Simple Cell Mapping Research Articles

A Geometric Numerical Integration with Simple Cell Mapping for Global Analysis of Nonlinear Dynamical Systems

In order to improve the efficiency in computing the global properties of nonlinear systems, some effective methods have been proposed for the global analysis of these systems. However, there are only few investigations focusing on the numerical algorithms of the system response trajectories. This paper presents a higher-efficiency geometric numerical integration method with simple cell mapping for solving the basins of attraction of nonlinear systems. This numerical algorithm is based on the rule of Lie derivative, using it to calculate the trajectories of the nonlinear systems. Then, the global structure is studied by the numerical algorithm associated with a simple cell mapping method. Compared with the traditional Runge–Kutta methods of orders 4 and 5, it is demonstrated that the proposed Lie derivative iterative algorithm has significant advantages in efficiency.

Multi-Objective Optimization of an Elastic Rod with Viscous Termination

In this paper, we study the multi-objective optimization of the viscous boundary condition of an elastic rod using a hybrid method combining a genetic algorithm and simple cell mapping (GA-SCM). The method proceeds with the NSGAII algorithm to seek a rough Pareto set, followed by a local recovery process based on one-step simple cell mapping to complete the branch of the Pareto set. To accelerate computation, the rod response under impulsive loading is calculated with a particular solution method that provides accurate structural responses with less computational effort. The Pareto set and Pareto front of a case study are obtained with the GA-SCM hybrid method. Optimal designs of each objective function are illustrated through numerical simulations.

Nonlinear oscillations and bifurcations of a multistable truss and dynamic integrity assessment via a Monte Carlo approach

In the present work, a truss model inspired by von Mises and Bergan trusses is considered. The resulting structure is a phenomenological model that represents the behavior of many engineering structures and may exhibit in-plane snap-through or pitchfork unstable bifurcation and/or lateral buckling, leading two a multiwell potential function. The topology of the resulting potential function generates a complex bifurcation scenario under harmonic forcing with multiple attractors. The first aim of this work is to investigate the nonlinear vibrations of the proposed model. The interaction of the bifurcation phenomena leads to motions in two-, four- and six-dimensional phase-space. The use of different integrity measures is essential for assessing the robustness of the driven structure to unpredictable finite disturbances of a both static and dynamic nature in such cases. However of the various numerical tools for global dynamic analysis, few are well suited for higher-dimensional systems. In particular the evaluation of integrity measures of multidimensional system requires significant computational resources, being a rather time-consuming procedure. Thus the second aim of the paper is to propose numerical methodologies for evaluating the dynamic integrity measures of multidimensional systems by a Monte Carlo approach. Dynamic integrity analysis of the resulting multidimensional system is performed using three integrity measures, global and local integrity measures (GIM and LIM, respectively) and the integrity factor. Three algorithms based on the Monte Carlo method are proposed to estimate these measures by means of sampling initial conditions. We demonstrate that the proposed approach has a major advantage in comparison to classical methods, since basins of attraction are not needed. Computationally expensive procedures, such as simple cell mapping, are not required. Therefore, the proposed approach has the potential to estimative the dynamic integrity of high dimensional systems with less computational effort, as shown by the obtained results.

Global Invariant Manifolds of Dynamical Systems with the Compatible Cell Mapping Method

An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

Six-dimensional basins of attraction computation on small clusters with semi-parallelized SCM method

In modern engineering applications it is needed to determine the robustness of stable steady states, which depends on the shape and size of the associated basins of attraction. Once the basins are known, the quantitative measures can be computed by means of dynamical integrity tools and arguments. Those numerical techniques applied to strongly nonlinear systems, with six of more phase-space variables, demand considerable amounts of computing power, available only on High Performance Computing platforms. With the aim to minimize utilization of computer resources, we developed a software to adapt basin computations to small, affordable, clusters. It is based on Simple Cell Mapping method, modified to reduce the memory load, adjust integration time and overcome discretization discontinuities. Resource intensive part of computations is parallelized with Message Passing Interface and less demanding operations are kept serial, due to their inherit sequential nature. As intended, the program computes full six-dimensional basins of attraction at the adequate accuracy to distinguish compact parts of basins, but not to fully disclose fractalities or chaos. Disadvantages of the SCM method are addressed and the effectiveness of proposed solutions are demonstrated and discussed by the paradigmatic example composed of three coupled Duffing oscillators.

An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

An indicator of a passive biped walker’s global stability is its domain of attraction, which is usually estimated by the simple cell mapping method. It needs to calculate a large number of cells’ Poincare mapping result in the estimating process. However, the Poincare mapping is usually computationally expensive and time-consuming due to the complex dynamical equation of the passive biped walker. How to estimate the domain of attraction efficiently and reliably is a problem to be solved. Based on the simple cell mapping method, an improved method is proposed to solve it. The proposed method uses the multiple iteration algorithm to calculate a stable domain of attraction and effectively decreases the total number of Poincare mappings. Through the simulation of the simplest passive biped walker, the improved method can obtain the same domain of attraction as that calculated using the simple cell mapping method and reduce calculation time significantly. Furthermore, this improved method not only proposes a way of rapid estimating the domain of attraction, but also provides a feasible tool for selecting the domain of interest and its discretization level.

Dynamic modeling of aircraft landing gear and multi-objective optimization with simple cell mapping method

To study the dynamic characteristics of aircraft landing gear and carry out successive optimizations, a mathematical model of flexible landing gear is established by the Hamiltonian principle. The dynamic model includes a tire force estimation derived from the impact model. Dynamic analysis with the flexible model is then conducted. Stress distribution is obtained from the dynamic analysis, which can be used for fatigue analysis, optimization design, etc. To achieve better dynamic characteristics in terms of vibration reduction, a multi-objective optimization problem is formulated and solved via a simple cell mapping algorithm. Optimal simulations indicate the quality of optimal structural designs. Compared with the baseline structure, candidate optimal designs can improve dynamic performance of fuselage vibration suppression, shock absorber efficiency, and stress settling time. The proposed multi-objective optimal parameter design provides a fast tuning procedure that saves considerable time compared to finite element method-based optimization. In addition, the optimal parameter set provides useful interface information for detailed landing gear structural modeling that serves other analysis purposes.

Multi‐objective optimal motion control of a laboratory helicopter based on parallel simple cell mapping method

AbstractThis paper presents a study of multi‐objective optimal design of nonlinear control systems and has validated the control design with a twin rotor model helicopter. The gains of the porportional integral differential (PID) control are designed in the framework of multi‐objective opitmization. Eight design paramaters are optimized to minimize six time‐domain objective objective functions. The study of multi‐objective optimal design of feedback control with such a number of design paramaters and objective functions is rare in the literature. The Pareto optimal solutions are obtained by the proposed parallel simple cell mapping method consisting of a robust Pareto set recovery algorithm and a rolling subdivision technique. The proposed parallel simple cell mapping algorithm has two features: the number of cells in the invariant set grows linearly with the rolling subdivisions, and the Pareto set is insensitive to the inital set of seed cells. The current control design is compared with the classical LQE control for linear systems, and is also experimentally validated. The current design provides improved control performance as compares with the LQR control, and is applicable to complex nonlinear systems.

The Hybrid Micro-chaos Map: Digitally Controlled Inverted Pendulum with Dry Friction

Digital effects (quantization, sampling and delay) can lead to small amplitude chaotic oscillations, called micro-chaos [1, 2]. Often, these vibrations are neglected due to their small amplitude or replaced by random noise, but doing so one might be unable to capture some important behavior of the digitally controlled system.One notable example is the PD-controlled inverted pendulum with quantization at the calculated control effort. Taking digital effects into account leads to separated chaotic attractors in the state space. While the amplitude of the chaotic oscillations is indeed small, these attractors are situated rather far from the desired position, introducing considerable control error. Micro-chaos is undoubtedly present in ideal models of computer-controlled mechanical systems, however an important question is still open: does it persist if a more complex model of reality is used? For instance, does it survive in the presence of dry friction?This paper answers the latter question analyzing the micro-chaos in a system with Coulomb friction. We introduce the so-called hybrid micro-chaos map that describes the behavior of a digitally controlled system with dry friction. Then, the theoretical analysis of this map is presented and numerical results are provided that were acquired using a new mathematical tool, the Clustered Simple Cell Mapping method.Lastly, we conclude, that the phenomena of micro-chaos can withstand the presence of Coulomb friction and chaotic attractors can coexist with sticking zones in the state space.

Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

A generalized compatible cell mapping (CCM) method is proposed in this paper to take advantages of the simple cell mapping (SCM) method, the generalized cell mapping (GCM) method together with a subdivision procedure. A coarse cell partition is first used to obtain a covering set of the global attractor. Then, a finer global attractor is obtained by the subdivision process. The probabilistic response of stochastic dynamic systems is obtained by the sparse matrix analysis algorithm applied to the covering set of the global attractor. Because the computational domain is the covering set of the global attractor rather than the whole state space, the numerical efficiency of the proposed method can be greatly improved as compared to the GCM. A three-dimensional and a four-dimensional dynamical system under Poisson white noise excitation are studied to demonstrate the effectiveness of the proposed method for the probabilistic response analysis. Monte Carlo simulations show a good agreement with the proposed method.

Building Vibration Control by Active Mass Damper With Delayed Acceleration Feedback: Multi-Objective Optimal Design and Experimental Validation

The building structural vibration control by an active mass damper (AMD) with delayed acceleration feedback is studied. The control is designed with a multi-objective optimal approach. The stable region in a parameter space of the control gain and time delay is obtained by using the method of stability switch and the numerical code of NDDEBIFTOOL. The control objectives include the setting time, total power consumption, peak time, and the maximum power. The multi-objective optimization problem (MOP) for the control design is solved with the simple cell mapping (SCM) method. The Pareto set and Pareto front are found to consist of two clusters. The first cluster has negative feedback gains, i.e., the positive acceleration feedback. We have shown that a proper time delay can enhance the vibration suppression with controls from the first cluster. The second cluster has positive feedback gains and is located in the region which is sensitive to time delay. A small time delay will deteriorate the control performance in this cluster. Numerical simulations and experiments are carried out to demonstrate the analytical findings.

Dynamic Cell Mapping Algorithm for Computing Basins of Attraction in Planar Filippov Systems

Discontinuities are a common feature of physical models in engineering and biological systems, e.g. stick-slip due to friction, electrical relays or gene regulatory networks. The computation of basins of attraction of such nonsmooth systems is challenging and requires special treatments, especially regarding numerical integration. In this paper, we present a numerical routine for computing basins of attraction (BA) in nonsmooth systems with sliding, (so-called Filippov systems). In particular, we extend the Simple Cell Mapping (SCM) method to cope with the presence of sliding solutions by exploiting an event-driven numerical integration routine specifically written for Filippov systems. Our algorithm encompasses a method for dynamic construction of the cell state space so that a lower number of integration steps are required. Moreover, we incorporate an adaptive strategy of the simulation time to render more efficiently the computation of basins of attraction. We illustrate the effectiveness of our algorithm by computing basins of attraction of a sliding control problem and a dry-friction oscillator.

Multi-objective Optimal Design of Vibration Absorber with Simple Cell Mapping Algorithm

In this paper, we investigate a multi-objective optimal design of a tuned vibration absorber for a two degrees-of-freedom linear system, in which we take white Gaussian noise as the excitation. We use the simple cell mapping method to obtain solutions to the multi-objective optimization problem. We optimize tuned vibration absorbers in two different configurations. As our objective, we take various features of the response of the power spectrum density function. Our results show that the primary mass vibration with the optimally designed absorber has a lower peak power spectrum value. In addition, the two natural frequencies of the optimized system are further apart than in the non-optimized system, which implies that the absorber has a wider effective region in the frequency domain.

An analytical control law of length rate for tethered satellite system

Based on the nonlinear dynamic equations of a tethered satellite system with three-dimensional attitude motion, an analytical tether length rate control law for deployment is derived from the equilibrium positions of the system and the scheme of the value range of the expected in-plane pitch angle. The proposed control law can guarantee that the tensional force acting on the end of the tether remains positive. The oscillation of the out-of-plane roll motion in conjunction with the in-plane pitch motion is effectively suppressed during deployment control. The analytical control law is still applicable, even if the system runs on a Keplerian elliptical orbit with a large eccentricity. The local stability of the non-autonomous system during deployment control is analyzed using the Floquet theory, and the global behavior is numerically verified using simple cell mapping. The numerical simulations in the paper demonstrate the proposed analytical control law.

Multi-objective optimal design of sliding mode control with parallel simple cell mapping method

This paper presents a study of the multi-objective optimal design of a sliding mode control for an under-actuated nonlinear system with the parallel simple cell mapping method. The multi-objective optimal design of the sliding mode control involves six design parameters and five objective functions. The parallel simple cell mapping method finds the Pareto set and Pareto front efficiently. The parallel computing is done on a graphics processing unit. Numerical simulations and experiments are done on a rotary flexible arm system. The results show that the proposed multi-objective designs are quite effective.

Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method

When a dynamical system has a complex structure of fixed points, periodic cycles or even chaotic attractors, Cell Mapping methods are excellent tools to discover and thoroughly analyse all features in the state space. These methods discretize a region of the state space into cells and examine the dynamics in the cell state space. By determining one or more image cells for each cell, the global behaviour within the region can be quickly determined. In the simplest case – Simple Cell Mapping (SCM) method – only one image corresponds to a cell and usually a rectangular grid of cells is used. In typical applications the grid of cells is refined at specific locations.This paper, however, introduces a different approach, which is useful to expand the analysed state space region to include all features which properly characterize the global dynamics of the system. Instead of refining the initial cell state space, we start with a small initial state space region, analyse other interesting regions of the state space and incorporate them into a cluster of cell mapping solutions. By this approach, trajectories escaping the original state space region can be followed automatically and additional objects in the state space can be discovered.To illustrate the benefits of the method, we present the exploration of the phase-space of the micro-chaos map – a simple model of digitally controlled systems.

A hybrid method of evolutionary algorithm and simple cell mapping for multi-objective optimization problems

A hybrid method is proposed to take advantages of evolutionary algorithms (EAs) and the simple cell mapping (SCM) for multi-objective optimization problems (MOPs). The hybrid method starts with a random search for Pareto optimal solutions with an EA, and follows up with a neighborhood based search and recovery algorithm using the SCM. The non-dominated sorting genetic algorithm-II (NSGA-II) is used as an example of EAs. It is found that the SCM based search and recovery algorithm can reconstruct the branches of the Pareto set even when only one point in the vicinity of the set is available from the random search by the EA. We have chosen several benchmark MOPs to compare NSGA-II and SCM separately with the EA\(+\)SCM hybrid method while using the Hausdorff distance as a performance metric, and applied the method to develop multi-objective optimal designs of PID controls for a nonlinear oscillator with time delay. The results show that the EA\(+\)SCM hybrid method is very promising.

Parallel simple cell mapping for multi-objective optimization

In this article the Parallel Simple Cell Mapping (pSCM) is presented, a novel method for the numerical treatment of multi-objective optimization problems. The method is a parallel version of the simple cell mapping (SCM) method which also integrates elements from subdivision techniques. The classical SCM method exhibits nice properties for parallelization, which is used to speed up computations significantly. These statements are underlined on some classical benchmark problems with up to 10 decision variables and up to 5 objectives and provide comparisons to sequential SCM. Further, the method is applied on illustrative examples for which the method is also able to find the set of local optimal solutions efficiently, which is interesting in multi-objective multi-modal optimization, as well as the set of approximate solutions. The latter is of potential interest for the decision maker since it comprises an extended set of possible realizations of the given problem.

Finding zeros of nonlinear functions using the hybrid parallel cell mapping method

Analysis of nonlinear dynamical systems including finding equilibrium states and stability boundaries often leads to a problem of finding zeros of vector functions. However, finding all the zeros of a set of vector functions in the domain of interest is quite a challenging task. This paper proposes a zero finding algorithm that combines the cell mapping methods and the subdivision techniques. Both the simple cell mapping (SCM) and generalized cell mapping (GCM) methods are used to identify a covering set of zeros. The subdivision technique is applied to enhance the solution resolution. The parallel implementation of the proposed method is discussed extensively. Several examples are presented to demonstrate the application and effectiveness of the proposed method. We then extend the study of finding zeros to the problem of finding stability boundaries of potential fields. Examples of two and three dimensional potential fields are studied. In addition to the effectiveness in finding the stability boundaries, the proposed method can handle several millions of cells in just a few seconds with the help of parallel computing in graphics processing units (GPUs).

Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems1

The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. The cell mapping methods have been applied to deterministic, stochastic, and fuzzy dynamical systems. Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space: the interpolated cell mapping (ICM) and the set-oriented method with subdivision technique. For a long time, the cell mapping methods have been applied to dynamical systems with low dimension until now. With the advent of cheap dynamic memory and massively parallel computing technologies, such as the graphical processing units (GPUs), global analysis of moderate- to high-dimensional nonlinear dynamical systems becomes feasible. This paper presents a parallel cell mapping method for global analysis of nonlinear dynamical systems. The simple cell mapping (SCM) and generalized cell mapping (GCM) are implemented in a hybrid manner. The solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses, followed by the subdivision technique to enhance the accuracy of the steady-state responses. When the cells are small enough, no further subdivision is necessary. We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database, which provides a basis for the ICM to generate the pointwise approximation of the solutions without additional numerical integrations of differential equations. A modified global analysis of nonlinear systems with transient states is developed by taking advantage of parallel computing without subdivision. To validate the parallelized cell mapping techniques and to demonstrate the effectiveness of the proposed method, a low-dimensional dynamical system governed by implicit mappings is first presented, followed by the global analysis of a three-dimensional plasma model and a six-dimensional Lorenz system. For the six-dimensional example, an error analysis of the ICM is conducted with the Hausdorff distance as a metric.