Consistent hierarchical economic NMPC for a class of hybrid systems using neighboring-extremal updates

A hierarchical two-layer control algorithm is developed for a class of hybrid (discrete-continuous dynamic) systems to support economically optimal operation of batch or continuous processes with a predefined production schedule. For this class of hybrid systems, the optimal control moves as well as the controlled switching times between two adjacent modes are determined online. In contrast to closely related schemes for integrated scheduling and control, the sequence of modes is not optimized. On the upper layer, the economic optimal control problem is solved rigorously by a slow hybrid economic model predictive controller at a low sampling rate. On the lower layer, a fast hybrid neighboring-extremal controller is based on the same economic optimal control problem as the slow controller to ensure consistency between both layers. The fast neighboring-extremal controller updates rather than tracks the optimal trajectories from the upper layer to account for disturbances. Consequently, the fast controller steers the process to its operational bounds under disturbances and the economic potential of the process is exploited anytime. The suggested two-layer control algorithm provides fully consistent control action on the fast and slow time-scale and thus avoids performance degradation and even infeasibilities which are commonly encountered if inconsistent optimal control problems are formulated and solved.     

A batch-to-batch iterative optimal control strategy based on recurrent neural network models

A batch-to-batch model-based iterative optimal control strategy for batch processes is proposed. To address the difficulties in developing detailed mechanistic models, recurrent neural networks are used to model batch processes from process operational data. Due to model-plant mismatches and unmeasured disturbances, the calculated optimal control profile may not be optimal when applied to the actual process. To address this issue, model prediction errors from previous batch runs are used to improve neural network model predictions for the current batch. Since the main interest in batch process operation is on the end of batch product quality, a quadratic objective function is introduced to track the desired qualities at the end-point of a batch. Because model errors are gradually reduced from batch-to-batch, the control trajectory gradually approaches the optimal control policy. The proposed scheme is illustrated on a simulated methyl methacrylate polymerisation reactor.     

An HMM approach for optimal investment of an insurer

In this paper, we introduce a Hidden Markov Model (HMM) for studying an optimal investment problem of an insurer when model uncertainty is present. More specifically, the financial price and insurance risk processes are modulated by a continuous‐time, finite‐state, hidden Markov chain. The states of the chain represent different modes of the model. The HMM approach is viewed as a ‘dynamic’ version of the Bayesian approach to model uncertainty. The optimal investment problem is formulated as a stochastic optimal control problem with partial observations. The innovations approach in the filtering theory is then used to transform the problem into one with complete observations. New robust filters of the chain and estimates of key parameters are derived. We discuss the optimal investment problem using the Hamilton–Jacobi–Bellman (HJB) dynamic programming approach and derive a closed‐form solution in the case of an exponential utility and zero interest rate. Copyright © 2011 John Wiley & Sons, Ltd.     

基于遗传算法的污水处理过程优化控制方法

分析和研究了在限制有机物排放总量条件下,污水处理运行费用最低的优化控制问题,给出污水处理过程中多变量最优控制的数学模型,提出一种采用遗传算法求解污水处理过程优化控制问题新的计算方法,论述了二进制编码、适应度函数计算的设计与实现方法.该方法避免了对迭代初值进行猜测的困难,提高了计算效率.数值仿真说明在污水处理过程中采用遗传算法寻优是可行和有效的.     

Average cost optimal policies for Markov control processes with Borel state space and unbounded costs

We show the existence of average cost optimal stationary policies for Markov control processes with Borel state space and unbounded costs per stage, under a set of assumptions recently introduced by L.I. Sennott (1989) for control processes with countable state space and finite control sets.     

The development of optimal ingot heating patterns in soaking pits

Based on the discrete state space model of the ingot-pit systems and the novel algorithm for solving the constrained optimal control problems developed by the authors, the optimal heating patterns are developed theoretically for minimizing the fuel consumption in the soaking pits.The computer simulation results show the relationships between the operational conditions and the optimal heating strategies in the soaking pits and the potential economic benefits to be obtained by using the optimal patterns. The method used in this paper can also be applied to the synthesis of the optimal control and/or state trajectories in the other batch processes.The results in this paper also show that the new optimal control algorithm developed by the authors is a powerful tool for solving the high-dimensional, time-varying, nonlinear system optimal control problems with the complicated equality and/or inequality constraints.     

Pollution's ambient problems and regularity of optimal cost function

We study pollution's ambient problems by using the optimal control theory applied to partial differential equations. We consider the problem to find the optimal way to eliminate pollution in the time, such that the concentration is close to a standard level which does not affect the ecological equilibrium when the source is pointwise. We consider a quadratic cost functional and we prove the existence and uniqueness of optimal control. We find a characterisation which makes possible the computing of optimal control. Additionally, we consider the problem moving the pointwise source. So, we define a function j(b) that associates to any point b ∈ Ω the optimal cost functional applied to the optimal control. We show that j is differentiable, provided that the controls are taken in a convenient subset of admissible functions satisfying the cone properties. We also find the point in Ω, for which the cost functional is minimum.     

Bilinear optimal control of a Kirchhoff plate

We consider the problem of optimal control of a Kirchhoff plate. A bilinear control is used as a force to make the plate close to a desired profile taking into the account, a quadratic cost of control. We prove the existence of an optimal control and characterize it uniquely through the solution of an optimality system.     

A numerical approach of optimal control for generalized delay systems by general Legendre wavelets

A general Legendre wavelets approach is presented. The delay function is expanded by general Legendre wavelets. The general operational matrix of integration is introduced. By the general Legendre wavelets, the linear-quadratic problem of generalized delay systems are transformed into the optimization problem of multivariate functions. The approximate solutions of the optimal control and state as well as the optimal value of the objective functional are derived. The numerical examples demonstrate that the algorithms are valid.     

SBR污水生化处理系统的最优控制及改进

针对序批式反应器污水处理工艺,提出一种以排水质量指标为约束条件,以能量消耗最小为日标的最优控制方案．由于生化过程机理复杂,导致这个最优问题求解困难．所构造的解法是首先建立通用的序批式反应器模型;然后借助仿真工具试差求取次优解．为解决搜索求解计算量大的问题,进一步结合进退法提出了一种改进的优化算法,显著提高了求解效率。     

An optimal control approach to robust control design

We propose an optimal control approach to robust control design. Our goal is to design a state feedback to stabilize a system under uncertainty. We translate this robust control problem into an optimal control problem of minimizing a cost. Because the uncertainty bound is reflected in the cost, the solution to the optimal control problem is a solution to the robust control problem. Our approach can deal with both linear and non-linear systems. Furthermore it can handle both matched and unmatched uncertainties. It can also handle uncertainty in the control input matrix.     

Dynamic buffer management using optimal control of hybrid systems

This paper studies a dynamic buffer management problem with one buffer inserted between two interacting components. The component to be controlled is assumed to have multiple power modes corresponding to different data processing rates. The overall system is modeled as a hybrid system and the buffer management problem is formulated as an optimal control problem. Different from many previous studies, the objective function of the proposed problem depends on the switching cost and the size of the continuous state space, making its solution much more challenging. By exploiting some particular features of the problem, the best mode sequence and the optimal switching instants are characterized analytically using a variational approach. Simulation results based on real data shows that the proposed method can significantly reduce the energy consumption compared with another heuristic scheme in several typical situations.     

A graph-theoretic optimal control problem for terminating discrete event processes

Most of the results to date in discrete event supervisory control assume a zero-or-infinity structure for the cost of controlling a discrete event system, in the sense that it costs nothing to disable controllable events while uncontrollable events cannot be disabled (i.e., their disablement entails infinite cost). In several applications however, a more refined structure of the control cost becomes necessary in order to quantify the tradeoffs between candidate supervisors. In this paper, we formulate and solve a new optimal control problem for a class of discrete event systems. We assume that the system can be modeled as a finite acylic directed graph, i.e., the system process has a finite set of event trajectories and thus is terminating. The optimal control problem explicitly considers the cost of control in the objective function. In general terms, this problem involves a tradeoff between the cost of system evolution, which is quantified in terms of a path cost on the event trajectories generated by the system, and the cost of impacting on the external environment, which is quantified as a dynamic cost on control. We also seek a least restrictive solution. An algorithm based on dynamic programming is developed for the solution of this problem. This algorithm is based on a graph-theoretic formulation of the problem. The use of dynamic programming allows for the efficient construction of an optimal subgraph (i.e., optimal supervisor) of the given graph (i.e., discrete event system) with respect to the cost structure imposed. We show that this algorithm is of polynomial complexity in the number of vertices of the graph of the system.Research supported in part by the National Science Foundation under grant ECS-9057967 with additional support from GE and DEC.     

Minimizing control variation in nonlinear optimal control

In any real system, changing the control signal from one value to another will usually cause wear and tear on the system’s actuators. Thus, when designing a control law, it is important to consider not just predicted system performance, but also the cost associated with changing the control action. This latter cost is almost always ignored in the optimal control literature. In this paper, we consider a class of optimal control problems in which the variation of the control signal is explicitly penalized in the cost function. We develop an effective computational method, based on the control parameterization approach and a novel transformation procedure, for solving this class of optimal control problems. We then apply our method to three example problems in fisheries, train control, and chemical engineering.     

On optimal control of non-autonomous switched systems with a fixed mode sequence

We consider differentiability with respect to the switch times of the value function of an optimal control problem for a non-autonomous switched system. The control variables are the switch times between the modes and the input in each mode. We provide a method to compute the derivative of the cost function given a nominal input. Then, we view the optimal control problem as a parametrized optimization problem in which the switch times are the parameters and the optimization is over the set of feasible inputs of each mode. From this point of view, we provide conditions under which the continuity and differentiability of the optimal value function, that is the cost function optimized over the inputs, can be guaranteed.     

A neural network approach for optimal software testing and maintenance

In this paper, we propose a neural network-based model for optimal software testing and maintenance policy, where the software testing environment and the operational environment are characterized by an environmental factor. We also present a systematic study for defect detection and correction processes. In our proposed approach, we consider the logistic growth curve model and the constant correction time for defect prediction. Then, we estimate the jointly optimal software testing period and maintenance limit via minimization of a software cost function that takes into account the environmental factor and the imperfect fault removal. More precisely, the total expected cost is formulated via a discrete-type software reliability model based on the difference between operational environments, imperfect defect removal, and defect correction process. Experimental results on a real software data set are presented to demonstrate the effectiveness of the proposed approach in defect prediction as well as in determining the jointly optimal testing period and planned maintenance limit.     

The generalized continuous algebraic Riccati equation and impulse-free continuous-time LQ optimal control

The purpose of this paper is to investigate the role that the so-called constrained generalized Riccati equation plays within the context of continuous-time singular linear–quadratic (LQ) optimal control. This equation has been defined following the analogy with the discrete-time setting. However, while in the discrete-time case the connections between this equation and the linear–quadratic optimal control problem has been thoroughly investigated, to date very little is known on these connections in the continuous-time setting. This note addresses this point. We show, in particular, that when the continuous-time constrained generalized Riccati equation admits a solution, the corresponding linear–quadratic problem admits an impulse-free optimal control. We also address the corresponding infinite-horizon LQ problem for which we establish a similar result under the additional constraint that there exists a control input for which the cost index is finite.     

Necessary and sufficient conditions for distributed constrained optimal consensus under bounded input

In this paper, we study a distributed constrained optimal consensus problem for discrete‐time first‐order integrator systems under bounded input. Each agent is assigned with a local convex cost function, and all agents are required to achieve consensus at the minimum of the aggregate cost over a common convex constraint set, which is only accessible by part of the agents. A 2‐step control protocol is designed to solve the problem under bounded input. Firstly, at each time step, each agent moves a bounded step of the subgradient descent from an individual cost and another one along the projection direction if it can access the constraint. The second movement is then given by a bounded average of the relative positions to neighbors. Specifically, to coordinate the subgradient step within the network without using global information, we introduce an estimate of the upper bound of all agents' subgradients, which is updated by a unilateral consensus mechanism. Under the given control protocol, we obtain the necessary and sufficient conditions to achieve the constrained optimal consensus for a fixed topology. Under similar conditions, we also solve the problem for switching topologies and conduct a convergence rate analysis for strongly convex costs.