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In our previous work, the authors presented an effective nonlinear proportional-integral-derivative (PID) controller by incorporating a nonlinear function. The proposed controller is based on a conventional PID control architecture, wherein a nonlinear gain is coupled in series with the integral action to scale the error. Three new tuning rules for processes represented as the first-order plus time delay (FOPTD) model were obtained by solving an optimization problem formulated to minimize three performance indices. The main feature of the proposed controller is that it preserves the numbers of tuning gains even though nonlinearity is introduced and remains easy implementation in real applications. However, due to the introduction of a nonlinear element, the stability problem of the proposed controller may be raised. This paper presents one sufficient condition in the frequency domain for the absolute stability of the nonlinear PID controller, based on circle stability theory. It is proved that the nonlinear gain used is in the sector [0, 1]. The condition of the linear block F(s) is derived for the overall feedback system to be stable. Checking the stability and the effectiveness and robustness of the feedback system for setpoint tracking are demonstrated through a set of simulation works on three processes with uncertainty.

Since transients can happen in just few tens of microseconds and the overvoltages can be multiple times the corresponding ratings, small time‐steps are required for the electro‐magnetic transient (EMT) simulation to accurately capture the transient behavior. The selection of time‐stepping scheme is related to the nonlinear element solution method, which provides certain variables for the time‐step control algorithm. The adaptive time‐stepping (ATS) process is determined by the properly selected discretization method and time‐step control scheme, and based on which of the specific equipment models are involved in the EMT simulation. Transmission lines and rotating machines that widely exist in power systems should be accurately modeled in real‐time EMT simulation for obtaining precise results for HIL applications. An ATS universal line model and universal machine model are presented for accurate EMT simulation of power systems in real‐time and faster‐than‐real‐time mode.

In this paper, a double-layer half-cycle generator (HCG) has been designed and built. The proposed HCG creates sinusoidal flow with 400 A peak and 5 µs pulse width which feeds RLC load with 100 kHz work frequency, within resonance mode. Previous HCGs have been designed based on saturable reactor (SR). It should be noted that SR is a nonlinear element, and it will be distanced from its desirable performance upon change of temperature and under impact; however, a double-layer HCG has been proposed with no such defects. In proposed HCG, thyristor has been used instead of SR which is controllable, and it has none of aforementioned problems. Also, when SR is eliminated, its reverse retrieval needs no minority carriers. This way, sweep-out circuit may be also eliminated. From among other advantages of proposed HCG, reduction in number of circuit layers may be referred to, comparing to previous HCG. This results in reduction in number of equipment which in turn optimizes proposed HCG, in terms of volume and manufacturing costs. Finally, proposed HCG has been simulated in Matlab, while it has been implemented practically. Comparing simulation results and laboratory sample confirms correctness of relationships, in analytical design of HCG.

The integrator is an important building block in circuit design and its transfer characteristic is inherently frequency-dependent. When it is implemented as an $RC$ op-amp circuit, a nonlinear element (the amplifier) is placed within a frequency-dependent feedback loop. Conventional techniques allow us to obtain the nonlinear error easily at certain frequencies only. Furthermore, when the integrator is periodically reset, the circuit is not in steady-state and conventional techniques assuming a steady-state behavior do not apply. It is not obvious how the actual error relates to the circuit parameters when the integrator is driven with a wideband signal. In this paper we present an approach to quickly estimate the normalized mean squared error (NMSE) as a function of the circuit and signal parameters. We show that, to first-order, the NMSE is proportional to the ratio of the signal bandwidth and the dominant pole of the integrator. For an integrator with reset, the signal builds up from zero and results in an additional, integration-time dependent NMSE reduction.

The memristor is a controllable nonlinear element that can easily generate chaotic signals. Currently, most researchers focus on the nonlinear properties of memristors, while the ability of memristor to control and adjust chaotic systems has rarely been addressed. Therefore, we design a memristor-based chaotic system to produce the upper-lower attractors by changing the polarity of the memristor. Dynamics phenomena, such as multi-stability, transient phenomena, sustained chaos, bi-stability and intermittent chaos, are observed in the system. Several fundamental dynamical behaviors, including the stability of equilibria, symmetry, and dissipation, are investigated. To prove the physical existence of the system, we give the PSPICE circuit simulation and the MCU (microcontroller unit) hardware implementation. Additionally, the theory of stochastic differential equations is applied to demonstrate the immunity of the system to zero-mean Gaussian noise. Based on the designed recursive back-stepping controller and the proposed chaotic system, we present a new method to detect weak multi-frequency signals embedded in Gaussian noise. Experimental results demonstrate that the proposed method is able to detect each frequency of a multi-frequency weakly periodic signal submerged in a Gaussian noise background, which is important for promoting the engineering application of the memristor.

For the nonlinear, time-variable and unpredictable environment (such as the manipulator workspace that has many random and moveable obstacles), a Hammerstein adaptive (HA) impedance controller based on a Hammerstein impedance model is proposed in this paper. The model consists of a static nonlinear element and a linear dynamic element, which can describe the nonlinearity of the environment. Then, an adaptive law is designed by a Lyapunov function, which can adjust the parameters of the static nonlinear element, so that the model has good adaptability and robustness for the environment. By introducing a parameters self-adjust model-free adaptive controller as the inner position controller, the HA impedance control system is built, which does not require the knowledge of the robot dynamics. In order to verify the force-control performance of the proposed control scheme, some simulations and experiments are done on a bionic wrist joint actuated by pneumatic muscles. Compared with the conventional impedance controller with constant parameters, experimental results demonstrate that HA impedance controller has better force tracking characteristic and can restrict the contact force effectively, no matter how the external environment changes, which increases the safety between robots and humans or operation objects.

В роботi представлена модифiкована хаотична схема Чуа, яка реалiзує хаотичну поведiнку. Приведена система диференцiйних рiвнянь, що описує дану схему при певних значеннях системних параметрiв та початкових умов, генерує хаотичний атрактор. Ця схема має простий нелiнiйний елемент, спроектований так, щоб мати кусково-лiнiйну характеристику, тобто комбiнацiю операцiйного пiдсилювача з двома дiодами, що ввiмкненi назустрiч один одному. Для реалiзацiї нелiнiйностi, для двох дiодiв не потрiбно окремого джерела живлення, достатньо лише одного двополярного живлення для операцiйного пiдсилювача. Приведено схему для дослiдження нелiнiйного елементу та результати комп’ютерного моделювання, тобто вольт-амперну характеристику (ВАХ) при певних значеннях номiналiв компонентiв схеми. Ця модифiкована схема Чуа, яка генерує хаотичний та контрольований атрактор з фiксованим перiодом, може використовуватися в сучасних системах передавання та приймання iнформацiї. Управлiння хаотичними коливаннями є досить важливим процесом в радiотехнiцi та телекомунiкацiях. Число видiлених окремих перiодичних (контрольованих) атракторiв може використовуватися як ключi для маскування iнформацiйного носiя. За допомогою програмного середовища MultiSim проведено схемотехнiчний аналiз i представлено результати моделювання нелiнiйного елемента та модифiкованої хаотичної схеми Чуа. Дослiджена поведiнка системи за допомогою чисельного моделювання, використовуючи вiдомi iнструменти нелiнiйної теорiї, такi як хаотичний атрактор i часовi розподiли хаотичних координат. Приведено опис порогового методу для здiйснення управлiння хаотичними коливаннями та представлено результати практичного застосування даного методу до модифiкованої хаотичної схеми Чуа. Практичними результатами є видiленi 2- та 3-перiоднi контрольованi орбiти iз хаотичного атрактора. Introduction. General scientific fields where can be used circuits that realize chaotic behavior and generate chaotic oscillations are presented.Methods for control of chaotic oscillations are also presented. For modelling, analysis and demonstrate results was selected MultiSim software environment. Modelling and Analysis of Non-Linear Element. This modified Chua’s circuit has a simple non-linear element, designed to have a piecewise-linear characteristic, that is, a combination of an opamp with two diodes that are mutually inline. For realization of nonlinearity, for two diodes do not need a separate power source, only one bipolar power source for the opamp is enough. The scheme for modelling of the nonlinear element and the results of computer simulation, i.e. the volt-ampere characteristic (VAC) at certain values of the components of the scheme’s nominal values, is presented. This modified Chua’s circuit, which generates a chaotic and controlled attractor with a fixed period, can be used in modern transmission and reception systems of information. Modeling and Analysis of the Modified Chaotic Chua’s Generator. System’s behavior is investigated through numerical simulations, by using well known tools of nonlinear theory, such as chaotic attractor and time distributions of the chaotic coordinates. Threshold Method for Control of Chaotic Oscillations. System of equations that realize chaotic oscillations of Chua’s circuit is presented. Using threshold method was practical realization of the control of chaotic attractor. This modified Chua’scircuit that generate a chaotic and controlled attractor with a fixed period can be used in modern systemstransmitting and receiving information. Number of periodic (controlled) attractor can be used as a keys formasking of information carrier. Conclusions. For the first time was used threshold method forcontrol of chaotic oscillations for modified Chua’schaotic generator. This modified Chua’s circuit thatgenerate a chaotic and controlled attractor with afixed period can be used in modern systems transmittingand receiving information. Number of periodic(controlled) attractor can be used as a keys for maskingof information carrier. В работе представлена модифицированная хаотичная схема Чуа, которая реализует хаотическое поведение. Эта схема имеет простой нелинейный элемент, спроектированный так, чтобы иметь кусочно-линейную характеристику. Эта модифицированная схема Чуа, которая генерирует хаотический и контролируемый аттрактор с фиксированным периодом, может использоваться в современных системах передачи и получения информации. Количество периодических (контролируемых) аттракторов может использоваться как ключи для маскировки информационного носителя. С помощью программной среды MultiSim проведен схемотехнический анализ и представлены результаты моделирования нелинейного элемента и модифицированной хаотической схемы Чуа. Исследовано поведение системы с помощью численного моделирования, используя известные инструменты нелинейной теории, такие как хаотичный аттрактор и временные распределения хаотических координат. Приведено описание порогового метода для осуществления управления хаотическими колебаниями и представлены результаты практического применения данного метода к модифицированной хаотической схеме Чуа. Практическими результатами являются выделенные 2- и 3-периодные контролируемые орбиты с хаотического аттрактора.

Recently there has been increased research interest in the study of the hybrid dynamical systems (Sun & Ge, 2005) and (Li et al., 2005). These systems involve the interaction of discrete and continuous dynamics. Continuous variables take the values from the set of real numbers and the discrete variables take the values from finite set of symbols. The hybrid systems have the behaviour of an analog dynamic system before certain abrupt structural or operating conditions are changed. The event driven dynamics in hybrid control systems can be described using different frameworks from discrete event systems (Cassandras & Lafortune, 2008) such as timed automata, max-plus algebra or Petry nets. For dynamic systems whose component are dominantly discrete event, main tools for analysis and design are representation theory, supervisory control, computer simulation and verification. From the clasical control theory point of view, hybrid systems may be considered as a switching control between analog feedback loops. Generally, hybrid systems can achieve better performance then non-switching controllers because they can to reconfigure and reorganize their structures. For that is necessery correct coordination of discrete and analog control variables. The mathematical model for real process, generally, has the Hammerstein-Wiener form (Crama & Atkins, 2001) and (Zhao & Chen, 2006). It means that on the input and output of the process are present nonlinear elements (actuator and sensor). Here we will consider Hammerstein model which has the input saturation as nonlinear element. That is the most frequent nonlinearity encountered in practice (Hippe, 2006). Also, unmodeled dynamics with matching condition is present. As a control strategy will be used switching control. The switched systems can be viewed as higher abstraction of hybrid systems. The design of switching controllers having guaranted stability, known as the picewise linear LQ control (PLC), is first considered in (Wredenhagen & Belanger, 1994). The picewise linear systems are systems that have different linear dynamics in different regions of the continuous state space (Johansson, 2003). The PLC control has the associated switching surfaces in form of positively invariant sets and yields a relatively low-gain controller. In the LHG (low-and-high gain) design a low gain feedback law is first designed in such a way

The integrating process with time delay (IPTD) is a fundamentally unstable open-loop system due to poles at the origin of the transfer function, and designing controllers with satisfactory control performance is very difficult because of the associated time delay, which is a nonlinear element. Therefore, this study focuses on the design of an intelligent proportional-integral-derivative (PID) controller to improve the regulatory response performance to disturbance in an IPTD, and addresses problems related to optimally tuning each parameter of the controller with a real coded genetic algorithm (RCGA). Each gain of the nonlinear PID (NPID) controller consists of a product of the gains of the linear PID controller and a simple nonlinear function. Each of these nonlinear functions changes the gains in the controller to on line by nonlinearly scaling the error signal. A lead-lag compensator or first-order filter is also added to the controller to mitigate noise, which is a disadvantage of ideal derivative action. The parameters in the controller are optimally tuned by minimizing the integral of time-weighted absolute error (ITAE) using a RCGA. The proposed method is compared with three other methods through simulation to verify its effectiveness.

In this paper we report the control and synchronisation of chaos in a memristive Murali–Lakshmanan–Chua (MLC) circuit. This circuit, introduced by the present authors in 2013, is basically a non-smooth system having two discontinuity boundaries by virtue of it having a flux-controlled active memristor as its nonlinear element. While the control of chaos has been effected using state feedback techniques, the concept of adaptive synchronisation and observer-based approaches have been used to effect synchronisation of chaos. Both these techniques are based on state space representation theory which is well known in the field of control engineering. As in our earlier works on this circuit, we have derived the Poincare discontinuity mapping (PDM) and zero time discontinuity mapping (ZDM) corrections, both of which are essential for realising the true dynamics of non-smooth systems. Further, we have constructed the observer- and controller-based canonical forms of the state-space representations, have set up the Luenberger observer, derived the controller gain vector to implement state feedback control and calculated the gain matrices for switch feed back and finally performed parameter estimation for effecting observer-based adaptive synchronisation. Our results obtained by numerical simulation include time plots, phase portraits, estimation of the parameters and convergence of error graphs and phase plots showing complete synchronisation.

This paper addresses a phenomenon caused by resetting only one of the two states of a so-called second order 'Constant in gain Lead in phase' (CgLp) element. CgLp is a recently introduced reset-based nonlinear element, bound to circumvent the well-known linear control limitation - the waterbed effect. The ideal behaviour of such a filter in the frequency domain is unity gain while providing a phase lead for a broad range of frequencies, which clearly violates the linear Bode's gain phase relationship. However, CgLp's ideal behaviour is based on a describing function, which is a first order approximation that neglects the effects of higher order harmonics in the output of the filter. Consequently, achieving the ideal behaviour is challenging when higher order harmonics are relatively large. It is shown in this paper that by resetting only one of the two states of a second order CgLp, the nonlinear filter will act as a linear one at a certain frequency, provided that some conditions are met. This phenomenon can be used to the benefit of reducing higher order harmonics of CgLp's output and achieving the ideal behaviour and thus better performance in terms of precision.

Summarization: The classical Chua’s circuit that realizes chaotic behavior is presented. This circuit having a simple nonlinear element designed to be accurately piecewise-linear modelled. The circuit was modelled by using MultiSim software environment. The system’s behavior is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as chaotic attractor and time distributions of the chaotic coordinates. Using threshold method was practical realization of the control of chaotic attractor. This classical Chua’s circuit that generates a chaotic and controlled attractor with a fixed period can be used in modern systems transmitting and receiving information. Number of periodic (controlled) attractor can be used as a key for masking of information carrier. Παρουσιάστηκε στο: 12th Chaotic Modeling and Simulation International Conference

This paper presents a practical implementation of a memristor based Duffing oscillator. We replace a diode based nonlinear element by a simple flux-controlled memristor in an existing circuit early proposed.We model the current-voltage characteristic and show experimentally that our circuit presents, for the same forcing amplitude, a route to chaos on the falling branch of the hysteresis loop, while its dynamics remains regular on the raising branch. This observation clearly highlights the memory effect of the memristor.

This paper revisits the stability analysis problem for feedback interconnections comprising a multivariable linear-time-invariant system and a static slope-restricted nonlinear element. The analysis is based on Zames-Falb multipliers but incorporates aspects of external positivity theory to improve the computational efficiency of the multiplier search. An appealing aspect of the work is that few user choices are required to define the stability conditions, making the algorithm straightforward to use. Some numerical examples taken from the literature illustrate the effectiveness of the approach and its competitiveness with the state-of-the-art.

The small parameter method, implemented in frequency domain, was used for the analytical analysis of nonlinear electrical circuits of electrotechnical systems. To have the possibility of calculations in frequency domain, the automated method of forming orthogonal harmonic components of electrical magnitudes based on the algorithm of discrete convolution was used. The characteristic of a nonlinear element was represented by the polynomial function of third degree. It was shown that application of the small parameter method with its realization in frequency domain makes it possible to simplify the process of analyzing electrical circuits with nonlinear elements by automating calculations in the mathematical package. Analytical and numerical calculations of a circuit with actively inductive load demonstrated sufficient accuracy of the proposed method, the relative error for the main harmonic of current did not exceed 6 %. The conducted comparative analysis of the proposed small parameter method with the classic small parameter method on the example of calculation of an electrical circuit with RL load showed that the developed method provides better adequacy of results and high calculation accuracy in comparison with the existing one. Relative error by amplitude of first and third harmonics of current does not exceed 2.5 %, and by phase, it does not exceed 1.042·10 -3 %. The method of numerical structural modeling was used to determine the reference values of current of the researched circuit. The results of the research can be used in calculations of electrotechnical devices, containing semiconductor components and electrical devices with nonlinear characteristics. In addition, the obtained results will make it possible to improve the processes of active compensation of harmonics of current in electrical networks with nonlinear load and to develop the tools of passive compensation

Structural decoupling problem, i.e. predicting dynamic behavior of a particular substructure from the knowledge of the dynamics of the coupled structure and the other substructure, has been well investigated for three decades and led to several decoupling methods. In spite of the inherent nonlinearities in a structural system in various forms such as clearances, friction and nonlinear stiffness, all decoupling studies are for linear systems. In this study, decoupling problem for nonlinear systems is addressed for the first time. A method, named as FRF Decoupling Method for Nonlinear Systems (FDM−NS), is proposed for calculating FRFs of a substructure decoupled from a coupled nonlinear structure where nonlinearity can be modeled as a single nonlinear element. Depending on where nonlinear element is, i.e., either in the known or unknown subsystem, or at the connection point, the formulation differs. The method requires relative displacement information between two end points of the nonlinear element, in addition to point and transfer FRFs at some points of the known subsystem. However, it is not necessary to excite the system from the unknown subsystem even when the nonlinear element is in that subsystem. The validation of FDM−NS is demonstrated with two different case studies using nonlinear lumped parameter systems. Finally, a nonlinear experimental test structure is used in order to show the real-life application and accuracy of FDM−NS.

The paper presents a hybrid time-frequency model, which is a combination of two types of models are typically used in a modern computer simulation of power systems: time domain models and frequency domain models. To simulate the nonlinear part of the system a Simulink time domain model of each nonlinear element is used, while the simulation of the linear part of the system is performed using a frequency domain model of the system applied in PCFLO. A well-designed programming interface allows seamless data exchange between the two environments and provides control over the simulation process. It is shown how the hybrid model compares to the time and frequency domain models using a 20 bus power supplying system with nonlinear loads (6-pulse rectifies). The comparison allows to examine the convergence and efficiency of the developed hybrid model and determine the directions for its further improvement.

In this work, a Proportional-Integral-Derivative (PID) controller based two-degree-of-freedom control scheme for Depth of Hypnosis (DoH) in general anesthesia is proposed. This approach uses the Bispectral Index Scale (BIS) as a controlled variable and propofol administration as a control variable. The developed structure applies a new compensation scheme, which reduces the influence of the nonlinear element. In this context, we exploit the linear part of the patient model, that can be obtained from the demographics of each individual patient. The parameters are tuned using the optimization procedure based on a genetic algorithm. The evaluation of the proposed technique is performed using intra-patient variability with a Monte Carlo method. Additionally, the performance of the analyzed system has been verified using several indexes. The simulation results show that desired characteristics are obtained for both induction and maintenance phases.

Non-parametric tuning was introduces by J. Ziegler and N. Nichols with the step test and continuous cycling tests and the respective tuning rules. Whereas the step test based tuning had some feature of parametric tuning, the continuous cycling test based tuning was purely non-parametric. The latter test involves bringing the system under proportional control into a sustained oscillation and measuring the frequency of these oscillations and fixing the gain value, that are called the ultimate frequency and ultimate gain. The test was intended for manual tuning of PID controllers. It is relatively hard to automate, and it is not used in auto-tuners. Another test, called relay feedback test (RFT), was proposed by K. Astrom and T. Hagglund to excite self-oscillations through the inclusion of a nonlinear element (relay) in the control loop. This test is easier to automate, and it found applications in industrial auto-tuners. In both these tests ultimate frequency coincides with the phase cross-over frequency of the process/plant. Yet, both the original continuous cycling test and the RFT suffer from such a drawback as the change of the phase cross-over frequency after PID controller introduction. A new test, the modified relay feedback test (MRFT), which can ensure that the test oscillations are excited at he frequency that becomes the phase cross-over frequency after introduction of a controller in the loop, was recently introduced. This property is attained through coordinated test and tuning, which is considered in the book chapter. Moreover, the tuning rules are optimized for a selected class of processes, which offers the paradigm of optimal non-parametric tuning.

The passive suppression mechanism of the vortex-induced vibrations (VIV) of the cylinder by means of an essentially nonlinear element, the nonlinear energy sink (NES) is investigated. The flow-induced loads on the cylinder are modeled using a prevalent van der Pol oscillator which is experimentally validated, coupling to the structural vibrations in the presence of the NES structure. Based on the coupled nonlinear governing equations of motion, the performed analysis indicates that the mass and damping of NES have significant effects on the coupled frequency and damping of the aero-elastic system, leading to the shift of synchronization region and mitigation of vibration responses. It is demonstrated that the coupled system of flow-cylinder-NES behaves resonant interactions, showing periodic, aperiodic, and multiple stable responses which depend on the values of the NES parameters. In addition, it is found that the occurrence of multiple stable responses can enhance the nonlinear energy pumping effect, resulting in the increment of transferring energy from the flow via the cylinder to the NES, which is related to the essential nonlinearity of the sink stiffness. This results in a significant reduction in the VIV amplitudes of the primary circular cylinder for appropriate NES parameter values.

Hysteresis is a cornerstone of threshold based switching circuits. The idea of pinched hysteresis has taken on a new relevance with the advent of the memory resistor. There are models in literature that employ nonlinear elements paired with passive phase shift to demonstrate pinched hysteresis. This work analyzes the suitability of passive models for demonstrating hysteresis with a representative passive circuit candidate. The computational model employed in this study analyzes lobe symmetry, lobe area, response at zero stimulus, linearity etc. to arrive at its conclusions. The computed results are validated against SPICE simulations with good agreement. This study shows that pinched hysteresis emulators implemented with passive, nonlinear element based phase shift may seem superficially satisfactory but fail analytical inspection.

Identification of the dynamic properties of the linear hydraulic actuator using the self-excited oscillations is the main topic of the paper. The linear hydraulic actuators which consist of the hydraulic cylinder controlled by the flow control valve using the closed-loop control system with the position feedback have a lot of applications in the industry. The dynamic behaviour of hydraulic actuators depends on the dimensioning of the cylinder and control valve, working conditions such as system pressure, moving mass etc. Although the hydraulic drives are non-linear systems for the controller design the linearized models are used and the linear hydraulic cylinder is described by the second order system characterized by the low damping and eigenfrequency depending on the cylinder parameters. Experimental identification of the eigenfrequency and damping ratio of the actuator can be done by the simple experiment using the nonlinear element in the feedback and evaluation of the self-excited oscillations. The identification method, its application on the linear hydraulic actuator and results from the experiment on the laboratory test rig are introduced in the paper.

Maintaining balance and recovering from unexpected thrust is an important performance for bipedal robots. In this paper, we use the simplified biped robot model to solve the ZMP point, the variable height inverted pendulum model and sliding mode controller to solve the CoM point. Then, a nonlinear element is designed to describe the stability criterion of robot attitude. Finally, the whole process of push recovery is simulated by computer and the results verify the effectiveness and rapidity of the proposed method.

In the present paper, a new memristor based oscillator is obtained from the autonomous Jerk circuit [Kengne et al., Nonlinear Dynamics (2016) 83: 751765] by substituting the nonlinear element of the original circuit with a first order memristive diode bridge. The model is described by a continuous time four-dimensional autonomous system with smooth nonlinearities. Various nonlinear analysis tools such as phase portraits, time series, bifurcation diagrams, Poincare section and the spectrum of Lyapunov exponents are exploited to characterize different scenarios to chaos in the novel circuit. It is found that the system experiences period doubling and crisis routes to chaos. One of the major results of this work is the finding of a window in the parameters’ space in which the circuit develops hysteretic behaviors characterized by the coexistence of four different (periodic and chaotic) attractors for the same values of the system parameters. Basins of attractions of various coexisting attractors are plotted showing complex basin boundaries. As far as the authors’ knowledge goes, the novel memristive jerk circuit represents one of the simplest electrical circuits (no analog multiplier chip is involved) capable of four disconnected coexisting attractors reported to date. Both PSpice simulations of the nonlinear dynamics of the oscillator and laboratory experimental measurements are carried out to validate the theoretical analysis.

An electronic model of Duffing oscillator with a characteristic memristive nonlinear element is proposed instead of the classical cubic nonlinearity. The memristive Duffing oscillator circuit system is mathematically modeled, and the stability analysis presents the evolution of the proposed system. The dynamical behavior of this circuit is investigated through numerical simulations, statistical analysis, and real-time hardware experiments, which have been carried out under the external periodic force. The chaotic dynamics of the circuit is studied by means of phase diagram. It is found that the proposed circuit system shows complex behaviors, like bifurcations and chaos, three tori, transient chaos, and intermittency for a certain range of circuit parameters. The observed phenomena and scenario are illustrated in detail through experimental and numerical studies of memristive Duffing oscillator circuit. The existence of regular and chaotic behaviors is also verified by using 0–1 test measurements. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio. The numerically observed results are confirmed from the laboratory experiment.

In this paper, a novel type of network called network sharing common nonlinearity comprising both autonomous and non-autonomous oscillators have been investigated. We propose that these networks are robust for operating at desired modes i.e., chaotic or periodic by altering the v–i characteristics of common nonlinear element alone. The dynamics of these networks were examined through numerical, analytical, experimental and Multisim simulations.

An inverse solution technique with the ability of obtaining complete steady-state primary harmonic responses of local nonlinear structures in the frequency domain is proposed in the present paper. In this method, the nonlinear dynamic equations of motion is first condensed from many to only one algebraic amplitude–frequency equation of relative motion. Then this equation is transformed into a polynomial form, and with its frequency as the unknown variable, the polynomial equation is solved by tracing all the solutions of frequency with the increase of amplitude. With this solution technique, some complicated dynamic behaviors such as sharp tuning, anomalous jumps, breaks in responses and detached resonance curves could be obtained. The proposed method is demonstrated and validated through a finite element beam under force excitations and a lumped parameter model with a local nonlinear element under base excitations. The phenomenon of detached resonance curves in the frequency response and its coupling effects with multiple linear modes in the latter example are observed.

We have constructed a simple second-order dissipative nonautonomous circuit exhibiting ordered and chaotic behaviour. This circuit is the well known Murali–Lakshmanan–Chua(MLC) circuit but with diode based nonlinear element. For chosen circuit parameters this circuit admits familiar MLC type attractor and also Duffing–van der Pol circuit type chaotic attractors. It is interesting to note that depending upon the circuit parameters the circuit shows both period doubling route to chaos and quasiperiodic route to chaos. In our study we have constructed two-parameter bifurcation diagrams in the forcing amplitude–frequency plane, one parameter bifurcation diagrams, Lyapunov exponents, 0–1 test and phase portrait. The performance of the circuit is investigated by means of laboratory experiments, numerical integration of appropriate mathematical model and explicit analytic studies.

A novel identification method of neuro-fuzzy based MIMO Hammerstein model by using the correlation analysis method is presented in this paper. A special test signal that contains independent separable signals and uniformly random multi-step signal is adopted to identify the MIMO Hammerstein process, resulting in the identification problem of the linear model separated from that of nonlinear part. As a result, the identification of the dynamic linear element can be separated from the static nonlinear element without any redundant adjustable parameters. Moreover, it can circumvent the problem of initialization and convergence of the model parameters discussed in the existing iterative algorithms used for identification of MIMO Hammerstein model. Examples are used to illustrate the effectiveness of the proposed method.

Synchronization of large acquisition bandwidth brings great challenges to the traditional second-order phase-locked loop (PLL). To address the contradiction between acquisition bandwidth and noise suppression capability of the traditional PLL, a new second-order PLL coupled with a nonlinear element is proposed. The proposed nonlinear second-order PLL regulates the loop noise bandwidth adaptively by the nonlinear module. When a large input–output phase error occurs, this PLL reduces the frequency offset quickly by taking advantage of the large bandwidth. When the phase error is reduced by the loop control, the proposed PLL suppresses noises by using the small bandwidth to increase the tracking accuracy. Simulation results demonstrate that the tracking speed of the proposed PLL is increased considerably, and its acquisition bandwidth is increased to 18.8 kHz compared with that of the traditional second-order PLL (4 kHz).

This paper presents the used of dither signal injection technique to stabilize the effect of nonlinearities in feedback control system. Various form of dither has been proposed including square wave, sine wave, and triangle wave dither. The analytical study of modified nonlinear element concept and simulation results show that square wave dither required the minimum amplitude to eliminate the ‘hunt’ (self-oscillation) or limit cycles but the triangle wave dither signal has the best ability to eliminate the limit cycle because its actual of output signal is closest to the reference signal and smoothest than others.

To develop a robot system for collaboration with humans, robots are required to perform accurate external force detection and flexible motion from the prespective of safety. Electro-hydrostatic actuators (EHAs) can achieve high backdrivability mechanically. However, in a hydraulic system, torque efficiency decreases due to nonlinear elements, including friction and the internal leakage of hydraulic fluid, which results in lower backdrivability. In other words, in EHAs that can achieve high backdrivability mechanically, further improvement of backdrivability can be achieved by performing force control through nonlinear element compensation. Therefore, herein, we modeled the torque efficiency and attempted to improve the force control performance by model-based pressure control.

This work presents a novel 'Constant in gain Lead in phase' (CgLp) element using nonlinear reset technique. PID is the industrial workhorse even to this day in high-tech precision positioning applications. However, Bode's gain phase relationship and waterbed effect fundamentally limit performance of PID and other linear controllers. This paper presents CgLp as a controlled nonlinear element which can be introduced within the framework of PID allowing for wide applicability and overcoming linear control limitations. Design of CgLp with generalized first order reset element (GFORE) and generalized second order reset element (GSORE) (introduced in this work) is presented using describing function analysis. A more detailed analysis of reset elements in frequency domain compared to existing literature is first carried out for this purpose. Finally, CgLp is integrated with PID and tested on one of the DOFs of a planar precision positioning stage. Performance improvement is shown in terms of tracking, steady-state precision and bandwidth.

This paper deals with common-mode (CM) electromagnetic interference (EMI) in an inverter-fed motor drive system. A simplified CM equivalent circuit including π-type equivalent circuit for the motor winding is proposed by using the waveform of the common-mode current flowing in the ground conductor. Moreover, a parameter identification method applying the computer-aided software modeFRONTIER is proposed to decide objectively the circuit parameters of the proposed simplified equivalent circuit. Validity and effectiveness of the proposed equivalent circuit are confirmed by comparing the measured impedance characteristics with the calculated ones. Finally, the possibility of a nonlinear phenomenon is discussed in this system. It is shown that the accuracy of simulation result can be improved by introducing the nonlinear element. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A variety of methods have been developed to identify the presence of stiction in linear closed-loop systems. One of the commonly used approaches is based on identification of a Hammerstein model (linear dynamic model preceded by static nonlinear element) between the controller output and process output. These techniques utilize the fact that control valve stiction introduces nonlinearities in the otherwise linear feedback system. However, the present work shows that these techniques could provide ambiguous results depending on the frequency response of the controller and the process of interest. Therefore, in this work, a reliability measure to validate the results from Hammerstein model-based stiction detection approaches is proposed. This measure of reliability is important from an industrial perspective because (i) it helps in reducing false alarms and (ii) it improves the computational speed by guiding the selection of search space in the linear model identification step. The applicability of this reliability measure in industrial setting is demonstrated using various simulation and industrial case studies. � 2015 American Chemical Society.

The paper deals with a crucial task of parametric optimization of an adaptive inertial sensor capable of varying the parameters in order to reduce errors which are caused by the influence of high-frequency transient disturbances. A method of parametric optimization of each component that makes it possible to vary the parameters of the nonlinear element of a compensating inertial sensor operating in the self-oscillation mode is developed. The method was tested with a pendulous accelerometer taken as an example. Numerical investigations confirmed the possibility of filtering external vibrations and internal noise of the inertial sensor while maintaining its sensitivity in the required measuring range due to the synthesis of the adaptive loop based on a non-linear element of the "hysteresis loop" type. The essence of the method is based on parametric synthesis of the adaptive loop with a nonlinear element. Numerical investigations show that the error of the pendulous accelerometer can be reduced, on the average, by an order of magnitude thanks to the adaptive parameter setting of the non-linear element, depending on the amplitude of the external vibrations. The above method of parametric optimization of each component can be applied to most compensating inertial sensors measuring motion parameters of spacecraft operating in conditions of non-stationary external and internal disturbances. The adaptation circuit in adaptive inertial sensors can be implemented through the use of modern microcontrollers.

Summary In this paper, two methods are proposed for investigating stability of fractional-order systems under saturated linear feedback. The first stability condition can be used for fractional-order linear systems with nonlinear element that is Lipschitz in state. The second stability condition has been achieved by exploring the special property of saturation using an auxiliary feedback matrix. For comparison of the two analyses, domains of attraction are applied using the proposed optimizations whose conditions can be expressed as Linear Matrix Inequalities in terms of all the varying parameters hence being appropriate for controller synthesis. Furthermore, the stability condition is achieved with regard to persistence disturbance. It is employed for determining the invariant sets of the given system by the proposed optimization problems. The results of the proposed analysis and methods are illustrated by three examples. Copyright © 2015 John Wiley & Sons, Ltd.

In this study, a new method which provides closed form expressions for the frequency domain solutions of multi-degree of freedom systems is introduced. The method is applicable for a multi-degree of freedom system with a single weak nonlinear element. Using this method a multi-degree of freedom system with a single nonlinear element can be put into a form such that solution of the linear part of the system along with the solution of a single nonlinear equation gives the nonlinear system response. The analytic solution of this nonlinear scalar equation is provided for cubic stiffness and Coulomb damping cases. The proposed approach is implemented on case studies of systems with cubic stiffness and dry friction elements. The results show that the method is accurate and completely eliminates the need for an iterative solution; hence, significant time savings can be obtained for large systems. The lack of need for iteration makes the method ideal for large finite element models with a single nonlinear element. © 2015 Elsevier Ltd. All rights reserved.

A new route to strange nonchaotic attractors (SNAs), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series $$\textit{LCR}$$ circuit with a simple nonlinear element. Upon increasing the control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighbourhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincare map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behaviour via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are charaterized by Poincare maps. Apart from the multilayered bubble route to SNA, Heagy–Hammel and fractalization routes to SNA have also been numerically observed and are characterized qualitatively interms of phase portraits, power spectrum. The above mentioned three routes are further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance.

The pinched hysteresis loops drawn in the voltage-current, voltage-charge, and flux-current characteristics belong to well-known fingerprints of the memory elements known as the memristor, memcapacitor, and meminductor. It is shown that generating such loops is in fact a natural attribute of all nonlinear elements from Chua's table, and that the hysteresis need not necessarily be a manifestation of the internal memory of the element. A generalized homothety theorem is introduced, which describes the regularities of such a hysteresis and its frequency dependence.

Reluctance actuators are thought to be a good solution for the nano-precision stage to provide high acceleration as a kind of feedforward actuator in the next generation lithography. However, hysteresis has a significant impact on the accuracy of reluctance actuators. There are many methods to model the hysteresis, among which Prandtl-Ishlinskii (PI) model is the most popular one. To effectively control the nonlinear element, the PI model requires precisely matching the hysteresis loop. The parameter identification of the PI model is of crucial importance in model accuracy. In this paper, we use teaching-learning based optimization (TLBO) method to identify these parameters. The performances of the TLBO method are compared with the other algorithms by simulation, and the simulation results show that TLBO have advantages of accuracy and efficiency, and the model parameters have been properly identified.

In nonlinear systems, chaos may exsist in some's parameters and may cause some unexpected damage. This paper proposes a design method of a reset controller for nonlinear chaotic systems. We consider a plant consisting of a linear system with a nonlinear feedback element and show a condition of the parameter for which chaos controls, using describing functions of the nonlinear element and the reset controller. We apply the proposed design method of the reset controller to the forced Van der Pol equation system and show its effectiveness.

The paper presents results of theoretical and practical researching the radiofrequency Chua's oscillator based on a tunnel diode. A mathematical model of Chua's oscillator based on a tunnel diode has been proposed, it differs from Chua's mathematical model by the changed system's third differential equation and by an approximation function of a nonlinear element. Lyapunov exponents have been calculated. Phase portraits of the oscillator have been obtained. Time, frequency and statistic characteristics of the generated oscillation have been obtained.

A two-stage decoupling identification method is developed for Hammerstein system containing the threshold-discrete Prandtl-Ishlinskii (TDPI) operator with non-local memories. In the first stage, a persistently exciting input is designed so that the linear subsystem can be made decoupled from the nonlinear element and linear subsystem identification is coped with using a least squares estimator. Then, the nonlinear subsystem is estimated through designing an appropriate exciting signal, which guarantees the estimator consistency. The effectiveness of the algorithm is verified by piezoelectric actuator (PEA) experiments.

The paper deals with an unconventional approach to the analysis of nonlinear RLC cell serving as a basic building block of nonlinear transmission line (NLTL) models. The only nonlinear element is considered, capacitor C represented by a semiconductor varactor with an analytically defined capacitance-voltage relation. A method based on the use of Volterra series expansion of voltage responses is investigated, while efficient way of getting respective kernels utilizing Taylor series expansion is shown. Computer simulation results in Matlab are presented and compared with ODEs built-in solver.

Structural decoupling problem has been well investigated for three decades and led to several decoupling methods. In spite of the inherent nonlinearities in a structural system in various forms all decoupling studies are for linear systems. In this study, decoupling problem for nonlinear systems is addressed for the first time and a method is proposed for calculating the frequency response functions of a substructure decoupled from a coupled nonlinear structure where nonlinearity can be modelled as a single nonlinear element. The method proposed is validated through simulated case studies.

Memristor-based systems and their potential applications, in which memristor is both a nonlinear element and a memory element, have been received significant attention in the control literature. In this work, we study a memristor-based hyperchaotic system with hidden attractors. First, we study the dynamic properties of the memristor-based hyperchaotic system such as equilibria, Lyapunov exponents, Poincare map, etc. We obtain the Lyapunov exponents of the memristor-based system as \(L_1 = 0.1244\), \(L_2 = 0.0136\), \(L_3 = 0\) and \(L_4 = -10.8161\). Since there are two positive Lyapunov exponents, the memristor-based system is hyperchaotic. Also, the Kaplan-Yorke fractional dimension of the memristor-based hyperchaotic system is obtained as \(D_{KY} = 3.0128\). Next, we design adaptive control and synchronization schemes for the memristor-based hyperchaotic system. The main adaptive control and synchronization results are established using Lyapunov stability theory. MATLAB simulations are shown to illustrate all the main results of this work. Finally, an electronic circuit emulating the memristor-based hyperchaotic system has been designed using off-the-shelf components.

Summary A general approach is presented to analyze the worst case input/output gain for an interconnection of a linear parameter varying (LPV) system and an uncertain or nonlinear element. The LPV system is described by state matrices that have an arbitrary, that is not necessarily rational, dependence on the parameters. The input/output behavior of the nonlinear/uncertain block is described by an integral quadratic constraint (IQC). A dissipation inequality is proposed to compute an upper bound for this gain. This worst-case gain condition can be formulated as a semidefinite program and efficiently solved using available optimization software. Moreover, it is shown that this new condition is a generalization of the well-known bounded real lemma type result for LPV systems. The results contained in this paper complement known results that apply IQCs for analysis of LPV systems whose state matrices have a rational dependence on the parameters. The effectiveness of the proposed method is demonstrated on simple numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.

The stability and L2 performance analysis of systems consisting of an interconnection of a linear-time-invariant (LTI) system and a static nonlinear element which is Lipschitz, slope restricted and sector bounded is revisited. The main thrust of the paper is to improve and extend an existing result in the literature to enable (i) concise and correct conditions for asymptotic stability of the interconnection and (ii) reasonably tight bounds on the L2 gain between an exogenous input and a given output to be obtained. Numerical examples indicate that the proposed algorithm performs well compared to competing results in the literature.