Lyapunov functions, in many cases it is almost impossible to get ones hands onto one such lyapunov function. Without restrictive assumptions found in previous approaches, a minmax optimization problem is formulated to solve for a quadratic lyapunov function. Structured and simultaneous lyapunov functions for system. Kellett and teel, on the robustness of klstability for difference inclusions.
R n is an equilibrium point of the system if fxe 0 xe is an equilibrium point xt xe is a trajectory suppose xe is an equilibrium point system is globally asymptotically stable g. Liapunov functions and stability in control theory springerlink. This is a paper in a series of investigations into the use of. It is shown that a pseudoboolean function in the proper form can play the role of lyapunov functions for bns, based on which several lyapunov based stability results are obtained. The core of the engineering theory shares many features with my. This may be discussed by the theory of aleksandr lyapunov. Related work lyapunov functions are convenient tools for stability i. Statebased lyapunov functions overview of lyapunov. Burton northwest research institute 732 caroline st. Lyapunov functions and feedback in nonlinear control. Control theory can provide useful insights into the properties of controlled.
Lyapunov functions and feedback in nonlinear control universite. Many physical laws are described in terms of such generally homogeneous functions. The paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Rninto rnand there is at least one equilibrium point x, that is f x 0. 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. 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. A proper form of pseudoboolean functions is found, and the concept of strict lyapunov functions is thus given. Nonmonotonic lyapunov functions for stability analysis 1569 the rest of this paper contains ve more sections. 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. These tools will be used in the next section to analyze the stability properties of a robot controller. This book introduces the basic principles of control theory in a concise selfstudy tutorial. In optimal control, stabilization and nonsmooth analysis, volume 301 of lecture notes in control and inform. Liapunov function and stability in control theory, springer, 2005. Computation of lyapunov functions and stability of.
In extending the technique of lyapunov functions to control systems, a number. Theory and applications of stochastic systems lecture 2 b60. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i. First, in the application of liapunovs direct method to problems of stability in delay di. The notion of exponential stability guarantees a minimal rate of decay, i. Lyapunov functions for nonlinear discretetime systems. Lyapunov introduced the special functions later named after him in the theory of stability the problem of finding or constructing them in the most general case has seemed desperately insurmountable.
Linearization methods and control of nonlinear systems monash university, australia. 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. 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. Liapunov functions and stability in control theory andrea. A transfer function of a linear system is defined as the ratio of the laplace transform of the output and the.
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. Classical stability theory is not applicable for systems described by discontinuous differential equations based on the local lipschitz assumption i. The ordinary lyapunov function is used to test whether a dynamical system is stable more restrictively, asymptotically stable. Modern control theory a historical perspective pdf. The use of vector lyapunov functions in control theory is not frequent. It must be kept in mind that a control engineers role is not merely one of designing control. An isolated equilibrium point x e is called locally stable in the sense of lyapunov if 80, 9 0 s. Lyapunov theory lyapunov theory is used to make conclusions about trajectories of a system x. 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.
Every proof ive read seems to mimic the proof for the continuous case. Stability analysis of nonlinear systems using lyapunov theory. Sometimes, mechanical methods are used to improve the stability of systems. The stability of such systems is a function of both the dynamics and the interconnection topology. In simple terms, if the solutions that start out near an equilibrium point stay near forever. But since the time, that is about a hundred years ago, when a. Nonlinear 2d systems, vector lyapunov functions, stability, iterative learning control. Lecture 12 basic lyapunov theory stanford university. We focus in particular on the problem of the existence of liapunov functions converse liapunov theorems and their regularity, whose interest is especially motivated by applications to automatic control. 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. This is lecture 7 in the series of nonlinear systems and control. The analysis includes systems where random failures modeled by a markov chain with a nite set of states can arise in an iterative learning control application. The system in 1 is autonomous time invariant if f fx.
I will assume knowledge of my autumn 2002 model theory lecture notes 1. Linearization methods and control of nonlinear systems. Linearization methods and control of nonlinear systems two. Xiii lyapunov design shuzhi ge encyclopedia of life support systems eolss function candidate v x, and ii evaluating its derivative v x along system dynamics 1 and checking its negativeness for stability analysis. Lecture notes stability theory math 414 spring 2003.
As an example, lets take a look at the proof given in this technical report, focusing on the underlined sentence note that there is a mistake there. Finally, we look at alternate feedback control laws and closed loop dynamics. Feedback control theory systems control group university of. Detecting new e ective families of lyapunov functions can be seen as a serious advance. Lyapunovlike characterization for the problem of inputtostate stability in the probability of nonautonomous stochastic control systems is established. In this case we say the lyapunov function vx xtpxestablishes stability of the di. In ran01, rantzer introduced a dual to the l yapuno v function, referred to by the author as a density function, to dene and study weak er notion of. A particular focus is on the problem of the existence of liapunov functions converse liapunov theorems and their regularity, whose.
The main theorems on stability based on nonmonotonic lyapunov function are presented in section 3. Classically, a clf enforces that the resulting closedloop state. Learning control lyapunov function to ensure stability of. F or nonlinear odes, two ideas ha ve appeared in recent literature tow ards ov ercoming this barrier. Lyapunov characterization for the stability of stochastic. Maintain stable gradient and phase in presence of field perturbations. In this paper, we use piecewise continuous vector lyapunov functions to study practical stability of the solutions of nonlinear impulsive functional differential equations. 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.
I havent been able to find a solid proof for the standard lyapunov theorem related to the stability of discrete systems. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. Note that all nine transfer functions above are affine functions of the free. Hence, the corresponding theory, known as the method of vector lyapunov functions, offers a very. Lyapunovbased stability and construction of lyapunov.
Vector lyapunov function based stability of a class of. Vector lyapunov functions for practical stability of. The lyapunov concept of stability from the standpoint of. We present a survey of the results that we shall need in the sequel, with no proofs. 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. This paper focuses on the design of control lyapunov function for control affine systems to guarantee the stability for the states of interest in a specified region. The most important type is that concerning the stability of solutions near to a point of equilibrium. In it the transfer function, also known as the system function or network function, is a. Control theory deals with the control of continuously operating dynamical systems in. Unesco eolss sample chapters control systems, robotics and automation vol. A central tool in systems theory for synthesizing control laws that achieve stability are control lyapunov functions clfs. This book presents a modern and selfcontained treatment of the liapunov method for stability analysis, in the framework of mathematical nonlinear control theory.
This lecture covers fundamentals of lyapunov theory, nonlinear systems and equilibrium points, concepts of stability. We then analyze and apply lyapunovs direct method to prove these stability properties, and develop a nonlinear 3axis attitude pointing control law using lyapunov theory. Haddad, senior member, ieee abstractvector lyapunov theory has been developed to weaken the hypothesis of standard lyapunov theory. We extend the wellknown artsteinsontag theorem to derive the necessary and sufficient conditions for the inputtostate stabilization of stochastic control systems.
For switched systems with generally homogeneous functions as nonlinear righthand sides, a method to construct the common lyapunov function clf from the lyapunov functions for partial systems is proposed. I am struggling to see how differentiation of the lyapunov function is carried out. In engineering and mathematics, control theory deals with the behaviour of. Stability notions and lyapunov functions for sliding mode. Nonlinear systems and control lecture 7 fundamentals of. In particular, for vx xtpxquadratic form, where pis a real symmetric matrix, vx is positive semide nite if and. Example of stability problem we consider the system x0 y x3. Lyapunov equation use of the lyapunov functions for control design by introducing unknown parameters in the operator a definition of the lyapunov function by expanding the nonlinear system using. General references for lyapunov functions in control include 2 and 14.
In control theory, a controllyapunov function is a lyapunov function for a system with control inputs. Stability of dynamical systems on a graph mohammad pirani, thilan costa and shreyas sundaram abstractwe study the stability of largescale discretetime dynamical systems that are composed of interconnected subsystems. In stability theory these functions as the widest class of the first approximation equations are used. 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. Optimal control theorywhich is playing an increasingly important role in the design of modern. Elements of filippov theory of differential equations with discontinuous righthand sides and its recent extensions are discussed. Common and multiple lyapunov functions in stability.
In 78, it is summarized that one can construct a scalar or vector lyapunov function for the whole system by imposing certain conditions on lyapunov functions for each free subsystem i. It is shown that a pseudoboolean function in the proper form can play the role of lyapunov functions for bns, based on which several lyapunovbased stability results are obtained. Jul 21, 2017 this is lecture 7 in the series of nonlinear systems and control. Control theory tutorial basic concepts illustrated by. Liapunov functions and stability in control theory. In this paper, we use piecewise continuous vector lyapunov functions to study practical stability of the solutions of nonlinear impulsive. More formally, suppose we are given an autonomous dynamical system. Stability notions from lyapunov stability 1982 to fixedtime stability 2012 are. 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.
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