On optimal control of constrained linear systems with imperfect state information and stochastic disturbances

We address the problem of finite horizon optimal control of discrete-time linear systems with input constraints and uncertainty. The uncertainty for the problem analysed is related to incomplete state information (output feedback) and stochastic disturbances. We analyse the complexities associated with finding optimal solutions. We also consider two suboptimal strategies that could be employed for larger optimization horizons.

Interacting motile agents : taking a mean-field approach beyond monomers and nearest-neighbor steps

We consider a discrete agent-based model on a one-dimensional lattice, where each agent occupies L sites and attempts movements over a distance of d lattice sites. Agents obey a strict simple exclusion rule. A discrete-time master equation is derived using a mean-field approximation and careful probability arguments. In the continuum limit, nonlinear diffusion equations that describe the average agent occupancy are obtained. Averaged discrete simulation data are generated and shown to compare very well with the solution to the derived nonlinear diffusion equations. This framework allows us to approach a lattice-free result using all the advantages of lattice methods. Since different cell types have different shapes and speeds of movement, this work offers insight into population-level behavior of collective cellular motion.

Collective motion of dimers

We consider a discrete agent-based model on a one-dimensional lattice and a two-dimensional square lattice, where each agent is a dimer occupying two sites. Agents move by vacating one occupied site in favor of a nearest-neighbor site and obey either a strict simple exclusion rule or a weaker constraint that permits partial overlaps between dimers. Using indicator variables and careful probability arguments, a discrete-time master equation for these processes is derived systematically within a mean-field approximation. In the continuum limit, nonlinear diffusion equations that describe the average agent occupancy of the dimer population are obtained. In addition, we show that multiple species of interacting subpopulations give rise to advection-diffusion equations. Averaged discrete simulation data compares very well with the solution to the continuum partial differential equation models. Since many cell types are elongated rather than circular, this work offers insight into population-level behavior of collective cellular motion.

Comparing multilevel and Bayesian spatial random effects survival models to assess geographical inequalities in colorectal cancer survival: a case study

Background Multilevel and spatial models are being increasingly used to obtain substantive information on area-level inequalities in cancer survival. Multilevel models assume independent geographical areas, whereas spatial models explicitly incorporate geographical correlation, often via a conditional autoregressive prior. However the relative merits of these methods for large population-based studies have not been explored. Using a case-study approach, we report on the implications of using multilevel and spatial survival models to study geographical inequalities in all-cause survival. Methods Multilevel discrete-time and Bayesian spatial survival models were used to study geographical inequalities in all-cause survival for a population-based colorectal cancer cohort of 22,727 cases aged 20–84 years diagnosed during 1997–2007 from Queensland, Australia. Results Both approaches were viable on this large dataset, and produced similar estimates of the fixed effects. After adding area-level covariates, the between-area variability in survival using multilevel discrete-time models was no longer significant. Spatial inequalities in survival were also markedly reduced after adjusting for aggregated area-level covariates. Only the multilevel approach however, provided an estimation of the contribution of geographical variation to the total variation in survival between individual patients. Conclusions With little difference observed between the two approaches in the estimation of fixed effects, multilevel models should be favored if there is a clear hierarchical data structure and measuring the independent impact of individual- and area-level effects on survival differences is of primary interest. Bayesian spatial analyses may be preferred if spatial correlation between areas is important and if the priority is to assess small-area variations in survival and map spatial patterns. Both approaches can be readily fitted to geographically enabled survival data from international settings

Insights into students' conceptual understanding using textual analysis: A case study in signal processing

Concept inventory tests are one method to evaluate conceptual understanding and identify possible misconceptions. The multiple-choice question format, offering a choice between a correct selection and common misconceptions, can provide an assessment of students' conceptual understanding in various dimensions. Misconceptions of some engineering concepts exist due to a lack of mental frameworks, or schemas, for these types of concepts or conceptual areas. This study incorporated an open textual response component in a multiple-choice concept inventory test to capture written explanations of students' selections. The study's goal was to identify, through text analysis of student responses, the types and categorizations of concepts in these explanations that had not been uncovered by the distractor selections. The analysis of the textual explanations of a subset of the discrete-time signals and systems concept inventory questions revealed that students have difficulty conceptually explaining several dimensions of signal processing. This contributed to their inability to provide a clear explanation of the underlying concepts, such as mathematical concepts. The methods used in this study evaluate students' understanding of signals and systems concepts through their ability to express understanding in written text. This may present a bias for students with strong written communication skills. This study presents a framework for extracting and identifying the types of concepts students use to express their reasoning when answering conceptual questions.

Output feedback control of networked systems with a stochastic communication protocol

This paper addresses an output feedback control problem for a class of networked control systems (NCSs) with a stochastic communication protocol. Under the scenario that only one sensor is allowed to obtain the communication access at each transmission instant, a stochastic communication protocol is first defined, where the communication access is modelled by a discrete-time Markov chain with partly unknown transition probabilities. Secondly, by use of a network-based output feedback control strategy and a time-delay division method, the closed-loop system is modeled as a stochastic system with multi time-varying delays, where the inherent characteristic of the network delay is well considered to improve the control performance. Then, based on the above constructed stochastic model, two sufficient conditions are derived for ensuring the mean-square stability and stabilization of the system under consideration. Finally, two examples are given to show the effectiveness of the proposed method.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Discrete choice modelling of combined mode and departure time

Typical daily decision-making process of individuals regarding use of transport system involves mainly three types of decisions: mode choice, departure time choice and route choice. This paper focuses on the mode and departure time choice processes and studies different model specifications for a combined mode and departure time choice model. The paper compares different sets of explanatory variables as well as different model structures to capture the correlation among alternatives and taste variations among the commuters. The main hypothesis tested in this paper is that departure time alternatives are also correlated by the amount of delay. Correlation among different alternatives is confirmed by analyzing different nesting structures as well as error component formulations. Random coefficient logit models confirm the presence of the random taste heterogeneity across commuters. Mixed nested logit models are estimated to jointly account for the random taste heterogeneity and the correlation among different alternatives. Results indicate that accounting for the random taste heterogeneity as well as inter-alternative correlation improves the model performance.

Real-time decision making by driverless city vehicles : a discrete event driven approach

This thesis addresses the topic of real-time decision making by driverless (autonomous) city vehicles, i.e. their ability to make appropriate driving decisions in non-simplified urban traffic conditions. After addressing the state of research, and explaining the research question, the thesis presents solutions for the subcomponents which are relevant for decision making with respect to information input (World Model), information output (Driving Maneuvers), and the real-time decision making process. TheWorld Model is a software component developed to fulfill the purpose of collecting information from perception and communication subsystems, maintaining an up-to-date view of the vehicle’s environment, and providing the required input information to the Real-Time Decision Making subsystem in a well-defined, and structured way. The real-time decision making process consists of two consecutive stages. While the first decision making stage uses a Petri net to model the safetycritical selection of feasible driving maneuvers, the second stage uses Multiple Criteria Decision Making (MCDM) methods to select the most appropriate driving maneuver, focusing on fulfilling objectives related to efficiency and comfort. The complex task of autonomous driving is subdivided into subtasks, called driving maneuvers, which represent the output (i.e. decision alternatives) of the real-time decision making process. Driving maneuvers are considered as implementations of closed-loop control algorithms, each capable of maneuvering the autonomous vehicle in a specific traffic situation. Experimental tests in both a 3D simulation and real-world experiments attest that the developed approach is suitable to deal with the complexity of real-world urban traffic situations.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Smoothing a Discrete Hazard Function for the Number of Patients Colonized with Methicillin-Resistance Staphylococcus Aureus in an Intensive Care Unit

A new method for estimating the time to colonization of Methicillin-resistant Staphylococcus Aureus (MRSA) patients is developed in this paper. The time to colonization of MRSA is modelled using a Bayesian smoothing approach for the hazard function. There are two prior models discussed in this paper: the first difference prior and the second difference prior. The second difference prior model gives smoother estimates of the hazard functions and, when applied to data from an intensive care unit (ICU), clearly shows increasing hazard up to day 13, then a decreasing hazard. The results clearly demonstrate that the hazard is not constant and provide a useful quantification of the effect of length of stay on the risk of MRSA colonization which provides useful insight.

Cell invasion with proliferation mechanisms motivated by time-lapse data

Cell invasion involves a population of cells which are motile and proliferative. Traditional discrete models of proliferation involve agents depositing daughter agents on nearest- neighbor lattice sites. Motivated by time-lapse images of cell invasion, we propose and analyze two new discrete proliferation models in the context of an exclusion process with an undirected motility mechanism. These discrete models are related to a family of reaction- diffusion equations and can be used to make predictions over a range of scales appropriate for interpreting experimental data. The new proliferation mechanisms are biologically relevant and mathematically convenient as the continuum-discrete relationship is more robust for the new proliferation mechanisms relative to traditional approaches.

Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.

An implicit RBF meshless approach for time fractional diffusion equations

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems

Biochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically difficult to solve, since the state-space involved can be very large or even countably infinite. Recently a finite state projection method (FSP) that truncates the state-space was suggested and shown to be effective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the final time at which the solution was computed, were relatively small. Presented here is a Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time. Additionally the new method computes the solution at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was first demonstrated. Also the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is significantly faster and extendable to larger biological models.