the results that we shall need in the sequel, with no proofs. Fixedtime stability looks promising if a controller. As shown in 12, the problem of checking if a given polynomial can be written as a sum of squares can be solved via convex optimization, in particular semide. Lyapunov stability theory research papers academia. 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. Learning control lyapunov function to ensure stability of.