Nonlinear Quadratic Systems Research Articles

Experimental and artificial intelligence optimization of paint wastewater (PWW) coagulation using novel Phaseolus vulgaris seed extract (PVSE)

This study investigated the application of artificial intelligence algorithms (AIA) in the coagulation treatment of paint wastewater anchored by novel Phaseolus vulgaris seed extract (PVSE). Untreated wastewater discharge harms the ecosystem, and therefore harmful industrial effluent, such as paint wastewater, must be brought to safe discharge levels before being released into the environment. In addition to AIA, comprehensive characterization tests, coagulation kinetics, and process optimization were also executed. Characterization results revealed that total solid in the PWW was above allowable standard, justifying the need for effective particle decontamination. The XRD and FTIR characterization indicated that PVSE structure is amorphous with abundant amine groups. Results of analysis of variance (ANOVA) obtained from process modeling indicated that the coagulation-flocculation process was a nonlinear quadratic system (F-value = 45.51) which was mostly influenced by PVSE coagulant dosage (F-value = 222.48; standardized effect = 14.85). Artificial intelligence indicated that neural network training effectively captured the nonlinear nature of the system in ANN (RMSE = 0.00040194; R = 0.98497), and ANFIS (RMSE = 0.003961) algorithms. Regression coefficient obtained from process modeling highlighted the suitability of RSM (0.9662), ANN (0.9739), and ANFIS (0.9718) in forecasting the coagulation-flocculation process, while comparative statistical appraisal authenticated the superiority of ANN model over RSM and ANFIS models. The coagulation kinetics experiment, which used a coagulation kinetic model, revealed a constant flocculation constant (Kf-value) for all jar test batches and a strong association between the Menkonu coagulation-flocculation constant (Km) and Kf values. Best removal efficiency of 97.01 % was obtained using ANN coupled genetic algorithm optimization (ANN-GA) at PVSE dosage of 4 g/L, coagulation time of 29 min and temperature of 25.1oC.

MONTE CARLO AND NUMERICAL METHODS TO SOLVE THE TIME SERIES MODEL

In this paper we will solve the nonlinear system of equations in the parameters of the time series model by Monte Carlo methods and by numerical methods. When we identify the variance and the inter-covariances of time series, we obtain, dividing by variance, a quadratic nonlinear system of equation that does not contain the variance of wihite noise. We use only the autocorrelation function. AMS Subject Classification: 62M10, 91B84, 65C05.

Control of a three-wheeled omnidirectional mobile robot via a mixed FTB/H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal H_\\infty $$\\end{document} approach

In this paper, the problem of guidance and motion control of mobile robots is addressed and solved within the novel framework of the mixed finite-time/H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal H_\\infty $$\\end{document} control theory of nonlinear quadratic systems (NLQSs). Starting from a NLQS describing the dynamics of omnidirectional mobile platforms, the main tasks performed for controlling in closed loop the motion of omnidirectional robots can be conveniently formulated as a mixed finite-time/H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal H_\\infty $$\\end{document} control problem. A robust motion controller, which can effectively rejects disturbances deviating the robot platform from a planned path, can be designed after choosing a linear state-feedback structure for the controller. The synthesis problem is solved through some sufficient conditions contemplating both norm-bounded disturbances and sets constraining initial and terminal conditions, together with a finite-time bound on the output transient. Therefore, for all the allowable uncertainties, in presence of nonzero initial conditions and exogenous disturbance inputs which are possible within an unstructured environment, the motion control tasks can be accomplished through optimal H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal H_\\infty $$\\end{document} performance by simultaneously guaranteeing that the NLQS, which governs in closed loop the robot platform, is finite-time bounded. Finally, the applicability and control performance of the design approach have been evaluated through numerical simulations.

Dynamics and Bifurcations in a Quadratic Nonlinear System with Univariate Product Vector Fields Part II

Dynamics and Bifurcations in a Quadratic Nonlinear System with Univariate Product Vector Fields Part II

Bispectral analysis and simulation modeling of quadratic nonlinear system with specific turbulent-fluctuation-excitation-response types

There exists a kind of quadratic nonlinear system with specific type of turbulent fluctuation excitation in nature, which belongs to a special non-Gaussian input signal system. Its characteristic is that the input signal spectrum is generated by turbulent fluctuations, and the power spectrum distribution of this turbulence fluctuation signal is close to Gaussian distribution. Starting with the work of Choi et al. (<ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1016/0022-460X(85)90370-0">1985 <i>J. Sound Vib</i><i>.</i> <b>99</b> 309</ext-link>) and Kim et al. [<ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://ieeexplore.ieee.org/document/1145286">1987 <i>IEEE J. Ocean. Eng.</i> <b>OE-12</b> 568</ext-link>), we extend the simulation of a specific turbulent fluctuation excited response-type quadratic nonlinear system represented by the wave excited mooring ship response, and fully develop the internal development of turbulence based on bispectral analysis technology. We also extend the simulation system and conduct systematic modeling analysis. The complete iterative method [<ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1088/1402-4896/ab725f">2020 <i>Phys. Scr.</i> <b>95</b> 055202</ext-link>] is used to solve the model, and calculate the linear transfer function and quadratic nonlinear transfer function. The comparison of simulation and modeling results with the real systems and their models confirms the correctness of the results from system simulation, system modeling, and model solving. The results obtained are all in line with expectations. The coherence analysis shows that the quadratic coherence of the random wave-ship swaying system is much greater than the linear coherence, but the linear coherence of the fully developed turbulence is greater for the near Gaussian input type. The reverse computation verification or comparison with real systems indicates that the turbulence simulation and system modeling method in this work have good accuracy and high efficiency in solving algorithms, and the research results can be effectively applied to the model description and system analysis of the quadratic nonlinear systems related to specific turbulent fluctuation excitation response.

Dynamics and Bifurcations in a Quadratic Nonlinear System with Univariate Product Vector Fields

Dynamics and Bifurcations in a Quadratic Nonlinear System with Univariate Product Vector Fields

State and adaptive disturbance observer co‐design for incrementally quadratic nonlinear descriptor systems with nonlinear outputs

AbstractThe problem of designing state and adaptive disturbance observers for a class of nonlinear descriptor systems with the disturbance generated by an unknown exogenous system is addressed in this paper. Especially, the nonlinear terms satisfy incremental quadratic constraints parameterized by a bunch of multiplier matrices, which can represent various classes of nonlinearities. For three different cases in the output equation, the state and adaptive disturbance observers are proposed. Some sufficient conditions formulated in terms of linear matrix inequalities and algebraic constraints are derived to guarantee that state and disturbance error systems, simultaneously, are asymptotically stable with performance level. The availability of proposed observers is validated by two practical examples.

Novel closed-loop controllers for fractional nonlinear quadratic systems

<abstract><p>A novel closed-loop optimal controller for fractional nonlinear quadratic optimal control problems is introduced. By using a new idea, the optimality conditions for the fractional nonlinear problems are derived. The linearized Riccati fractional order differential equation is derived and a new solution method is given for the first time, which can be applied to integer order nonlinear optimal control problems. The proposed closed-loop controller is applied to illustrative examples. Novel unprecedented processes of designing a variable linear controller and of finding the optimal performance index for integer order nonlinear systems are presented.</p></abstract>

Quantum algorithm for solving a quadratic nonlinear system of equations

Solving a quadratic nonlinear system of equations (QNSE) is a fundamental, but important, task in nonlinear science. We propose an efficient quantum algorithm for solving $n$-dimensional QNSE. Our algorithm embeds QNSE into a finite-dimensional system of linear equations using the homotopy perturbation method and a linearization technique; then we solve the linear equations with a quantum linear system solver and obtain a state which is $\epsilon$-close to the normalized exact solution of the QNSE with success probability $\Omega(1)$. The complexity of our algorithm is $O({\rm polylog}(n/\epsilon))$, which provides an exponential improvement over the optimal classical algorithm in dimension $n$, and the dependence on $\epsilon$ is almost optimal. Therefore, our algorithm exponentially accelerates the solution of QNSE and has wide applications in all kinds of nonlinear problems, contributing to the research progress of nonlinear science.

A Novel Highly Nonlinear Quadratic System: Impulsive Stabilization, Complexity Analysis, and Circuit Designing

This work introduces a three‐dimensional, highly nonlinear quadratic oscillator with no linear terms in its equations. Most of the quadratic ordinary differential equations (ODEs) such as Chen, Rossler, and Lorenz have at least one linear term in their equations. Very few quadratic systems have been introduced and all of their terms are nonlinear. Considering this point, a new quadratic system with no linear term is introduced. This oscillator is analyzed by mathematical tools such as bifurcation and Lyapunov exponent diagrams. It is revealed that this system can generate different behaviors such as limit cycle, torus, and chaos for its different parameters’ sets. Besides, the basins of attractions for this system are investigated. As a result, it is revealed that this system’s attractor is self‐excited. In addition, the analog circuit of this oscillator is designed and analyzed to assess the feasibility of the system’s chaotic solution. The PSpice simulations confirm the theoretical analysis. The oscillator’s time series complexity is also investigated using sample entropy. It is revealed that this system can generate dynamics with different sample entropies by changing parameters. Finally, impulsive control is applied to the system to represent a possible solution for stabilizing the system.

Emerging approaches for nonlinear parameter varying systems

Emerging approaches for nonlinear parameter varying systems

Periodic event‐triggered control for incrementally quadratic nonlinear systems

AbstractPeriodic event‐triggered control (PETC) evaluates triggering conditions only at periodic sampling times, based on which it is decided whether the controller needs to be updated. This article investigates the global stabilization of nonlinear systems that are affected by external disturbances under PETC mechanisms. Sufficient conditions are provided to ensure the resulting closed‐loop system is input‐to‐state stable (ISS) for the state feedback and the observer‐based output feedback configurations separately. The sampling period and the triggering functions are chosen such that the ISS‐Lyapunov function of continuous dynamics is also the ISS‐Lyapunov function of the overall system. Based on that, sufficient conditions in the form of linear matrix inequalities are provided for the PETC design of incrementally quadratic nonlinear systems. Two simulation examples are provided to illustrate the effectiveness of the proposed method.

On the Convergence, Dynamics and Applications of a New Class of Nonlinear System Solvers

In this paper, a uni-parametric class of third-order iterative algorithms for solving systems of nonlinear equations is proposed. The local convergence of the suggested schemes is studied using generalized Lipschitz-type condition on the first-order Frechet derivative. Furthermore, we analyze the numerical stability of the new methods applying complex dynamics tool. The nonlinear systems related to the equation of molecular interaction, a boundary value problem, the integral equation from Chandrasekhar’s work, etc. are discussed. The most interesting fact about the proposed third-order family is that it generates a super convergent scheme (for $$\gamma =2$$ ) for solving quadratic nonlinear systems (QNS). This particular method produces much better results for QNS in comparison with other third-order schemes. Also, it is observed that the approximate computational order of convergence (ACOC) of this scheme (for $$\gamma =2$$ ) is approximately four (3.99–4.00) while solving QNS.

A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

In this paper, we consider the quadratic nonlinear Schrodinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrodinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.

Observer-Based Controllers for Incrementally Quadratic Nonlinear Systems With Disturbances

Robust global stabilization of nonlinear systems by observer-based feedback controllers is a challenging task. This article investigates the problem of designing observer-based stabilizing controllers for incrementally quadratic nonlinear systems with external disturbances. The nonlinearities considered in the system model satisfy the incremental quadratic constraints, which are characterized by incremental multiplier matrices and encompass many common nonlinearities. The simultaneous search for the observer and the controller gain matrices is formulated as a feasibility problem of linear matrix inequalities, for two parameterizations (i.e., the block diagonal parameterization and the block anti-triangular parameterization) of the incremental multiplier matrices, respectively. The closed-loop system implementing the observer-based feedback controller is proven to be input-to-state stable with respect to external disturbances. Using the proposed continuous-time observer-based controllers, event-triggered controllers with time regularization are constructed for globally Lipschitz systems, such that the closed-loop system is Zeno-free and input-to-state practically stable.

Improved Delay-Dependent Stability of Nonlinear Quadratic TS Fuzzy Systems

This paper focuses on the problem of delay-dependent stability for nonlinear quadratic Takagi–Sugeno (TS) fuzzy systems with time-varying delay using the input–output approach. The results are based on the model transformation by employing a three-terms approximation of delayed state vector. By applying the scaled small-gain theorem and Lyapunov–Krasovskii functional, the stability criteria is obtained in terms of linear matrix inequalities. Furthermore, the Wirtinger-based integral inequality approach has been employed to derive less conservative results. Finally, the numerical examples are provided to demonstrate the effectiveness of the obtained results and for comparison with previous work.

Global dynamics below the ground state for the focusing Schrödinger equation with a potential

In this paper, we consider the nonlinear Schrodinger equation with a real-valued potential $$V=V(x)$$ . We study global behavior of solutions to the equation with data below the ground state under some conditions for the potential V and prove a scattering result and a blowing-up result in mass-supercritical and energy-subcritical. Our proof of the scattering result is based on an argument by Dodson–Murphy (Proc Am Math Soc 145(11):4859–4867, 2017). The proof of the blowing-up or growing-up result without radially symmetric assumption is based on the argument by Du–Wu–Zhang (Discrete Contin Dyn Syst 36(7):3639–3650, 2016). We can exclude the possibility of the growing-up result by the argument (Inui et al. in Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrodinger system without mass resonance, arXiv:1810.09153.; Nakanishi and Schlag in Calc Var Partial Differ Eq 44(1-2):1–45, 2012; Glassey in J Math Phys 18(9):1794–1797, 1977) if “the data and the potential are radially symmetric” or “the data has finite variance.”

Analytical solutions of periodic motions in a first-order quadratic nonlinear system

In this paper, analytical solutions of period-1 motion of a first-order quadratic nonlinear system with both parametric excitation and external excitation are investigated through the generalized harmonic balance method. The analytical solutions of the system are obtained via solving the coefficients of all harmonic terms at the equilibrium position. The precision of analytical solutions is guaranteed via convergence study of harmonic balance terms. Stability analysis is carried out via eigenvalue analysis. The analytical solutions are different from the perturbation analysis solutions. Moreover, the trajectories of periodic motions obtained from analytical solutions can better explain the dynamics of the system. To verify the accuracy of analytical solutions, numerical simulation are performed and the simulation results are compared with the analytical solutions. The harmonic spectra are also presented to illustrate of the contribution of each harmonic term on a periodic motion.

Adaptive State Observers for Incrementally Quadratic Nonlinear Systems with Application to Chaos Synchronization

This paper is concerned with the problem of adaptive state observer design for a class of incrementally quadratic nonlinear systems with parameter uncertainty. Its nonlinearity satisfies the condition of incremental quadratic constraints which contains many commonly encountered nonlinearities in existing literature as some special cases. A circle criterion-based adaptive observer is first designed to expand the freedom of observer design. Based on the incremental quadratic constraint condition, sufficient conditions ensuring asymptotic stability of estimation error dynamics are established in terms of linear matrix inequalities. Finally, the adaptive observer is applied to the synchronization design of chaotic systems with parameter uncertainty, which fully demonstrates the effectiveness of the proposed observer.

H∞ Filter Design for T–S Fuzzy Nonlinear Quadratic Systems with Time-Varying Delay

This paper focuses on the problem of H∞ filtering for continuous-time T–S fuzzy nonlinear quadratic systems with a varying time delay. To ensure H∞ performance bound for the augmented nonlinear quadratic system constituted by the original system and the filter, we use a quadratic Lyapunov–Krasovskii functional and some inequality conditions. We aim to derive a less conservative result than the existing ones by designing the filter parameters via linear matrix inequalities. Free parameters provide extra degrees of freedom in optimizing the guaranteed H∞ performance and lead to a less conservative design.