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Finite Element Approximation and Input parameterization for the Optimal control of Current Profiles in Tokamak Plasmas

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IFAC Proceedings Volumes 2014年第3期47卷 7892-7897页

作者： Zhigang Ren Chao Xu Qun Lin Ryan Loxton Kok Lay Teo Jian Chu The State Key Laboratory of Industrial Control Technology and Institute of Cyber-Systems & Control Zhejiang University Hangzhou Zhejiang China Department of Mathematics & Statistics Curtin University Perth Western Australia Australia

In this paper, we consider a simplified dynamic model describing the evolution of the poloidal flux during the ramp-up phase of the tokamak discharge. We first use the Galerkin method to obtain a finite-dimensional model based on the original PDE system. Then, we apply the control parameterization method to obtain an approximate optimal parameter selection problem governed by a lumped parameter system. Computational optimization techniques are subsequently deployed to solve this approximate problem. To validate our approach, we perform numerical simulations using experimental data from the DIII-D tokamak in San Diego, California. The results show that our numerical optimization procedure can generate optimal controls that drive the current profile to within close proximity of the desired profile at the terminal time, thus demonstrating that the Galerkin and control parameterization methods are effective tools for current profile control in tokamak plasmas.

关键词： Nuclear Fusion Current Profile control Galerkin Method control parameterization Sequential Quadratic Programming (SQP)

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Dynamic Optimal control of a One-Dimensional Magnetohydrodynamic System With Bilinear Actuation

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IEEE ACCESS 2018年 6卷 24464-24474页

作者： Ren, Zhigang Zhao, Zhijia Wu, Zongze Chen, Tehuan Guangdong Univ Technol Sch Automat Guangzhou 510006 Guangdong Peoples R China Guangdong Univ Technol Guangdong Key Lab IoT Informat Technol Guangzhou 510006 Guangdong Peoples R China Guangzhou Univ Sch Mech & Elect Engn Guangzhou 510006 Guangdong Peoples R China Ningbo Univ Fac Mech Engn & Mech Ningbo 315211 Zhejiang Peoples R China

The manipulation of the magnetic field has been a proven effective method to change the flow velocity in magnetohydrodynamics (MHD) flow systems. In this paper, we consider a novel bilinear magnetic control problem arising in a 1-D MHD flow system modeled by a set of coupled partial differential equations (PDEs). We formulate the control of the magnetic field as a finite-time PDE-constrained dynamic optimal control problem and our aim is to realize the desired stationary state of the flow velocity at a specific terminal time. A model order reduction technique, based on proper orthogonal decomposition and Galerkin projection procedure, is first adopted to approximate the original complex optimization problem governed by PDEs into a semi-discrete approximation problem governed by a low-dimensional reduced-order model, and therefore can efficiently reduce the computational burden of the dynamic system. Then, the piecewise-linear control parameterization method is used to obtain an approximate optimal parameter selection problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming. The exact formulas for the gradients of the defined cost functional with respect to the decision parameters are analytically derived via state sensitivity method. Numerical simulation results verify the effectiveness of our proposed computational method. The methodology proposed in this paper is a potential implementation of a real-time control strategy in a number of MHD flow systems.

关键词： Distributed parameter systems PDE-constrained optimization control parameterization Galerkin projection proper orthogonal decomposition (POD) MHD flow

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Learning-PDE-Based Approximate Optimal control for an MHD System With Uncertainty Quantification

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IEEE TRANSACTIONS ON SYSTEMS MAN CYBERNETICS-SYSTEMS 2022年第11期52卷 7185-7192页

作者： Chen, Tehuan Ren, Zhigang Lin, Guang Xu, Chao Ningbo Univ Sch Mech Engn & Mech Ningbo 315211 Zhejiang Peoples R China Shanghai Jiao Tong Univ Ningbo Artificial Intelligence Inst Ningbo 315000 Zhejiang Peoples R China Guangdong Univ Technol Sch Automat Guangzhou 510006 Guangdong Peoples R China Chongqing Univ Posts & Telecommun Minist Educ Key Lab Ind Internet Things & Networked Control Chongqing 400065 Peoples R China Purdue Univ Dept Math W Lafayette IN 47907 USA Purdue Univ Sch Mech Engn W Lafayette IN 47907 USA Zhejiang Univ State Key Lab Ind Control Technol Hangzhou 310027 Zhejiang Peoples R China Zhejiang Univ Inst Cyber Syst & Control Hangzhou 310027 Zhejiang Peoples R China

Handling uncertainty is one of the most important challenges in real physical systems. In this article, we study an approximate optimal magnetic control strategy for a one-dimensional (1-D) magnetohydrodynamic (MHD) system with uncertainty quantification within the learning framework of the underlying MHD model. First, the MHD flow system is modeled by coupled partial differential equations (PDEs) wherein the Reynolds number is not deterministic but random. Then, the optimal magnetic control problem is formulated and reduced to a parameter selection problem with stochastic PDE constraints by means of the control parameterization method. Significantly different from the conventional sensitivity analysis and adjoint methods, a polynomial chaos expansion (PCE) based on a multifidelity model is developed to construct the underlying PDEs model by polynomial functions. Thus, the relationship between the objective function and the control sequence is derived explicitly, which, in turn, transforms the optimal parameter selection problem with stochastic PDE constraints into a typical algebraic optimization problem that can be easily solved by the existing nonlinear programming algorithm. Numerical simulations are illustrated to demonstrate the high performance of our proposed method.

关键词： Magnetohydrodynamics Mathematical models Optimal control Magnetic separation Uncertainty Linear programming Numerical models control parameterization multifidelity probabilistic collocation method optimal control uncertainty quantification

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A numerical method for a class of mixed switching and impulsive optimal control problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2006年第5期52卷 625-636页

作者： Liu, Yanqun Eberhard, A. Teo, Kok Lay Royal Melbourne Inst Technol Dept Math & Stat Melbourne Vic 3001 Australia Curtin Univ Technol Dept Math & Stat Perth WA 6845 Australia

In this paper, we consider a class of optimal control problems in which the dynamical system involves both switching and impulsive controls. The number of switching times are supposed to be finite and, without loss of generality, to occur together with the impulses. We show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems. The gradient formulae for the cost functional and the constraint functional are derived. On this basis, the control parameterization is shown to be applicable for numerical solution. A numerical example is solved for illustration. (c) 2006 Elsevier Ltd. All rights reserved.

关键词： impulsive control switching control state jump optimal parameter selection control parameterization

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Optimal control computation for nonlinear systems with state-dependent stopping criteria

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AUTOMATICA 2012年第9期48卷 2116-2129页

作者： Lin, Qun Loxton, Ryan Teo, Kok Lay Wu, Yong Hong Curtin Univ Technol Dept Math & Stat Perth WA 6845 Australia

In this paper, we consider a challenging optimal control problem in which the terminal time is determined by a stopping criterion. This stopping criterion is defined by a smooth surface in the state space;when the state trajectory hits this surface, the governing dynamic system stops. By restricting the controls to piecewise constant functions, we derive a finite-dimensional approximation of the optimal control problem. We then develop an efficient computational method, based on nonlinear programming, for solving the approximate problem. We conclude the paper with four numerical examples. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Nonlinear optimal control control parameterization Nonlinear programming Time-scaling transformation

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Computational bilinear optimal control for a class of one-dimensional MHD flow systems

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ISA TRANSACTIONS 2019年 85卷 129-140页

作者： Ren, Zhigang Zhou, Zhongcheng Xu, Chao Wu, Zongze Chen, Tehuan Guangdong Univ Technol Sch Automat Guangzhou Guangdong Peoples R China Guangdong Key Lab IoT Informat Technol Guangzhou Guangdong Peoples R China Zhejiang Univ State Key Lab Ind Control Technol Hangzhou Zhejiang Peoples R China Zhejiang Univ Inst Cyber Syst & Control Hangzhou Zhejiang Peoples R China Southwest Univ Sch Math & Stat Chongqing Peoples R China Ningbo Univ Fac Mech Engn & Mech Ningbo Zhejiang Peoples R China

In this paper, we consider a bilinear optimal control problem arising in a one-dimensional (1-D) MHD flow modeled by an array of coupled partial differential equations (PDEs). The external control input (external induction of magnetic field) in the model takes the multiplicative effect on both state variables (i.e., momentum and magnetic components). Our aim is to drive the flow velocity to within close proximity of a desired target flow velocity at the pre-indicated terminal time. We first use the Galerkin method combined with a set of basis quadratic B-spline functions to obtain a semi-discrete approximation problem. Next, the convergence of the semi-discrete approximation problem is proved. Then the control parameterization method combined with the time-scaling transformation technique is utilized to obtain an approximate optimal parameter selection problem, in which the exact gradients of the cost function with respect to the decision parameters are computed based on our analytical equations. The approximate problem are then solved by using the gradient-based optimization techniques such as the sequential quadratic programming (SQP). Finally, the numerical results validate the effectiveness of our method. (C) 2018 ISA. Published by Elsevier Ltd. All rights reserved.

关键词： Magnetohydrodynamic Flow control Galerkin method control parameterization Time-scaling transformation Bilinear control system

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Robust multi-objective optimal switching control arising in 1,3-propanediol microbial fed-batch process

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NONLINEAR ANALYSIS-HYBRID SYSTEMS 2017年 25卷 1-20页

作者： Liu, Chongyang Gong, Zhaohua Teo, Kok Lay Sun, Jie Caccetta, Louis Shandong Inst Business & Technol Sch Math & Informat Sci Yantai 264005 Shandong Peoples R China Changsha Univ Sci & Technol Sch Math & Stat Changsha 410114 Hunan Peoples R China Curtin Univ Dept Math & Stat Perth WA 6845 Australia

This paper considers optimal control of glycerol producing 1,3-propanediol (1,3-PD) via microbial fed-batch fermentation. The fed-batch process is formulated as a nonlinear switched time-delay system. In general, the time-delay in the fed-batch process cannot be exactly estimated. Our goal is to design an optimal switching control scheme to simultaneously maximize 1,3-PD productivity and 1,3-PD yield under time-delay uncertainty. Accordingly, we propose a robust multi-objective optimal switching control model, in which two objectives, i.e., 1,3-PD productivity and 1,3-PD yield, and their sensitivities with respect to uncertain time-delay are considered in the objective vector. The control variables in this problem are the feeding rate of glycerol, the switching instants and the terminal time of the process. By introducing an auxiliary dynamic system to calculate the objective sensitivities and performing a time-scaling transformation, we obtain an equivalent multi-objective optimal switching control problem in standard form. We then convert the equivalent multi-objective optimal control problem into a sequence of single-objective optimal switching control problems by using a modified normal boundary intersection method. A novel gradient-based single-objective solver combining control parameterization with constraint transcription technique is developed to solve these resulting single-objective optimal control problems. Finally, numerical results are provided to verify the effectiveness of the proposed solution approach. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Switched time-delay system Multi-objective optimal control Normal boundary intersection control parameterization Fed-batch process

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Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2016年 32卷 31-48页

作者： Ren, Zhigang Xu, Chao Lin, Qun Loxton, Ryan Teo, Kok Lay Zhejiang Univ State Key Lab Ind Control Technol Hangzhou 310003 Zhejiang Peoples R China Zhejiang Univ Inst Cyber Syst & Control Hangzhou 310003 Zhejiang Peoples R China Curtin Univ Dept Math & Stat Perth WA 6845 Australia

Establishing a good current spatial profile in tokamak fusion reactors is crucial to effective steady-state operation. The evolution of the current spatial profile is related to the evolution of the poloidal magnetic flux, which can be modeled in the normalized cylindrical coordinates using a parabolic partial differential equation (PDE) called the magnetic diffusion equation. In this paper, we consider the dynamic optimization problem of attaining the best possible current spatial profile during the ramp-up phase of the tokamak. We first use the Galerkin method to obtain a finite-dimensional ordinary differential equation (ODE) model based on the original magnetic diffusion PDE. Then, we combine the control parameterization method with a novel time-scaling transformation to obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques such as sequential quadratic programming (SQP). This control parameterization approach involves approximating the tokamak input signals by piecewise-linear functions whose slopes and break-points are decision variables to be optimized. We show that the gradient of the objective function with respect to the decision variables can be computed by solving an auxiliary dynamic system governing the state sensitivity matrix. Finally, we conclude the paper with simulation results for an example problem based on experimental data from the DIII-D tokamak in San Diego, California. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Nuclear fusion Current profile control control parameterization Time-scaling transformation

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Insulin injections and exercise scheduling for individuals with diabetes: An optimal control model

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OPTIMAL control APPLICATIONS & METHODS 2018年第2期39卷 663-681页

作者： Al Helal, Zahra Rehbock, Volker Loxton, Ryan Curtin Univ Dept Math & Stat Perth WA 6102 Australia

The aim of this paper is to promote the development of new biological models and the application of optimization methods to these models. Fast and cheap computing power allows for the ready implementation of increasingly complex dynamic models in biology. However, these models are normally developed in isolation, and their highly coupled nature can make it difficult to incorporate features from one model into another. In addition, there are many recent advances in numerical optimal control, which have not yet been applied to biological models. In this paper, we illustrate how an existing biological model can be extended to incorporate features from other models, and we demonstrate that numerical optimal control techniques can readily determine optimal strategies for managing the resulting system. In particular, we develop a new composite dynamic model for the blood glucose regulatory system by incorporating the effects of exercise and insulin injections into an existing model with 8 state variables. We formulate an optimal control problem, in which the aim is to determine optimal injection times, optimal injection volumes, and an optimal exercise regime to regulate the blood glucose level. A numerical approach, based on the concept of control parameterization and a time-scaling transformation, is then developed for solving the optimal control problem. Numerical results for 5 scenarios show that optimal treatment regimes can be readily determined via the proposed approach.

关键词： blood glucose control parameterization diabetes optimal control time-scaling transformation

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Trajectory Planning of Autonomous Vehicles Based on Parameterized control Optimization in Dynamic on-Road Environments

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JOURNAL OF INTELLIGENT & ROBOTIC SYSTEMS 2020年第3-4期100卷 1055-1067页

作者： Zhu, Sheng Aksun-Guvenc, Bilin Ohio State Univ Automated Driving Lab Columbus OH 43212 USA Ohio State Univ Dept Mech & Aerosp Engn Columbus OH 43212 USA

This paper presents a trajectory planning framework to deal with the highly dynamic environments for on-road driving. The trajectory optimization problem with parameterized curvature control was formulated to reach the goal state with the vehicle model and its dynamic constraints considered. This in contrast to existing curve fitting techniques guarantees the dynamic feasibility of the planned trajectory. With generation of multiple trajectory candidates along the Frenet frame, the vehicle is reactive to other road users or obstacles encountered. Additionally, to deal with more complex driving scenarios, its seamless interaction with an upper behavior planning layer was considered by having longitudinal motion planning responsive to the desired goal state. The trajectory evaluation and selection methodologies, along with the low-level tracking control, were also developed under this framework. The potential of the proposed trajectory planning framework was demonstrated under different dynamic driving scenarios such as lane-changing or merging with surrounding vehicles with its computation efficiency proven in real-time simulations.

关键词： Autonomous vehicles Trajectory planning control parameterization Moving objects

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