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A novel series of damped multistep formulas of matrix coefficients are developed to find the solutions of dynamic problems with negligible stiff components or eigenmodes. They are derived from the concept of eigenmode and the development details are presented. A total of six multistep formulas is developed and they have an order of accuracy from 1 to 6 corresponding to the 1-step to 6-step multistep formula. Each formula has damping to suppress or remove the stiff components or eigenmodes that are of no importance or of no interest while the slow eigenmode solutions can be still accurately calculated. The 1-step and 2-step formulas of the series are L-stable. Besides, each formula can have a non-iterative solution procedure and hence it is of high computational efficiency in contrast to the corresponding conventional backward differentiation method. This series can give a better quality of numerical solutions for dynamic problems with negligible stiff components or eigenmodes when compared to the methods with no damping. Numerical tests reveal that this series performs well like the backward differentiation method and it can save many efforts in computing if a non-iterative solution procedure is used when compared to the backward differentiation method.

In this paper multilayer neural networks (MNNs) are used to control the balancing of a class of inverted pendulums. Unlike normal inverted pendulums, the pendulum discussed here has two degrees of rotational freedom and the base-point moves randomly in three-dimensional space. The goal is to apply control torques to keep the pendulum in a prescribed position in spite of the random movement at the base-point. Since the inclusion of the base-point motion leads to a non-autonomous dynamic system with time-varying parametric excitation, the design of the control system is a challenging task. A feedback control algorithm is proposed that utilizes a set of neural networks to compensate for the effect of the system's nonlinearities. The weight parameters of neural networks updated on-line, according to a learning algorithm that guarantees the Lyapunov stability of the control system. Furthermore, since the base-point movement is considered unmeasurable, a neural inverse model is employed to estimate it from only measured state variables. The estimate is then utilized within the main control algorithm to produce compensating control signals. The examination of the proposed control system, through simulations, demonstrates the promise of the methodology and exhibits positive aspects, which cannot be achieved by the previously developed techniques on the same problem. These aspects include fast, yet well-maintained damped responses with reasonable control torques and no requirement for knowledge of the model or the model parameters. The work presented here can benefit practical problems such as the study of stable locomotion of human upper body and bipedal robots.

In this work we present a data-driven modeling of the insulin dynamics in different in silico patients using a recurrent neural network with output feedback. The inputs for the identification is the rate of insulin (μU / dl / min) applied to the patient, and blood glucose concentration. The output is insulin concentration (μU / ml) present in the blood stream. Once completed the off-line modeling, this model could be used for on-line monitoring of the insulin concentration for a better treatment. The learning law of the recurrent neural network is inspired by adaptive observer theory, and proven to be convergent in the parameters and stable in the Lyapunov sense, even with only 13 samples available. Simulation results are shown to validate the presented modeling.

A novel cryptography method based on the Lorenz's attractor chaotic system is presented. The proposed algorithm is secure and fast, making it practical for general use. We introduce the chaotic operation mode, which provides an interaction among the password, message and a chaotic system. It ensures that the algorithm yields a secure codification, even if the nature of the chaotic system is known. The algorithm has been implemented in two versions: one sequential and slow and the other, parallel and fast. Our algorithm assures the integrity of the ciphertext (we know if it has been altered, which is not assured by traditional algorithms) and consequently its authenticity. Numerical experiments are presented, discussed and show the behavior of the method in terms of security and performance. The fast version of the algorithm has a performance comparable to AES, a popular cryptography program used commercially nowadays, but it is more secure, which makes it immediately suitable for general purpose cryptography applications. An internet page has been set up, which enables the readers to test the algorithm and also to try to break into the cipher.

This paper presents the analogue simulation of a nonlinear liquid level system composed by two tanks; the system is controlled using the methodology of exact linearization via state feedback by cellular neural networks (CNNs). The relevance of this manuscript is to show how a block diagram representing the analogue modeling and control of a nonlinear dynamical system, can be implemented and regulated by CNNs, whose cells may contain numerical values or arithmetic and control operations. In this way the dynamical system is modeled by a set of local-interacting elements without need of a central supervisor.

Numerical procedure plays a key role in tackling the solutions of nonlinear dynamical systems. With the advent of the age of big data and high-power computing, developing efficient and fast numerical algorithms is an urgent task. This paper extends the Lie derivative discretization algorithm to the nonautonomous nonlinear systems and investigates the numerical solutions of the systems. The periodic solutions of three different classical nonlinear systems are calculated, and the results are compared to those values calculated from the Runge–Kutta fourth-order algorithm, which demonstrated that the Lie derivative algorithm has the advantages of large time step and short computation time.

We show that Green function methods can be straightforwardly applied to nonlinear equations appearing as the leading order of a short time expansion. Higher-order corrections can be then computed giving a satisfactory agreement with numerical results. The relevance of these results relies on the possibility of fully exploiting a gradient expansion in both classical and quantum field theory granting the existence of a strong coupling expansion. Having a Green function in this regime in quantum field theory amounts to obtain the corresponding spectrum of the theory.

We investigate synchronization between two unidirectionally linearly coupled chaotic multifeedback Mackey–Glass systems and find the existence and stability conditions for complete synchronization. Numerical simulations fully support the theory. We also present generalization of the approach to the wider class of nonlinear systems.

In this paper, a class of uncertain nonlinear systems is investigated and a sliding mode control (SMC) design is presented. The method is proposed for uncertain systems with model uncertainties, nonlinear dynamics and external disturbances. Using nominal system and related bounds of uncertainties, a chattering alleviating scheme is also proposed, which can ensure the robust SMC law. The performance and the significance of the controlled system are investigated under variation in system parameters and also in presence of an external disturbance. The simulation results indicate that performance of the proposed controller is effective compared to the existing controllers.

This study presents an astrophysics-inspired transit search optimization (TSO) algorithm based on exoplanet search divided into five phases: galaxy phase, star phase, transit phase, neighbor phase and exploitation phase for effective parameter estimation of fractional Hammerstein control autoregressive (Fr-HCAR) systems. Various physical phenomena and real processes can be modeled with Fr-HCAR systems and estimating the Fr-HCAR parameters becomes a vital task. The mean-square error (MSE)-based criterion function is developed, and efficacy of the TSO for Fr-HCAR identification is deeply analyzed for different fractional orders, disturbance levels and degrees of freedom. The TSO remained accurate, convergent, robust and stable for all variations in Fr-HCAR but the accuracy level degrades a little bit for high disturbance and increased degrees of freedom. The reliability and trustworthiness of the TSO for Fr-HCAR identification are endorsed through statistical analyses conducted on sufficient autonomous executions of the scheme.

A novel methodology is proposed for the development of neural network models for complex engineering systems exhibiting nonlinearity. This method performs neural network modeling by first establishing some fundamental nonlinear functions from a priori engineering knowledge, which are then constructed and coded into appropriate chromosome representations. Given a suitable fitness function, using evolutionary approaches such as genetic algorithms, a population of chromosomes evolves for a certain number of generations to finally produce a neural network model best fitting the system data. The objective is to improve the transparency of the neural networks, i.e. to produce physically meaningful "white box" neural network model with better generalization performance. In this paper, the problem formulation, the neural network configuration, and the associated optimization software are discussed in detail. This methodology is then applied to a practical real-world system to illustrate its effectiveness.

A two-stage design method for artificial neural networks is presented. The first stage is an evolutionary RLS (recursive least squares) algorithm which determines the optimal configuration of the net based on the concept of optimal interpolation. During this stage, the members of a given sample set are processed sequentially and a small consistent subset, constituting what we call prototypes, is selected to form the building blocks of the net. The synaptic weights as well as the internal dimensions of the net are updated recursively as each new prototype is selected. The evolving net at each intermediate step is a modified version of the Optimal Interpolative (OI) net derived in a recent paper by one of the authors. The concept of an evolving network configuration is attractive since it does not require the prescription of a fixed configuration in order to learn the optimal synaptic weights. This can eventually lead to a network architecture which is only as complex as it needs to achieve a required interpolation function.

The second stage is for the fine adjustment of the synaptic weights of the network structure acquired during the first stage. This stage is a two-step iterative optimization procedure using the method of steepest descent. The initial values of the synaptic weights in the iterative search are obtained from the first stage. It is seen that they are indeed very close to the optimal values. Hence, fast convergence during the second stage is guaranteed.

When a CDMA signal is passed through an RF transmitter, nonlinear elements cause spectral regrowth which result in reduction of spectral efficiency. CDMA signal has a pseudo noise nature; hence its mathematical treatment is too complex to analysis. In this paper, first it will be shown how to simplify the complex mathematics of CDMA signal. Then, a deterministic signal replaces the CDMA signal and the system response to both of them is calculated. In this paper, for the first time, ACPR is explicitly calculated for both the CDMA and deterministic signals. ACPR is calculated in terms of the nonlinear system coefficients and input power, and therefore, can be used in design objectives. In addition, it will be shown that if input power of the deterministic signal is multiplied by (i.e., correction factor), ACPR error of these kinds of signals in -55 dBc is less than 2.2% for the system nonlinearity orders up to 13. This correction factor is obtained by both theoretical and simulation methods.

This paper proposes an adaptive design of nonlinear feedback controllers for a class of complex nonlinear applications that have ill-defined mathematical models due to the effects of uncertainties and external disturbances. The design is aimed at estimating the uncertain parameters of the system while using a feedforward-like technique to cancel the effect of disturbances and unwanted nonlinearities. This is being achieved using a combination of state feedback and Lyapunov-based techniques that can guarantee the asymptotic stability of the closed loop system. The controller is synthesized such that it will follow the performance of a reference model via satisfying a certain criterion. The control law is demonstrated to be easily achievable for applications that can be modeled by low-order dynamics, e.g., industrial processes (level, flow, pressure, etc.) and some automotive applications (active suspension). The key factor in the design is arriving at the best parameter update law that guarantees both stability and satisfactory transient performance. The application of the proposed controller is extended to higher-order systems via proposing a low-order nonlinear model that is capable of encapsulating the dominant dynamics of the system without using linearization techniques. Trade-offs between stability and performance are carefully studied along with comparisons with a nonmodel-based PID controller to highlight the superiority of the proposed design. A simulated nonlinear Duffing oscillator and a continuous stirred tank reactor are used to exemplify the suggested technique. Finally, a conclusion is submitted with comments regarding feasibility of the controller along with its advantages and limitations.

Experimental results of a SPICE-compatible macromodel to model the nonlinear behavior of second generation current conveyors (CCIIs) at low frequency are presented. The derived macromodel includes not only those real physical performance parameters more important for CCII like the dynamic range, slew rate, DC gain and gain–bandwidth product, but parasitic resistors and capacitors associated to the input and output terminals are also included. To validate the derived macromodel, a saturated nonlinear function series (SNFS) was built by using AD844AN commercially available active device configured as CCII and embedded in a chaotic system. After that, an experimentally generated chaotic signal was applied as excitation signal to SNFS built with AD844AN foundry-provided macromodel and the proposed herein. Our results indicate that the derived macromodel can be used in the time-domain for forecasting the behavior of nonlinear circuits without worsening the accuracy and at less time compared with the foundry-provided macromodel.

Many literatures have discussed fractional order memristor and memcapacitor-based chaotic oscillators but the entire oscillator model is considered to be of fractional order. My interest is to propose a nonlinear oscillator with considering only the memcapacitor element of fractional order. Hence, I propose a fractional order memcapacitor (FMC)-based novel chaotic oscillator. The complete mathematical model for the proposed oscillator is derived and presented in this paper. The dimensionless state equations are then analyzed by using the equilibrium points and their stability, Eigen values, Kaplan–Yorke dimensions and Lyapunov exponents. To understand the complete dynamical behavior, bifurcation graphs are obtained and presented. Finally, the proposed fractional memcapacitor oscillator is implemented by using the shelf components.

This paper is concerned with the global asymptotic stability (GAS) problem of fixed-point Lipschitz nonlinear digital filters employing 2’s complement overflow arithmetic. Nonlinear digital filtering finds immense applications in various fields such as adaptive systems and controllers, digital controllers and observers for nonlinear systems, realization of neural networks using digital hardware, controllers for feedback linearization, etc. Lipschitz nonlinear digital filter is considered in this paper as it is frequently employed in nonlinear digital filtering, state filtering, neural networks, feedback control, digital controllers, decision-taking systems and so on. Based on Lyapunov theory, the property of 2’s complement overflow arithmetic and Lipschitz condition associated with system nonlinearities, a new criterion for the suppression of overflow oscillations in 2’s complement state variable realizations of digital filters is established. Several examples along with simulation results are provided to highlight the utility of the criterion.

This paper presents novel and low computational complexity spline adaptive filtering (SAF) algorithms based on partial coefficients selection and adaption. They are addressed partial update SAF normalized least mean squares (PU-SAF-NLMS) algorithms. The PU-SAF-NLMS equations are established via the steepest gradient descent constraint to the proposed cost function. In the proposed algorithms, the filter coefficients are updated based on periodic, sequential, and selective rules, leading to reduction in computational complexity. Furthermore, the convergence speed of PU-SAF-NLMS algorithms is close to the conventional SAF-NLMS. Moreover, the convergence analysis of algorithms is studied. Ultimately, the performance of the presented algorithms is appraised through several experimental results.

This paper studies the single-input controllable nonlinear system and explores its invariant structure in terms of internal feedback loops. The invariant structure provides a useful framework for the analysis and design of nonlinear systems. An application of the invariant structure to controlling Lorenz chaos is presented.

This paper proposes a new iteration method for chaotifying and controlling dynamical systems. By applying this iteration method, the dimension of the given dynamical system can be reduced from to n to n-1. Moreover, the chaotified system is not necessarily Hurwitz stable originally. The iteration method is applied to three-dimensional systems for demonstration, for which a sufficient condition is obtained for chaotification. In addition, the iteration method can be used to control a class of chaotic systems. These results are illustrated via simulations on the Duffing oscillator and the Chen system.