Review on computational methods for lyapunov functions. Liapunov functions and stability in control theory. Basic results on lyapunov functions are now covered in every textbook on nonlinear. The problems on stability in the first approximation are given here. Haddad, senior member, ieee abstractvector lyapunov theory has been developed to weaken the hypothesis of standard lyapunov theory. Dec 01, 2005 liapunov functions and stability in control theory, second ed.
Ifac papersonline 501 2017 7456a7461 7457 lyapunov functions. Nonlinear dynamical systems and control presents and develops an extensive treatment of stability analysis and control design of nonlinear dynamical systems, with an emphasis on lyapunovbased methods. Lyapunov functions and feedback in nonlinear control. In the theory of ordinary differential equations odeslyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ode. Lyapunov functions and stability in control theory request pdf. Lyapunov matrix equation in system stability and control pdf.
The main objective of control is to modify the behavior of a dynamical system, typically with the purpose of regulating certain variables or of tracking desired signals. Problems to understand lyapunov stability nonlinear control. Presentation of lyapunov and lagrange stability theory for dynamical systems defined on general metric spaces involving monotonic and nonmonotonic lyapunov functions specialization of this stability theory to finitedimensional dynamical systems specialization of this stability theory to infinitedimensional dynamical systems. The coefficient r is an exponential function of the temperature and the reagent. Below, we restrict ourselves to the autonomous systems \\ \\mathbfx. Helmholtz free energy function, proposed by horn and jackson 18. So, the idea behind lyapunov stability is pretty straight forward. Lecture notes in control and information sciences 267. In this regard piecewise quadratic lyapunov functions 810 and fuzzy lyapunov functions 1216 are introduced to replace the single quadratic lyapunov function. On lyapunov stability of scalar stochastic timedelayed.
How to use a lyapunov function to test stability of a nonlinear system. The celebrated lyapunov function method or the direct lyapunov method introduced in the ph. 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. Lyapunovs direct method appears ubiquitously in control theory. Stability is a classical issue in dynamical system theory. Piecewise linear in rates lyapunov functions for stability analysis. Sulem, applied stochastic control of jump diffusions, vol. Liapunov functions and stability in control theory, second ed. A particular focus is on the problem of the existence of liapunov functions converse liapunov theorems and their regularity, whose. Request pdf on jan 1, 2005, andrea bacciotti and others published lyapunov functions.
Nonmonotonic lyapunov functions for stability of nonlinear. View lyapunov stability theory research papers on academia. The work in this dissertation is based on lyapunov stability theory a common tool in nonlinear control and presents several control strategies for open problems in nonlinear control literature. By using lyapunov functions it is possible to estimate the region of attraction for an equilibrium point. We continue this line of work in the current paper.
Lyapunov in 1892 is a simple effective tool for stability analysis of differential. We study switched nonlinear differential algebraic equations daes with respect to existence and nature of solutions as well as stability. November 15, 2009 1 1 lyapunov theory of stability introduction. 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. Web of science you must be logged in with an active subscription to view this. A central tool in systems theory for synthesizing control laws that achieve stability are control lyapunov functions clfs. We present a survey of the results that we shall need in the sequel, with no proofs. One of the powerful tools for stability analysis of systems of differential equations, including nonlinear systems, are lyapunov functions.
Learning control lyapunov function to ensure stability of. Lyapunov functions for nonlinear discretetime systems. 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. Additionally to the advantage that lyapunovs direct method is a general method for stability analysis it can also be used for linear control design for nonlinear plants. Lyapunov functions and global stability for agestructured. So it means that at best youre always converging to the origin, at worst its negative semidefinite.
An optimization theory perspective a boris polyak a. Rninto rnand there is at least one equilibrium point x, that is f x 0. Kokotovi6 department of electrical and computer enoineerino, university of california, santa barbara, ca 93106, usa. Finally, the issue of stabilizing feedback design must be considered, for this is one of the main reasons to introduce control lyapunov functions. Control lyapunov functions for adaptive nonlinear stabilization. It only means that such stability property cannot be established by using this lyapunov function. 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. Defining stability using lyapunov functions energy. Overview of lyapunov stability theory lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. Pdf liapunov functions and stability in control theory. Lyapunov function partial differential equations for. Stability of the interconnected systems is then analyzed by the small gain theorem in comparison form and the small gain theorem in dissipative form, respectively. Recently, a novel method to construct lyapunov functions for odes via linear programming was presented 5. 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.
Converse lyapunov theorems converse lyapunov theorems are of the form. How to convert pdf to word without software duration. Solution theory, lyapunov functions, and stability. Computation of lyapunov functions and stability of interconnected systems. This lyapunov function can be derived from the microscopic level using potential theory 2. 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. It only means that such stability property cannot be. In this thesis, we investigate the problems of computation of lyapunov functions and stability analysis of interconnected systems. Systems with inputs we now consider systems with inputs, i. Since common quadratic lyapunov functions tend to be conservative, it is thus desirable to develop less conservative stability results for ts fuzzy systems. Lyapunov functions and feedback in nonlinear control universite.
Kellett and teel, on the robustness of klstability for difference inclusions. Detecting new e ective families of lyapunov functions can be seen as a serious advance. This book presents a modern and selfcontained treatment of the liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. 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. The model allows the production rate of viral particles and the death rate of productively infected cells to vary and depend on the infection age. The results show that the global asymptotic stability of the infection free steady state and the infected steady. Liapunov functions and stability in control theory ebook. Can anyone provide clarity on this or a more detailed step by step method to using lyapunov functions to test for stability. Rninto rnand there is at least one equilibrium point x. In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. Lyapunov stability theory research papers academia. In the classical lyapunov stability theory, we assume that the ode in consideration has a smooth at. Computation of piecewise quadratic lyapunov functions for.
Lyapunov functions and stability in control theory. 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. 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. Stability of dynamical systems on the role of monotonic. Lyapunov functions, in many cases it is almost impossible to get ones hands onto one such lyapunov function. Institute of control sciences, russian academy of sciences. 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. 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. Lecture 12 basic lyapunov theory stanford university. Additionally to the advantage that lyapunov s direct method is a general method for stability analysis it can also be used for linear control design for nonlinear plants. For an iiss dynamic system with perturbation, we introduce an auxiliary system which is uniformly ve stable.
Lecture notes stability theory math 414 spring 2003. I will assume knowledge of my autumn 2002 model theory lecture notes 1. We utilize piecewisesmooth distributions introduced in earlier work for linear switched daes to establish a solution framework for switched nonlinear daes. Nonlinear dynamical systems and control presents and develops an extensive treatment of stability analysis and control design of nonlinear dynamical systems, with an emphasis on lyapunov based methods. Inputtostate stability iss applies lyapunov notions to systems with inputs. By using the direct lyapunov method and constructing suitable lyapunov functions, dynamical properties of the agestructured model without or with drug treatment are established.
Lionel rosier the main purpose of this work is to present a modern and selfcontained treatment of the liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. Lecture notes on nonlinear systems and control spring. General references for lyapunov functions in control include 2 and. Liapunov functions and stability in control theory andrea. Overview of lyapunov stability theory lyapunov s direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. 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. Example of stability problem we consider the system x0 y x3.
Switched nonlinear differential algebraic equations. 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. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. Classically, a clf enforces that the resulting closedloop state. We propose new methods for learning control policies and neural network lyapunov functions for nonlinear control problems, with provable guarantee of stability. Lyapunovbased safe policy optimization for continuous. The intuitive picture is that of a scalar output function, often thought. Lyapunov theory is used to make conclusions about trajectories of a system x. Stability theory developed historically in the work of shelah as a chunk of machinery intended to help generalize morleys theorem to a computation of the possible \spectra of complete rst order theories. You plug in your differential equations, and its basically the gradient of your lyapunov functions mapped on to your dynamical system has to be negative.
Liapunov functions and stability in control theory request pdf. 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. This technique is discussed in detail in the separate web page method of lyapunov functions. Theory and applications of stochastic systems lecture 2 b60. Lyapunovbased safe policy optimization for continuous control. Liapunov functions and stability in control theory, second. The possible function definiteness is introduced which forms the building block of lyapunovs direct method. The case for less regular lyapunov functions strong stability is most often of interest when f arises from a perturbation of an ordinary uncontrolled di.
Lyapunov function and relate it to these v arious stabilit y notions. Lyapunovs second or direct method provides tools for studying asymp. Lyapunovs theory for characterizing and studying the stability of equilibrium points is presented for timeinvariant and timevarying systems modeled by ordinary differential equations. The possible function definiteness is introduced which forms the building block of lyapunov s direct method. Request pdf liapunov functions and stability in control theory this book presents a modern. In control theory, a control lyapunov function is a lyapunov function for a system with control inputs. His research interests include switched and hybrid systems, nonlinear control theory, control with limited information, and uncertain and stochastic systems. However, it is a remarkable fact that the converse of theorem 4. Lyapunov asymptotic stability overview of lyapunov. Hi, like jordan mentioned, wikipedia has a very good description about lyapunov stability.
The lyapunov function method is applied to study the stability of various differential equations and systems. Computation of lyapunov functions and stability of. 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. These tools will be used in the next section to analyze the stability. Dynamical system theory lies at the heart of mathematical sciences and engineering. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable.
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