Class Of Nonlinear Dynamical Systems Research Articles

An online model-free adaptive learning control solution for robotic arms

This paper focuses on the online control of a class of nonlinear dynamical systems, specifically robotic manipulators. Solutions utilizing Proportional–Integral–Derivative (PID) control schemes are employed to control the joints of robotic manipulators. However, the existing control strategies utilize fixed gains, which do not fully account for the inherent nonlinearity of the dynamical structure or the dynamics of reference-tracking error. Additionally, the individual joint’s dynamic performance is optimized independently from the performance of other joints. This work introduces an adaptive integral Reinforcement Learning algorithm to control a four-DoF robotic arm in real time. This is done using a model-free Value Iteration process implemented in a continuous-time mode. The solution does not assume any knowledge of the dynamics of the robot arm and does not require any initial admissible control strategy to proceed with the adaptive learning solution. The self-learning algorithm provides adaptable strategies to control the turntable, forearm, bicep, and wrist joints of the robotic arm. The performance of the adaptive learning solution is compared with those of Proportional–Integral–Derivative and high-order model-free adaptive control schemes to highlight its effectiveness.

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Dual-wavelength channel GHz repetition rate mode-locked VECSEL cavities sourced from a common gain medium.

Mode-locked vertical external cavity semiconductor lasers are a unique class of nonlinear dynamical systems driven far from equilibrium. We present a novel, to the best of our knowledge, experimental result, supported by rigorous microscopic simulations, of two coexisting mode-locked V-cavity configurations sourced by a common gain medium and operating as independent channels at angle controlled separated wavelengths. Microscopic simulations support pulses coincident on the common gain chip extracting photons from a nearby pair of coexisting kinetic holes burned in the carrier distributions.

Actuator and sensor fault isolation in a class of nonlinear dynamical systems

Fault isolation in dynamical systems is a challenging task due to modeling uncertainty and measurement noise, interactive effects of multiple faults and fault propagation. This paper proposes a unified approach for isolation of multiple actuator or sensor faults in a class of nonlinear uncertain dynamical systems. Actuator and sensor fault isolation are accomplished in two independent modules, that monitor the system and are able to isolate the potential faulty actuator(s) or sensor(s). For the sensor fault isolation (SFI) case, a module is designed which monitors the system and utilizes an adaptive isolation threshold on the output residuals computed via a nonlinear estimation scheme that allows the isolation of single/multiple faulty sensor(s). For the actuator fault isolation (AFI) case, a second module is designed, which utilizes a learning-based scheme for adaptive approximation of faulty actuator(s) and, based on a reasoning decision logic and suitably designed AFI thresholds, the faulty actuator(s) set can be determined. The effectiveness of the proposed fault isolation approach developed in this paper is demonstrated through a simulation example.

Hamiltonian-Driven Adaptive Dynamic Programming With Efficient Experience Replay.

This article presents a novel efficient experience-replay-based adaptive dynamic programming (ADP) for the optimal control problem of a class of nonlinear dynamical systems within the Hamiltonian-driven framework. The quasi-Hamiltonian is presented for the policy evaluation problem with an admissible policy. With the quasi-Hamiltonian, a novel composite critic learning mechanism is developed to combine the instantaneous data with the historical data. In addition, the pseudo-Hamiltonian is defined to deal with the performance optimization problem. Based on the pseudo-Hamiltonian, the conventional Hamilton-Jacobi-Bellman (HJB) equation can be represented in a filtered form, which can be implemented online. Theoretical analysis is investigated in terms of the convergence of the adaptive critic design and the stability of the closed-loop systems, where parameter convergence can be achieved under a weakened excitation condition. Simulation studies are investigated to verify the efficacy of the presented design scheme.

Predefined‐time adaptive backstepping control for a class of nonlinear dynamical systems with parametric uncertainties

SummaryIn this article, we address the problem of achieving predefined‐time control for a class of nonlinear dynamical systems with parameter uncertainties. We introduce an adaptive predefined‐time control algorithm based on integrator backstepping for th order nonlinear systems. Using Lyapunov's stability theory, we establish that the proposed predefined time control approach, combined with an adaptive law, ensures the boundedness of all signals within the closed‐loop system. A notable advantage of our method is its potential to achieve convergence within a predetermined time frame. To validate our approach's efficacy, we present a practical example involving a 2 DOF helicopter, demonstrating the effectiveness of the proposed scheme.

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences.

The Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars-classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism-and its foundations are grounded on the optimization of the BG (additive) entropic functional [Formula: see text]. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional [Formula: see text]. The index [Formula: see text] and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

Observer normal form design for the nonlinear MIMO systems using coupled auxiliary dynamics

This paper presents a transformation of a class of nonlinear multiple-input and multiple-output (MIMO) dynamical systems into a class of extended nonlinear normal forms. The extended normal forms are compatible with the widely used high-gain observer, which enables the estimation of the system’s states. It is realized by using a series of transformations which are performed by means of the so-called dynamics extension method, which explicitly supplies a set of auxiliary dynamics and a change of coordinates. Many researchers have focused their efforts on transforming nonlinear dynamical systems into normal forms of nonlinear observers. However, few of proposed transformations have been specifically developed for MIMO systems using an extended dynamics transformation. The main objective of this work is to fill this gap and provide a class of nonlinear dynamical systems that support the extended dynamics method, which can lead to a better understanding of their structures. Additionally, the results are highlighted by studying a class of nonlinear systems that describe the Dengue fever infection status of both human and mosquito populations. Finally, the effectiveness of the proposed high-gain observer is verified through a numerical example.

Introducing some classes of stable systems without any smooth Lyapunov functions

This paper proves that certain classes of stable nonlinear systems do not admit any smooth Lyapunov functions. In fact, the stability analysis of two different classes of nonlinear dynamical systems is provided, and it is demonstrated that their stability properties cannot be established by any convex Lyapunov functions or smooth Lyapunov functions. In this regard, we begin by studying our first class of autonomous dynamical systems and prove that, despite stability, there are no convex Lyapunov functions in the form of V(x) or V(t,x) to establish the stability properties. For the second class, we consider a much more general form of nonlinear autonomous dynamical systems with a stable origin, and prove that this class, too, does not admit any smooth Lyapunov functions in the form of V(x) or V(t,x). Furthermore, another general class of stable non-autonomous dynamical systems is presented, and it is demonstrated that there is no smooth Lyapunov function in the form of V(x) to establish the stability properties. Finally, for the second class of more general stable non-autonomous systems, it is proved that the class does not admit even a continuous Lyapunov function in the form of V(x) or V(t,x).

Dynamical system of a time-delayed ϕ6-Van der Pol oscillator: a non-perturbative approach

A remarkable example of how to quantitatively explain the nonlinear performance of many phenomena in physics and engineering is the Van der Pol oscillator. Therefore, the current paper examines the stability analysis of the dynamics of ϕ6-Van der Pol oscillator (PHI6) exposed to exterior excitation in light of its motivated applications in science and engineering. The emphasis in many examinations has shifted to time-delayed technology, yet the topic of this study is still quite significant. A non-perturbative technique is employed to obtain some improvement and preparation for the system under examination. This new methodology yields an equivalent linear differential equation to the exciting nonlinear one. Applying a numerical approach, the analytical solution is validated by this approach. This novel approach seems to be impressive and promising and can be employed in various classes of nonlinear dynamical systems. In various graphs, the time histories of the obtained results, their varied zones of stability, and their polar representations are shown for a range of natural frequencies and other influencing factor values. Concerning the approximate solution, in the case of the presence/absence of time delay, the numerical approach shows excellent accuracy. It is found that as damping and natural frequency parameters increase, the solution approaches stability more quickly. Additionally, the phase plane is more positively impacted by the initial amplitude, external force, damping, and natural frequency characteristics than the other parameters. To demonstrate how the initial amplitude, natural frequency, and cubic nonlinear factors directly affect the periodicity of the resulting solution, many polar forms of the corresponding equation have been displayed. Furthermore, the stable configuration of the analogous equation is shown in the absence of the stimulated force.

Physics-informed recurrent neural network modeling for predictive control of nonlinear processes

In this work, we present a physics-informed recurrent neural network (PIRNN) modeling approach, and a PIRNN-based predictive control scheme for a general class of nonlinear dynamic systems. Specifically, we first develop a hybrid data-driven and physics-guided modeling framework that integrates measurement data and mechanistic mathematical models to construct high-fidelity RNN models. Then, we derive a generalization error bound of the PIRNN model based on a nominal system model via the Rademacher complexity technique from statistical machine learning theory. Subsequently, the PIRNN model is utilized in Lyapunov-based model predictive controllers and applied to a chemical reactor example with Gaussian measurement noise to demonstrate its improved noise rejection and generalization performance in comparison to the purely data-driven and the purely physics-guided RNN-based predictive control schemes.

New Approach to the Synthesis of Optimal Terminal Control of Nonlinear Dynamic Systems

The problem of constructing common solutions to terminal control problems of nonlinear systems is considered here. Previously proven positions are used that the optimal trajectory is an envelope of a parametric family of surfaces (a parametric family of singular curves), and that optimal control can be found on this family. The fact that at each point of the optimal trajectory the vector-function of Lagrange factors is tangent to it, but also tangent to the singular curve, is played out here. A constructive method of constructing singular curves based on conditional separation of variables in the Hamilton-Jacobi equation is given. The " free" parameters of singular curves are based on the condition of minimizing the terminal functionality, which avoids an explicit solution to the boundary problem for a class of nonlinear dynamic systems, and simplifies computational algorithms. Singular curves are described by a reduced (abbreviated) mathematical model. Thus, to synthesize the law of optimal control, we must use the complete (original) mathematical model of the dynamic system, but to calculate it at one time or another, it is enough reduced model. This consideration defines the principle of informational dualism. An illustrative example is given. It has been shown that this approach can be used to solve some classes of differential games.

ПОСТРОЕНИЕ ОПТИМАЛЬНЫХ ТРАЕКТОРИЙ ЛЕТАТЕЛЬНОГО АППАРАТА НА МНОЖЕСТВЕ СИНГУЛЯРНЫХ КРИВЫХ

The problem of constructing general solutions to the problems of terminal control of nonlinear systems is considered. It is proved that: 1) the optimal trajectory is the envelope of a parametric family of surfaces and, accordingly, a parametric family of singular curves defined on them, 2) optimal control can be found on this family. A constructive method for constructing singular curves is given. The "free" parameters of singular curves are found from the condition of minimization of the terminal functional. Such an approach in some cases avoids the explicit solution of the boundary value problem for a class of nonlinear dynamical systems, and simplifies computational algorithms.

Neural network-based sensor fault estimation and active fault-tolerant control for uncertain nonlinear systems

In this work, we developed a novel active fault-tolerant control (FTC) design scheme for a class of nonlinear dynamic systems subjected simultaneously to modelling imperfections, parametric uncertainties and sensor faults. Modelling imperfections and parametric uncertainties are dealt with using an adaptive radial basis function neural network (RBFNN) that estimates the uncertain part of the system dynamics. For sensor fault estimation (FE), a nonlinear observer based on the estimated dynamics is designed. A scheme to estimate sensor faults in real-time using the nonlinear observer and an additional RBFNN is developed. The convergence properties of the RBFNN, used in the fault FE part, are improved by using a sliding surface function. For FTC design, a sliding surface is designed that incorporates the real-time sensor FE. The resulting sliding mode control (SMC) technique-based FTC law uses the estimated dynamics and real-time sensor FE. A double power-reaching law is adopted to design the switching part of the control law to improve the convergence and mitigate the chattering associated with the SMC. The FTC works well in the presence and absence of sensor faults without the requirement for controller reconfiguration. The stability of the proposed active FTC law is proved using the Lyapunov method. The developed scheme is implemented on a nonlinear simulation of an unmanned aerial vehicle (UAV). The results show good performance of the proposed unified FE and the FTC framework.

Periodic solutions to perturbed nonlinear oscillators with memory

We consider a general class of nonlinear dynamical systems with memory. We prove under which conditions periodic perturbations allow solutions with the same period. In particular, we consider memory kernels of Γ-type, which give prominence to critical phenomena in the past.

New criteria for predefined finite time synchronization of memristor-based neural networks

The fast synchronization problem of memristor-based neural networks is studied in this article. Firstly, a novel predefined finite time stability of a class of nonlinear dynamical systems is investigated under the incomplete beta function, complete beta function, and inequality technology. Then, an active predefined finite time controller is designed that guarantees finite time stability of synchronization errors of two memristor-based neural networks with/without proportional delay. Some simulation results are presented to show the effectiveness of theoretical results.

Characteristics of Stability Boundary of Nonlinear Continuous Dynamical Systems

The theory of differential equations has been widely known and developed in recent years. One of the issues that many authors give their undivided attention to is the stability boundary of nonlinear dynamical systems. In this work, we first review several properties of equilibrium points on the stability boundary. We next extend the characteristics of the stability boundary for a fairly large class of nonlinear dynamical systems. These characteristics are the key to completely determine the stability boundary of nonlinear dynamical systems.

Dark Type Dynamical Systems: The Integrability Algorithm and Applications

Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities.

THE SYNTHESIS OF MATHEMATICAL MODELS OF NONLINEAR DYNAMIC SYSTEMS USING VOLTERRA INTEGRAL EQUATION

The problem of creating mathematical models of nonlinear dynamical systems does not have an unambiguous solution and requires the creation of a separate synthesis method for each such object. To develop a method for synthesizing mathematical models of an extensive class of nonlinear dynamical systems with polynomial nonlinearities. The work uses a method based on the solution of the Volterra integral equation in the ideology set forth in Van Trees H.L., according to which the structure of a nonlinear dynamical object present47s a series connection of the linear part, characterizing the inertial properties of the system, and the nonlinear element, given by static characteristic. The difference of the suggested version of the method from the classical one, proposed in the works of Van Trees H.L., is an expansion of their input and output signals into Fourier series and a representation of the inertial part of these systems by their Bode plots, connected into one structure with input and output signals and non-linearity by Volterra integral equation. The algorithm of the proposed method is disclosed by the example of solving the problem of identifying a nonlinear dynamical system which impulse response of the inertial part satisfies the separability requirement, the order of the polynomial nonlinearity is three, and the model of the input signal has the form of a sinusoid "raised" over the time axis on a priori given constant level. A computational experiment was carried out on the example of nonlinear dynamical systems with the third order of the nonlinear characteristic and the first and second orders of the model of the inertial part of these systems with the specified algorithms of their parametric identification. The suggested method allows to synthesis the mathematical model of a nonlinear dynamical system with the polynomial static characteristic to the case when the input signal has an arbitrary number of harmonics, and the model of the inertial part and the nonlinear polynomial function have an arbitrary order.

Universal constraint on nonlinear population dynamics

Ecological and evolutionary processes show various population dynamics depending on internal interactions and environmental changes. While crucial in predicting biological processes, discovering general relations for such nonlinear dynamics has remained a challenge. Here, we derive a universal information-theoretical constraint on a broad class of nonlinear dynamical systems represented as population dynamics. The constraint is interpreted as a generalization of Fisher’s fundamental theorem of natural selection. Furthermore, the constraint indicates nontrivial bounds for the speed of critical relaxation around bifurcation points, which we argue are universally determined only by the type of bifurcation. Our theory is verified for an evolutionary model and an epidemiological model, which exhibit the transcritical bifurcation, as well as for an ecological model, which undergoes limit-cycle oscillation. This work paves a way to predict biological dynamics in light of information theory, by providing fundamental relations in nonequilibrium statistical mechanics of nonlinear systems.