Designing priors for robust Bayesian optimal experimental design

Building mathematical models is a common task in process systems engineering, which requires estimation of model parameters as the final step of modeling exercise. Model based experimental design has evolved as a potential statistical tool for reducing uncertainties in parameter estimates. Often a huge volume of process information is generated as an end result of an experimental design. Designing optimal experiments based on current or prior process knowledge is still an open research problem. This paper deals with how information, available a priori, can be organized and systematically used for designing robust Bayesian dynamic experiments, in the presence of process constraints. The designed experiments are ‘robust’ to a poor choice of nominal parameter values. Several novel techniques for organizing a priori process knowledge are explored from a theoretical view point. The influence of proposed prior designs on parameter estimates is demonstrated on a semi-continuous baker's yeast fermenter problem.     

Simulation and experimental study in determining injection molding process parameters for thin-shell plastic parts via design of experiments analysis

This paper deals with the application computer-aided engineering integrating with statistical technique to reduce warpage variation depended on injection molding process parameters during production of thin-shell plastic components. For this purpose, a number of Mold-Flow analyses are carried out by utilizing the combination of process parameters based on three level of L₁₈ orthogonal array table. In the meantime, apply the design of experiments (DOE) approach to determine an optimal parameter setting. In addition, a side-by-side comparison of two different approaches of simulation and experimental is provided. In this study, regression models that link the controlled parameters and the targeted outputs are developed, and the identified models can be utilized to predict the warpage at various injection molding conditions. The melt temperature and the packing pressure are found to be the most significant factors in both the simulation and the experimental for an injection molding process of thin-shell plastic parts.     

Model-based temperature estimation of power electronics systems

This paper proposes a method for accurate temperature estimation of thermally-aware power electronics systems. The duality between electrical systems and thermal systems was considered for thermal modeling. High dimensional thermal models present a challenge for online estimation. Therefore, the complexity of the thermal network was reduced by applying a structure-preserving model order reduction technique. An optimal number and placement of temperature sensors were used in a Kalman filter to accurately estimate the dynamic spatial thermal behavior of the system. The optimal number of temperature sensors was found by comparing the actual values of the states obtained from the thermal model to the estimated values of the states obtained from the Kalman filter. The optimal placement of temperature sensors was found by maximizing the trace of the observability Gramian. Simulation and experimental results validate the approach on a prototype inverter.     

On the Experiment Design for Direct Dynamic Parameter Identification of Parallel Robots

Accurate robot dynamic models require the estimation and validation of the dynamic parameters through experiments. To this end, when performing the experiments, the system has to be properly excited so that the unknown parameters can be accurately estimated. The experiment design basically consists of optimizing the trajectory executed by the robot during the experiment. Due to the restricted workspace with parallel robots this task is more challenging than for serial robots; thus, this paper is focused on the experiment design aimed at dynamic parameter identification of parallel robots. Moreover, a multicriteria algorithm is proposed in order to reduce the deficiencies derived from the single-criterion optimization. The results of the identification using trajectories based on a single criterion and the multicriteria approaches are compared, showing that the proposed optimization can be considered as a suitable procedure for designing exciting trajectories for parameter identification.     

Optimal experiment design for linear systems with input-output constraints

The problem of optimal experiment design for parameter estimation in linear dynamic systems is studied. Results relating to both constrained input and output variances are established. For the case of constrained input variance, it is shown that a D-optimal experiment exists in which the system input is generated externally provided the system and noise transfer functions have no common parameters. For the case of constrained output variance, it is shown that an experiment in which the system input is generated by a combination of a minimum variance control law together with an external set point perturbation is D-optimal for certain classes of systems. Other related results are also presented which illustrate the role of feedback in optimal experiment design.     

Constrained parameter estimation with applications to blending operations

The classical least squares approach to parameter estimation for dynamic models ignores a priori information about the feasible values of the estimated parameters. But in many practical problems, such information is available in the form of upper and lower limits. In this paper, two alternative techniques are evaluated for this important class of constrained parameter estimation problems for dynamic systems. Simulation results for two blending problems illustrate that more accurate parameter estimates and better predictions can be obtained by using a quadratic programming approach.     

Optimal allocation of testing effort during testing and debugging phases: a control theoretic approach

Allocation of efforts to a software development project during the testing phase is a multifaceted task for software managers. The challenges become stiffer when the nature of the development process is considered in the dynamic environment. Many software reliability growth models have been proposed in last decade to minimise the total testing-effort expenditures, but mostly under static assumption. The main purpose of this article is to investigate an optimal resource allocation plan to minimise the cost of software during the testing and operational phase under dynamic condition. An elaborate optimisation policy based on the optimal control theory is proposed and numerical examples are illustrated. This article also studies the optimal resource allocation problems for various conditions by examining the behaviour of the model parameters and also suggests policy for the optimal release time of the software. The experimental results greatly help us to identify the contribution of each selected parameter and its weight.     

考虑退化模式动态转移的健康状态自适应预测

为了更加精确地在设备退化过程中对其健康状态进行预测,本文深入研究了设备处于不同健康状态时的数据特点,针对现有单一预测方法的特点与不足,引入了退化模式的划分方法,并对不同的预测模型与退化模式的关系进行分析. 进而建立“模式-模型”关联表,并通过关联表优选预测模型,实现了考虑退化模式动态转移的健康状态自适应预测以及剩余寿命估计.最后,以滚动轴承实验为实例,对该轴承进行了健康状态预测与剩余寿命估计.实验结果表明本方法较精确地预测了轴承的剩余寿命,证明了方法的有效性.     

Design of experiments for bivariate binary responses modelled by Copula functions

Optimal design for generalized linear models has primarily focused on univariate data. Often experiments are performed that have multiple dependent responses described by regression type models, and it is of interest and of value to design the experiment for all these responses. This requires a multivariate distribution underlying a pre-chosen model for the data. Here, we consider the design of experiments for bivariate binary data which are dependent. We explore Copula functions which provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. We present methods for deriving optimal experimental designs for dependent bivariate binary data using Copulas, and demonstrate that, by including the dependence between responses in the design process, more efficient parameter estimates are obtained than by the usual practice of simply designing for a single variable only. Further, we investigate the robustness of designs with respect to initial parameter estimates and Copula function, and also show the performance of compound criteria within this bivariate binary setting.     

Data-driven,using design of dynamic experiments,versus model-driven optimization of batch crystallization processes

A new data-driven experimental design methodology, design of dynamic experiments (DoDE), is proposed as a means of developing a response surface model that can be used to effectively optimize batch crystallization processes. This data-driven approach is especially useful for complex processes for which it is difficult or impossible to develop a knowledge-driven model in a timely fashion for the optimization of an industrial process. Design of dynamic experiments [1] generalizes the formulation of time-invariant design variables from design of experiments, allowing for consideration of time-variant design variables in the experimental design. When combined with response surface modeling and an appropriate optimization algorithm, a data-driven optimization methodology is produced, which we call DoDE optimization. The method is used here to determine the optimal cooling rate profile, which integrates to give the optimum temperature profile, for a batch crystallization process. To examine the effectiveness of the DoDE optimization method, the data-driven optimum temperature profile is compared to the optimum temperature profile obtained using a model-based optimization technique for the potassium nitrate–water batch crystallization model developed by Miller and Rawlings [2]. The temperature profiles calculated using DoDE optimization yield response values within a few percent of the true model-based optimum values. A sensitivity analysis is performed on one case study to evaluate the distribution of the response variable from each method in the presence of parameter and initial seed distribution variability. It is demonstrated that there is partial overlap in the distributions when only variability in the model parameters is evaluated and there is substantial overlap when variability is included in both the model and initial seed distribution parameters. From this evidence, it can be concluded that the DoDE optimization method has the potential to be a useful data-driven optimization tool for batch crystallization processes where a first-principles model is not available or cannot be developed due to time and/or cost constraints.     

Faster power estimation of CMOS designs using vector compaction - a fractal approach

Low power digital complementary metal oxide semiconductor (CMOS) circuit design requires accurate power estimation. In this paper, we present a compaction algorithm for generating compact vector sets to estimate power efficiently. Power can be estimated using dynamic (simulation) or static (statistical/probabilistic) techniques. Dynamic power estimation techniques simulate the design using a large input vector set for accurate estimation. However, the simulation time is prohibitively long for bigger designs with larger vector sets. The statistical methods, on the other hand, use analytical tools that make them faster but less accurate. To achieve the accuracy of dynamic power estimation and the speed of statistical methods, one approach is to generate a compact, representative vector set that has the same switching transition behavior as the original larger vector set. The compaction algorithm presented in this paper uses fractal concepts to generate such a compact vector set. The fractal technique quantifies correlation by a fractal parameter which can be determined faster than calculating correlation explicitly. Experimental results on circuits from the ISCAS85 and ISCAS89 benchmark suites, with correlated input vector sets, resulted in a maximum compaction ratio of 65.57X (average 38.14X) and maximum power estimation error of 2.4% (average 2.06%). Since the size of the compact vector set used for simulation is smaller, the simulation time will be shorter and will significantly speed up the design cycle.     

Robust, reduced-order, nonstrictly proper state estimation via theoptimal projection equations with guaranteed cost bounds

A state estimation design problem involving parametric plant uncertainties is considered. An estimation error bound suggested by multiplicative white-noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties. Necessary conditions that generalize the optimal projection equations for reduced-order state estimation are used to characterize the estimator that minimizes the error bound. The design equations thus effectively serve as sufficient conditions for synthesizing robust estimators. Additional features include the presence of a static estimation gain in conjunction with the dynamic (Kalman) estimator to obtain a nonstrictly proper estimator     

Robot dynamic calibration: Optimal excitation trajectories and experimental parameter estimation

Advanced robot control schemes require an accurate knowledge of the dynamic parameters of the manipulator. This article examines various issues related to robot dynamic calibration, from generation of optimal excitation trajectories to data acquisition and filtering, and experimental inertial and friction parameter estimation. In particular, a new method is developed for the determination of optimal joint trajectories for the calibration experiment, which is based on evolutionary optimization techniques. A genetic algorithm is used to determine excitation trajectories that minimize either the condition number of the regression matrix or the logarithmic determinant of the Fisher information matrix. All the calibration steps have been carried out on a SCARA two‐link planar manipulator, and the experimental results are discussed. © 2001 John Wiley & Sons, Inc.     

序贯实验设计法在吸附平衡实验中的应用

采用序贯实验设计法对正葵烷在5A分子筛上的吸附等温线实验进行了实验设计。根据Box和Lucas所提出的使待估参数联合置信域的容积趋于最小的参数估算的序贯实验设计准则，推导得出了正葵烷吸附等温线实验观测点的安排原则，发现对于一元线性模型，实验观测点应安排在可操作区的两端。对序贯设计实验和未经设计实验做了比较，经过序贯设计的实验所得到的模型参数的联合置信域比未经设计实验的小得多，表明序贯实验设计可提高模型参数估算的精度，减少实验工作量。     

Identification of a rice drying model with an improved sequential optimal design of experiments

Getting relevant parameter estimation of a non-linear model is often a hard task from both an experimental and numerical point of view. The objective of optimally designed experiments procedure is to diminish the experimental effort needed such that the identification is within acceptable confidence ranges. After each experiment, the next experiment is optimally designed, taking into account all past experimental results. It allows quality information to be extracted from the experimental data with less experimental time and resource consumption.In this paper, we present an original approach and implementation of the classical A-, D- and E-optimality on the estimation of 5 unknown (transfer related) coefficients in a compartmental model used to describe the convective drying of rice. The originality of our method is that it uses reparameterization of both parameter and protocol vectors which permits to avoid using a global optimization algorithm. The presented method is implemented in Matlab as a Toolbox and fully tested on a pilot plant. The case study (drying of rice) is typical in the field of process engineering: the dynamic model is strongly non-linear in its parameters and cannot be analytically solved. In addition, the specific technical constrains (inertias, limits, etc.) on the pilot are explicitly taken into account for improved experimental feasibility.In this drying application, three experiments with non-constant drying conditions are shown to be quite as effective as a two-factor three-level grid of nine experiments at constant conditions, with only one third of the experimental effort.     

Sensitivity analysis of microbial growth parameter distributions with respect to data quality and quantity by using Monte Carlo analysis

Nowadays, most of the mathematical models used in predictive microbiology are deterministic, i.e. their model output is only one single value for the microbial load at a certain time instant. For more advanced exploitation of predictive microbiology in the context of hazard analysis and critical control points (HACCP) and risk analysis studies, stochastic models should be developed. Such models predict a probability mass function for the microbial load at a certain time instant. An excellent method to deal with stochastic variables is Monte Carlo analysis. In this research, the sensitivity of microbial growth model parameter distributions with respect to data quality and quantity is investigated using Monte Carlo analysis. The proposed approach is illustrated with experimental growth data. There appears to be a linear relation between data quality (expressed by means of the standard deviation of the normal distribution assumed on experimental data) and model parameter uncertainty (expressed by means of the standard deviation of the model parameter distribution). The quantity of data (expressed by means of the number of experimental data points) as well as the positioning of these data in time have a substantial influence on model parameter uncertainty. This has implications for optimal experiment design.     

Optimal experiment design for regression polynomial models identification

In this paper the problem of optimal experimental design for parameter identification of static non-linear blocks is addressed. Non-linearities are assumed to be polynomial and represented according to the Vandermonde base. The optimality problem is formulated in a set membership context and the cost functions to be minimized are the worst case parameter uncertainties. Closed form optimal input sequences are derived when the input u is allowed to vary on a given interval [ u a, u b ]. Since optimal input sequences are, in general, not invariant to base changes, results and criteria for representing polymomials with different bases, still preserving the optimal set of input levels derived from the Vandermonde parameterization, are introduced as well. Finally numerical results are reported showing the effectiveness of using optimal input sequences especially when identifying some block described dynamic models that include in their structure static non-linearities (such as Hammerstein and LPV models). In such cases the improvement achieved in the confidence of the estimates can add up to a factor of several hundreds with respect to the case of random generated inputs.     

Structural identifiability and indistinguishability analyses of the Minimal Model and a Euglycemic Hyperinsulinemic Clamp model for glucose-insulin dynamics

Many mathematical models have been developed to describe glucose-insulin kinetics as a means of analysing the effective control of diabetes. This paper concentrates on the structural identifiability analysis of certain well-established mathematical models that have been developed to characterise glucose-insulin kinetics under different experimental scenarios. Such analysis is a pre-requisite to experiment design and parameter estimation and is applied for the first time to these models with the specific structures considered. The analysis is applied to a basic (original) form of the Minimal Model (MM) using the Taylor Series approach and a now well-accepted extended form of the MM by application of the Taylor Series approach and a form of the Similarity Transformation approach. Due to the established inappropriate nature of the MM with regard to glucose clamping experiments an alternative model describing the glucose-insulin dynamics during a Euglycemic Hyperinsulinemic Clamp (EIC) experiment was considered. Structural identifiability analysis of the EIC model is also performed using the Taylor Series approach and shows that, with glucose infusion as input alone, the model is structurally globally identifiable. Additional analysis demonstrates that the two different model forms are structurally distinguishable for observation of both glucose and insulin.     

An approach to the selection of optimal sensor locations in distributed parameter systems

This paper presents an approach to the selection of optimal sensor locations in distributed parameter systems, which distinguishes the purposes of state estimation from the purposes of parameter estimation. In the first case, the optimality criterion is based on a measure of independence between the sensor responses, while in the second case, it is based on a measure of independence between the parameter sensitivity functions. The procedure, which is general and can be applied to models with any degree of complexity, is illustrated with the optimal placement of temperature sensors in a catalytic fixed-bed reactor. Some numerical results for the on-line estimation of temperature and concentration profiles as well as for the estimation of unknown model parameters are discussed.     

Digital parameter-adaptive control of an air-conditioning plant

Two parameter-adaptive (self-tuning) control algorithms for multivariable systems are described. The algorithms are designed on the basis of linear input-output system models by the combination of recursive parameter estimation and control algorithms: a parameter-adaptive deadbeat controller and a parameter-adaptive optimal state controller. These controllers are applied to a two-input two-output air-conditioning pilot plant, which consists of an air heater and an air humidifier and whose output variables are the temperature and the relative humidity of the air measured at the air outlet. The air-conditioning plant is a nonlinear system and its linearized static and dynamic behaviour is strongly dependent on the operating point characterized mainly by the output variables and by the air flow rate through the plant. The results of the real-time control experiments indicate that it is possible to use the self-tuning features of the parameter-adaptive controllers to stabilize the controlled system after a short adaption phase and to achieve at least a satisfactory control performance for time varying air flow rates and for time varying setpoints of the output variables.