Stochastic Trajectory Planning Considering Parameter and Initial State Uncertainties

Errors in numerical weather forecasts can be attributed to two causes: deficiencies in the modeling system and inaccurate initial conditions. Understanding of the characteristics of the growth of forecast spread related to model uncertainty is less developed than that for initial condition uncertainty. In this research, the authors aim to construct a theoretical basis for describing such forecast error growth resulting from model uncertainty using mostly an empirical modeling approach. Primitive equation models with different vertical discretization and different horizontal resolutions are used to investigate the impacts of model uncertainties on the predictability of extratropical cyclones. Three sets of initial perturbations related to an upper-level trigger, with slightly different amplitudes, are designed for representing the situation when the initial condition uncertainty leads to significant forecast error growth. Forecast error growth is here estimated by following the properties of a developing cyclone in the simulations. Generally, there are three phases for forecast error growth in the experiments with initial condition and model uncertainties. For the experiments with the structured initial condition uncertainties, the errors grow rapidly at the earlier transient stage, with the growth rate well above the fastest growing normal mode. Afterward the error grows exponentially at approximately the same growth rate as the cyclone, followed by a saturation period, when the growth rate starts to decline. For the experiments with the model uncertainties, the forecast errors are initially zero and increase as time to a power of μ, which is between 0.5 and 3 depending on the strength of the cyclone at the time the simulation is initiated. After a certain time interval, the exponential growth phase and saturation period start as in the initial error experiments. Starting an integration with a stronger initial cyclone, the forecast error associated with the model uncertainty takes a shorter time to reach the exponential growth period and the forecast error grows more rapidly initially with a smaller value of μ. Also, when the initial cyclone is strong enough, then the exponential growth phase may only last for a very short time.

Rhamnolipids are bio surfactants which are extra-cellular glycolipids composed of l-rhamnose and 3-hydroxyalkanoics. Rhamnolipids are produced through fermentation process by using Pseudomonas sp. as the species. An alteration to the traditional procedures in order to achieve increase in the production of biosurfactants, a numerous process technologies have been adopted in fed batch mode. Fed batch mode facilitates the high production of product by avoiding the substrate or product inhibition line of attack. To overcome the controlling parameters which reduce the product yield, optimal control profiles are designed. In order to develop viable control methods for fed-batch fermentation of Rhamnolipid production, multiple substrate feeding strategies were employed and their efficiencies were compared with different substrates concentration of glucose, nitrogen and phosphorous. The product formation depends upon the substrate feeding strategy and so, the fed-batch fermentation was carried out by using P. aeruginosa providing substrates at manifold rates. With the obtained experimental data, using the kinetic models (logistic equation and by Luedeking Piret), the kinetic parameters were estimated. These kinetic parameters were implemented in tabu search algorithm and this programme was executed in Dev-C++, optimal control profiles were generated as a result. These obtained optimal control profiles have shown an increase in productivity of rhamnolipid with a decline in computational time. Through this procedure, the optimal control profiles of substrate feeding strategies of glucose, nitrogen and phosphorous were estimated. In comparison with other algorithms like genetic algorithm, Tabu Search algorithm was able to generate an accurate optimal control profiles with a reduction in their intricacy.

Process reliability and safety under uncertainties

A multiresolution approach is presented to study the uncertainty propagation problem in dynamic systems. The proposed approach is an extension of the generalized polynomial chaos and involves a separation of random variables from deterministic ones in the solution algorithm for a stochastic dierential equation. The random variables are expanded in various local polynomial expansions which are further blended together to obtain a global approximation for the random state variables. These polynomials are associated with the assumed probability density functions for the input variables. Galerkin projection is used to generate a system of deterministic dierential equations for the expansion coecients. We rst present this method to linear systems with parametric uncertainties, and subsequently we generalize it to nonlinear systems with parametric and initial condition uncertainties. The eectiveness of the proposed method is demonstrated by considering dierent numerical examples.

The performance of empirical model based fed-batch process optimal control is strongly affected by the model prediction reliability at the end-point of a batch. An optimal control profile calculated from an empirical model may not give the best performance when applied to the actual process due to model-plant mismatches. To tackle this issue, a new method for improving the reliability of fed-batch process optimal control by incorporating model prediction confidence bounds is proposed. Multiple neural networks (MNN) are used to build an empirical model of fed-batch process based on process operation data. Model prediction confidence bounds are calculated based on predictions of all component networks in an MNN model and the model prediction confidence bound at the end-point of a batch is incorporated into the optimization objective function. The modified objective function penalizes wide prediction confidence bounds in order to obtain a reliable optimal control profile. The non-linear optimization problem based on MNN with augmented objective function is solved by iterative dynamic programming. The proposed control strategy is illustrated on a simulated fed-batch ethanol fermentation process. The results demonstrate that the optimal control profile calculated from the proposed approach is reliable in the sense that its performance degradation is limited when applied to the actual process.

This paper considers the design of open-loop fuel/time optimal controllers for flexible space struc tures using a frequency domain approach. The control system consists of a time-delay filter whose output signal is the optimal control profile when it is subject to a step input. A constrained parameter optimization problem is formulated to minimize a weighted combination of fuel consumed and total maneuver time for a rest-to-rest maneuver. The parameters to be optimized for are the delays of a time-delay filter. The number of switches of the fuel/time optimal control profile is shown to be a function not only of number of flexible modes but also of the weighting parameter, which is illustrated via numerical examples.

This paper considers a method for designing a compensation law for constrained linear systems with initial state uncertainty. By introducing a decomposition technique for finite-horizon linear systems, we derive a compensation law that satisfies system constraints under initial state uncertainty of a plant. The compensation law is characterized by a piecewise affine function of the available plant state. A numerical example is presented for illustrating the proposed compensation law.

Summer-time short- to medium-range predictability of precipitation, 500-hPa geopotential height, and wind fields over East Asia are investigated by comparing three ensemble forecast configurations: multi-analysis, multi-convection, and multi-model. These three systems are used in this study in order to assess initial condition uncertainties, model uncertainties, and a combination of initial condition and model uncertainties in an ensemble forecast approach. Each system has a set of six members. Ensemble forecast skill is verified in both deterministic and probabilistic senses using the European Center for Medium-range Weather Forecasting analyses and the Tropical Rainfall Measuring Mission Microwave Imager 2A12 rain estimates. The multi-model configuration, which considers both the initial condition and model uncertainties to predict weather phenomena over East Asia, is an optimal set of ensemble members. The bias-corrected ensemble and the superensemble (SE) show similar predictability, but slightly better skill is obtained from the SE forecasts.

In this paper we consider infinite dimensional optimization problems with equality constraints. The basic underlying algorithm is a reduced SQP method. We give a global convergence proof in Hilbert space and extend this analysis to inexact reduced SQP methods. These methods are useful when discretizing the infinite dimensional problems and solving the resulting large scale discretized optimization problems. The convergence analysis takes into account the Maratos effect which occurs for nonsmooth norms. The inexact reduced SQP method is applied to a discretized parabolic control problem

Mars entry trajectory planning using robust optimization and uncertainty quantification

As the food industry is largely batch-oriented, many optimization problems in food engineering are actually optimal control problems. A computer package, ICRS/DS (Integrated Controlled Random Search for Dynamic Systems), for the optimal control of batch processes has been developed. It is based on the reduction of the optimal control problem to a constrained nonlinear optimization problem via an adequate transformation of the control function. This problem is then solved using an stochastic optimization algorithm which assures convergence with reasonable computation times. This package has been successfully used for the optimization of two important batch processes in the food industry: thermal processing of canned foods and air dehydration of foodstuffs. Different objective functions (overall nutrient retention, quality factor retention, process time, energy consumption, etc.) and constraints were considered for these processes. In all cases, ICRS/DS proved to be a reliable, useful and easy-to-use computational tool. Results show that, in some situations, the optimal non-linear control profiles have significant advantages over the traditional constant ones.

The focus of this paper is on the development of an open loop controller for type 1 diabetic patients which is robust to meal and initial condition uncertainties in the presence of hypo- and hyperglycemic constraints. Bernstein polynomials are used to parametrize the evolving uncertain blood-glucose. The unique bounding properties of these polynomials are then used to enforce the desired glycemic constraints. A convex optimization problem is posed in the perturbation space of the model and is solved repeatedly to sequentially converge on a sub-optimal solution. The proposed approach is demonstrated on the classic Bergman model for Type 1 diabetic patients.

As simulation continues to replace experimentation in the design cycle, the need to quantify uncertainty in model outputs due to uncertainties in the model parameters becomes critical. For distributed parameter models, current approaches assume the mean and variance of parameters are known, then use recently developed efficient numerical methods for approximating stochastic partial differential equations. However, the statistical descriptions of the model parameters are rarely known. A number of recent works have investigated adapting existing variational methods for parameter estimation to account for parametric uncertainty. In this paper, we formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers and the first order necessary conditions. A spectral approximation of the uncertain observations (via a truncated Karhunen-Loève expansion) allows an approximation of the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called `finite noise' problem, in the space of functions with bounded mixed derivatives. We prove convergence of `finite noise' minimizers to the corresponding infinite dimensional solutions, and devise a gradient based strategy for locating these numerically. Lastly, we illustrate our method with a numerical example.

Optimal control of batch electrochemical reactor using K–L expansion

Optimal control of semi-batch reactors