A Lyapunov-based direct adaptive controller for the suppression and synchronization of a perturbed nuclear spin generator chaotic system

doi:10.1016/j.amc.2020.125858

Applied Mathematics and Computation

Volume 395, 15 April 2021, 125858

https://doi.org/10.1016/j.amc.2020.125858 Get rights and content

Highlights

•
This paper design and synthesizes a novel robust direct adaptive control technique.
•
It investigates the suppression and synchronization of chaos in the nuclear spin generator chaotic system.
•
It accomplishes oscillation free, faster and smooth convergence behavior of the state variables (error vectors) to the steady-state.
•
It demonstrates robust performance in the presence of unknown model uncertainties and external disturbances. It shows higher precision performance.

Abstract

This article designs and synthesizes a new Lyapunov-based robust direct-adaptive controller (RDAC) and investigates the control and synchronization of chaos in the nuclear spin generator chaotic (NSGC) system. The inevitable time-varying external disturbances and model uncertainties perturb the NSGC system. The nonlinear terms, external disturbances, model uncertainties, and plant's parameters are unknown and bounded. Avoiding the cancelation of the nonlinear terms of the plant by the controller makes the closed-loop robust stable in the presence of unknown parametric uncertainties; this concept blooms base for efficient control law design. The proposed RDAC eradicates the effects of the time-varying unknwon external disturbances and model uncertainties and accomplishes quick and smooth convergence of the state varaible (error vector) trajectories to the origin with reduced oscillations. Based on the Lyapunov function principle, the article describes a detailed analysis of the closed-loop stability. It provides suitable adaptive laws that estimate the upper bound of unknown controller parameters, external disturbances, and model uncertainties. The computer simulation results endorse the theoretical analysis, and the comparative study highlights the benefits.

Introduction

Chaos theory discusses the empirical and qualitative analysis of complex and irregular dynamic behavior generated by chaotic systems [1]. In general, chaotic dynamical systems exhibit random-like noise, parameter variations, sensitivity to initial conditions, and topological mixing [1]. Being these generic features, the investigation of chaotic dynamical systems has emerged as an intrinsic framework to model various real-world problems [2,3]. The control (suppression) and synchronization of chaotic systems are two significant applications of chaos theory. The study of chaos control and synchronization has attracted the attention of the research community due to their successful applications in economics, chemistry, biology, physics, and other engineering fields [4], [5], [6], [7], [8], [9], [10]–11]. For example, to handle various motion planning tasks such as patrolling, grass cutting, and cleaning (among others), mobile(navigate) robots encounter several complex issues, including vibrations, noise-sensing generation, and highly erratic robot-environmental interfaces. These nonlinearities create chaos in mobile robots [4]. The main objective of chaos control and synchronization in mobile robots is to perform multiple tasks simultaneously [4]. The Single Machine Infinite Bus (SMIB) oscillator represents an electric power system. Load disturbances, lack of power, and capacitor's switching create instability in the bus's voltage; it causes chaotic behavior in the SMIB [5]. The aim of suppressing and synchronizing chaos in the SMIB oscillators is to tackle load disturbances and avoid the black-out of power to ensure the supply of electric power to different nodes [6]. Perturbation torque acts on the spacecraft during the orbital motion that generates chaos in the spacecraft. It moves the spacecraft away from the desired stable orbit [7]. For the control of chaos in the spacecraft, the primary purpose is to design a suitable controller that forces the spacecraft's attitude motion to a stable orbit [7]. Synchronization of spacecraft means that the attitudes of multiple spacecraft reach an agreement at a particular orbit of interest [8]. The coronary artery system exhibits both regular and chaotic dynamics. The coronary artery system's chaotic phenomenon occurs due to myocardial infraction that gives birth to cardiopathy and other arrhythmia diseases [9]. The core objective of synchronization of chaos in the coronary artery system is to diagnose and propose a treatment that reduces cardiopathy risk and diminishes other arrhythmia problems [9]. The DC-DC converter is a type of electric power circuits used in everyday electronic equipment. The external disturbances and fluctuations in the voltage produce instability; it creates chaos in the DC-DC converter [10]. The existence of chaos affects the performance of the DC-DC converter in practice, including actuator fault. The investigation of suppression and synchronization of chaos in the DC-DC converter is to run the system at a desirable low frequency and rejecting fault tolerance [10,11]. Similarly, chaotic behavior, control, and synchronization have been investigated in many physical and natural systems [12,13].

The aforementioned applications motivate the researchers in various fields that developed different state-of-the-art feedback control methodologies to address several challenges in science and engineering applications by controlling and synchronizing chaotic systems. These include adaptive control techniques [14,15], sample data controller [16], sliding mode control approach [17], backstepping controller [18], H_∞ control method [19], and intermittent control scheme [20], among others. The adaptive control strategy is an effective control technique for designing feedback control signals that successfully tackle the effects of uncertain (unknown) time-varying parameters. The adaptive feedback controller engenders control signals, which are functions of the input, output of the system, and estimation(s) of the uncertain (unknown) parameters [21]. The adaptive control techniques treat complex systems that have variations in their uncertainties and parameters.

The nuclear spin generator is a high-frequency oscillator, which is used to investigate the nuclear magnetic resonance (NMR) in a device (machine) for sensing rotation [22]. NMR spectrum gives the local structure, while the spin relaxation time gives the local dynamics of microscopic materials [23]. The nuclear spin generator system either eliminates or reduces the damping and keeps the precession of the nuclear magnetization vector related to the magnetic field [23]. The NSGC system displays very complex nonlinear phenomena, including chaos, and exhibits an archetypal behavior than the traditional chaotic Lorenz system [24]. The NSGC system was first proposed in [22] for describing the dynamics of a high-frequency oscillator using the phenomena of the NMR. After the pioneering work of [22], colossal research has been devoted to investigating the dynamics, control, and synchronization of chaos in the NSGC system. A brief review of some of the available results as follows. Sachdev and Sarathy [23] studies the existence and stability of the NSGC system's periodic solutions. Yuan and Yang [24] investigates further the chaotic behavior of the NSGC system, including bifurcations, a route to chaos, and strange attractors for random fluctuations in the system's parameters. Starkov [25] computes bounds for a compact domain that contains all compact invariant sets of the system describing the NSGC system's behavior. Nikolov et al. [26] studies the chaotic phenomenon of the NSGC system due to the sensitivity of the parameter variations. Fiaz et al. [27] discusses the existence of Si'lnikov chaos in the NGSC system. Based on the Routh-Hurwitz criterion, Lyapunov stability theory, and using the linear feedback controller, Hegazi et al. [28] investigates the control of chaos in the NSGC system. The paper [28] discusses the effects of parameter variations on the closed-loop dynamical system's stability. Sun and Zhang [29] proposes a modified impulsive controller to realize the suppression and synchronization of chaos in the NSGC system. The nonlinear control technique based on the passive control theory is developed in [30] to suppress the NSGC system's chaotic behavior. Hegazi et al. [31] proposes a modified adaptive control technique and studies the synchronization of two identical NSGC systems with unknown parameters. In the paper [31], adaptive laws are designed to estimate the uncertain controller parameter. Based on the Routh-Hurwitz criterion and employing the synchronization control strategy proposed in [32], the paper [33] studies the generalized synchronization of two linearly coupled NSGC systems. The synchronization of two identical NSGC systems is investigated in [34] by using the impulsive controller. Moreover, Khan [35] investigates the generalized synchronization of two coupled NSGC systems with application in secure communication systems.

Though the proposed suppression (synchronization) schemes [28], [29], [30]–31,[33], [34]–35] bear some challenges, the controllers establish suppression (synchronization) of chaos in the NSGC system. In the reported results [28–31,33–35], the convergence speed is slow, and the state variables (error vectors) exhibit large oscillations that give birth to various undesired behaviors in practice. Further, the proposed closed-loop dynamical system is sensitive to the parametric uncertainties present in the NGSC system. The following items (i) to (iii) describe an overview of the challenges.

(i)
In the process of chaos control (synchronization), nonlinearity performs a vital role; a complex closed-loop system improves the feedback controller's performance. The majority of the control algorithms [30,33–35] assume that the states are measurable, and the parameters are known in advance. Further, these controllers eliminate most of the plant's nonlinear (or residual) components and form a linearized closed-loop. Nevertheless, it may not be possible to cancel these terms since it is hard to measure the states and the plant's parameters are unknown (uncertain) in practice [21]. Insufficient compensation of the nonlinear terms generates erroneous control (synchronization) phenomenon and can blow the system. Also, a linearized closed-loop dynamical system increases the energy consumption that puts an extra burden on the controller design; it produces oscillations when the state variable (error vector) trajectories lie in the vicinity of zero due to the hefty control input signals [21]. This reality creates a motive for developing control law, which does not use the term cancelation philosophy.
(ii)
The controllers in [28–31,33–35] show slower convergence rates of the state variable (error vector) trajectories to the origin. Such state-feedback controllers take a long time to produce the desired spin. This lag possibly turns out unstable closed-loop, oscillations, and other malfunctionings in the system.
(iii)
During the suppression (synchronization) process, different types of inevitable external disturbances and unstructured uncertainties may act on the NSGC system, including the magnetic loads, heat, and incorrect sensor calibration [38]. These disturbances and unstructured uncertainties harm the closed-loop system. They create unwanted oscillations in the state variable (error vector) trajectories, preventing them from converging to the origin, precisely. Most of the reported works [28–31,33–35] investigate the suppression (synchronization) of the NSGC system without considering the effects of time-varying unknown external disturbances and unstructured uncertainties.

These challenges motivate to develop fast and efficient chaos suppression (synchronization) methodologies for the NSGC system that develop a closed-loop structure with the following properties.

(i)
It aims to design a controller that establishes suppression (synchronization) of chaos in the NSGC system without cancelation of the plant's nonlinear terms. The motivation behind this concept is the development of a robust closed-loop, which is insensitive to the parameter uncertainties of the nonlinear terms.
(ii)
It accomplishes the fast convergence rates of the state variable (error vector) trajectories to the steady-state.
(iii)
It demonstrates robust performance in the presence of time-varying unknown model uncertainties and external disturbances.
(iv)
It shows higher precision performance.

The literature survey reveals that the problem of control and synchronization of chaos in the NSGC system with unknown parameters, external disturbances, and model uncertainties has not been solved.

This article designs and synthesizes a new RDAC algorithm to address the above challenges with the properties given in motivation items (i) to (iv). The proposed RDAC consists of three components; one linear adaptive, a nonlinear, and one nonlinear adaptive terms. The linear adaptive component keeps the closed-loop system stable; it ensures that the state varaible (error vector) trajectories converge to zero. The nonlinear component heavily penalizes the large state variables (error vectors) in the transient; its effect becomes minimal in the steady-state. Fundamentally, this property reduces oscillations in the state varaible (error vector) trajectories during the transient. The nonlinear adaptive component eradicates the effects of time-varying external disturbances and model uncertainties. The external disturbances, model uncertainties, and nonlinear terms of the NSGC system are unknown and bounded. The proposed controller does not need the cancelation of the nonlinear (or residual) terms of the plant and is independent of the system parameters. The existence of the nonlinear terms significantly improves the performance of the controller. The proposed RDAC accomplishes the rapid convergence of the state variable (error vector) trajectories to the origin with reduced oscillations. Using the direct method of Lyapunov [36], the article describes a detailed analysis that assures the robust stability of the closed-loop dynamical system and adaptive laws to estimate the upper bound of unknown feedback gain, external disturbances, and model uncertainties. Computer-based simulations validate the theoretical analysis. The paper performs detailed comparative performance assessments with some related published results to verify the efficiency of the proposed RDAC. These assessments show that the proposed controller achieves quick stabilization (synchronization) with reduced oscillations, and the closed-loop is robustly stable for all admissible uncertainties.

The remaining paper comprises five sections to accomplish the stabilization and synchronization of a perturbed NSGC system with unknown parameters.

Section 2 describes the symbol and notations used in this paper and presents the mathematical model of the NSGC system. Section 3 formulates the chaos stabilization and synchronization problems for the NSGC system with unknown parameters. In Sections 4 and 5, the article discusses the controller design for stabilizing and synchronizing the NSGC system with unknown parameters, respectively. They provide a detailed stability analysis of the closed-loop, furnish the numerical simulations, and perform comparative studies. The paper concludes with some remarks in Section 6.

Section snippets

Notations and symbols

Table 1 shows the notations and symbols that are used in this paper. The nuclear spin generator system is a high-frequency oscillator used to define the precedence of magnetization (M_x, M_y, M_y) in a simple NMR experiment. An NMR active nuclei (usually a liquid containing Hydrogen-1, Phosphorus-1, or Carbon-13 nuclei) is surrounded by one or more radiofrequency (RF) coils, which are placed at the center of a strong static magnetic field B₀ [22]. In the laboratory frame of reference, an exciting

Problem 1: suppression of chaos in the NSGC system with unknown parameters

This sub-section formulates the problem of chaos suppression in the NSGC system with unknown parameters. When external disturbances ϕ_i(t), model uncertainties F(χ_i(t)), and control efforts u_i(t) act on the NSGC system (1), then (2) represents the closed-loop dynamics. ${\begin{matrix} {\dot{χ}}_{1} (t) = - ϱ_{2} χ_{1} (t) + χ_{2} (t) + F (χ_{1} (t)) + φ_{1} (t) + u_{1} (t) \\ {\dot{χ}}_{2} (t) = - ϱ_{2} χ_{2} (t) - χ_{1} (t) + ϱ_{2} ϱ_{3} χ_{2} (t) χ_{3} (t) + F (χ_{2} (t)) + φ_{2} (t) + u_{2} (t) \\ {\dot{χ}}_{3} (t) = ϱ_{1} ϱ_{2} - ϱ_{1} ϱ_{2} χ_{3} (t) - ϱ_{2} ϱ_{3} χ_{2}^{2} (t) + F (χ_{3} (t)) + φ_{3} (t) + u_{3} (t) \end{matrix}$ where ϱ₁ > 0, ϱ₂ > 0, and ϱ₃ > 0 are unknown.

Assumption 1

[25,37]. Computing bounds exist

Solution to problem 1

The main objective of this section is to design a state feedback controller u(t) ∈ R^{3 × 1} without cancelation of the nonlinear terms of the plant. The motivation behind this concept is the development of a robust closed-loop, which is insensitive to unknown model uncertainties, unknown external disturbances, and slow and smooth variations in the plant parameters. It converges the state variables of the system (2) to the equilibrium point at the origin and establishes the global asymptotic

Solution to problem 2

This section furnishes the solution to Problem 2. It introduces a new RDAC and provides the global asymptotic stability analysis of the closed-loop (8a).

Conclusion

This article develops a novel Lyapunov based direct adaptive control algorithm. It does not cancel the nonlinear terms appearing in the closed-loop. This structure of the closed-loop is robust to the parametric uncertainties and smooth variations in the plant parameters. The proposed controller accomplishes the suppression and synchronization of chaos in the nuclear spin generator chaotic system with unknown parameters. This controller achieves faster convergence of the perturbed state

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A Lyapunov-based direct adaptive controller for the suppression and synchronization of a perturbed nuclear spin generator chaotic system

Highlights

Abstract

Introduction

Section snippets

Notations and symbols

Problem 1: suppression of chaos in the NSGC system with unknown parameters

Solution to problem 1

Solution to problem 2

Conclusion

Chaos Solit. Fract.

App. Math. Comp.

App. Math. Comp.

App. Math. Comp.

Physica A

App. Math. Comp.

Commun. Nonlinear Sci. Numer. Simulat.

Chaos. Solit. Fract.

J. Comp. Appl. Math.

Chaos Solit. Fract.

Chaos Solit. Fract.

Chaos Solit. Fract.

Nonlinear Ana: Real World App.

J. Math. Anal. Appl.

Poincaré, Deterministic Chaos (Sensitivity to Initial Conditions), in C. Ruhla, The Physics of Chance: From Blaise Pascal to Niels Bohr

Controlling chaos using superior feedback technique with applications in discrete traffic models

Int. J. Fuzzy Syst.

Application of chaos synchronization technique and pattern clustering for diagnosis analysis of partial discharge in power cables

IEEE Access

Applications of chaotic dynamics in robotics

Int. J. Adv. Robot. Syst.

Chaotic dynamics for a class of single-machine-infinite bus power system

J. Vib. Chaos.

Non-identical synchronization, anti-synchronization and control of single machine infinite-bus power system via active control

Inform. Tech. Cont.