Evaluation of the parameters used in iterative dynamic programming

To apply iterative dynamic programming (IDP) to optimal control problems having a very large number of control variables the use of randomly chosen values for control at each grid point is required. To gain insight into the effect of the number of allowable values for control, the region contraction factor, and the number of grid points for the state vector to be used, computational results are presented for two nonlinear systems, one of which possesses numerous local optima. The reliability of obtaining the global optimum for the bifunctional catalyst blend optimization problem was found to be somewhat higher by using randomly chosen values for control rather than by choosing the control values over a uniform distribution. The global optimum is obtained even when a small number of allowable values for control at each grid point and a small number of grid points for the states are used. There is a wide range of the region contraction factor for which rapid convergence to the optimum is obtained. Also the number of grid points for the state can be very small without adversely affecting convergence to the optimum.     

On solving optimal control problems with free initial condition using iterative dynamic programming

For solving optimal control problems where the initial conditions of some of the state variables are not specified, a procedure based on iterative dynamic programming (IDP) is presented. In this procedure, the free initial conditions are taken to be additional control variables for the first time stage only; then the search for the optimal initial conditions and also the optimal control policy is carried out simultaneously using IDP. The procedure is straightforward, and as illustrated with two nonlinear optimal control problems, for each case the optimum performance index is readily obtained.     

Further developments in the new approach to boundary condition iteration in optimal control

In solving the boundary value problem resulting from the use of Pontryagin's maximum principle, a transformation matrix is used to relate the sensitivity of the final state to the initial state. This avoids the need to solve the (n × n) differential equation to give the transition matrix, and yields very rapid convergence to the optimum. To ensure convergence, iterative dynamic programming (IDP) is used for a number of passes to yield good starting conditions for this boundary condition iteration procedure. Clipping technique is used to handle constraints on control. Five optimal control problems are used to illustrate and to test the procedure.     

Optimal spectra for double object‐colour solids

Optimal colours for human vision occur on the boundary of a three‐dimensional object‐colour solid, and result from optimal reflectance spectra that take on only the values 0 and 1, with at most two transitions between those values. Different illuminants lead to different solids. If there are two illuminants and a single sensing device, then we can construct a six‐dimensional double object‐colour solid by concatenating colour signals from both illuminants. Colours on the boundary of a double‐object solid, and the spectra that generate them, can also be called optimal. This article shows that, while optimal spectra for double solids take on only the values 0 and 1, there is no maximum number of transitions between those values: given a device, we can always construct two illuminants such that the resulting double object‐colour solid has an optimal reflection spectrum with as many transitions as desired.     

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two‐dimensional (2‐D) parabolic partial differential equation with time‐varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time‐varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite‐dimensional plant in terms of a linear time‐varying system. This reduced‐order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2‐D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. © 2014 American Institute of Chemical Engineers AIChE J, 61: 494–502, 2015     

Application of iterative dynamic programming to optimal control of a distiltation column

Iterative Dynamic Programming (IDP) is proven to be a useful technique for solving constrained dynamic optimisation problems. A high purity binary distillation column model has been chosen to investigate some of the IDP properties as well as its applicability. The investigated problems cover transitions from one steady state to another with the minimization of a quadratic cost function with associated terminal constraints.     

A data‐driven rolling‐horizon online scheduling model for diesel production of a real‐world refinery

A rolling‐horizon optimal control strategy is developed to solve the online scheduling problem for a real‐world refinery diesel production based on a data‐driven model. A mixed‐integer nonlinear programming (MINLP) scheduling model considering the implementation of nonlinear blending quality relations and quantity conservation principles is developed. The data variations which drive the MINLP model come from different sources of certain and uncertain events. The scheduling time horizon is divided into equivalent discrete time intervals, which describe regular production and continuous time intervals which represent the beginning and ending time of expected and unexpected events that are not restricted to the boundaries of discrete time intervals. This rolling‐horizon optimal control strategy ensures the dimension of the diesel online scheduling model can be accepted in industry use. LINGO is selected to be the solution software. Finally, the daily diesel scheduling scheme of one entire month for a real‐world refinery is effectively solved. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1160–1174, 2013     

Analysis of Pre‐filter Based Closed‐loop Control‐relevant Identification Methodologies

Control‐relevant identification strategies have been variously proposed for open loop model building. In this paper, issues related to control‐relevant model building in closed‐loop schemes are discussed. Various important aspects such as pre‐filter design and plant friendliness of the perturbation signals have been examined. Simulations involving representative problems have been considered from chemical engineering literature to highlight the applicability of the proposed methods.     

On‐line optimizing control of bulk polymerization of methyl methacrylate: Some experimental results for heater failure

An experimental unit has been assembled to carry out on‐line optimizing control of the bulk polymerization of methyl methacrylate (MMA). A rheometer‐reactor assembly is used. Temperature and viscosity measurements are used to describe the state of the system. The polymerization is carried out under an off‐line computed optimal temperature history, T_op(t). A planned disturbance (heating system failure) is introduced at time t₁. This disturbance leads to a fall in the temperature of the reaction mass. A new optimal temperature history, T_reop(t), is re‐computed on‐line and is implemented on the reaction mass at time t₂, when the heating is resumed. This procedure helps ‘save the batch’. A genetic algorithm is used to compute this reoptimized temperature history in a short period of ～2 min of real time. The feasibility of the on‐line optimizing control scheme has been demonstrated experimentally. Replicable results for the viscosity history, η(t), of the polymerizing mass under several non‐isothermal conditions have been obtained. These experimental results are quite trustworthy, even though the model predictions are only in approximate agreement with them, perhaps because of the extreme sensitivity of results to the values of the model parameters. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 85: 2350–2360, 2002     

Four‐transition 0‐1 functions for reflectance spectra

It is known that a reflectance spectrum for an optimal colour takes on the value 0 or 1 at every wavelength, with at most two transitions between those values. This article shows that any non‐optimal colour can be produced by a reflectance spectrum that takes on the value 0 or 1 at every wavelength, with at most four transitions. While the two‐transition optimal spectrum is unique, the four‐transition non‐optimal spectrum is not unique.     

Robust batch‐to‐batch optimization in the presence of model‐plant mismatch and input uncertainty

In model‐based optimization in the presence of model‐plant mismatch, the set of model parameter estimates which satisfy an identification objective may not result in an accurate prediction of the gradients of the cost‐function and constraints. To ensure convergence to the optimum, the predicted gradients can be forced to match the measured gradients by adapting the model parameters. Since updating all available parameters is impractical due to estimability problems and overfitting, there is a motivation for adapting a subset of parameters for updating the predicted outputs and gradients. This article presents an approach to select a subset of parameters based on the sensitivities of the model outputs and of the cost function and constraint gradients. Furthermore, robustness to uncertainties in initial batch conditions is introduced using a robust formulation based on polynomial chaos expansions. The improvements in convergence to the process optimum and robustness are illustrated using a fed‐batch bioprocess. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2660–2670, 2017     

CO2 methanation: Optimal start‐up control of a fixed‐bed reactor for power‐to‐gas applications

Utilizing volatile renewable energy sources (e.g., solar, wind) for chemical production systems requires a deeper understanding of their dynamic operation modes. Taking the example of a methanation reactor in the context of power‐to‐gas applications, a dynamic optimization approach is used to identify control trajectories for a time optimal reactor start‐up avoiding distinct hot spot formation. For the optimization, we develop a dynamic, two‐dimensional model of a fixed‐bed tube reactor for carbon dioxide methanation which is based on the reaction scheme of the underlying exothermic Sabatier reaction mechanism. While controlling dynamic hot spot formation inside the catalyst bed, we prove the applicability of our methodology and investigate the feasibility of dynamic carbon dioxide methanation. © 2016 American Institute of Chemical Engineers AIChE J, 63: 23–31, 2017     

Integrated batch‐to‐batch and nonlinear model predictive control for polymorphic transformation in pharmaceutical crystallization

Polymorphism, a phenomenon in which a substance can have more than one crystal form, is a frequently encountered phenomenon in pharmaceutical compounds. Different polymorphs can have very different physical properties such as crystal shape, solubility, hardness, color, melting point, and chemical reactivity, so that it is important to ensure consistent production of the desired polymorph. In this study, an integrated batch‐to‐batch and nonlinear model predictive control (B2B‐NMPC) strategy based on a hybrid model is developed for the polymorphic transformation of L ‐glutamic acid from the metastable α‐form to the stable β‐form crystals. The hybrid model comprising of a nominal first‐principles model and a correction factor based on an updated PLS model is used to predict the process variables and final product quality. At each sampling instance during a batch, extended predictive self‐adaptive control (EPSAC) is employed as a NMPC technique to calculate the control action by using the current hybrid model as a predictor. At the end of the batch, the PLS model is updated by utilizing the measurements from the batch and the above procedure is repeated to obtain new control actions for the next batch. In a simulation study using a previously reported model for a polymorphic crystallization with experimentally determined parameters, the proposed B2B‐NMPC control strategy produces better performance, where it satisfies all the state constraints and produces faster and smoother convergence, than the standard batch‐to‐batch strategy. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Nonlinear model predictive control for the polymorphic transformation of L‐glutamic acid crystals

Polymorphism, a phenomenon where a substance can have more than one crystal forms, has recently become a major interest to the food, speciality chemical, and pharmaceutical industries. The different physical properties for polymorphs such as solubility, morphology, and dissolution rate may jeopardize operability or product quality, resulting in significant effort in controlling crystallization processes to ensure consistent production of the desired polymorph. Here, a nonlinear model predictive control (NMPC) strategy is developed for the polymorphic transformation of L ‐glutamic acid from the metastable α‐form to the stable β‐form crystals. The robustness of the proposed NMPC strategy to parameter perturbations is compared with temperature control (T‐control), concentration control (C‐control), and quadratic matrix control with successive linearization (SL‐QDMC). Simulation studies show that T‐control is the least robust, whereas C‐control performs very robustly but long batch times may be required. SL‐QDMC performs rather poorly even when there is no plant‐model mismatch due to the high process nonlinearity, rendering successive linearization inaccurate. The NMPC strategy shows good overall robustness for two different control objectives, which were both within 7% of their optimal values, while satisfying all constraints on manipulated and state variables within the specified batch time. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Volume‐of‐fluid‐based model for multiphase flow in high‐pressure trickle‐bed reactor: Optimization of numerical parameters

Aiming to understand the effect of various parameters such as liquid velocity, surface tension, and wetting phenomena, a Volume‐of‐Fluid (VOF) model was developed to simulate the multiphase flow in high‐pressure trickle‐bed reactor (TBR). As the accuracy of the simulation is largely dependent on mesh density, different mesh sizes were compared for the hydrodynamic validation of the multiphase flow model. Several model solution parameters comprising different time steps, convergence criteria and discretization schemes were examined to establish model parametric independency results. High‐order differencing schemes were found to agree better with the experimental data from the literature given that its formulation includes inherently the minimization of artificial numerical dissipation. The optimum values for the numerical solution parameters were then used to evaluate the hydrodynamic predictions at high‐pressure demonstrating the significant influence of the gas flow rate mainly on liquid holdup rather than on two‐phase pressure drop and exhibiting hysteresis in both hydrodynamic parameters. Afterwards, the VOF model was applied to evaluate successive radial planes of liquid volume fraction at different packed bed cross‐sections. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Extension of multiunit global optimisation to three‐input systems

Finding the global optimum of an objective function has been of interest in many disciplines. Recently, a global optimisation technique based on multiunit extremum seeking has been proposed for scalar and two‐input systems. The idea of multiunit extremum‐seeking is to control the gradient evaluated using finite difference between two identical units operating with an offset. For scalar systems, it was shown that the global optimum could be obtained by reducing the offset to zero. For two‐input systems, the univariate global optimisation is performed on the circumference of a circle of reducing radius. In this study, the concept is extended to three‐input systems where the circle of varying radius sits on a shrinking sphere. The key contribution lies in formulating the rotation required to keep the best point found in the search domain. The theoretical concepts are illustrated on the global optimisation of several examples. Comparison results with other competitive methods show that the proposed technique performs well in terms of number of function evaluations and accuracy. © 2011 Canadian Society for Chemical Engineering     

Mixed integer optimal control of an intermittently aerated sequencing batch reactor for wastewater treatment

Optimal aeration control strategies for sequencing batch reactors in WWT with bypass nitrification are hereby studied. The operation is defined alternating aerobic and anoxic phases with high frequency. The controlled variable, the aeration, can only adopt fixed values, on and off, leading to a discrete trajectory of bang–bang type. The problem is to compute the number of switches and individual length of each aerobic and anoxic stage. This leads to a mixed integer nonlinear optimal control problem (MINTOC). The solution is challenging, since both integer and continuous variables ought to be considered in the optimization. In contrast to previous work, where optimization is performed based on the separation and independent solution of the integer and continuous problem, we apply an algorithm originally proposed by Sager (2005). The optimization program minimizes operation time and energy consumption. Effluent concentrations are considered as nonlinear constraints in accordance to environmental regulations.