On the analysis of time-periodic nonlinear dynamical systems

In this study, a general technique for the analysis of time-period nonlinear dynamical systems is presented. The method is based on the fact that all quasilinear periodic systems can be replaced by similar systems whose linear parts are time invariant via the well-known Liapunov-Floquet (L-F) transformation. A general procedure for the computation of L-F transformation in terms of Chebyshev polynomials is outlined. Once the transformation has been applied, a periodic orbit in original coordinates has a fixed point representation in the transformed coordinates. The stability and bifurcation analysis of the transformed equations are studied by employing thetime-dependent normal form theory and time-dependent centre manifold reduction. For the two examples considered, the three generic codimension-one bifurcations, viz, Hopf, flip and tangent, are analysed. The methodology is semi-analytic in nature and provides a quantitative measure of stability even under critical conditions. Unlike the perturbation or averaging techniques, this method is applicable even to those systems where the periodic term in the linear part does not contain a small parameter or a generating solution does not exist due to the absence of the time-invariant term in the linear part.     

Normal form analysis of stochastically forced dynamical systems

The possibility of reducing the dynamics of a system undergoing a Hopf bifurcation and forced by multiplicative noise to a universal normal form is examined on a simple model of geophysical interest. It is shown that the normal form equations and the stationary probability distribution depend on the way the noise is coupled to the original system. The universality of the normal forms is thus severely limited in the presence of noise.     

Wedge simulation method for calculating the reliability of linear dynamical systems

In this paper the problem of calculating the probability of failure of linear dynamical systems subjected to random excitations is considered. The failure probability can be described as a union of failure events each of which is described by a linear limit state function. While the failure probability due to a union of non-interacting limit state functions can be evaluated without difficulty, the interaction among the limit state functions makes the calculation of the failure probability a difficult and challenging task. A novel robust reliability methodology, referred to as Wedge-Simulation-Method, is proposed to calculate the probability that the response of a linear system subjected to Gaussian random excitation exceeds specified target thresholds. A numerical example is given to demonstrate the efficiency of the proposed method which is found to be enormously more efficient than Monte Carlo Simulations.     

An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability

This paper presents a simulation technique for reliability analysis of linear dynamical systems. It is based on simple additive rules of probability (in contrast to other probabilistic approaches such as importance sampling). It is shown that the proposed appoach is identical to a newly developed approach, Importance Sampling using Elementary Events (ISEE) [Au SK, Beck JL. First excursion probabilities for linear sytems by very efficient importance sampling. Probabl Eng Mech 2001;16(3):193–208]. A simple formula for the coefficient of variation of the estimator of the failure probability using the samples is also given. A 10-story building model with nonstationary excitation is utilized to demonstrate the accuracy and efficiency of the proposed method.     

Stability analysis and controller synthesis for hybrid dynamical systems

Wherever continuous and discrete dynamics interact, hybrid systems arise. This is especially the case in many technological systems in which logic decision-making and embedded control actions are combined with continuous physical processes. Also for many mechanical, biological, electrical and economical systems the use of hybrid models is essential to adequately describe their behaviour. To capture the evolution of these systems, mathematical models are needed that combine in one way or another the dynamics of the continuous parts of the system with the dynamics of the logic and discrete parts. These mathematical models come in all kinds of variations, but basically consist of some form of differential or difference equations on the one hand and automata or other discrete-event models on the other hand. The collection of analysis and synthesis techniques based on these models forms the research area of hybrid systems theory, which plays an important role in the multi-disciplinary design of many technological systems that surround us. This paper presents an overview from the perspective of the control community on modelling, analysis and control design for hybrid dynamical systems and surveys the major research lines in this appealing and lively research area.     

Tradeoff analysis of non-linear dynamical systems under stochastic excitation

An efficient methodology to carry out multi-objective optimization of non-linear structural systems under stochastic excitation is presented. Specifically, an efficient determination of particular Pareto or non-inferior solutions is implemented. Pareto solutions are obtained by compromise programming which is based on the minimization of the distance between the point that contains the individual optima of each of the objective functions and the Pareto set. The response of the structural system is characterized in terms of the first two statistical moments of the response process, i.e. the mean and variance. An efficient sensitivity analysis of non-inferior solutions with respect to the design variables becomes possible with the proposed formulation. Such information is used for decision making and tradeoff analysis. The compromise programming problem is solved by an efficient procedure that combines a local statistical linearization approach, modal analysis, global approximation concepts, and a sequential optimization scheme. Numerical results show that the total number of stochastic analyses required during the multi-objective optimization process is in general very small. Hence, different compromise solutions including the design that best represents the outcome that the designer considers potentially satisfactory are obtained in an efficient manner. In this way, the analyst can conduct a decision-making analysis through an efficient interactive procedure.     

Assessment of low probability events of dynamical systems by controlled Monte Carlo simulation

Various procedures to extend the applicability and to increase the efficiency of Monte Carlo simulation (MCS) for the analysis of complex dynamical systems are discussed. In particular, the capabilities of the methods denoted Russian Roulette and Splitting (RR&S) and Double and Clump (D&C) are reviewed with regard to their capabilities to analyze such systems. In this context, the difficulties in identifying the ‘important' regions for simulation are detailed. It is shown that these difficulties may be circumvented by a newly introduced ‘distance controlled' MCS. This procedure, which allows the prediction of very low probability events and the analysis of systems of higher dimension, is applicable not only to mechanical systems and structures but also to complex dynamical systems encountered, for example, in economics, physics, etc. The procedure is shown to be particularly suited to cases where exact analytical methods and direct Monte Carlo simulation are infeasible, hence, being well suited for practical application.     

Frontiers and symmetries of dynamical systems

The frontiers of boundedness ? _b of the orbits of dynamical systems X defined on ? ⁿ are studied. When X is completely integrable some topological properties of ? _b are found and, in certain cases, ? _b is localized with the help of symmetries of X. Several examples in dimensions 2 and 3 are provided. In case the number of known first integrals of the vector field X is less than n ? 1, an interesting connection of ? _b with the frontier of boundedness of the level-sets of the first integrals of X is proved. This result also applies to Hamiltonian systems.     

Periodic orbit analysis of two dynamical systems for electrical engineering applications

Mathematical properties of two nonlinear adaptive filters for electrical engineering applications are presented. These filters are designed to extract a desired sinusoidal component of a given periodic signal and estimate its amplitude, phase angle and frequency. Two sets of non-autonomous ordinary differential equations govern the dynamics of the filters. It is shown that each of the filters possesses a unique and stable periodic orbit. The averaging theorem is used to prove the uniqueness and stability of the periodic orbit of one of the filters. Uniqueness and stability of the periodic orbit of the other filter are proven using the Poincaré map theorem. Computer simulations and numerical results are presented to provide numerical verification of the theoretical proofs, and finally experimental results of the laboratory implementation of the filters are presented.     

The kinetics of ostwald ripening in stressed solids: The low volume fraction limit

The thermodynamics of non-hydrostatically stressed solids has been used to examine the influence of misfit strains, interfacial stress, and coherency stress on the kinetics of coarsening in an elastic solid with a vanishingly small volume fraction of noninteracting spherical precipitates. The following results are obtained:

• 1.1. The dimensionless form of the governing equations for Ostwald ripening in a coherent solid, when expressed in the comparison state, is isomorphic with that of the classical LS theory.
• 2.2. The exponents of the temporal power laws for the mean radius, the number density of precipitate particles, and the supersaturation obtained in the long-time asymptotic limit are identical to those of the LS theory.
• 3.3. The amplitudes of the temporal power laws are different from those of LS and are dependent on the signs and magnitudes of the precipitate misfit, interfacial stress, and solute expansion coefficient.
• 4.4. The kinetics of coarsening are altered by the elastic stress primarily through the coupling of misfit and interfacial stress. Only the coefficient for the decay of the supersaturation depends on the solute expansion coefficient.
• 5.5. A new expression for the dependence of the precipitate misfit on time during coarsening has been obtained. If the solute expansion coefficient of the matrix is not zero, the misfit is predicted to approach its equilibrium value as t^−1/3.
• 6.6. The analysis shows that decreasing the magnitude of the precipitate misfit may either increase or decrease the coarsening kinetics.

     

Flowsheet simulation of solids processes: Current status and future trends

Complex manufacturing processes are nowadays applied for production of various solid products. It is very common that for production of particles with desired properties several transformation steps like drying, milling, classification, granulation, etc. should be involved. This leads to the process structures consisting of different apparatuses or transformation substeps connected with material and energy balances. Consequently, development of new processes or optimization of already existing, as well as an optimal control, is a very challenging task, which can be partially solved using numerical modelling.For the simulation of modern production processes, the flowsheet calculations can be effectively used. Starting from the 80 s a lot of work focused on the flowsheet simulation of liquid-vapor systems has been done and as result various well-established systems exist today. With respect to the solid processes the intensive research has been started much later. In this contribution we present our view about a current role of flowsheet simulation for modeling of particulate materials and specify the open fields which can be covered in future research.