Application of dynamic programming to high-dimensional systems described by difference equations

The application of dynamic programming using an interative procedure, which employs domain contraction and accessible grid points for the state vector, is investigated for solving optimal control problems where the system is described by a set of difference equations. A model of a gas absorber with six state variables and two control variables is used to examine the feasibility of the proposed algorithm and to analyze the effects of the grid size and of the domain contraction factor. This algorithm yields rapid convergence to the optimum even when a relatively coarse grid with as few as 11 grid points is used.     

Global optimization of the bifunctional catalyst problem

As shown by Luus et al. (1992), determination of the optimum catalyst blend along a tubular reactor to maximize the yield of a particular component is a difficult problem due to the large number of local optima. By using iterative dynamic programming, the effects of the region reduction factor and the choice of the initial region size are analyzed in detail for this problem. Even if a single grid-point for the state is used, the likelihood of obtaining the global optimum is very high if the initial region size is chosen to be sufficiently large.     

Bang‐Bang solution of nonlinear time‐optimal control problems using a semi‐exhaustivesearch

At times, the objective is to seek a bang‐bang control policy for nonlinear time‐optimal control problems. The usefulness of iterative dynamic programming (IDP) has been shown in the literature for solving such problems. However, the convergence to the optimal solution has been obtained from about 50% of the guessed values near the optimum. In this paper, we present a semiexhaustive search method for seeking such solutions and a comparison is made with the IDP. The results show that the convergence can be obtained from a significantly higher number of guessed values chosen over a much wider region around the optimum.     

Optimal control by direct search on feedback gain matrix

For a large class of optimal control problems the optical control law assumes the form of the product of a feedback gain matrix and the state vector. Although in general the gain matrix has time-depenent elements, results very close to the optimum can be obtained if constant elements are used, provided that these are chosen in optimal manner.Three examples are chosen to test the feasibility of direct search optimization in determining the best values for the elements in the gain matrix. The method is attractive due to the ease of programming and the assurance of the global optimum.     

Incorporation of gradient into random search optimization

To improve the rate of convergence of random search optimization procedures, the incorporation of a gradient-oriented one-dimensional search is investigated. With this modification to the random search procedure based on uniform sampling and region contraction [1], convergence to within 0.01% of the global optimum was obtained substantially faster for typical chemical engineering problems. It was also found that the reliability of obtaining the global optimum was improved.     

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.     

Observer-based nonlinear control law for a continuous stirred tank reactor with recycle

This paper proposes new results concerning the problem of the control of a continuous stirred tank reactor with recycle. The novelty of the proposed results consists of a new nonlinear observer-based controller which is found by means of recent results of differential geometry for time-delay nonlinear systems, without using linear approximations of the model. Local convergence of the system state to the arbitrarily chosen operating point is theoretically proved. The significance of the proposed control law is shown by many simulations, which show high performances with any initial conditions, even at the start-up, and with critical cases of mismatched parameter values.     

Time optimal control of high dimensional systems by iterative dynamic programming

For time optimal control, the problem is first transformed into a finite dimensional optimization problem by using time stages of varying lengths to enable accurate switching, and then solved by iterative dynamic programming. In high dimensional systems care is necessary to ensure convergence to the global optimum. Incorporating penalty functions into the performance index and using an adequate number of stages can yield the global optimum. The use of a continuation approach, where the number of stages in an intermediate solution is systematically increased, appears to be more effective. Three linear systems are used to develop and to test the approaches.     

Comparative Analysis of Single‐Cascade Five‐Zone and Two‐Zone SMB Systems for the Separation of a Ternary Amino Acid Mixture

A ternary separation usually requires the use of two simulated moving bed (SMB) units in series. Since an increase in the number of SMB units leads to a significant increase in capital and operational costs, the use of a single SMB unit is preferred if its structure can be modified to treat a ternary separation. Such a modified single SMB unit has been typified by a five‐zone SMB or a two‐zone SMB so far. The separation performances‐of a five‐zone SMB and a two‐zone SMB are compared in this paper by using the ternary amino acid mixture as a model system. A five‐zone SMB is designed with the safety margin method while a two‐zone SMB is optimized using genetic algorithm. A five‐zone SMB based on the maximum allowable safety margin, although it may not guarantee the global optimum solution, results in much better separation performance than a two‐zone SMB at its global optimum state.     

Electrochemical digital simulations with an exponentially expanding grid: General expressions for higher order approximations to spatial derivatives: The special case of four-point formulas and their application to multipulse techniques in planar and any size spherical electrodes

This paper deals with the consideration, in digital simulation, of a grid in which there is a geometrical relationship between the distances between neighbouring pairs of points, leading to an exponentially expanding sequence of intervals. We show that the expressions for derivatives for higher order spatial discretisations take a very simple form, with coefficients which are dependent on the number of points taken to the left and right of the point considered, but independent of the absolute position of the point in the grid. This enables convenient and efficient manipulation and incorporation in computer programs of these derivatives in the context of finite difference approach. Moreover, the explicit expressions obtained lead to very interesting particular situations. The most interesting is the optimal behaviour of the four-point spatial discretisation for very high expansion factors, which, in combination with fully implicit time-integration schemes, leads to very fast and accurate calculations both for planar and any size spherical electrodes, including microelectrodes. Applications of these coefficients in the simulation of some multipulse and square wave experiments are also presented.     

多块修正技术——块修正技术的一般化推广

块修正技术可以大大加速数值法求解传热传质与流动问题时的收敛速度,双块修正技术使许多问题的收敛速度进一步提高.本文将块修正技术作了一般化推广,推导了多块(任意块数)修正技术的计算公式,研究了修正块数对不同问题收敛速度的影响,并找得了最佳修正块数与所研究问题的性质及网格结点数之间的一般关系.文中还研究了块修正与行TDMA迭代扫描次数之间的相互配合关系,并确定了每次块修正技术之后最佳的行(或列)TDMA迭代扫描次数.     

Solution of population balances by high order moment-conserving method of classes: reconstruction of a non-negative density distribution

A method is proposed for reconstruction of a non-negative piecewise continuous density distribution from a solution obtained by the high order moment conserving method of classes (HMMC), presented in our earlier papers [Alopaeus, V., Laakkonen, M., Aittamaa, J., 2006]. Solution of population balances with breakage and agglomeration by high order moment-conserving method of classes. Chemical Engineering Science 61, 6732-6752; Alopaeus, V., Laakkonen, M., Aittamaa, J., 2007. Solution of population balances with growth and nucleation by high order moment-conserving method of classes. Chemical Engineering Science 62, 2277-2289.]. The resulting distribution is strictly non-negative, and the overall distribution moments obtained by HMMC are conserved very accurately. The distribution is reconstructed by using the positive region of HMMC as an initial estimate for the continuous density distribution. This estimate is then refined by transforming the abscissas and scaling the ordinates of the density distribution so that the distribution moments are as close to the original solution as possible. The method is tested with a number of numerical examples. These cases are chosen due to their tendency to produce oscillations in HMMC, or due to their nature of producing multimodal distributions. It is shown that when the reconstruction algorithm is applied to the HMMC solution, the analytical density distribution is captured extremely well. This is true even in cases where the width of the analytical distribution is only a fraction of the original HMMC grid size. Multimodal population density distributions can also be reconstructed accurately from a HMMC solution with a reasonably small number of grid points, without a need to assume any basis functions for the reconstruction.     

Continuous global optimization of structured process systems models

Nonconvex models for the optimization of process systems in chemical engineering give rise to multiple suboptimal solutions, and a number of complications that often cause failure of standard local optimization techniques. A deterministic branch and bound algorithm is presented in this paper for the global optimization of structured process systems models that include non-convexities introduced by concave univariate, bilinear and linear fractional terms. The proposed branch and contract algorithm relies on a bounds contraction operation, which consists of the solution of a finite sequence of convex bounds-contraction subproblems for the subset of nonconvex variables in a problem. The application of the proposed algorithm is illustrated with several numerical examples, which include heat exchanger networks, chemical reactors, simplified process flowsheets, and waste-water treatment systems. The results show that by executing the contraction operation at selected branch and bound nodes, large portions of the search region over which the objective function takes only values above a known upper bound are eliminated. It is shown that with the proposed approach the total number of nodes in the solution tree is kept relatively small, and for some problems, no branching is required at all.     

Receding horizon control using modified iterative dynamic programming and neural network models

The basic idea of the contribution is to replace a mathematical model of the process by an equivalent neural network (NN) that mimics the phenomenological model and it is used as a process predictor in the modified iterative dynamic programming control (IDP) algoritlun. IDP is a very useful teclmique for solving unconstrained and constrained dynamic optimisation problems. The original IDP method is developed for continuous systems within state space formulation. The modified algoritlun uses a learned NN as a process predictor. The algoritlun modifications resulting from this type of the models include several important issues that arise from the use of discrete-time and input-output model formulations. Moreover, there are also some significant problems that are not to be overlooked stemming from the receding horizon implementation of the method. The contribution discusses all these issues. The benefits of the proposed approach are small number of iterations required to converge to global optimum, ability to handle multivariable constrained systems and significant time reduction compared to the original IDP method.     

Optimization of the Design and Operation of Extractive Distillation Processes

The achievement of the optimal operating point of extractive distillation systems involves determining the values of the process variables, such as the solvent flowrate, the reflux ratio of the extractive, and recovery columns. From the point of view of design, the optimum involves defining the number of stages of extractive and recovery columns, as well as the feed stage positions of these columns. The above-mentioned columns are coupled through a recycle stream, which makes obtaining the optimal operating and design points a more complex task. This study arose from a new procedure for the analysis of extractive distillation columns, in which the solvent mole fraction in the solvent feed stage is the primary variable to be analyzed. The procedure allows for determining the values of the process and design variables that provide the global minimum for the total annual cost and the specific energy consumption of the extractive distillation processes (extractive and recovery columns). Furthermore, it is possible to determine the minimum solvent flowrate and the minimum reflux ratio for separation. Obtaining anhydrous ethanol using ethylene glycol as solvent is the case study of this work.     

A simultaneous approach for singular optimal control based on partial moving grid

Singular optimal control plays an important role in process engineering, including optimal operation of batch and semi-batch reactions. However, for many practical applications, accurate solution of singular optimal control profiles is still an open issue. In particular, numerical optimization must deal with an ill-conditioned problem that often leads to very slow convergence or failure. Starting from the nested approach in our previous work in 2016, this study develops a more efficient strategy for singular control through a heuristic approach for the outer problem. The approach includes three stages. Starting from a coarse distribution of finite elements, sufficiently many finite elements are inserted where control profiles are steep and fixed gridpoints are inserted on the basis of error estimation of state profiles. Then, moving gridpoints are inserted where the modified switching function is violated. Initial junctions are obtained by moving the latest inserted gridpoints. Moreover, further mesh refinements are considered based on switching point detection and a moving grid point update strategy, until modified switching conditions are satisfied over the whole-time span. A key feature of this approach is that only a subset of finite elements needs to move during optimization. Complexity of the optimization formulation is considerably decreased compared to our previous work. This approach is demonstrated on eight classical singular control problems with known solutions, as well as six complex singular control problems drawn from the chemical engineering literature.     

Backstepping control of chemical tubular reactors

In this paper, a globally stabilizing boundary feedback control law for an arbitrarily fine discretization of a nonlinear PDE model of a chemical tubular reactor is presented. A model that assumes no radial velocity and concentration gradients in the reactor, the temperature gradient described by use of a proper value of the effective radial conductivity, a homogeneous reaction, the properties of the reaction mixture characterized by average values, the mechanism of axial mixing described by a single parameter model, and the kinetics of the first order is considered. Depending on the values of the nondimensional Peclet numbers, Damköhler number, the dimensionless adiabatic temperature rise, and the dimensionless activation energy, the coupled PDE equations for the temperature and concentration can have multiple equilibria that can be either stable or unstable. The objective is to stabilize an unstable steady state of the system using boundary control of temperature and concentration on the inlet side of the reactor. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs we transform the original coupled system into two uncoupled target systems that are asymptotically stable in l²-norm with appropriate homogeneous boundary conditions. In the real system, the designed control laws would be implemented through small variations of the prescribed inlet temperature and prescribed inlet concentration. The control design is accompanied by a simulation study that shows the feedback control law designed with sensing only on a very coarse grid (using just a few measurements of the temperature and concentration fields) can successfully stabilize the actual system for a variety of different simulation settings (on a fine grid).     

Analytical design of proportional-integral controllers for the optimal control of first-order processes with operational constraints

A novel analytical design method of industrial proportional-integral (PI) controllers was developed for the optimal control of first-order processes with operational constraints. The control objective was to minimize a weighted sum of the controlled variable error and the rate of change in the manipulated variable under the maximum allowable limits in the controlled variable, manipulated variable and the rate of change in the manipulated variable. The constrained optimal servo control problem was converted to an unconstrained optimization to obtain an analytical tuning formula. A practical shortcut procedure for obtaining optimal PI parameters was provided based on graphical analysis of global optimality. The proposed PI controller was found to guarantee global optimum and deal explicitly with the three important operational constraints.     

Model predictive control for the reactant concentration control of a reactor

The reactant concentration control of a reactor using Model Predictive Control (MPC) is presented in this paper. Two major difficulties in the control of reactant concentration are that the measurement of concentration is not available for the control point of view and it is not possible to control the concentration without considering the reactor temperature. Therefore, MIMO control techniques and state and parameter estimation are needed. One of the MIMO control techniques widely studied recently is MPC. The basic concept of MPC is that it computes a control trajectory for a whole horizon time minimising a cost function of a plant subject to a dynamic plant model and an end point constraint. However, only the initial value of controls is then applied. Feedback is incorporated by using the measurements/estimates to reconstruct the calculation for the next time step. Since MPC is a model based controller, it requires the measurement of the states of an appropriate process model. However, in most industrial processes, the state variables are not all measurable. Therefore, an extended Kalman filter (EKF), one of estimation techniques, is also utilised to estimate unknown/uncertain parameters of the system. Simulation results have demonstrated that without the reactor temperature constraint, the MPC with EKF can control the reactant concentration at a desired set point but the reactor temperator is raised over a maximum allowable value. On the other hand, when the maximun allowable value is added as a constraint, the MPC with EKF can control the reactant concentration at the desired set point with less drastic control action and within the reactor temperature constraint. This shows that the MPC with EKF is applicable to control the reactant concentration of chemical reactors.     

基于混合遗传算法的人工神经网络模型及其对有机化合物熔点的预测

为了避免BP神经网络在训练过程中收敛于局部极小的缺陷,采用自适应交叉变异、最优保存的混合遗传算法对BP网络的权值和阈值进行优化,从而提出一种新的基于混合遗传算法的神经网络模型.该算法首先对一给定的网络结构,采用混合自适应交叉变异和最优保存策略,取各自的长处,用尽可能少的搜索代数找到问题的最优解,从而既防止算法陷入局部最优,又保证算法有较好的平均适应值和最佳的适应值个体.采用上述优化策略的人工神经网络可明显改善收敛的稳定性和收敛速度,并确保网络收敛于全局极小点.人工神经网络运用于物性数据的预测是一个具有潜力和有待开发的领域.运用该模型,根据有机化合物的分子量、临界密度、正常沸点和偶极矩,对其熔点进行预测.预测结果表明：提出的混合遗传算法神经网络优于其他算法神经网络,而且预测结果优于文献上已有的Joback方程和许氏方程的计算值.