J. Fraser Forbes

Real-time optimization under parametric uncertainty: a probability constrained approach

Abstract Uncertainty is an inherent characteristic in most industrial processes, and a variety of approaches including sensitivity analysis, robust optimization and stochastic programming have been proposed to deal with such uncertainty. Uncertainty in a steady state nonlinear real-time optimization (RTO) system and particularly making robust decisions under uncertainty in real-time has received little attention. This paper discusses various sources of uncertainty within such closed loop RTO systems and a method, based on stochastic programming, that explicitly incorporates uncertainty into the RTO problem is presented. The proposed method is limited to situations where uncertain parameters enter the constraints nonlinearly and uncertain economics enter the objective function linearly. Our approach is shown to significantly improve the probability of a feasible solution in comparison to more conventional RTO techniques. A gasoline blending example is used to demonstrate the proposed robust RTO approach.

Brief paper: LQ control design of a class of hyperbolic PDE systems: Application to fixed-bed reactor

A general linear controller design method for a class of hyperbolic linear partial differential equation (PDEs) systems is presented. This is achieved by using an infinite-dimensional Hilbert state-space description with infinite-dimensional (distributed) input and output. A state LQ-feedback operator is computed via the solution of a matrix Riccati differential equation in the space variable. The proposed method is applied to a fixed-bed chemical reactor control problem, where one elementary reaction takes place. An optimal controller is designed for linearized fixed-bed reactor model, it is applied to the original nonlinear model and the resulting closed-loop stability is analyzed. Numerical simulations are performed to show the performance of the designed controller.

Extended design cost: a performance criterion for real-time optimization systems

Abstract This paper presents a performance metric and design criterion for model-based real-time optimization (RTO) systems, named extended design cost. Extended design cost is a systematic, comprehensive approach for evaluating the performance of RTO design alternatives based on fundamental principles from optimization, control and statistics theory. This metric is defined as the total loss in performance of the RTO system during some pre-specified evaluation period due to design imperfections, and is shown to consist of contributions from both the transient and steady-state behavior of the closed-loop RTO system. The proposed method is demonstrated using several examples. This paper concludes by presenting a systematic approach for selecting which model parameters to adjust on-line using extended design cost and the method is illustrated in the William–Otto reactor case study.

Dantzig–Wolfe decomposition and plant-wide MPC coordination

Abstract Due to the enormous success of model predictive control (MPC) in industrial practice, the efforts to extend its application from unit-wide to plant-wide control are becoming more widespread. In general, industrial practice has tended toward a decentralized MPC architecture. Most existing MPC systems work independently of other MPC systems installed within the plant and pursue a unit/local optimal operation. Thus, a margin for plant-wide performance improvement may be available beyond what decentralized MPC can offer. Coordinating decentralized, autonomous MPC has been identified as a practical approach to improving plant-wide performance. In this work, we propose a framework for designing a coordination system for decentralized MPC which requires only minor modification to the current MPC layer. This work studies the feasibility of applying Dantzig–Wolfe decomposition to provide an on-line solution for coordinating decentralized MPC. The proposed coordinated, decentralized MPC system retains the reliability and maintainability of current distributed MPC schemes. An empirical study of the computational complexity is used to illustrate the efficiency of coordination and provide some guidelines for the application of the proposed coordination strategy. Finally, two case studies are performed to show the ease of implementation of the coordinated, decentralized MPC scheme and the resultant improvement in the plant-wide performance of the decentralized control system.

Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs

This contribution addresses the development of a linear quadratic (LQ) regulator for a set of hyperbolic PDEs coupled with a set of ODEs through the boundary. The approach is based on an infinite-dimensional Hilbert state-space description of the system and the well-known operator Riccati equation (ORE). In order to solve the optimal control problem, the ORE is converted to a set of matrix Riccati equations. The feedback operator is found by solving the resulting matrix Riccati equations. The performance of the designed control policy is assessed by applying it to a system of interconnected continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) through a numerical simulation.

Integrity of systems under decentralized integral control

A decentralized controller that maintains closed loop stability, when the individual controllers fail or are taken out of service, provides fault tolerance and flexibility in operation. Recently, necessary and sufficient conditions [Gundes, Kabuli, 2001. IEEE Transactions on Automatic Control, 46(2), 296-301] for the existence of a block decentralized controller with integral action for a system to possess integrity against controller failures were proposed. In this paper, these conditions are equivalently expressed using the well-known notions of Niederlinski index (NI) and block relative gain (BRG). The alternate representation implies that under minor assumptions, the available necessary conditions based on NI and BRG are actually both necessary and sufficient. We also show that confirming the existence of a block decentralized controller with integral action such that the system has integrity is NP-hard.

Vitrification of intact human articular cartilage.

Articular cartilage injuries do not heal and large defects result in osteoarthritis with major personal and socioeconomic costs. Osteochondral transplantation is an effective treatment for large joint defects but its use is limited by the inability to store cartilage for long periods of time. Cryopreservation/vitrification is one method to enable banking of this tissue but decades of research have been unable to successfully preserve the tissue while maintaining cartilage on its bone base - a requirement for transplantation. To address this limitation, human knee articular cartilage from total knee arthroplasty patients and deceased donors was exposed to specified concentrations of 4 different cryoprotective agents for mathematically determined periods of time at lowering temperatures. After complete exposure, the cartilage was immersed in liquid nitrogen for up to 3 months. Cell viability was 75.4 ± 12.1% determined by membrane integrity stains and confirmed with a mitochondrial assay and pellet culture documented production of sulfated glycosaminoglycans and collagen II similar to controls. This report documents successful vitrification of intact human articular cartilage on its bone base making it possible to bank this tissue indefinitely.

Cryoprotectant agent toxicity in porcine articular chondrocytes.

Large articular cartilage defects have proven difficult to treat and often result in osteoarthritis of the affected joint. Cryopreservation of articular cartilage can provide an increased supply of tissues for osteochondral allograft but cryoprotective agents are required; however, few studies have been performed on the toxicity of these agents. This study was designed to determine the order of toxicity of five commonly used cryoprotectant agents as well as interactions that occur between them. Isolated porcine articular chondrocytes were exposed to individual cryoprotectant agents and combinations of these agents at 1M and 3M concentrations for 5 min and 120 min. Cell viability was determined using membrane integrity dyes and a metabolic activity assay. Subsequently, a regression analysis based study was undertaken to extract the maximum amount of information from this data. Results of this study demonstrated that all 1M solutions were minimally toxic. The 3M solutions demonstrated varying toxicity after 120 min. Ethylene glycol and glycerol were less toxic than propylene glycol, dimethyl sulfoxide, and formamide. Combinations of cryoprotectant agents were less toxic than single cryoprotectant agents at the same concentration. This is the most comprehensive study investigating cryoprotectant agent toxicity in articular chondrocytes and has resulted in important information regarding the order of toxicity and interactions that occur between these agents.

Results analysis for trust constrained real-time optimization

Abstract In real-time optimization (RTO), results analysis is used to ensure that RTO predictions can be implemented and are not the result of the unnecessary variance transmission around the RTO loop. Miletic and Marlin [I. Miletic, T.E. Marlin, On-line statistical results analysis in real-time operations optimization, Ind. Eng. Chem. Res. 37 (1998) 3670–3684.] proposed a statistical framework for analyzing RTO results; however, their method cannot effectively deal with inequality constraints. Many industrial RTO implementations include bounds on the changes that the RTO system can make to the process operation (i.e. trust-region constraints). Such trust-region constraints can seriously degrade the performance of existing results analysis methods. In this paper, a results analysis procedure is proposed that incorporates statistical testing on both the primal and dual variables of the optimization problem to effectively analyze steady-state RTO results in the presence of trust-region constraints. The proposed method is illustrated using two small case studies, one of which is the same Williams and Otto [T.J. Williams, R.E. Otto, A generalized chemical processing model for the investigation of computer control, AIEE Trans 79 (1960) 458–473.] reactor example used in Miletic and Marlin (1998) .

Coordination of Distributed Model Predictive Controllers for Constrained Dynamic Processes

Abstract Abstract In this paper, a coordinated-distributed model predictive control (MPC) scheme is presented for large-scale discrete-time linear process systems. Coordinated-distributed MPC control aims at enhancing the performance of fully decentralized MPC controllers by achieving the plant-wide optimal operations. The ‘price-driven’ decomposition-coordination method is used to adjust the operations of the individual processing units in order to satisfy an overall plant performance objective. Newtons method, together with a sensitivity analysis technique, are used to efficiently update the price in the price-driven decomposition-coordination method. The efficiency of the proposed control scheme is evaluated using a model of a fluid catalytic cracking process.