Sequential linear quadratic control of bilinear parabolic PDEs based on POD model reduction

We present a framework to solve a finite-time optimal control problem for parabolic partial differential equations (PDEs) with diffusivity-interior actuators, which is motivated by the control of the current density profile in tokamak plasmas. The proposed approach is based on reduced order modeling (ROM) and successive optimal control computation. First we either simulate the parabolic PDE system or carry out experiments to generate data ensembles, from which we then extract the most energetic modes to obtain a reduced order model based on the proper orthogonal decomposition (POD) method and Galerkin projection. The obtained reduced order model corresponds to a bilinear control system. Based on quasi-linearization of the optimality conditions derived from Pontryagin’s maximum principle, and stated as a two boundary value problem, we propose an iterative scheme for suboptimal closed-loop control. We take advantage of linear synthesis methods in each iteration step to construct a sequence of controllers. The convergence of the controller sequence is proved in appropriate functional spaces. When compared with previous iterative schemes for optimal control of bilinear systems, the proposed scheme avoids repeated numerical computation of the Riccati equation and therefore reduces significantly the number of ODEs that must be solved at each iteration step. A numerical simulation study shows the effectiveness of this approach.     

Robust stabilization of the current profile in tokamak plasmas using sliding mode approach in infinite dimension

This paper deals with the robust stabilization of the spatial distribution of tokamak plasmas current profile using a sliding mode feedback control approach. The control design is based on the 1D resistive diffusion equation of the magnetic flux that governs the plasma current profile evolution. The feedback control law is derived in the infinite dimensional setting without spatial discretization. Numerical simulations are provided and the tuning of the controller parameters that would reject uncertain perturbations is discussed. Closed loop simulations performed on realistic test cases using a physics based tokamak integrated simulator confirm the relevance of the proposed control algorithm in view of practical implementation.     

Optimal boundary control of coupled parabolic PDE–ODE systems using infinite-dimensional representation

The optimal boundary control problem is studied for coupled parabolic PDE–ODE systems. The linear quadratic method is used and exploits an infinite-dimensional state-space representation of the coupled PDE–ODE system. Linearization of the nonlinear system is established around a steady-state profile. Using appropriate state transformations, the linearized system has been formulated as a well-posed infinite-dimensional system with bounded input and output operators. It has been shown that the resulting system is a Riesz spectral system. The linear quadratic control problem has been solved using the corresponding Riccati equation and the solution of the corresponding eigenvalue problem. The results were applied to the case study of a catalytic cracking reactor with catalyst deactivation. Numerical simulations are performed to illustrate the performance of the proposed controller.     

基于特征线法的双曲型分布参数模型预测控制及其应用

本文针对双曲型分布参数系统提出基于特征线法的模型预测控制算法. 通过特征线变换将描述分布参数模型的偏微分方程转化为常微分方程; 进而求解得到分布参数系统状态变量的解析式; 离散化后作为预测模型用于模型预测控制. 以循环流化床烟气脱硫系统中SO2浓度控制为例, 进行仿真研究, 结果表明基于特征线法的模型预测控制算法可以实现对双曲型分布参数系统的有效控制, 并且该算法的控制效果优于目前工程应用的前馈反馈 控制策略.     

Delay compensated control of the Stefan problem and robustness to delay mismatch

This paper presents a control design for the one‐phase Stefan problem under actuator delay via a backstepping method. The Stefan problem represents a liquid‐solid phase change phenomenon which describes the time evolution of a material's temperature profile and the interface position. The actuator delay is modeled by a first‐order hyperbolic partial differential equation (PDE), resulting in a cascaded transport‐diffusion PDE system defined on a time‐varying spatial domain described by an ordinary differential equation (ODE). Two nonlinear backstepping transformations are utilized for the control design. The setpoint restriction is given to guarantee a physical constraint on the proposed controller for the melting process. This constraint ensures the exponential convergence of the moving interface to a setpoint and the exponential stability of the temperature equilibrium profile and the delayed controller in the norm. Furthermore, robustness analysis with respect to the delay mismatch between the plant and the controller is studied, which provides analogous results to the exact compensation by restricting the control gain.     

随机运动目标搜索问题的最优控制模型

提出了Rn空间中做布朗运动的随机运动目标的搜索问题的最优控制模型.采用分析的方法来研究随机运动目标的最优搜索问题,并将原问题转化为由一个二阶偏微分方程(HJB方程)所表示的确定性分布参数系统的等价问题,推导出随机运动目标的最优搜索问题的HJB方程,并证明了该方程的解即是所寻求的最优搜索策略.由此给出了一个计算最优搜索策略的算法和一个实例.     

Optimal control of convection–diffusion process with time-varying spatial domain: Czochralski crystal growth

This paper considers the optimal control of convection–diffusion systems modeled by parabolic partial differential equations (PDEs) with time-dependent spatial domains for application to the crystal temperature regulation problem in the Czochralski (CZ) crystal growth process. The parabolic PDE model describing the temperature dynamics in the crystal region arising from the first principles continuum mechanics is defined on the time-varying spatial domain. The dynamics of the domain boundary evolution, which is determined by the mechanical subsystem pulling the crystal from the melt, are described by an ordinary differential equation for rigid body mechanics and unidirectionally coupled to the convection–diffusion process described by the PDE system. The representation of the PDE as an evolution system on an appropriate infinite-dimensional space is developed and the analytic expression and properties of the associated two-parameter semigroup generated by the nonautonomous operator are provided. The LQR control synthesis in terms of the two-parameter semigroup is considered. The optimal control problem setup for the PDE coupled with the finite-dimensional subsystem is presented and numerical results demonstrate the regulation of the two-dimensional crystal temperature distribution in the time-varying spatial domain.     

Simulation technique of free-boundary equilibrium evolution in plasma ramp-up phase

We describe a new scheme for a consistent linkage between the transport and equilibrium solvers in the integrated transport code TOPICS for a transport simulation in tokamaks during a plasma volume ramp-up phase. Although the flux conserving tokamak (FCT) equilibrium solver implemented in TOPICS is well consistent with the transport solver, it cannot evaluate a toroidal magnetic flux associated with an evolving equilibrium by itself. The evolution of the equilibrium is then modeled with a combination of the determination of the toroidal magnetic flux by the iterative solution of the Grad-Shafranov equation and the FCT scheme. The new scheme developed is applied to a predictive simulation with the volume expansion in a tokamak comparable to JT-60U for both the limiter and the divertor cases.     

A galerkin/neural-network-based design of guaranteed cost control for nonlinear distributed parameter systems.

This paper presents a Galerkin/neural-network- based guaranteed cost control (GCC) design for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities. A parabolic PDE system typically involves a spatial differential operator with eigenspectrum that can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this, in the proposed control scheme, Galerkin method is initially applied to the PDE system to derive an ordinary differential equation (ODE) system with unknown nonlinearities, which accurately describes the dynamics of the dominant (slow) modes of the PDE system. The resulting nonlinear ODE system is subsequently parameterized by a multilayer neural network (MNN) with one-hidden layer and zero bias terms. Then, based on the neural model and a Lure-type Lyapunov function, a linear modal feedback controller is developed to stabilize the closed-loop PDE system and provide an upper bound for the quadratic cost function associated with the finite-dimensional slow system for all admissible approximation errors of the network. The outcome of the GCC problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal guaranteed cost controller in the sense of minimizing the cost bound is obtained. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

Modeling and vibration control for a flexible pendulum inverted system based on a PDE observer

This study addresses the problem of trajectory control of a flexible pendulum inverted system on the basis of the partial differential equation (PDE) and ordinary differential equation (ODE) dynamic model. One of the key contributions of this study is that a new model is proposed to simplify the complex system. In addition, this study proposed a nonlinear PDE observer to estimate distributed positions and velocities along flexible pendulum. Singular perturbation method is proposed to solve the coupling system of nonlinear PDE observer. The nonlinear PDE observer is divided into a fast subsystem and a slow subsystem by the use of the singular perturbation method. To stabilise this fast subsystem, a boundary controller is proposed at the free end of the beam. The sliding-mode control method is proposed to design controller for slow subsystems. The asymptotic stability of both the proposed nonlinear PDE observer and controller is validated by theoretical analysis. The results are illustrated by simulation.     

基于数据的网络化系统最优控制

针对网络介入导致的系统动态复杂性增加及建模困难等问题,提出了基于数据驱动控制的思想,独立于系统模型设计控制器。利用在线获取的系统输入/输出当前数据和历史数据,分别构造系统的输入数据矩阵和输出数据矩阵,建立两者之间的线性关系,获得当前采样时刻系统的Markov参数,将该参数代入到Markov参数形式表示的Riccati方程解中,获取最优控制增益。同时利用输入与输出之间的线性关系,构造一个控制器状态观测器,利用估计的控制器状态参与计算最优控制律。最后,在Truetime1.5和Matlab仿真平台上验证了提出的控制器是有效的。     

Optimal controller tuning for nonlinear processes

The present work proposes a systematic methodology for the optimal selection of controller parameters in the sense of minimizing a performance index, which is a quadratic function of the tracking error and the control effort. The performance index is calculated explicitly as an algebraic function of the controller parameters by solving Zubov's partial differential equation (PDE). Standard optimization techniques are then employed for the calculation of the optimal values of the controller parameters. The solution of Zubov's PDE is also used to estimate the closed-loop stability region for the chosen values of the controller parameters. The proposed approach is finally illustrated in a chemical reactor control problem.     

Determination Of The Phase Current Waveform For A Disc‐Type Axial‐Flux Wheel Motor

An optimal design of the driving current pattern for a disc‐type axial‐flux brushless DC wheel motor of an electric vehicle is proposed in this paper. The electro‐magnetic dynamic model of the motor is established with magnetic circuits, describing the relationship between the output torque and excitation current. The optimal current pattern, in terms of magnitude and phase angle, is then obtained by maximizing the output torque with respect to the rotor shift. Compared with the traditional three‐phase‐on current pattern of fixed 120–degree phase shift, both the average torque and efficiency with the driving current of an optimal advanced switching angle are seen to be improved under various loading conditions. The motor performance with the optimal driving waveform is simulated and verified by experiments.     

基于感应电机状态方程的定子电阻辨识方法的研究

基于感应电机两相同步旋转坐标系状态方程,提出了一种新的定子电阻辨识方案。在分别利用励磁电流微分方程和转矩电流微分方程推算转速的基础上,利用二者的等量关系实现了定子电阻在线辨识。该方案的优点是,定子电阻估算仅需要电机终端可测的定子电压、电流,无须借助转子或定子磁链。仿真实验将其应用于无速度闭环矢量调速系统,实时更新转速估算模块中的定子电阻,论证了该方案对电机参数变化鲁棒性强及对改善转速估算精度的有效性。     

The piecewise parametric optimal control of uncertain linear quadratic models

The optimal control of linear quadratic model is given in a feedback form and determined by the solution of a Riccati equation. However, the control-related Riccati equation usually cannot be solved analytically such that the form of optimal control will become more complex. In this paper, we consider a piecewise parametric optimal control problem of uncertain linear quadratic model for simplifying the form of optimal control. By introducing a piecewise control parameter, a piecewise parametric optimal control model is established. Then we present a parametric optimisation method for solving the optimal piecewise control parameter. Finally, an uncertain inventory-promotion optimal control problem is discussed and a comparison is made to show the effectiveness of proposed piecewise parametric optimal control model.     

Optimal Valve Closure Operations for Pressure Suppression in Fluid Transport Pipelines

When a valve is suddenly closed in fluid transport pipelines, a pressure surge or shock is created along the pipeline due to the momentum change. This phenomenon, called hydraulic shock, can cause major damage to the pipelines. In this paper, we introduce a hyperbolic partial differential equation (PDE) system to describe the fluid flow in the pipeline and propose an optimal boundary control problem for pressure suppression during the valve closure. The boundary control in this system is related to the valve actuation located at the pipeline terminus through a valve closing model. To solve this optimal boundary control problem, we use the method of lines and orthogonal collocation to obtain a spatial-temporal discretization model based on the original pipeline transmission PDE system. Then, the optimal boundary control problem is reduced to a nonlinear programming (NLP) problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming (SQP). Finally, we conclude the paper with simulation results demonstrating that the full parameterization (FP) method eliminates pressure shock effectively and costs less computation time compared with the control vector parameterization (CVP) method.      

MHD channel flow control in 2D: Mixing enhancement by boundary feedback

A nonlinear Lyapunov-based boundary feedback control law is proposed for mixing enhancement in a 2D magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, which is electrically conducting, incompressible, and subject to an external transverse magnetic field. The MHD model is a combination of the Navier-Stokes PDE and the Magnetic Induction PDE, which is derived from the Maxwell equations. Pressure sensors, magnetic field sensors, and micro-jets embedded into the walls of the flow domain are employed for mixing enhancement feedback. The proposed control law, designed using passivity ideas, is optimal in the sense that it maximizes a measure related to mixing (which incorporates stretching and folding of material elements), while at the same time minimizing the control and sensing efforts. A DNS code is developed, based on a hybrid Fourier pseudospectral-finite difference discretization and the fractional step technique, to numerically assess the controller.     

Dynamic restoration of the unknown function in the linear stochastic differential equation

For the linear stochastic differential equation with diffusion, consideration was given to the problem of restoration of the unknown parameter characterizing the level of noise. Equations of this kind are used, in particular, in the problem of optimal portfolio selection in order to describe the time profile of the asset prices under risk investments. Restoration must be based on the measurements of the current phase state. The problem under study comes to the inverse problem of the ordinary matrix differential equation satisfied by the covariance matrix of the initial random process. The proposed algorithm which is based on a combination of the methods of the theory of improperly posed problems and the theory of positional control with model is constructed in the class of finite-step algorithms counting on computer-aided realization. The algorithm is stable to the errors in information and computations. The algorithm’s precision was estimated in terms of the measurable realizations of the initial process.     

Optimization of Wave Processes in Inhomogeneous Media

An optimal control problem for parabolic Schroedinger-type wave equation with a complex non-self conjugate operator is considered. An optimality criterion is formulated. A numerical method is proposed for solution of the optimization problem. A stability of a difference scheme is analyzed.