On the scalability and convergence of simultaneous parameter identification and synchronization of dynamical systems
📝 Abstract
The synchronization of dynamical systems is a method that allows two systems to have identical state trajectories, appart from an error converging to zero. This method consists in an appropriate unidirectional coupling from one system (drive) to the other (response). This requires that the response system shares the same dynamical model with the drive. For the cases where the drive is unknown, Chen proposed in 2002 a method to adapt the response system such that synchronization is achieved, provided that (1) the response dynamical model is linear with a vector of parameters, and (2) there is a parameter vector that makes both system dynamics identical. However, this method has two limitations: first, it does not scale well for complex parametric models (e.g., if the number of parameters is greater than the state dimension), and second, the model parameters are not guaranteed to converge, namely as the synchronization error approaches zero. This paper presents an adaptation law addressing these two limitations. Stability and convergence proofs, using Lyapunov’s second method, support the proposed adaptation law. Finally, numerical simulations illustrate the advantages of the proposed method, namely showing cases where the Chen’s method fail, while the proposed one does not.
💡 Analysis
The synchronization of dynamical systems is a method that allows two systems to have identical state trajectories, appart from an error converging to zero. This method consists in an appropriate unidirectional coupling from one system (drive) to the other (response). This requires that the response system shares the same dynamical model with the drive. For the cases where the drive is unknown, Chen proposed in 2002 a method to adapt the response system such that synchronization is achieved, provided that (1) the response dynamical model is linear with a vector of parameters, and (2) there is a parameter vector that makes both system dynamics identical. However, this method has two limitations: first, it does not scale well for complex parametric models (e.g., if the number of parameters is greater than the state dimension), and second, the model parameters are not guaranteed to converge, namely as the synchronization error approaches zero. This paper presents an adaptation law addressing these two limitations. Stability and convergence proofs, using Lyapunov’s second method, support the proposed adaptation law. Finally, numerical simulations illustrate the advantages of the proposed method, namely showing cases where the Chen’s method fail, while the proposed one does not.
📄 Content
On the scalability and convergence of simultaneous parameter identification and synchronization of dynamical systems Bruno Nery Rodrigo Ventura Institute for Systems and Robotics Instituto Superior T´ecnico, Lisbon, Portugal Abstract The synchronization of dynamical systems is a method that allows two systems to have identical state trajectories, appart from an error converging to zero. This method consists in an appropriate unidirec- tional coupling from one system (drive) to the other (response). This requires that the response system shares the same dynamical model with the drive. For the cases where the drive is unknown, Chen proposed in 2002 a method to adapt the response system such that synchronization is achieved, provided that (1) the response dynamical model is linear with a vector of parameters, and (2) there is a param- eter vector that makes both system dynamics identical. However, this method has two limitations: first, it does not scale well for complex parametric models (e.g., if the number of parameters is greater than the state dimension), and second, the model parameters are not guar- anteed to converge, namely as the synchronization error approaches zero. This paper presents an adaptation law addressing these two limitations. Stability and convergence proofs, using Lyapunov’s sec- ond method, support the proposed adaptation law. Finally, numerical simulations illustrate the advantages of the proposed method, namely showing cases where the Chen’s method fail, while the proposed one does not. 1 Introduction Consider two identical continuous time dynamical systems, designated drive (D) and response (R). It is well known that the state evolution of each system, when taken separately, may differ radically if the initial condition for each system differ, namely in the case of chaotic dynamical systems [6, 4]. However, in the presence of a unidirectional coupling from the drive to the response system, synchronization of their state trajectories is known to occur [10, 5, 8]. In this paper we limit the discussion to the simplest coupling scheme, in which the response system receives the full state vector from the 1 arXiv:1108.1066v1 [cs.SY] 4 Aug 2011 drive. In this situation it is easy to design a controller that synchronizes both systems, using feedback linearization (Section 2). Such synchronization assumes that both drive and response have the same dynamical model. This paper addresses the problem of achieving syn- chronization of a response system, when the dynamical model of the drive is unknown. In particular, we target the problem of simultaneous adap- tation and synchronization of a response system, given an unknown drive. Two assumptions are made: (1) the response dynamical model depends lin- early on a parameter vector, and (2) there is a value for this vector that makes both systems identical. In 2002, Chen and L¨u proposed a method to simultaneously adapt this parameter vector and to make both systems synchronized [3]. Lyapunov second method was used to prove the feasibility of this method, however, due to the construction of the Lyapunov function employed, convergence of the response parameters is not guaranteed. This has two consequences that prevent the general usage of this method. Firstly, it does not scale in complexity: if the dimension of the parameter vector is greater than the dimension of the state vector, convergence is not guaran- teed. And secondly, even with a small number of parameters, Chen’s proof does not guarantee effective convergence of the parameters. In this paper we address both of these problems, presenting a convergence proof for the simultaneous synchronization and adaptation of the response to an arbitrary drive system. Moreover, numerical simulations comparing the proposed approach with Chen’s method illustrate the benefits of the approach. Chaotic synchronization was first introduced by Pecora and Carrol in 1990 [7]. Since then, many publications have deepend our knowledge about this concept [1, 5, 8, 2]. A method for synchronizing the R¨ossler and the Chen chaotic systems using active control was proposed by Agiza and Yassen [2], however the approach is specific to these particular systems. Chen and L¨u proposed a method to perform simultaneous identification and synchronization of chaotic systems [3], but the results show some limitations, which are discussed in length and addressed in this paper. The paper is structured a follows: section 2 states formally the problem, followed by the proposed solution in section 3; experimental results are presented in section 4, and section 5 concludes the paper. 2 Problem statement Consider two dynamical systems, called drive and response, with a unidi- rectional coupling between them. Throughout this paper we will assume that both drive and response systems are identical, apart from a parameter vector, which is unknown. The goal of the adaptation law is to determine 2 this parameter vector. Consider the drive system mod
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