Third-order Nonlinear Systems Research Articles

Event-based consensus for third-order nonlinear multi-agent systems

Event-based consensus problem of third-order nonlinear multi-agent systems is considered. Event-based controllers are proposed and MASs with event-based controllers are viewed as nonlinear systems with disturbances. By a degenerate transformation, an error system is derived from multi-agent systems, and consensus for multi-agent systems is converted into stability of the error system. Based on Lyapunov theory and Kronecker product method, several sufficient conditions on event-based consensus are obtained. It is shown that multi-agent systems can reach event-based consensus if the trigger parameter is sufficiently small and the control gain is sufficiently large under some simple conditions. Furthermore, the Zeno-behavior of the controllers is abstained. An example is presented to show the effectiveness of the derived results.

A Finite-Time Observer-Based Identification of Sinusoidal Signal With Unknown Frequency

In this article, we studied the parameters identification problem for the single sinusoidal signal based on the finite-time observer technique. After representing the sinusoidal signal with known frequency as a second-order time-invariant system, a finite-time observer-based estimation strategy is designed based on the homogeneous system theory to obtain unknown amplitude and phase angle. When the signal frequency is unknown, the original signal will be constructed into a third-order nonlinear system, which is challenging to design the corresponding state estimator. Based on the Lyapunov analysis method and adding a power integrator technique, a finite-time estimation scheme is developed and analyzed strictly. Finally, for a single-phase grid signal, some experimental results are provided to illustrate the efficacy of the proposed method.

Calculating Parametric Oscillation of Third-Order Nonlinear System With Dynamic Friction and Fractional Damping

AbstractIn this paper, the parametric resonance of third-order parametric nonlinear system with dynamic friction and fractional damping is investigated using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the resonance solution is then obtained by means of Lyapunov theory. Additionally, the effect of the fractional derivative on the system dynamics is analyzed. The effects of the two parameters of the fractional-order derivative, i.e., the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated.

Partial component consensus analysis of second-order and third-order nonlinear multi-agent systems

Partial component consensus is a phenomenon that some components of agents reach identity, but the others components are not the same. In this paper, the partial component consensus problem of second-order and third-order nonlinear multi-agent systems has been investigated. Some concepts and results of partial component consensus are given through partial component stability theory and pinning control protocol. Numerical simulations are finally given to verify the theoretical results.

Event-Based Consensus for Third-Order Nonlinear Multi-Agent Systems

Event-Based Consensus for Third-Order Nonlinear Multi-Agent Systems

Methods of Ensuring Invariance with Respect to External Disturbances: Overview and New Advances

In this paper, we carry out a demonstration and comparative analysis of known methods of the synthesis of various control laws ensuring the invariance of the output (controlled) variable with respect to external disturbances under various assumptions about their type and channels of acting on the control plant. Methods of the synthesis are presented on the example of a third-order nonlinear system with single input and single output (SISO-systems), dynamic feedback synthesis is presented at a descriptive level and the focus is on procedures of static feedback synthesis. For the systems in which the matching conditions are not satisfied, it is concluded that it is expedient to introduce smooth and bounded nonlinear local feedbacks. Within the framework of the block control principle, we developed an iterative procedure of synthesis of S-shaped sigmoid feedbacks for such systems. Nonlinear local feedbacks ensure stabilization of the output variable with the given accuracy and settling time as in a system with traditionally used linear local feedbacks with high gains. However, in contrast to it, sigmoid functions do not lead to a large overshoot of state variables and control actions.

On the convexity of the reachable set with respect to a part of coordinates at small time intervals

We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

Integrated intelligent computing paradigm for nonlinear multi-singular third-order Emden–Fowler equation

In this study, an advance computational intelligence scheme is designed and implemented to solve third-order nonlinear multiple singular systems represented with Emden–Fowler differential equation (EFDE) by exploiting the efficacy of artificial neural networks (ANNs), genetic algorithms (GAs) and active-set algorithm (ASA), i.e., ANN–GA–ASA. In the scheme, ANNs are used to discretize the EFDE for formulation of mean squared error-based fitness function. The optimization task for ANN models of nonlinear multi-singular system is performed by integrated competency GA and ASA. The efficiency of the designed ANN–GA–ASA is examined by solving five different variants of the singular model to check the effectiveness, reliability and significance. The statistical investigations are also performed to authenticate the precision, accuracy and convergence.

Coexisting modes and bifurcation structure in a pair of coupled detuned third order oscillators

The bifurcation structure of a system of two non-linear coupled oscillators is considered here. This system is motivated by coupled micro-oscillators proposed for microelectromechanical systems (MEMS) applications. The model starts from previous work in which the authors considered the bifurcation structure in a pair of identical coupled, oscillators, each of which is modelled by a third-order nonlinear system. In the present paper, the case is considered where the two oscillators are non-identical i.e. detuned from each other. The series of bifurcations obtained is described in detail. It is seen that the symmetry of the bifurcation structure present in the identical oscillators is lost in the detuned oscillators. A very surprising homoclinic bifurcation is seen in one region of the parameter space, and this is examined in more detail through consideration of a simplified system.

Efficient telecom-to-visible spectral translation through ultra-low power nonlinear nanophotonics.

The ability to spectrally translate lightwave signals in a compact, low-power platform is at the heart of the promise of nonlinear nanophotonic technologies. For example, a device to link the telecommunications band with visible and short near-infrared wavelengths can enable a connection between high-performance chip-integrated lasers based on scalable nanofabrication technology with atomic systems used for time and frequency metrology. While second-order nonlinear () systems are the natural approach for bridging such large spectral gaps, here we show that third-order nonlinear () systems, despite their typically much weaker nonlinear response, can realize spectral translation with unprecedented performance. By combining resonant enhancement with nanophotonic mode engineering in a silicon nitride microring resonator, we demonstrate efficient spectral translation of a continuous-wave signal from the telecom band (≈ 1550 nm) to the visible band (≈ 650 nm) through cavity-enhanced four-wave mixing. We achieve such translation over a wide spectral range >250 THz with a translation efficiency of (30.1 ± 2.8) % and using an ultra-low pump power of (329 ± 13) μW. The translation efficiency projects to (274 ± 28) % at 1 mW and is more than an order of magnitude larger than what has been achieved in current nanophotonic devices.

On Calculation of the First Integrals for Three-dimensional ODE Systems

The search for solutions of nonlinear stationary systems of ordinary differential equations (ODE) is sometimes very complicated. It is not always possible to obtain a general solution in an analytical form. As a consequence, a qualitative theory of nonlinear dynamical systems has been developed. Its methods allow us to investigate the properties of solutions without finding a general solution. Numerical methods of investigation are also widely used.In the case when it is impossible to find an analytically general solution of the ODE system, sometimes, nevertheless, it is possible to find its first integral. There is a number of known results that make it possible to obtain the first integral for certain special cases.The article deals with the method for obtaining the first integrals of ODE systems of the third order, based on the fact of integrability of the involutive distribution.The method proposed in the paper allows us to obtain the first integral of a nonlinear ODE system of the third order in the case when a vector field, which generates an involutive distribution of dimension 2 together with the vector field of the right-hand side of a given ODE system, is known. In this case, the solution of a certain sequence of Cauchy problems allows us to construct a level surface of the function of the first integral containing the given point of the state space of the system. Using the method of least squares, in a number of cases it is possible to obtain an analytic expression for the first integral.The article gives examples of the method application to two ODE systems, namely to a simple nonlinear third-order system and to the Lorentz system with special parameter values. The article shows how the first integrals can be obtained analytically using the method developed for the two systems mentioned above.

A physical SBT-memristor-based Chua's circuit and its complex dynamics.

SBT memristor is a physical memristor built on nanometer film, described by a deterministic flux-controlled mathematical model. A new physical SBT-memristor-based Chua's circuit is presented in this paper, using the memristor to replace the Chua diode to connect with a negative conductance in parallel. The circuit can be well modeled by a fourth-order nonlinear system in the voltage-current domain and a third-order nonlinear system in the flux-charge domain, respectively. The dependences of the circuit stability on both the initial state value of the SBT memristor and the linear resistance in the circuit are analyzed theoretically, verified by numerical simulations, which shows that the circuit system has complex dynamics with multiple behaviors, including sink, periodic cycle, limit cycle, chaos, and some complicated transient dynamical behaviors. The new findings shed light on future research on the physical realization of the SBT-memristor-based Chua's circuit and other similar devices.

Analytical properties of solutions to a class of third-order nonlinear dynamical dissipative systems with no chaotic behaviour

The singular analysis was performed for solutions of one class of third-order nonlinear dynamical systems with no chaotic behaviour. It is established that all systems of the given class (except one) are not the systems of Painlevé-type.

Robust Flat Filtering Control of a Nonlinear Manipulator-Direct Current Motor System

This paper presents a linear robust output reference trajectory tracking controller, addressed here as a flat filtering controller (FFC), for nonlinear differentially flat systems. Here, we illustrate the controller's performance, via digital computer simulations and, also, via laboratory experiments, carried out on a single link-direct current (DC) motor driven robot manipulator undergoing a reference trajectory tracking task. The proposed linear FFC only requires the output to be regulated of the composite system and none of the internal states of the resulting third-order nonlinear system. The controller is designed on the basis of a drastic simplification of the combined single link-DC motor dynamics to a, perturbed, third-order pure integration system. This demonstrates the robustness of the proposed scheme with respect to ignored nonlinear state-dependent, endogenous, disturbances and, also, to independent unstructured exogenous disturbances inevitable in an experimental setup. Simulation and experimental results, as well as comparisons with other controllers, are presented.

Adaptive near-optimal consensus of high-order nonlinear multi-agent systems with heterogeneity

In this paper, the near-optimal distributed consensus of high-order nonlinear multi-agent systems consisting of heterogeneous agents is investigated. The consensus problem is formulated as a receding-horizon optimal control problem. Under the condition that the dynamics of all agents are fully known, a nominal near-optimal protocol is designed and proposed via making approximation of the performance index. For the situation with fully unknown system parameters, sliding-mode auxiliary systems, which are independent for different agents, are built to reconstruct the input–output properties of agents. Based on the sliding-mode auxiliary systems, an adaptive near-optimal protocol is finally presented to control high-order nonlinear multi-agent systems with fully unknown parameters. Theoretical analysis shows that the proposed protocols can simultaneously guarantee the asymptotic optimality of the performance index and the asymptotic consensus of multi-agent systems. An illustrative example about a third-order nonlinear multi-agent system consisting of 10 heterogeneous agents with fully unknown parameters further substantiates the efficacy and superiority of the proposed adaptive near-optimal consensus approach.

Nonsingular fast terminal sliding-mode stabilizer for a class of uncertain nonlinear systems based on disturbance observer

This paper investigates a novel nonsingular fast terminal sliding-mode control method for the stabilization of the uncertain time-varying and nonlinear thirdorder systems. The designed disturbance observer satis es the nite-time convergence of the disturbance approximation error and the suggested nite-time stabilizer assuresthe presence of the switching behavior around the switching curve in the nite time. Furthermore, this approach can overcome the singularity problem of the fast terminal sliding-mode control technique. Moreover, knowledge about the upper bounds of the disturbances is not required and the chattering problem is eliminated. Usefulness and e ffectiveness of the o ffered procedure are con rmed by numerical simulation results.

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

Generation of amplitude death and rhythmogenesis in coupled hidden attractor system with experimental demonstration

Third-order nonlinear dynamical systems with attractors (one with no fixed point and the other with a stable fixed point) are conjugately coupled. It is observed that the combined system gives rise to nontrivial fixed points, and the corresponding long-time dynamics leads to amplitude death. On the other hand, if the coupled system starts from the neighborhood of these generated fixed points, there exists a very novel path to the generation of oscillations. Such a phenomenon is called rhythmogenesis. The characterization of such dynamics is done with the help of bifurcation diagrams and Lyapunov exponents. Another important outcome is an observed bifurcation with respect to the initial condition variation. This last phenomenon may be attributed to the fact that usual route to chaos is absent in the uncoupled system. In the second part of our paper, we have constructed the electronic circuit pertaining to the coupled system, collected the data via NI 6009 DAC, and have shown the existence of the amplitude death experimentally. The case of rhythmogenesis is done similarly, which only requires an extra circuit to fix the initial condition, and was devised in our earlier paper. It has been used to construct a new circuit to detect rhythmogenesis.

Global stability of a third-order nonlinear system of difference equations with period-two coefficients

The main objective of this paper is to study the global behavior of the following third-order rational system of difference equations $$\begin{aligned} x_{n+1}=\frac{p_n+y_n}{p_n+y_{n-2}}, \ \ y_{n+1}=\frac{q_n+x_n}{q_n+x_{n-2}}, \qquad n=0,1,2,\ldots , \end{aligned}$$ where \(\{p_n\}\) and \(\{q_n\}\) are 2-periodic sequences of positive numbers, and the initial values \(x_{-i},\, y_{-i} \in [0, \infty )\), for \(i=0,1,2\).

An adaptive fast terminal sliding mode control combined with global sliding mode scheme for tracking control of uncertain nonlinear third-order systems

In this paper, an adaptive fast terminal sliding mode control technique combined with a global sliding mode control scheme is investigated for the tracking problem of uncertain nonlinear third-order systems. The proposed robust tracking controller is formulated based on the Lyapunov stability theory and guarantees the existence of the sliding mode around the sliding surface in a finite time. Under the uncertainty and nonlinearity effects, the reaching phase is removed and the chattering phenomenon is eliminated. This scheme guarantees robustness against nonlinear functions, parameter uncertainties and external disturbances. The derivative of the state variable is replaced by a delay term in the form of an Euler approximation of the derivative function. Furthermore, the knowledge of upper bounds of the system uncertainties is not required, which is more flexible in the real implementations. Simulation results are presented to show the effectiveness of the suggested method.