On constrained infinite-time nonlinear optimal control

In this work, we consider the infinite-time optimal control of input affine nonlinear systems subject to point-wise in time inequality constraints on both the process inputs and outputs. Fundamental to solving this constrained infinite-time nonlinear optimal control (CITNOC) problem is the ability to calculate the value function of it's unconstrained counterpart, the infinite-time nonlinear optimal control (ITNOC) problem. Unfortunately, the traditional ITNOC solution procedure of specifying an objective function and then solving for the optimal control policy and corresponding value function is computationally intractable in all but the simplest of examples. However, in many cases one can easily identify a stabilizing feedback for near operating point regulation. Building from this local policy, the proposed method is to construct a meaningful optimal control objective function as well as its corresponding value function. These functions are then used to analyze the closed-loop stability of the proposed policy. Upon return to the constrained case the constructed objective and value functions are again used to develop a self-consistent constrained finite-time scheme that will, for the first time, provide an exact solution to the CITNOC problem. The mechanics of the proposed method are then illustrated by an example from chemical reactor control.     

全局自优化控制策略及其测量变量子集选择

针对过程系统的优化运行问题,介绍一种基于Monte Carlo模拟的全局自优化控制策略。利用非线性模型计算整个操作空间内的平均经济损失,通过对某些条件进行合理假设,得到全局被控变量的解析表达形式。为了平衡传感器成本和系统性能,在全局自优化控制策略的基础上,引入混合整数约束,对测量变量子集进行选择。通过求解混合整数规划问题,能够同时获得最优的测量变量子集以及由其构成的全局被控变量,此外上述子集选择方法还可以处理附加的结构性约束问题。通过对蒸发过程的研究表明,该方法可以更加高效地处理测量变量子集选择问题,通过对精馏塔案例的研究,进一步验证了该方法在处理结构性约束问题中的优势。     

Analytical optimization and sensitivity analysis of forced periodic chemical processes

Intentional periodic manipulation of the inputs to a chemical reactor can in some cases improve the amount of product and its composition. A variational method for analyzing the effects of single and multivariable periodic forcing functions on process performance is presented in this work and illustrated in several examples. While the method discussed here is not theoretically justified for large-amplitude input variations, comparisons of analytical results obtained with the method and simulation results for large-amplitude reactor cycling show good qualitative agreement. The approximate analytical method provides a reasonable initial estimate of the structure of an advantageous periodic control.     

Discrete-time contraction constrained nonlinear model predictive control using graph-based geodesic computation

Modern chemical processes need to operate around time-varying operating conditions to optimize plant economy, in response to dynamic supply chains (e.g., time-varying specifications of product and energy costs). As such, the process control system needs to handle a wide range of operating conditions whilst optimizing system performance and ensuring stability during transitions. This article presents a reference-flexible nonlinear model predictive control approach using contraction based constraints. Firstly, a contraction condition that ensures convergence to any feasible state trajectories or setpoints is constructed. This condition is then imposed as a constraint on the optimization problem for model predictive control with a general (typically economic) cost function, utilizing Riemannian weighted graphs and shortest path techniques. The result is a reference flexible and fast optimal controller that can trade-off between the rate of target trajectory convergence and economic benefit (away from the desired process objective). The proposed approach is illustrated by a simulation study on a CSTR control problem.     

Nonlinear predictive control using multi-rate sampling

A nonlinear predictive control (NLPC) strategy based on a nonlinear, lumped parameter model of the process is developed in this paper. A constrained optimization approach is used to estimate unmeasured state variables and load disturbances. Additional model/process mismatch is handled by using an additive output term which is equivalent to the Internal Model Control approach. Similar to linear predictive control methods, an optimal sequence of future control moves is determined in order to minimize an objective function based on a desired output trajectory, subject to manipulated variable constraints (absolute and velocity). Deadtime is explicitly included in the model formulation, giving NLPC the same deadtime compensation feature of linear model-predictive techniques. The multi-rate sampling nature of most chemical processes is also used to improve estimates of process disturbances. Infrequent composition measurements in conjunction with frequent temperature measurements are used to improve the “inferential” control of the composition in a continuous flow stirred tank reactor (CSTR).     

Systematic performance analysis of continuous processes subject to multiple input cycling

Potential improvement in performance of continuous processes via weak periodic perturbations in one or more process inputs is systematically analyzed using the generalized π-criterion. At very low forcing frequencies, the performance of a periodically forced continuous process is tied to the second total differentials of the objective function at steady state. Periodic forcing of multiple inputs and the role of binary interactions between multiple inputs are analyzed in detail. Where permissible, conditions for properness of periodic control are derived and expressions for optimum phase differences (between input pairs), optimum amplitude ratios, and the maximum performance improvement vis-a-vis steady state operation are obtained. The powerful analytical results are applicable for any continuous process. Two illustrations, one dealing with antibiotic selection in recombinant cell cultures and the other dealing with series-parallel reactions, are considered for numerical study. Portions of the operating parameter space (base values of input variables and forcing frequency) where periodic operations involving weak variations in one or more inputs are superior to optimal steady state operation are identified for each illustration. An increase in the number of inputs perturbed is demonstrated to lead to broadening of the regions in the operating parameter space where forced periodic operations are superior to steady state operation and also to improved process performance. Both analytical and numerical results reveal the significant benefits of binary input interactions.     

Generic model controller tuning for chemical processes with input saturation

Control in the face of process input constraints is very common and of great practical importance in the processing industries. Generic Model Control (GMC) is a model‐based control framework for both linear and nonlinear systems. In this paper, a constrained GMC controller tuning approach using a nonlinear least squares technique is proposed. This tuning approach is simple to apply. For a SISO GMC control system with input saturation, the tracking performance is significantly improved by adding a simple heuristic switching strategy. The effectiveness of the proposed controller tuning approach is demonstrated using dynamic simulations and MIMO real‐time experiments.     

离散粒子群算法的波纹管优化设计研究

为了解决波纹管所涉及的非线性约束离散变量的优化设计问题,文中尝试将粒子群算法与惩罚函数法相结合,建立新型离散粒子群算法,实现离散变量与连续变量之间的转化。提出了一种新的离散惩罚因子更新策略,以保证离散解的精度及算法的收敛性。通过著名的容器设计算例验证,文中方法优于文献方法。用该方法对波纹管工程实例进行优化设计,优化目标值比在用产品提高了79.96%,且与理论值接近,离散解的精度满足要求,进一步证明了该方法在求解波纹管工程非线性约束离散变量优化设计问题时的有效性。     

On the tuning of predictive controllers: Application of generalized Benders decomposition to the ELOC problem

This work investigates the computational procedures used to obtain global solution to the economic linear optimal control (ELOC) problem. The proposed method employs the generalized Benders decomposition (GBD) algorithm. Compared to the previous branch and bound approach, a naive application of GBD to the ELOC problem will improve computational performance, due to less frequent calls to computationally slow semi-definite programming (SDP) routines. However, the reverse-convex constraints of the original problem will reappear in the relaxed master problem. In response, a convexification of the relaxed master constraints has been developed and proven to preserve global solution characteristics. The result is a multi-fold improvement in computational performance. A technological benefit of decomposing the problem into steady-state and dynamic parts is the ability to utilize nonlinear steady-state models, since the relaxed master problem is free of SDP type constraints and can be solved using any global nonlinear programming algorithm.     

Suboptimal control of nonlinear dynamic systems via moving model

Suboptimal feedback control with a quadratic performance index is considered for nonlinear dynamic systems. The systems are represented by a moving model which is derived by using block pulse functions. This approach permits the linear feedback control law to be applied to nonlinear systems. The suboptimal feedback control of a nonisothermal CSTR with control constraints is presented to illustrate the considerable promise that the method exhibits.     

Optimal operation and tracking control of vapor-recompressed batch distillation

Vapor recompressed batch distillation (VRBD) is an energy-integrated configuration which works on the principle of a heat pump. Operation of such a column is challenging due to unsteady, nonlinear dynamics and strong interplay between separation and energy efficiency. In this paper, a two-step approach is proposed for optimal operation and control of such a column. Initially, an openloop optimal operation policy is generated for maximization of an overall performance index using offline optimization. To this end, three performance indices are proposed to capture interplay between separation and energy efficiency. Subsequently, a model-based output feedback controller is designed to track this optimal performance trajectory. The effectiveness of the proposed approach is demonstrated using a benzene-toluene separation case study wherein it is shown that the proposed approach helps to achieve optimal operation in the presence of operational disturbances.     

Periodic control of continuous stirred tank reactors—III: Case of multistage reactors

The optimal periodic control of a multistage CSTR in which a nonisothermal parallel reaction is taking place is investigated. By applying the infinite-frequency pi criterion it is examined in what condition the optimal periodic control is proper. Then, the optimal steady control and the suboptimal periodic control are determined by numerical calculation. The results indicate that the improvement in the reactor performance for the periodic control relative to that for the steady control is very much increased with increase in the number of stages.     

Sustainable design and synthesis of algae‐based biorefinery for simultaneous hydrocarbon biofuel production and carbon sequestration

The superstructure optimization of algae‐based hydrocarbon biorefinery with sequestration of CO₂ from power plant flue gas is proposed. The major processing steps include carbon capture, algae growth, dewatering, lipid extraction and power generation, and algal biorefinery. We propose a multiobjective mixed‐integer nonlinear programming (MINLP) model that simultaneously maximizes the net present value (NPV) and minimizes the global warming potential (GWP) subject to technology selection constraints, mass balance constraints, energy balance constraints, technoeconomic analysis constraints, and environmental impact constraints. The model simultaneously determines the optimal decisions that include production capacity, size of each processing unit, mass flow rates at each stage of the process, utility consumption, economic, and environmental performances. We propose a two‐stage heuristic solution algorithm to solve the nonconvex MINLP model. Finally, the bicriteria optimization problem is solved with ε‐constraint method, and the resulting Pareto‐optimal curve reveals the trade‐off between the economic and environmental criteria. The results show that for maximum NPV, the optimal process design uses direct flue gas, a tubular photobioreactor for algae growth, a filtration dewatering unit, and a hydroprocessing pathway leading to 47.1 MM gallons of green diesel production per year at $6.33/gal corresponding to GWP of 108.7 kg CO₂‐eq per gallon. © 2013 American Institute of Chemical Engineers AIChE J, 59: 1599–1621, 2013     

基于最优控制方法的聚合物驱注入浓度优化

针对聚合物驱提高原油采收率技术,建立了确定最佳注入浓度的最优控制模型。性能指标为一定时间内原油开采所获得的利润,约束条件包括非线性渗流力学偏微分方程组、积分不等式约束和控制变量的边界约束。利用二维分布参数系统最优控制的必要条件获得了原最优控制问题的伴随问题以及目标泛函的梯度。给出了求解该最优控制问题的一种基于梯度的数值方法,并通过研究实例表明了所提出方法的有效性。     

Altering oscillatory dynamics of an electrochemical system using external forcing

We report results (numerical and experimental) indicating control of observed oscillatory dynamics in an electrochemical system under the influence of superimposed sinusoidal forcing. By varying the frequency and amplitude of the external forcing (control parameters) not only were the chaotic dynamics converted to periodic states (controlling chaos via entrainment), but the period-1 (P1) behavior could also be transformed to oscillatory dynamics of higher periodicity (for example: P2). Therefore for appropriate values of control constants and nature of the unperturbed (periodic and/or chaotic) dynamics, a final state of either lower (chaos→P1) or higher complexity (P1→P2) can be attained. Since no prior information regarding the system dynamics is required for implementation of the forcing control it is thus relevant for application to real systems.     

Constraint handling optimal PI control of open-loop unstable process: Analytical approach

This paper proposes the closed-form analytical design of proportional-integral (PI) controller parameters for the optimal control of an open-loop unstable first order process subject to operational constraints. The main idea of the design process is not only to minimize the control performance index, but also to cope with the constraints in the process variable, controller output, and its rate of change. To derive an analytical design formula, the constrained optimal control problem in the time domain was transformed to an unconstrained optimization in a parameter space associated with closed-loop dynamics. By taking advantage of the proposed analytical approach, a convenient shortcut algorithm was also provided for finding the optimal PI parameters quickly, based on the graphical analysis for the optimal solution of the corresponding optimization problem in the parameter space. The resulting optimal PI controller guarantees the globally optimal closed-loop response and handles the operational constraints precisely.     

Evolutionary search for globally optimal constrained stable cycles

The problem of finding globally optimal constrained stable periodic control processes (globally optimal cycles) arising from the optimization of some chemical, biotechnological, and flight processes is considered. An evolutionary optimization algorithm dealing with a population of cycles evolving by the crossing, and the mutation operations is developed. It incorporates various constraints imposed on the optimized cycles such as the averaged control constraints, the pointwise state constraints, and the stability requirements formulated as the maximal admissible level of the modulus of the Floquet's multipliers. The proposed method is applied to the global periodic optimization of some constrained chemical production processes.     

PERIODIC OPERATION OF CHEMICAL REACTOR SYSTEMS: ARE GLOBAL IMPROVEMENTS ATTAINABLE?

Application of forced oscillations to chemical reactor systems is investigated for three systems based on: (1)a homogeneous/heterogeneous reaction mechanism proposed by Horn and Bailey (1968) (2)amine and alcohol reactions on oxide catalysts and (3)cobalt catalyzed CO methanation. Previous analyses of these systems have demonstrated experimentally and/or theoretically, significant enhancements in reactor performance upon periodic operation. For the three cases investigated, computer-simulated modelling confirms great improvements in reactor performance upon periodic concentration forcing: (1)30% enhancement in selectivity, (2)50 fold increase in reaction rate and (3)13 fold improvement in conversion, respectively. However, it is also observed that yield improvements in these systems do not exceed or even match the performance obtained under optimal steady state conditions.     

Modelling laminar pulsed flow for the enhancement of cleaning

A Green function method is presented which enables computation of laminar flow of an incompressible Newtonian fluid in circular and annular pipes, subject to an arbitrary forcing periodic pressure gradient, in terms of Bessel functions. The response to a step change in pressure gradient in an annular pipe is presented. The method allows direct calculation of wall shear stress and flow rates generated by pulsed flows, which are of interest in fouling mitigation and cleaning-in-place systems.     

一种约束一般模型控制新方法

在引入近似预测模型的基础上 ,应用基于二次规划的滚动优化算法 ,处理被控量、操作量及其变化速率的线性约束 .将此优化算法与经典的一般模型控制 (GMC)方法相结合 ,给出了一种基于二次规划的约束一般模型控制新方法 .