Three-dimensional State Space Research Articles

Hybrid Optimal Control of a Flying+Sailing Drone

Abstract This paper studies the combined maneuver of flying and sailing for a robotic system which is referred to as a flying+sailing drone. Due to the emergence of hybrid systems behavior in tasks which involve both the flying and sailing modes, a hybrid systems formulation of the robotic system is presented. Key characteristics of the system are (i) changes in the dimension of the state space as the system switches from flying to sailing and vice versa and (ii) the presence of autonomous switchings triggered only upon the landing of the drone on the water surface. For the scenario in which the drone's initial state is given in the flying mode and a fixed terminal state is specified in the sailing mode, the associated optimal control problems are studied within the vertical plane passing through the given points, hence the dynamics of the drone in the flying mode are represented in a five-dimensional state space (associated with three degrees-of-freedom) and in a three-dimensional state space in the sailing mode (associated with two degrees-of-freedom). In particular, the optimal control problems for the minimization of time and the minimization of the control effort are formulated, the associated necessary optimality conditions are obtained from the Hybrid Minimum Principle (HMP), and the associated numerical simulations are presented.

Transient response of energy harvesting systems with multi-well potential under Poisson white noise excitations

This paper studies the transient response of vibration-based energy harvesting (VEH) systems characterized by multi-well potential functions exposed to Poisson white noise excitations. With the recently developed radial basis function neural networks (RBFNN) approach, the generalized Fokker–Planck–Kolmogorov (GFPK) equation in three-dimensional state space for the transient probability density function (PDF) is solved. The trial solution is composed of a series of radial basis Gaussian functions, each multiplied by a time-varying weight coefficient. The method of Lagrange multiplier is introduced for the constraint of weight coefficients to normalized the trial solution of the PDF solution. As examples to validate the proposed solution approach, the mono-stable and tri-stable energy harvester systems are studied. Monte Carlo simulations (MCS) have been applied to check the analytical solution. Some theoretical trends of the harvested mean square voltage (MSV) as a function of system parameters are examined. The non-Gaussian excitation and time-varying nature of the system can significantly affect the MSV output. It is noted that this work provides researchers and engineers a tool to optimize energy harvesting systems operated in random environment.

Expression Recognition Algorithm Enhancements for Students Who Integrate Artificial Intelligence Games into Classrooms to Gain Sensitivity

This paper proposes an evaluation index system for students' learning performance in colleges and universities, which is composed of 16 indicators in four categories with the integration of artificial intelligence games into the classroom as the basic framework. In terms of functions and evaluation, the theoretical structure and analysis path have been improved. Guided by the improvement of residents' sense of acquisition, the index of students' sense of acquisition is selected. Combining the three-dimensional learning state space with the emotional dimension theory, by analyzing the students' expressive states in the pleasant dimension, combined with the intensity of arousal and valence, an effective classroom state division method is realized, based on behavior theory.

Stable and unstable curves in vehicle three-dimensional state space

In the field of vehicle system dynamics, revealing the dynamic characteristics of nonlinear vehicle models with high degrees of freedom is a key point in current research. In the classic 2-DOF (degrees of freedom) model, the dynamics is analysed and represented in two-dimensional state space. However, it is necessary to reveal the three-dimensional state characteristics of the system when analysing high-DOF vehicle models. The concept of equilibrium points in two-dimensional space will be extended to curves in three-dimensional space, including stable curves and unstable curves, which can qualitatively reveal the evolvement characteristics of the system under different control parameters. In addition, because the dissipation of energy method can quantitatively reveal the global dynamic characteristics of the vehicle, this paper combines the (un)stable curves in the three-dimensional space with dissipation of energy to better analyse the global dynamics characteristics and mutually verify the correctness of conclusions. In this paper, first, the connection between (un)stable curves and region boundary is established. Second, the theoretical analysis of relationship between dissipation of energy and phase trajectories convergence characteristics is perfected. Third, the static and dynamic characteristics of vehicle system are analysed. Therefore, this paper proposed the concept of stable and unstable curves to intuitively reveal the global dynamics characteristics and system evolvement characteristics in three-dimensional state space.

Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions

We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are th first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original double optimal stopping problems to sequences of single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state space. We show that the optimal stopping boundaries are determined as the extremal solutions of the associated first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual real double lookback options with floating sunk costs in the Black-Merton-Scholes model.

Thermodynamic geometry of spin-one lattice models. I. Spin and quadrupolar orders and critical scaling functions in one dimension.

State space Riemannian geometry is obtained for the one-dimensional Blume-Emery-Griffiths model and its Blume-Capel and Griffiths model limits, and its (pseudo)critical as well as noncritical parameter regimes are extensively investigated. Two codimension one geometries are obtained by taking suitable hypersurfaces in the three-dimensional state space manifold, and the induced thermal metrics are accordingly interpreted in terms of constrained fluctuations. The three-dimensional scalar curvature and the two two-dimensional curvatures are shown to be consistent with Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature. Moreover, they are found to be in an excellent agreement over a greater part of the noncritical region with the corresponding correlation lengths for the spin and the quadrupolar order parameters. The scaling function for the free energy near the pseudocritical and tricritical points is obtained thermodynamically by using Ruppeiner's conjecture. A connection is made between the sign change in the curvatures and the change in fluctuation patterns of the order parameters. In the accompanying paper we shall analyze the geometry of the spin-one model in its mean field approximation.

Pricing and inventory control decisions in the stochastic hybrid production systems with multiple recovery options

Although pricing and inventory control are crucial decisions in each production system, these decisions are usually investigated separately. This paper considers pricing and inventory control decisions simultaneously in a hybrid production system. The hybrid production system has two recovery options, remanufacturing and refurbishing sites. The demand follows Poisson distribution, which depends on the sale price of each product. Returned products arrive according to a Poisson process. Each returned product can be remanufactured, refurbished, or disposed. The time to manufacturing, refurbishing, and remanufacturing a product also follows an exponential distribution. By modeling the system as a Markov chain, the long-run expected profit function is obtained in terms of the dispose-down-to level of returned products and the order-up-to level and the sale price of serviceable products. A three-dimensional state space of the Markov Chain dependent to the sale price is developed considering pricing and inventory control decisions simultaneously with remanufacturing and refurbishing returned products. Since the model is a mixed integer nonlinear programming and known as complex models, the Artificial Bee Colony (ABC) algorithm, simulation and complete search method are used to solve the problem. The results show that by increasing the purchase price of the returned products, the amount of returned products will increase. If the refurbishing cost of the returned products is high or the disposal cost is low, less inventory should be kept in the system with a high price of serviceable products. If the lost sale cost is high, the more inventory should be maintained. Moreover, by decreasing the price elasticity of demand, the customer’s demand increases, and then more inventory should be maintained in the system.

Feature Extraction and Identification of Alzheimer's Disease based on Latent Factor of Multi-Channel EEG.

Alzheimer's disease is a neurodegenerative disease in old age, early diagnosis will help to delay the progression of the disease. Presently, the features of brain functional diseases can be obtained with EEG analysis, but the relationship between characteristics of EEG and Alzheimer's disease has not been clearly clarified. In this work, we hypothesize that there exist default brain variables (latent factors) across subjects in disease processes, decoding latent factor from brain activity contributes to the study of cognitive impairment. To that end, this work proposes to extract characteristics of Alzheimer's disease by combing latent factors of EEG with variational auto-encoder to realize disease identification. Primarily, power spectrum characteristics is investigated and it is found that the dominant frequency of two groups is different. Further analysis reveals that latent factor distribution of Alzheimer's disease exists obvious differences with normal group in the theta frequency band. Moreover, the latent factors are projected onto the three-dimensional state space and the transient rotation of neural state is found, which shows the dynamic characteristics of latent factors. In addition, Takagi-Sugeno-Kang classifier is adopted and multiple latent factors are fed into Takagi-Sugeno-Kang classifier for decoding. Compared with linear classifier, Takagi-Sugeno-Kang fuzzy classifier has better performance in classification of energy feature from sub-frequency bands of latent factors. The accuracy of identification could up to 98.10% when the combination of energy features of four frequency bands is used as model input. Collectively, this work provides a feasible tool for identification of neurological dysfunction from the view of latent factors, especially contributing to the diagnosis of Alzheimer's disease.

Dynamical systems in perturbative scalar field cosmology

We derive a new regular dynamical system on a three-dimensional compact state space describing linear scalar perturbations of spatially flat Robertson–Walker geometries for relativistic models with a minimally coupled scalar field with an exponential potential. This enables us to construct the global solution space, illustrated with figures, where known solutions are shown to reside on special invariant sets. We also use our dynamical systems approach to obtain new results about the comoving and uniform density curvature perturbations. Finally we show how to extend our approach to more general scalar field potentials. This leads to state spaces where the state space of the models with an exponential potential appears as invariant boundary sets, thereby illustrating their role as building blocks in a hierarchy of increasingly complex cosmological models.

Impacts of Inventory Record Inaccuracy on Retailers’ Internal Operations

We present a continuous-time stochastic formulation of the standard Economic Order Quantity (EOQ) model incorporating all major costs incurred in carrying a product, including the cost of restocking, the cost of auditing the shelf stock level, the opportunity cost of capital, the cost of loss process, and the costs caused by not serving demand. In this formulation, the loss process is an unobservable point process while the demand point process is observed only when the item is in stock. The retailer’s control problem is reduced to an optimal stopping problem (corresponding to the restocking time) during which the shelf stock level can be audited finite number of times. The solution to the Hamilton-Jacobi-Bellman (HJB) equation of this stopping problem is fully characterized and constructed by applying a differential operator finite number of times. In particular, a closed-form expression for the optimal value function is obtained when auditing is infeasible or suboptimal. It is proved that the optimal restocking policy is a threshold policy in a three-dimensional state space that defines the Bayesian belief dynamics of the shelf stock level. Complete comparative statics analysis of this probability threshold along with its corresponding stopping time equivalent in the time domain are provided. We show that the jointly optimal auditing policy is not a threshold-based policy, that is the state trajectory can enter and leave the optimal audit region, leading to a finite time window during which auditing is optimal. We present a numerical example to illustrate the functional form of the optimal solution and estimate the impacts of inventory record inaccuracy on increasing the net present value of the expected total cost of carrying a product. The performance of the optimal policy is also compared with the best performing threshold-based heuristic policies in both time and probability domains. We show in this example that when the distribution of the shelf stock level is dynamically tracked, auditing the shelf stock level has a negligible impact on reducing the net present value of the expected total cost of carrying a product.

Variants of the Nosé–Hoover oscillator

The Nose–Hoover oscillator is a well-studied chaotic system originally proposed to model a harmonic oscillator in equilibrium with a heat bath at constant temperature. Although it is a simple three-dimensional system with five terms and two quadratic nonlinearities, it displays a rich variety of unusual dynamics, but it falls considerably short of its original purpose. This review describes two simple variants of the Nose–Hoover oscillator, the first of which satisfies the original goal exactly, and the second of which exhibits a hidden global chaotic attractor that fills all of its three-dimensional state space.

Dynamical systems analysis of the Maasch–Saltzman model for glacial cycles

This article is concerned with the internal dynamics of a conceptual model proposed by Maasch and Saltzman (1990) to explain central features of the glacial cycles observed in the climate record of the Pleistocene Epoch. It is shown that, in most parameter regimes, the long-term system dynamics occur on certain intrinsic two-dimensional invariant manifolds in the three-dimensional state space. These invariant manifolds are slow manifolds when the characteristic time scales for the total global ice mass and the volume of the North Atlantic Deep Water are well separated, and they are center manifolds when these characteristic time scales are comparable. In both cases, the reduced dynamics on these manifolds are governed by Bogdanov–Takens singularities, and the bifurcation curves associated to these singularities organize the parameter regions in which the model exhibits glacial cycles. In addition, knowledge of the reduced systems and their bifurcations is useful for understanding the effects of slowly varying parameters, which cause passage through Hopf bifurcations, and of orbital (Milankovitch) forcing. Both are central to the mechanism proposed by Maasch and Saltzman for the mid-Pleistocene transition in their model.

Simple chaotic 3D flows with surfaces of equilibria

Using a systematic computer search, twelve simple three-dimensional chaotic flows were found that have surfaces of equilibria. Although there are some four-dimensional systems with surfaces of equilibria, there is no such system in three-dimensional state space reported in the literature. Such systems are not difficult to design, but they can have some practical benefits. Study of chaotic flows with surfaces of equilibria provides a good reference for building systems with attractors that are protected from external influences, which can increase the safety of engineering.

Finding optimal solutions for vehicle routing problem with pickup and delivery services with time windows: A dynamic programming approach based on state–space–time network representations

Optimization of on-demand transportation systems and ride-sharing services involves solving a class of complex vehicle routing problems with pickup and delivery with time windows (VRPPDTW). This paper first proposes a new time-discretized multi-commodity network flow model for the VRPPDTW based on the integration of vehicles’ carrying states within space–time transportation networks, so as to allow a joint optimization of passenger-to-vehicle assignment and turn-by-turn routing in congested transportation networks. Our three-dimensional state–space–time network construct is able to comprehensively enumerate possible transportation states at any given time along vehicle space–time paths, and further allows a forward dynamic programming solution algorithm to solve the single vehicle VRPPDTW problem. By utilizing a Lagrangian relaxation approach, the primal multi-vehicle routing problem is decomposed to a sequence of single vehicle routing sub-problems, with Lagrangian multipliers for individual passengers’ requests being updated by sub-gradient-based algorithms. We further discuss a number of search space reduction strategies and test our algorithms, implemented through a specialized program in C++, on medium-scale and large-scale transportation networks, namely the Chicago sketch and Phoenix regional networks.

Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat Friedmann–Lemaître–Robertson–Walker cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the ‘attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Padé approximants and compare with the slow-roll approximation.

Allosteric regulation of phosphofructokinase controls the emergence of glycolytic oscillations in isolated yeast cells.

Oscillations are widely distributed in nature and synchronization of oscillators has been described at the cellular level (e.g. heart cells) and at the population level (e.g. fireflies). Yeast glycolysis is the best known oscillatory system, although it has been studied almost exclusively at the population level (i.e. limited to observations of average behaviour in synchronized cultures). We studied individual yeast cells that were positioned with optical tweezers in a microfluidic chamber to determine the precise conditions for autonomous glycolytic oscillations. Hopf bifurcation points were determined experimentally in individual cells as a function of glucose and cyanide concentrations. The experiments were analyzed in a detailed mathematical model and could be interpreted in terms of an oscillatory manifold in a three-dimensional state-space; crossing the boundaries of the manifold coincides with the onset of oscillations and positioning along the longitudinal axis of the volume sets the period. The oscillatory manifold could be approximated by allosteric control values of phosphofructokinase for ATP and AMP. The mathematical models described here have been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.mib.ac.uk/webMathematica/UItester.jsp?modelName=gustavsson5. [Database section added 14 May 2014 after original online publication].

Three-dimensional state space spline finite strip analysis of angle-plied laminates

In this paper, a combined spline finite strip and state space approach is introduced to obtain three-dimensional solutions of laminated composite plates with general boundary conditions. Spline and linear polynomial functions are used, respectively, as the shape functions in the longitudinal and transverse directions of the strips. The variations of the displacements and stresses through the thickness of a strip are calculated through solving differential equations in the form of the state space equations. This method allows the displacements and stresses to be computed simultaneously at all nodes along the nodal lines. In comparison with three dimensional finite element solutions, the current method guarantees the continuity of all the displacements and through-thickness stresses across all interfaces of material layer and, therefore, provides more reliable and accurate results. It can also reduce the computational effort when compared with the traditional three-dimensional finite element and finite strip methods.

A cyclic time-dependent Markov process to model daily patterns in wind turbine power production

Wind energy is becoming a top contributor to the renewable energy mix, which raises potential reliability issues for the grid due to the fluctuating nature of its source. To achieve adequate reserve commitment and to promote market participation, it is necessary to provide models that can capture daily patterns in wind power production. This paper presents a cyclic inhomogeneous Markov process, which is based on a three-dimensional state-space (wind power, speed and direction). Each time-dependent transition probability is expressed as a Bernstein polynomial. The model parameters are estimated by solving a constrained optimization problem: The objective function combines two maximum likelihood estimators, one to ensure that the Markov process long-term behavior reproduces the data accurately and another to capture daily fluctuations. A convex formulation for the overall optimization problem is presented and its applicability demonstrated through the analysis of a case-study. The proposed model is capable of reproducing the diurnal patterns of a three-year dataset collected from a wind turbine located in a mountainous region in Portugal. In addition, it is shown how to compute persistence statistics directly from the Markov process transition matrices. Based on the case-study, the power production persistence through the daily cycle is analyzed and discussed.

An information integration modelling architecture for product family life cycle in mass customisation

To support integrating, managing, sharing and reusing information throughout the entire life cycle of product family in mass customisation (MC), this paper puts forward an information integration modelling architecture for product family life cycle (I2MAfPFL). As an architecture built from the comprehensive perspective of the product family rather than the single product, the I2MAfPFL is centred on the model of three-dimensional (3-D) state space with integration of the model of unit information and the model of the 3-D evolution space. In the I2MAfPFL, the model of unit information is used to represent the features of unit information produced throughout all phases in the life cycle of product family in different depth and level. The model of 3-D state space provides the structure and framework to support integrating unit information, and the model of 3-D evolution space describes the dynamic evolution processes of unit information and reveals the internal correlated, interacted and evolved relationships among them. The expression of the three models in the I2MAfPFL is explained in detail. Finally, a case is used to demonstrate how the I2MAfPFL is used to integrate information for the product family life cycle in MC.

Transactional Innovation and the De-Commoditization of the Brazilian Coffee Trade

Recent research into international trade has highlighted the role of trade costs in determining the geography of trade. We propose that the notion of comparative advantage has to be expanded to include different capacities to transact and the pertinent factor endowments (organizational and social capital). In a ‘transactional regime’, sets of transaction-enabling tasks (TETs) are arranged to overcome a given pattern of trade resistances. We argue that transactional regimes impact the flows and contents of information that guide the decision making of producers and therefore also shape the resulting international division of labour. Transactional regimes can be classified according to a three-dimensional state space of trade resistances covering certain most generic aspects of information. We apply this approach to analyze the case of the Brazilian coffee business that has evolved into a bifurcated ‘commoditized’ and ‘de-commoditized’ regime in the past twenty years, triggered by the entrepreneurial action of a single company, illycaffe. An entirely new business model has resulted into a shift of the transactional regime in coffee trade, with implications for the distribution of the gains from trade.