Synchronization of reaction–diffusion Hopfield neural networks with s-delays through sliding mode control

Synchronization of reaction–diffusion Hopfield neural networks with s-delays via sliding mode control (SMC) is investigated in this paper. To begin with, the system is studied in an abstract Hilbert space C([–r; 0];U) rather than usual Euclid space Rn. Then we prove that the state vector of the drive system synchronizes to that of the response system on the switching surface, which relies on equivalent control. Furthermore, we prove that switching surface is the sliding mode area under SMC. Moreover, SMC controller can also force with any initial state to reach the switching surface within finite time, and the approximating time estimate is given explicitly. These criteria are easy to check and have less restrictions, so they can provide solid theoretical guidance for practical design in the future. Three different novel Lyapunov–Krasovskii functionals are used in corresponding proofs. Meanwhile, some inequalities such as Young inequality, Cauchy inequality, Poincaré inequality, Hanalay inequality are applied in these proofs. Finally, an example is given to illustrate the availability of our theoretical result, and the simulation is also carried out based on Runge–Kutta–Chebyshev method through Matlab

Projective synchronization analysis for BAM neural networks with time-varying delay via novel control

In this paper, the projective synchronization of BAM neural networks with time-varying delays is studied. Firstly, a type of novel adaptive controller is introduced for the considered neural networks, which can achieve projective synchronization. Then, based on the adaptive controller, some novel and useful conditions are obtained to ensure the projective synchronization of considered neural networks. To our knowledge, different from other forms of synchronization, projective synchronization is more suitable to clearly represent the nonlinear systems’ fragile nature. Besides, we solve the projective synchronization problem between two different chaotic BAM neural networks, while most of the existing works only concerned with the projective synchronization chaotic systems with the same topologies. Compared with the controllers in previous papers, the designed controllers in this paper do not require any activation functions during the application process. Finally, an example is provided to show the effectiveness of the theoretical results

Robust Control

The need to be tolerant to changes in the control systems or in the operational environment of systems subject to unknown disturbances has generated new control methods that are able to deal with the non-parametrized disturbances of systems, without adapting itself to the system uncertainty but rather providing stability in the presence of errors bound in a model. With this approach in mind and with the intention to exemplify robust control applications, this book includes selected chapters that describe models of H-infinity loop, robust stability and uncertainty, among others. Each robust control method and model discussed in this book is illustrated by a relevant example that serves as an overview of the theoretical and practical method in robust control

Synchronization of complex dynamical networks with fractional order

Complex dynamical networks (CDN) can be applied to many areas in real world, from medicine, biology, Internet to sociology. Study on CDNs has drawn great attention in recent years. Nodes in a CDN can be modelled as systems represented by differential equations. Study has shown that fractional order differential equations (DF) can better represent some real world systems than integer-order DFs. This research work focuses on synchronization in fractional CDNs.  A literature review on CDNs with fractional order has summarized the latest works in this area.  Fractional chaotic systems are studied in our initial investigation.  Fractional calculus is introduced and the relevant fundamentals to model, describe and analyse dynamical networks are presented. It is shown that the structure and topological characteristics of a network can have a big impact on its synchronizability. Synchronizability and its various interpretations in dynamical networks are studied. To synchronize a CDN efficiently, controllers are generally needed. Controller design is one of the main tasks in this research. Our first design is a new sliding mode control to synchronize a dynamical network with two nodes. Its stability has been proven and verified by simulations.  Its convergence speed outperforms Vaidyanathan's scheme, a well-recognized scheme in this area. The design can be generalized to CDNs with more nodes.  As many applications can be modelled as CDNs with node clustering, a different sliding mode control is designed for cluster synchronization of a CDN with fractional order. Its stability is proven by using Lyapunov method. Its convergence and efficiency is shown in a simulation. Besides these nonlinear methods mentioned, linear control is also studied intensively for the synchronization.  A novel linear method for synchronization of fractional CDNs using a new fractional Proportional-Integral (PI) pinning control is proposed.  Its stability is proven and the synchronization criteria are obtained. The criteria have been simplified using two corollaries so the right value for the variables can be easily assigned. The proposed method is compared with the conventional linear method which uses Proportional (P) controller. In the comparison, the mean squared error function is used. The function measures the average of the squared errors and it is an instant indicator of the synchronization efficiency. A numerical simulation is repeated 100 times to obtain the averages over these runs. Each simulation has different random initial values for both controllers. The average of the errors in all the 100 simulations is obtained and the area under the function curve is defined as an overall performance index (OPI), which indicates the controller's overall performance. In control, small overshoot is always desired. In our work, the error variation is also used as a measure.  The maximum variation from the average of 100 simulations is calculated and compared for both methods. With all the statistical comparisons, it is clear that with the same power consumption, the proposed method outperforms the conventional one and achieves faster and smoother synchronization. Communication constraints exist in most real world CDNs. Communication constraints and their impact on control and synchronization of CDNs with fractional order are investigated in our study. A new adaptive method for synchronizing fractional CDN with disturbance and uncertainty is designed. Its stability is proven and its synchronization criteria are obtained for both fractional CDN with known and unknown parameters. Random disturbance is also included in both cases. Our results show that the new method is efficient in synchronizing CDNs with presence of both disturbance and uncertainty

Optimized state feedback regulation of 3DOF helicopter system via extremum seeking

In this paper, an optimized state feedback regulation of a 3 degree of freedom (DOF) helicopter is designed via extremum seeking (ES) technique. Multi-parameter ES is applied to optimize the tracking performance via tuning State Vector Feedback with Integration of the Control Error (SVFBICE). Discrete multivariable version of ES is developed to minimize a cost function that measures the performance of the controller. The cost function is a function of the error between the actual and desired axis positions. The controller parameters are updated online as the optimization takes place. This method significantly decreases the time in obtaining optimal controller parameters. Simulations were conducted for the online optimization under both fixed and varying operating conditions. The results demonstrate the usefulness of using ES for preserving the maximum attainable performance

Robust hybrid synchronization control of chaotic 3-cell CNN with uncertain parameters using smooth super twisting algorithm

This paper presents the control design framework for the hybrid synchronization (HS) and parameter identification of the 3-Cell Cellular Neural Network. The cellular neural network (CNN) of this kind has increasing practical importance but due to its strong chaotic behavior and the presence of uncertain parameters make it difficult to design a smooth control framework. Sliding mode control (SMC) is very helpful for this kind of environment where the systems are nonlinear and have uncertain parameters and bounded disturbances. However, conventional SMC offers a dangerous chattering phenomenon, which is not acceptable in this scenario. To get chattering-free control, smooth higher-order SMC formulated on the smooth super twisting algorithm (SSTA) is proposed in this article. The stability of the sliding surface is ensured by the Lyapunov stability theory. The convergence of the error system to zero yields hybrid synchronization and the unknown parameters are computed adaptively. Finally, the results of the proposed control technique are compared with the adaptive integral sliding mode control (AISMC). Numerical simulation results validate the performance of the proposed algorithm