Robust Asynchronous Fuzzy Predictive Fault-Tolerant Tracking Control for Nonlinear Multiphase Batch Processes with Time-Varying Tracking Trajectories

A T-S model-dependent robust asynchronous fuzzy predictive fault-tolerant tracking control scheme is developed for nonlinear multiphase batch processes with time-varying tracking trajectories and actuator faults. Firstly, considering the influence of the mismatch between the previous phase state and the current phase controller during the switching, a T-S fuzzy switching model including the match and mismatch case is established. Depending on the fuzzy switching model, a robust fuzzy predictive fault-tolerant tracking controller is designed by considering the situation whether the model rules correspond to the controller rules. Secondly, using the related theories and methods, the system stability conditions presented by the linear matrix inequality considering the above conditions are provided to guarantee the stability of the system. By solving these stability conditions, the T-S fuzzy control law gain of each phase, the minimum running time of each match case and the maximum running time of each mismatch case are obtained. Then, through the maximum running time, the strategy of the switching signal in advance is adopted to avoid the occurrence of asynchronous switching, so as to ensure that the system can operate stably. Finally, the simulation results verify that the designed controller is effective and feasible.

Relay interlayer synchronisation: invariance and stability conditions

In this paper, the existence (invariance) and stability (locally and globally) of relay interlayer synchronisation (RIS) are investigated in a chain of multiplex networks. The local dynamics of the nodes in the symmetric positions layers on both sides of the non-identical middlemost layer(s) are identical. The local and global stability conditions for this synchronisation state are analytically derived based on the master stability function approach and by constructing a suitable Lyapunov function, respectively. We propose an appropriate demultiplexing process for the existence of the RIS state. Then the variational equation transverse to the RIS manifold for demultiplexed networks is derived. In numerical simulations, the impact of interlayer and intralayer coupling strengths, variations of the system parameter in the relay layers and demultiplexing on the emergence of RIS in triplex and pentaplex networks are explored. Interestingly, in this multiplex network, enhancement of RIS is observed when a type of impurity via parameter mismatch in the local dynamics of the nodes is introduced in the middlemost layer. A common time-lag with small amplitude shift between the symmetric positions and central layers plays an important role for the enhancing of relay interlayer synchrony. This analysis improves our understanding of synchronisation states in multiplex networks with nonidentical layers.

Analyzing Stability Conditions and Ore Dilution in Open Stope Mining

Ore dilution is a fundamental problem for the production process in underground mining operations. Especially in open stoping methods of underground mining, the continuous estimation, monitoring and treatment of instability issues is considered necessary in order to maintain the consistency of the production process. This paper aims to combine empirical nomograms of stability estimation and numerical approaches and thus link the extensive experiences of the empirical design and the quantitative data derived by numerical analyses. To facilitate this, a large number of different geomechanical conditions were modeled and analyzed in the pursuit of obtaining valid and applicable relationships between the empirical stability graphs’ approaches and the numerical simulation models. The parametric analysis was made to express the stability conditions and the dilution with specific design characteristics, using prevalent stability-graph approaches while the numerical models were tested using the RS2 software package. The obtained results include direct and easy-to-use mathematical expressions that can be applied during the initial design of the stoping process, especially for the case of sidewalls (hanging walls and foot walls). Furthermore, through the research, an initial proposal is made for a dilution-based stability graph that could be utilized for the early identification of dilution.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

On the courant-friedrichs-lewy condition for numerical solvers of the coagulation equation

Evolving the size distribution of solid aggregates challenges simulations of young stellar objects. Among other difficulties, generic formulae for stability conditions of explicit solvers provide severe constrains when integrating the coagulation equation for astrophysical objects. Recent numerical experiments have recently reported that these generic conditions may be much too stringent. By analysing the coagulation equation in the Laplace space, we explain why this is indeed the case and provide a novel stability condition which avoids time over-sampling.

Stability and bifurcation analysis of super- and sub-harmonic responses in a torsional system with piecewise-type nonlinearities

The nonlinear dynamic behaviors induced by piecewise-type nonlinearities generally reflect super- and sub-harmonic responses. Various inferences can be drawn from the stability conditions observed in nonlinear dynamic behaviors, especially when they are projected in physical motions. This study aimed to investigate nonlinear dynamic characteristics with respect to variational stability conditions. To this end, the harmonic balance method was first implemented by employing Hill’s method, and the time histories under stable and unstable conditions were examined using a numerical simulation. Second, the super- and sub-harmonic responses were investigated according to frequency upsweeping based on the arc-length continuation method. While the stability conditions vary along the arc length, the bifurcation phenomena also show various characteristics depending on their stable or unstable status. Thus, the study findings indicate that, to determine the various stability conditions along the direction of the arc length, it is fairly reasonable to determine nonlinear dynamic behaviors such as period-doubling, period-doubling cascade, and quasi-periodic (or chaotic) responses. Overall, this study suggests analytical and numerical methods to understand the super- and sub-harmonic responses by comparing the arc length of solutions with the variational stability conditions.

Linking Stability Conditions and Ore Dilution in Open Stope Mining

The estimation of the stability conditions, over-breaks, and spalling failures, which could inflict potential external dilution, is a key parameter so as to ensure the optimal design of the exploitation and its cost effectiveness The research undertaken aims at correlating established empirical approaches for the estimation of the stability condition with numerical analysis that identifies and measures the depth of failure. A number of analyses have been conducted and the results obtained yield promising results that can be transformed to direct mathematical expressions applied for the early estimation of dilution rates. Furthermore, through the research, an initial proposal is made for a dilution-based stability graph that could be utilized for the early identification of dilution.

Mathematical modeling in problems about dynamics and stability of elastic elements of wing profiles

The mathematical models describing the dynamics of elastic elements of wing structures and representing the initial-boundary value problems for systems of partial differential equations are proposed. The dynamics and stability of elastic elements of wings, flown around by a gas or liquid stream in a model of an incompressible medium, are investigated. To study the dynamics of elastic elements and a gas-liquid medium, both linear and nonlinear models of the mechanics of a solid deformable body and linear models of the mechanics of liquid and gas are used. On the basis of the constructed functionals for partial differential equations, the sufficient stability conditions are obtained in analytical form. The conditions impose restrictions on the parameters of mechanical systems. The obtained stability conditions are necessary for solving the problems of controlling the parameters of the aeroelastic system. On the basis of the Galerkin method, a numerical study of the dynamics of elastic elements was carried out, the reliability of which is confirmed by the obtained analytical results.