We call a tree parameter additive if it can be determined recursively as the sum of the parameter values of all branches, plus a certain toll function. In this paper, we prove central limit theorems for very general toll functions, provided that they are bounded and small on average. Simply generated families of trees are considered as well as Pólya trees, recursive trees and binary search trees, and the results are illustrated by several examples of parameters for which we prove normal or log-normal limit laws.

A univariate polynomial f over a field is decomposable if it is the composition f = g ○ h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposables over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing relative error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker.

We prove that several extensions of the classic Erlang loss function to non-integral numbers of servers are scalable: the blocking probability as described by the extension decreases when the offered load and the number of servers s are increased with the same relative amount, even when scaling up from integral s to non-integral s. We use this to prove that when several Erlang loss systems pool their resources for efficiency, various corresponding cooperative games have a non-empty core.

This paper proposes a model, the linear model, for randomly generating logic programs with low density of rules and investigates statistical properties of such random logic programs. It is mathematically shown that the average number of answer sets for a random program converges to a constant when the number of atoms approaches infinity. Several experimental results are also reported, which justify the suitability of the linear model. It is also experimentally shown that, under this model, the size distribution of answer sets for random programs tends to a normal distribution when the number of atoms is sufficiently large.

We present a concept of robust optimisation design for the spring actuator in a 10 kV/12.5 kA vacuum circuit breaker. We assume the breaking and closing velocity characteristics, which are derived form the technical data of the interrupter, as the specifications for the problem, and take the lengths of the connecting rods of the actuator and the stiffness coefficients of the breaking and closing springs as the optimisation variables. The variance between the specifications and the velocities calculated at each breaking and closing point and the maximal variation allowed by the design variables within acceptable tolerances make up the multiple objective function. The optimal parameters for the spring actuator are given by solving a non-linear programming problem with multiple targets and two-level optimisation.

It is difficult to design a controller directly for non-square multi-variable systems with time delay. In the current paper, we propose a new design method for an Internal Model Control PID controller based on a modified effective open-loop transfer function (MEOTF) for non-square processes with time delay. The MEOTF method is used to decompose the complex non-square process into several equivalent independent single-input/single-output processes. Using the Taylor Particle Swarm Optimisation (Taylor-PSO) model reduction method, the MEOTF of the non-square process is approximated by a reduced order form. The reduced form of the MEOTF is then used to design the Internal Model Control PID controller, which is then used for the original non-square process. To improve the robust stability, a first-order filter is added in the feedback loop. Finally, we present simulation results showing the validity and reliability of this method. In particular, our method has a strong anti-interference characteristic and retains its good control performance in the presence of model perturbation and interference.

Active-Passive-Jamming (APJ) is formed by a chaff decoy reflecting jamming signals from an aircraft's active-ECM jammer. The effectiveness of APJ depends on many factors, one of which is the jamming signal's frequency feature. In this paper, we discuss and analyse APJ's frequency feature in a typical given offense-defense scenario for the case where the ECM jammer adopted is velocity gate pull off (VGPO), which is a kind of deception jamming technique. Our results show that the APJ's frequency feature depends not only on the jamming techniques of the jammer, but also on the Native Doppler Frequency (NDF), which is related to the relative positions, distances, angles and velocities of the jammer, radar and chaff decoy. We also carried out simulations, and the results show that the APJ mode can be useful for jamming an enemy radar.

The flow shop scheduling problem based on ideal and precise conditions has been a focus of considerable research since the first easy scheduling problem was formulated. In reality, some uncertain factors always restrict the scheduling optimisation problem. In this paper, taking uncertain processing time as an example, we use generalised rough sets theory to transform the rough flow shop scheduling model into the precise scheduling model. We adopt a cooperative co-evolutionary particle swarm optimisation algorithm based on a niche sharing scheme (NCPSO) to minimise the makespan in comparison with the particle swarm optimiser (PSO) and co-evolution particle swarm optimiser (CPSO) algorithms. The new algorithm is characterised by a strengthening of the ability to reserve excellent particles and searching the optimal solution. Experimental results show that the new algorithm is more effective and efficient than the others.

Numerical simulation is the generally used method for studying stochastic resonance (SR), which is a kind of non-linear phenomenon that usually occurs in non-linear bistable systems. It has been found that the input signal needs to be over-sampled during the numerical simulation of SR. In this paper we provide an explanation of this phenomenon based on a stability analysis of the bistable system. We begin by studying the stability of a discrete model of a bistable system in numerical simulations. We then give a theoretical derivation of the stability conditions for the simulation model with different parameters, and carry out numerical experiments to show that the results coincide with the predictions of the theory. We explain why the input signal needs to be over-sampled in the simulation and provides guidelines for the choice of system parameters for the bistable system and the sampling time step in the numerical simulation of SR. Finally, we present the results of simulations showing an example of SR occurring in a bistable system and an example of weak periodic signal detection when it is processed by a bistable system.

Wind power projects face an uncertain external environment, they are complex projects in themselves and the capabilities of the designers, erectors and operators are limited. All this makes the identification of investment risks for wind power projects extremely complicated. In this paper, we propose a method for identifying the investment risk scientifically and accurately using a back propagation (BP) neural network. Specifically, we propose a hybrid wavelet transform fuzzy BP neural network (WT-FBPNN) optimisation model based on the construction of a risk evaluating index system. This improved model can not only exploit the time frequency localisation characteristic of wavelet transforms (WT), but also enhance the fit precision and algorithm convergence speed. The simulation results show that this model is reliable, and that this method of identifying the investment risk of wind power projects is feasible.

The process of river health evaluation is subject to uncertainty and complexity. To address this, we establish a river health evaluation system based on entropy and multi-objective space theory. We apply an analytic hierarchy process (AHP) to calculate the weight of the first grade indexes, and then apply the entropy to calculate the weight of the second indexes. In this way, the health measure can be considered in terms of objective data and a subjective classification. The results of the computation show that the whole system provides a good measure of the health. The results express the measurement of the health of the river on different levels and for different aspects, which indicates that the evaluation method is feasible.

We present a novel robust control scheme that deals with multi-body spacecraft attitude tracking problems. The control scheme consists of a radial basis function network (RBFN) and a robust controller. By using the finite time convergence property of the terminal sliding mode (TSM), we derive a new online learning algorithm for updating all the parameters of the RBFN that ensures the RBFN has fast approximation for the parameter uncertainties and external disturbances. We design a robust controller to compensate RBFN approximation errors and realise the anticipative stability and performance properties. We can also achieve closed-loop system stability using Lyapunov stability theory.

No detailed knowledge of the non-linear dynamics of the spacecraft is required at any point in the entire design process, and the proposed robust scheme is simple and effective and can be applied to more complex systems. Simulation results demonstrate the good tracking characteristics of the proposed control scheme in the presence of inertial uncertainties and external disturbances.