Sprott b a computer sciences department, university of wisconsin, 15 university avenue, madison, wi 53706, united states. Researcharticle adaptive neural networks control using barrier lyapunov functions for dc motor system with timevarying state constraints leima1 anddapengli 2 collegeofscience,liaoninguniversityoftechnology,jinzhou,liaoning,china. To guarantee that state constraints always remain in the asymmetric timevarying constraint regions, the asymmetric timevarying barrier lyapunov function blf is employed to structure an adaptive nn controller. Based on the approximation property of cnn 3033, there exist ideal weights. Evaluating lyapunov exponent spectra with neural networks a. In extending the technique of lyapunov functions to control systems, a number of new issues arise. In the variable structure neural network, the number of basis functions can be either increased or decreased with time according to speci. In sk84 is shown that a quadratictype lyapunov function for a singularly per.
Results have also been extended to recurrent neural networks 5, 6. Lyapunov analysis of neural network stability in an. Study on tcpaqm network congestion with adaptive neural. Among the significant prior research works reported in the literature, we focus on methods that construct or approximate lyapunov function by neural networks. The asymptotic and practical stabilization for the affine in the control nonlinear systems, which extends the results of artstein, sontag, and tsinias is explored. There is a neighborhood that if youre staying within that one, this v function is positive definite about x r. Neural networks are now a subject of interest to professionals in many fields, and also a tool for many areas of. Lyapunov functions to caputo fractional neural networks with. Lyapunov function based on approximation theory and the abilities of artificial neural networks. There are two types of architectures of multilayer neural networks. Pdf neural lyapunov model predictive control semantic scholar.
The study is based on the application of the lyapunov method. However, realistic neural networks of cognitive processes and physiological rhythm regulations are asymmetrically connected. Such neural networks are static inputoutput mapping schemes that can approximate a continuous function to an arbitrary degree of accuracy. A lyapunov function in the hnn is set up from the velocity picking problem. Integral barrier lyapunov functionsbased neural control for. Lyapunov functions to caputo fractional neural networks. In order to guarantee the exponential stability of our considered recurrent neural networks, two distinct types of sufficient conditions are derived on the basis of the lyapunov functional and coefficient of our given system and also to construct a lyapunov function for a large scale system a novel graphtheoretic approach is considered, which. Hopfield neural network for seismic velocity picking.
This paper presents neural lyapunov mpc, an algorithm to alternately train a lyapunov neural network and a stabilising constrained model predictive controller mpc, given a neural network model of the system dynamics. We write to denote the parameter vector for a lyapunov function candidate v. The constraints are tackled by extending the control input as an extended state and introducing an integral barrier lyapunov function iblf to each step in a backstepping procedure. Assignments introduction to neural networks brain and. Jul 25, 2018 a new robust tracking control approach is proposed for strictfeedback nonlinear systems with state and input constraints. The intuitive picture is that of a scalar outputfunction, often thought. May 27, 2003 a lyapunov function for the neural network is constructed and used in presenting a formal mathematical proof that verifies the following claim. Lyapunov functional approach to stability analysis of riemann.
In this sense we apply in this paper the results of this valuable analysis tool on the dynamics of laterally inhibited networks. Usually quadratic lyapunov functions are used and it leads to a restriction to lipschitz activation functions. Lyapunov conditions in order to leverage the representational power of neural networks to approximate a lyapunov function, the learned function must satisfy two conditions. A barrier lyapunov functions blfsbased localized adaptive neural network nn control is proposed for a class of uncertain nonlinear systems with state and asymmetric control constraints to trac. The resulting neural networks are lyapunov functions on the basis of. Finitetime stability of fractionalorder neural networks with constant transmission delay and constant. State evaluation functions and lyapunov functions for. Neural networks with lyapunov functions as one of the simplest examples, we treat the case where f,s are linear functions.
In the double inverted pendulum model for humanoid robot balancing section 4, the. Theoretical framework gursel serpen, phd electrical engineering and computer science department the university of toledo toledo, oh 43606, usa abstract an artificial neural network is proposed as a function approximator for empirical modeling of a. Existence of control lyapunov functions and applications. A lyapunov theorybased neural network approach for face recognition. Define the neural network with random parameters for lyapunov function and initialize controllers parameters to the solution of. Learning stable deep dynamics models nips proceedings. The principal lyapunov stability results for such systems are presented.
Here, we construct a lyapunov function from the probabilistic potential landscape. The lyapunov function method is applied to study the stability of various differential equations and systems. We use the gradient descent method to decrease the lyapunov function. Recently, feedforward neural networks have been shown to obtain successful results in system identification and control. In this paper, we present a method to learn accurate safety certificates for nonlinear, closedloop dynamical systems. Please go through the document to explore more all the best, abhishek. Neural networks stability of hopfield network are the memories stable. Lyapunovbased constrained control design is still open. Neural networks with continuous local transition func tions have been recently used for a variety of applications, especially in learning tasks and combinatorial optimization. Roma, 56 53100 siena, italy received 31 august 2005. Search for a lyapunov function through empirical approximation by. These derivatives are applied to various types of neural networks. Generation of lyapunov functions by neural networks.
Results demonstrate that, when a neural network trained on short sequences is used for predictions, a onestep horizon neural lyapunov mpc can successfully reproduce the expert behaviour and. Further details on examples in section 3 example 1. Barrier lyapunov functionsbased localized adaptive neural. The proposed scheme is successfully implemented on the real time control of the tq ma3000 robotic manipulator. Neural lyapunov model predictive control where v netx is a lipschitz feedforward network that produces a n v n xmatrix. Multiple lyapunov functions for adaptive neural tracking control of switched nonlinear nonlowertriangular systems. Fractional order lyapunov stability theory was applied for various types of fractional neural networks using quadratic lyapunov functions see 2,1214. This paper proposes an adaptive neural network nn control approach for a directcurrent dc system with full state constraints. Consistent with the philosophy of viewing the neural network 1 as an intercon nection of n free subsystems 2, we think of the lyapunov function 4 as consisting of a weighted sum of lyapunov functions for each free subsystem 2 with uit 0. A perspective on graph theorybased stability analysis of. There is a lyapunov function, this lyapunov function has continuous partial derivatives, thats one of the requirements.
Then we apply the results for networks of mccullochpitts type model neurons to see when there can be lyapunov functions. In connection with the lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of lyapunov functions among caputo fractional delay differential equations. Quadratictype lyapunov functions for competitive neural networks 339 neurodynamical problems. This chapter presents a new face recognition system comprising of feature extraction and the lyapunov theorybased neural network. Application of lyapunov stability theory for model identi. Pdf construction of neural network based lyapunov functions. A lyapunov function for the neural network is constructed and used in presenting a formal mathematical proof that verifies the following claim. A modified lyapunov functional with application to stability of neutral. Study on tcpaqm network congestion with adaptive neural network and barrier lyapunov function. Prokhorov in 1 suggests a lyapunov machine, which is a specialdesign artificial neural network, for approximating lyapunov function. A hopfield network is a form of recurrent artificial neural network popularized by john hopfield in 1982, but described earlier by little in 1974.
Homeomorphism mapping based neural networks for finite time. Multiple lyapunov functions for adaptive neural tracking. Adaptive neural networks control using barrier lyapunov. Previous works have shown that lyapunov or energy functions could be derived for networks of binary elements, thus allowing a rather complete char. Pdf neural lyapunov model predictive control semantic. Dec 10, 2019 in order to guarantee the exponential stability of our considered recurrent neural networks, two distinct types of sufficient conditions are derived on the basis of the lyapunov functional and coefficient of our given system and also to construct a lyapunov function for a large scale system a novel graphtheoretic approach is considered, which. Hopfield nets serve as contentaddressable associative memory systems with binary threshold nodes.
So it is recent yet a unique and accurate method for face recognition. A lyapunov based stable online learning algorithm for. Periodically intermittent stabilization of delayed neural. This paper proposes two straightforward methods for determining or approximating a lyapunov function based on approximation theory and the abilities of artificial neural networks. Face recognition using neural networks free download as powerpoint presentation. Lyapunov function other training issues adaptation generalization ability summary. Introduction stability of nonlinear dynamic systems plays an important. We found the landscape topography is critical in determining the global stability and function. Robust lyapunov functions for complex reaction networks. Lyapunov function an overview sciencedirect topics. Generalized lyapunov approach for convergence of neural networks with discontinuous or nonlipschitz activations m. Stabilization of unknown nonlinear discretetime delay. The ability to model the behaviour of arbitrary dynamic system is one of the most useful properties of recurrent networks. For notational convenience, we write uto denote both the control function and the.
Lyapunov finite time stable theory is used to guarantee the closedloop system signals with prescribed performance in finite time. This extends recent works on lyapunov networks to be able to train solely from expert demonstrations of onestep transitions. Although it is true that polynomials, trigonometric series and orthogonal functions can also be used as function approximator, neural networks have been found to be particularly useful for controlling highly. Nonaffine purefeedback nonlinear systems finite time adaptive neural networks control is considered in section 3, and a new finite time adaptive law is developed for training neural networks. The nonlinear character of the activation functions used in neural networks has a great impact in the. Lyapunov functions and feedback in nonlinear control. Lyapunov theorybased fusion neural networks for the. Feedforward neural networks are expressive in that they can. Quantifying the main thermodynamical functions such as energy, entropy, and free energy is helpful for addressing global properties and functions of neural networks. Lyapunov function based neural networks for adaptive tracking.
State evaluation functions and lyapunov functions for neural networks. A lyapunov function for the neural network is constructed and used in. The potential of the proposed methods are demonstrated by simulation examples. This paper constructs a generalized lyapunov functional by introducing new terms into the wellknown. Pdf a straightforward method for the construction of lyapunov functions represented by neural networks is presented in this paper. Quadratictype lyapunov functions for competitive neural. Find materials for this course in the pages linked along the left. For instance, a lyapunov approach has been used for identi.
Stability analysis of nonlinear systems using lyapunov theory i. The falsifier takes the learned control function and the neural lyapunov function from the learner and checks whether there is a state vector violating the lyapunov conditions. Ali alradhawi david angeli abstract we present a framework to transform the problem of nding a lyapunov function for a complex reaction network crn in concentration coordinates with arbitrary monotone kinetics into nding a common lyapunov function for a. Porod2 department of electrical and computer engineering university of notre dame notre dame, in 46556 abstract in the present paper we survey and utilize results from the qualitative theory of large. Two cases of timevarying bounded delays are considered. Below, we restrict ourselves to the autonomous systems \\ \\mathbfx.
A straightforward method for the construction of lyapunov functions represented by neural networks is presented in this paper. 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. Generation of lyapunov functions by neural networks iaeng. Evaluating lyapunov exponent spectra with neural networks.
Generalized lyapunov approach for convergence of neural. The key to this is the design of a proper lyapunov function, based on input convex neural networks 1, which ensures global exponential stability to an. The results show the promise of neural networks and deep learning in improving. Finding, for a given supply rate, a valid storage function or at least proving that one exists is a major challenge in constructive analysis of nonlinear systems. 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. Promising experimental results show the effectiveness of the proposed. Its kind of like x squared is the bowl shape, right. We constructed nonequilibrium thermodynamics for the neural networks. Neural networks for regression statistical learning models that can learn a very diverse class of realvalued functions a potential energy function e could it be learned from labeled molecular data. Dynamic ridge polynomial neural network drpnn is a recurrent neural network used for time series forecasting.
The most com mon approach is based on considering a linearly parameterized subset of storage function. Professor sebastian seung problem set 4 due march 3 lyapunov functions march 1, 2005 in lecture, you were told that the stability of. Previous works have shown that lyapunov or energy functions could be derived for networks of binary elements, thus allowing a rather complete char acterization. A lyapunov theorybased neural network approach for face. Pdf lyapunov analysis of neural network stability in an adaptive.
In 7, authors designed a novel feedback controller to realize the. Lecture 33 stability analysis of nonlinear systems using lyapunov theory i dr. Lecture 12 basic lyapunov theory stanford university. Research on the stability of recurrent neural networks in the early days was for. We write to denote the parameter vector for a lyapunov. Index termsapproximation theory, lyapunov function, nonlinear system, neural network. Neural network potentials are statistical learning models.
Based on the gaussian radial basis function grbf variable neural network, an adaptive state feedback controller is. The corresponding feedback laws are smooth, except possibly at the equilibrium of the system. Sprott b a computer sciences department, university of wisconsin, 15 university avenue, madison, wi 53706, united states b physics department, university of wisconsin, 1150 university avenue, madison, wi 53706, united states article info article history. Lyapunov functional approach to stability analysis of. Control lyapunov functions for adaptive nonlinear stabilization. The hopfield neural network hnn is adopted for velocity picking in the timevelocity semblance image of seismic data.
This paper introduces a novel fusion neural architecture and the use of a novel lyapunov theorybased algorithm, for the online approximation of the dynamics of nonlinear systems. The resulting neural networks are lyapunov functions on the basis of which asymptotic stability or instability of a nonlinear systems equilibrium. Lyapunov function based neural networks for adaptive. They are guaranteed to converge to a local minimum and, therefore, may converge to a false pattern wrong local minimum rather than the stored. Sufficient conditions for the existence of control lyapunov functions are presented guaranteeing stabilization. The output of single layer neural network is given by. This paper is concerned with the stabilization problem of delayed neural networks via a periodically intermittent controller. Later, in the classical works of massera, barbashin and krasovskii, and kurzweil, this su. Stability results for neural networks nips proceedings.
Specifically, we construct a neural network lyapunov function and a training algorithm that adapts it to the shape of the largest safe region in the state space. Energy functions in neural networks with continuous local. While learning from a fixed input manifold, the neural network is selfstabilizing in a globally asymptotically stable manner. We propose new methods for learning control policies and neural network lyapunov functions for nonlinear control problems, with provable guarantee of stability. In the following section, we find more direct form of the first and secondorder conditions and corre sponding lyapunov functions for some specific fks.
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