Multidimensional Nonlinear Systems Research Articles

Control of large amplitude limit cycle of a multi-dimensional nonlinear dynamic system of a composite cantilever beam

For the first time, a control strategy based on Fuzzy Sliding Mode Control is implemented in the control of a large amplitude limit cycle of a composite cantilever beam in a multi-dimensional nonlinear form. In the dynamic model establishment of the investigated structure, the higher-order shearing effect is applied, as well as the second-order discretization. Numerical simulation demonstrates that a multi-dimensional nonlinear dynamic system of the investigated structure is demanded for accurate estimation of large amplitude limit cycle responses. Therefore, a control strategy is employed to effectively suppress such responses of the beam in multi-dimensional nonlinear form.

Periodic on-off operations of the self-cycling ethanol fermentation process: Stability analysis and optimal operation strategies

Fuel ethanol is highlighted for being renewable and carbon neutral, but a multi-scale challenge exists that hampers its role as a future energy substitute, and with the breakthrough of cellulose pretreatment and enzyme hydrolysis technology, fermentation itself has become one of the main limiting factors for process enhancement. To tackle the problem, this paper studies periodic operation strategies to improve process performance and avoid possible instabilities. Firstly, superiority by periodic operation over the corresponding steady-state operation is verified. Then, to settle the problem of possible instabilities under impulsive control, stability of state-impulsive control is conducted and for the studied case, any self-cycling fermentation starts from batch mode with 50 % discharge-and-refill portion exhibits superior repeatability property, which allows continuous fermentation to operate under very slow overall dilution rate to cope with the extreme slow fermentation rate. Finally, on-off operation problem is formulated that is governed by a multidimensional nonlinear system, affine to the manipulated variable that is under the isoperimetric constraint; with the use of Pontryagin maximum principle, the optimal synthesis of the optimal strategies under integral constraint is computed out, and a practically plausible on-off operation strategy for ethanol fermentation is formulated with guaranteed periodic stability and optimality.

Effective Estimation of Trapping/Stability Regions and Bilateral Solutions’ Bounds for Some Multidimensional Nonlinear Systems with Time-Varying Coefficients

Effective Estimation of Trapping/Stability Regions and Bilateral Solutions’ Bounds for Some Multidimensional Nonlinear Systems with Time-Varying Coefficients

NOVEL UNIFIED STABILITY CRITERION FOR FRACTIONAL-ORDER TIME DELAY SYSTEMS WITH STRONG RESISTANCE TO FRACTIONAL ORDERS

In this study, a novel unified stability criterion is first proposed for general fractional-order systems with time delay when the fractional order is from 0 to 1. Such a new unified criterion has the advantage of having an initiative link with the fractional orders. A further advantage is that the corresponding asymptotic stability theorem, derived from the proposed criterion used to analyze the asymptotic stability, is only slightly affected by the change of the fractional order. In addition, the unified stability criterion is applied to general multi-dimensional nonlinear fractional-order systems with time delays, the corresponding asymptotic stability criterion is applied by combining the vector Lyapunov function with the M-matrix method. Compared with the traditional stability criterion, the unified stability criterion is slightly influenced by the changing fractional order and large time delays. The reliability and effectiveness of the novel uniform stability criterion were verified through three representative examples.

Chaotic vibration control of a composite cantilever beam

In this research, an adaptive control strategy adapted from fuzzy sliding mode control is established and applied in chaotic vibration control of a multiple-dimension nonlinear dynamic system of a laminated composite cantilever beam. The third order shearing effect on the vibration of the beam is considered in the nonlinear dynamic model establishment, and the Hamilton principle as well as the Galerkin method is employed. It is discovered that a multi-dimensional nonlinear dynamic system of the cantilever beam needs to be considered for accurate vibration estimation. Therefore, the control strategy appropriate for the chaotic vibration control of a multiple-dimension system of the laminated composite beam is necessary, and then proves to be effective in chaotic vibration control in numerical simulation.

Objective diagnosis of machine learning method applicability to land comprehensive carrying capacity evaluation: A case study based on integrated RF and DPSIR models

The evaluation of land comprehensive carrying capacity (LCCC) is a popular topic in regional sustainable development and land science research. However, these evaluations are often not objective due to the complexity and nonlinear characteristics of the LCCC evaluation indicators. This study provides a new framework for using machine learning methods to objectively evaluate LCCC. The Driver-Pressure-State-Impact-Response (DPSIR) indicator conceptual framework was used for the spatial visualisation of indicators and was then combined with the random forest (RF) model to optimise the LCCC evaluation. The results showed the following: (1) The accuracy of the integrated RF and DPSIR model was better than that of traditional support vector machine (SVM) and principal component analysis (PCA) methods, indicating that RF model was more suitable for LCCC evaluations. (2) The contribution of the DPSIR indicators after RF optimisation was significantly different from that of the traditional analytic hierarchy process (AHP), the contribution of the influence subsystem (I) (48.7%) was enhanced, while the contribution of the drive subsystem (D) (6.8%) was weakened, demonstrating that after RF optimisation, the DPSIR was more conducive for handling complex nonlinear systems and objectively reflecting indicator contributions. (3) The objective attributes of the DPSIR indicators were optimised through the processing of spatial visualisation models, indicating that RF was suitable for processing evaluations with a multidimensional spatial heterogeneity index system. The LCCC evaluation model can be applied to other carrying capacity case studies and operational processes and provide a scientific method for handling the multidimensional nonlinear evaluation index system and objective evaluation of LCCC.

Novel methods for reliability study of multi-dimensional non-linear dynamic systems

This research presents two unique techniques for engineering system reliability analysis of multi-dimensional non-linear dynamic structures. First, the structural reliability technique works best for multi-dimensional structural responses that have been either numerically simulated or measured over a long enough length to produce an ergodic time series. Second, a novel extreme value prediction method that can be used in various engineering applications is proposed. In contrast to those currently used in engineering reliability methodologies, the novel method is easy to use, and even a limited amount of data can still be used to obtain robust system failure estimates. As demonstrated in this work, proposed methods also provide accurate confidence bands for system failure levels in the case of real-life measured structural response. Additionally, traditional reliability approaches that deal with time series do not have the benefit of being able to handle a system's high dimensionality and cross-correlation across several dimensions readily. Container ship that experiences significant deck panel pressures and high roll angles when travelling in bad weather was selected as the example for this study. The main concern for ship transportation is the potential loss of cargo owing to violent movements. Simulating such a situation is difficult since waves and ship motions are non-stationary and complicatedly non-linear. Extreme movements greatly enhance the role of nonlinearities, activating effects of second and higher order. Furthermore, laboratory testing may also be called into doubt due to the scale and the choice of the sea state. Therefore, data collected from actual ships during difficult weather journeys offer a unique perspective on the statistics of ship movements. This work aims to benchmark state-of-the-art methods, making it possible to extract necessary information about the extreme response from available on-board measured time histories. Both suggested methods can be used in combination, making them attractive and ready to use for engineers. Methods proposed in this paper open up possibilities to predict simply yet efficiently system failure probability for non-linear multi-dimensional dynamic structure.

Direct computation of critical plume quantities required for initial assessment of contaminated sites

Estimates of plume extremes such as maximum plume width (Wmax) and its location (Xwmax), maximum plume area (Amax), and maximum plume length (Lmax), which are vital indicators of any site assessment work, mostly requires the use of rather complicated numerical approaches and a large number of site information. This paper focuses on simpler (semi)analytical quantification of Wmax and Xwmax, and their characterization with respect to source geometry (source width Sw and source thickness St) for estimating Amax. A direct computation of Wmax and Xwmax rather becomes a multi-dimensional nonlinear system problem for which only a few iterative methods are available. The challenge magnifies further as limited information on Wmax and Xwmax are available for obtaining an initial guess of the solution. Solving over 1000 synthetic problems, this work finds the Newton–Krylov method as the most suitable nonlinear system solver. Among the most important were the limiting results such as Lmax/8 ¡Xwmax¡Lmax/8 and 1×Sw¡Wmax¡1.4×Sw. These limiting results simplify the finding of initial guesses for Wmax and Xwmax. Further characterization of Wmax and Xwmax with Lmax, St, and Sw provided an empirical relation between plume and source areas, in addition to an approximation of Amax. A field site data illustrate the potential applications of the methods developed in this work.

Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: Applications to the Serre-Green-Naghdi equations

In this paper we study two multidimensional nonlinear dispersive systems: the Serre-Green-Naghdi (SGN) equations describing dispersive shallow water flows, and the Iordanskii-Kogarko-Wijngaarden (IKW) equations describing fluids containing small compressible gas bubbles. These models are Euler-Lagrange equations for a given Lagrangian and share common mathematical structure, namely the dependence of the pressure on material derivatives of macroscopic variables. We develop a generic dispersive model such that SGN and IKW systems become its special cases if only one specifies the appropriate Lagrangian, and then use the extended Lagragian approach proposed in Favrie and Gavrilyuk (2017) to build its hyperbolic approximation. The new approximate model is unconditionally hyperbolic for both SGN and IKW cases, and accurately describes dispersive phenomena, which allows to impose discontinuous initial data and study dispersive shock waves. We consider the 2-D hyperbolic version of SGN system as an example for numerical simulations and apply a second order implicit-explicit scheme in order to numerically integrate the system. The obtained 1-D and 2-D results are in close agreement with available exact solutions and numerical tests.

Optimal dichotomy of temporal scales and boundedness/stability of time-varying multidimensional nonlinear systems

This paper develops a new approach to the estimation of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficients—an essential task in various engineering and natural science applications. Known approaches to assessing the stability of such systems rest on the utility of Lyapunov functions and Lyapunov first approximation methodologies that typically provide conservative and computationally elaborate criteria for multidimensional systems of this category. Adequate criteria of boundedness of solutions to nonhomogeneous systems of this kind are rare in contemporary literature. Lately, we develop a new approach to these problems which rests on bounding the evolution of the norms of solutions to initial systems by matching solutions of a scalar auxiliary equation we introduced in Pinsky and Koblik (Math Probl Eng 2020), Pinsky (J Nonlinear Dyn 103:517–539, 2021, Stability and boundedness of solutions to some multidimensional nonautonomous nonlinear systems, 2022). Still, the technique advanced in Pinsky (Math. Problems in Eng, 2022:5098677, https://doi.org/10.1155/2022/5098677) rests on the assumption that the average of the linear components of the underlying system is defined by a stable matrix of general position. The current paper substantially amplifies the application domain of this approach. It is merely assumed that the time-dependent linear block of the underlying system can be split into slow- and fast-varying components by application of any smoothing technique. This dichotomy of temporal scales is determined by the optimal criterion reducing the conservatism of our estimates. In turn, we transform the linear subsystem with a slow-varying matrix in a diagonally dominant form by successive applications of the Lyapunov transforms. This prompts the development of novel scalar auxiliary equations embracing the estimation of the norms of solutions to our initial systems. Next, we formulate boundedness/stability criteria and estimate the relevant regions of the underlying systems using analytical and abridged numerical reasoning. Lastly, we authenticate the developed methodology in inclusive simulations.

Active Vibration Control of Nonlinear Elastic Wire Rope in Mine Hoisting System

Based on the importance of the safe operation of the mine hoist mechanism, an adaptive control method based on Fuzzy Sliding Mode Control (FSMC) was applied to synchronize the large-amplitude response of geometrically nonlinear elastic wire rope in mine hoist systems. To efficiently and precisely synchronize the lateral vibration of a wire rope under the adaptive synchronizing method, the control equation of the elastic wire rope was given considering the large von Karman geometrical deformation, and it was converted to a six-dimensional nonlinear dynamical system using third-order Galerkin discretization. Based on this dynamic system, the existing FSMC is improved, so that the improved control strategy will be available for the active dynamic control of multidimensional nonlinear dynamical systems. The research shows that the control strategy can effectively synchronize the large-amplitude response of wire rope, thus providing the basis for dynamic design, fault diagnosis, and safe operation of the mine hoist system.

Novel methods for wind speeds prediction across multiple locations

This article provides two unique methodologies that may be coupled to study the dependability of multidimensional nonlinear dynamic systems. First, the structural reliability approach is well suited for multidimensional environmental and structural reactions and is either measured or numerically simulated over sufficient time, yielding lengthy ergodic time series. Second, a unique approach to predicting extreme values has technical and environmental implications. In the event of measurable environmental loads, it is also feasible to calculate the probability of system failure, as shown in this research. In addition, traditional probability approaches for time series cannot cope effectively with the system's high dimensionality and cross-correlation across dimensions. It is common knowledge that wind speeds represent a complex, nonlinear, multidimensional, and cross-correlated dynamic environmental system that is always difficult to analyze. Additionally, global warming is a significant element influencing ocean waves throughout time. This section aims to demonstrate the efficacy of the previously mentioned technique by applying a novel method to the Norwegian offshore data set for the greatest daily wind cast speeds in the vicinity of the Landvik wind station. This study aims to evaluate the state-of-the-art approach for extracting essential information about the extreme reaction from observed time histories. The approach provided in this research enables the simple and efficient prediction of failure probability for the whole nonlinear multidimensional dynamic system.

Novel reliability method for multidimensional nonlinear dynamic systems

The paper describes a novel structural reliability method, particularly suitable for multi-dimensional structural responses, either measured or numerically simulated over sufficient period of time, resulting in sufficiently long ergodic time series. Unlike other reliability methods the new method does not require to re-start simulation each time system fails, in case of numerical simulation. In case of measured structural response, an accurate prediction of system failure probability is also possible as illustrated in this study. Moreover, classic reliability methods, dealing with time series do not have an advantage of dealing easily with system high dimensionality and cross-correlation between different dimensions.As an example for this reliability study was chosen container ship subjected to large deck panel stresses and extreme roll angles occurring during sailing in harsh weather. Risk of losing containers due to extreme motions is primary concern for ship transport. Due to non-stationarity and complicated nonlinearities of both waves and ship motions, it is a considerable challenge to model such a phenomenon. In case of extreme motions, the role of nonlinearities dramatically increases, activating effects of second and higher order. Moreover, laboratory tests may also be questioned because of the scaling and the sea state choice. Therefore, data measured on actual ships during their voyages in harsh weather provides a unique insight into statistics of ship motions.The aim of this work is to benchmark state of art method, which makes it possible to extract the necessary information about the extreme response from onboard measured time histories. The method proposed in this paper opens up the possibility to predict simply and efficiently failure probability for nonlinear multi-dimensional dynamic system as a whole.

Globally-evolving-based generalized density evolution equation for nonlinear systems involving randomness from both system parameters and excitations

It has long been one of the main challenges in science and engineering to capture the probabilistic response of high-dimensional nonlinear stochastic dynamic systems involving double randomness, i.e. randomness in both system parameters and excitations. For this purpose, a globally-evolving-based generalized density evolution equation (GE-GDEE) is established. Generally, for a multi-dimensional nonlinear system involving double randomness, if one single physical quantity as a response of the system is of interest, a GE-GDEE, as a two-dimensional partial differential equation (PDE) governing the probability density function (PDF), can be derived. The effective drift coefficients, which represent the physically driving force function in the GE-GDEE, can be determined based on the data from some representative deterministic dynamic analyses of the underlying physical system. A new estimator for effective drift coefficients is developed based on the vine copulas. Once the effective drift coefficients are determined, the GE-GDEE can be solved to capture the probability distributions of the quantities of interest. Several numerical examples, including linear and nonlinear multi-degree-of-freedom (MDOF) systems subjected to white noise or non-stationary earthquake ground motions, are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Stability and Boundedness of Solutions to Some Multidimensional Time-Varying Nonlinear Systems

Assessment of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent and nonperiodic coefficients is an important problem in various applied areas which has no adequate resolution yet. Most of the known techniques provide computationally intensive and conservative stability criteria in this field and frequently fail to estimate the regions of boundedness/stability of solutions to the corresponding systems. Recently, we outlined a new approach to this problem which is based on the analysis of solutions to a scalar auxiliary equation bounded from the above time histories of the norms of solutions to the original system. This paper develops a novel technique casting the auxiliary equation in a modified form which extends the application domain and reduces the computational burden of our prior approach. Consequently, we developed more general boundedness/stability criteria and estimated trapping/stability regions for some multidimensional nonlinear systems with nonperiodic time-dependent coefficients that are common in various application domains. This lets us to assess in target simulations the extent of boundedness/stability of some multidimensional, nonlinear, and time-varying systems which were intractable with our prior technique.

A unified formalism of the GE-GDEE for generic continuous responses and first-passage reliability analysis of multi-dimensional nonlinear systems subjected to non-white-noise excitations

The stochastic response analysis and first-passage reliability evaluation of multi-dimensional nonlinear systems subjected to non-white-noise engineering dynamic excitations have long been challenging problems. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE) provides a promising tool and is extended in the present paper for these purposes. Firstly, a more general derivation of the GE-GDEE is given for generic Markov or non-Markov path-continuous stochastic processes. In this unified formalism, the GE-GDEE can be established with respect to one- or two-dimensional path-continuous response(s) of interest in a multi-dimensional system subjected to non-white-noise excitations. Further, for the first-passage reliability evaluation, a new process absorbed at the boundary (ABP) corresponding to failure criterion is constructed, and the GE-GDEE of ABP can be established as a one- or two-dimensional partial differential equation (PDE). This equation can then be solved to obtain the probability density in the safety domain, and the first-passage reliability can thereby be obtained via an integral. Meanwhile, the eligibility of imposing an absorbing boundary condition for the first-passage reliability is rigorously proved. Numerical algorithms are elaborated. Several examples of first-passage reliability analysis of multi-dimensional linear/nonlinear systems subjected to white/non-white noise are illustrated, demonstrating the efficiency and accuracy of the proposed method.

Physically Driven Exact Dimension Reduction of a Class of Nonlinear Multidimensional Systems Subjected to Additive White Noise

Physically Driven Exact Dimension Reduction of a Class of Nonlinear Multidimensional Systems Subjected to Additive White Noise

Design of Three Control Algorithms for an Averaging Tank with Variable Filling

Abstract An averaging tank with variable filling is a nonlinear multidimensional system and can thus be considered a complex control system. General control objectives of such object include ensuring stability, zero steady-state error, and achieving simultaneously shortest possible settling time and minimal overshoot. The main purpose of this research work was the modeling and synthesis of three control systems for an averaging tank. In order to achieve the intended purpose, in the first step, a mathematical model of the control system was derived. The model was adapted to the form required to design two out of three planned control systems by linearization and reduction of its dimensions, resulting in two system variants. A multivariable proportional-integral-derivative (PID) control system for the averaging tank was developed using optimization for tuning PID controllers. State feedback and output feedback with an integral action control system for the considered control system was designed using a linear-quadratic regulator (LQR) and optimization of weights. A fuzzy control system was designed using the Mamdani inference system. The developed control systems were tested using theMATLAB environment. Finally, the simulation results for each control algorithm (and their variants) were compared and their performance was assessed, as well as the effects of optimization in the case of PID and integral control (IC) systems.

A Multi-Simplex Imperialist Competitive Paradigm for Solving Nonlinear Physical Systems

This paper proposes a novel gradient free multi-simplex topology fabric aided imperialist competitive algorithm (ICA) for solving nonlinear systems of algebraic equations by transforming them to equivalent global optimization problems. The high dependence of traditional gradient based solvers of such systems on initial guesses and the Jacobeans resulting in false convergence is the main motivation behind the present work. The present work provides a mechanism for enhancing exploitation powers of imperialist search phase of the algorithm and hence improves the convergence speed. The variants emerging from the proposed approach are applied to diverse nonlinear systems arising in different scientific areas and the results so obtained are analyzed in details. Based on the analysis of empirical results on complex benchmark models, it is observed that the proposed enhancement of ICA has not only boosted the problem solving power of the imperialist system when applied multidimensional nonlinear physical systems but also emerged as an efficient gradient/initial guess free alternative solver with high accuracy for nonlinear systems.

Decay rate estimates for a new class of multidimensional nonlinear Bresse systems with time-dependent dissipations

This paper studies the existence of weak solutions and their asymptotic behaviour to the initial boundary value problem for a multidimensional nonlinear Bresse system. The existence of the solutions of the problem is obtained by appliying Galerkin method. Then, we obtain an explicit decay rate estimation dependent on both the strain-caused stress and the damping terms by using the multiplier method with integral.