Reduced order modeling of a two-dimensional flow with moving shocks

The objective of this paper is to demonstrate the ability of proper orthogonal decomposition, in combination with domain decomposition, to produce accurate reduced order models (ROMs) for two-dimensional high-speed flows with moving shock waves. To demonstrate this ability, a blunt body flow with quasi-steady shock motion is considered. The blunt body flow contains a strong bow shock that is moved via a change in inlet Mach number and angle of attack. Accuracy is quantified by comparing surface pressures obtained from the ROMs with those from the full order simulation under the same free stream conditions. The order reduction, and computational performance of the ROM is also quantified relative to the full order simulation. The robustness of the ROM to varying flow parameters is explored. A non-Galerkin quasi-implicit steady state implementation is considered.     

模型的快速降阶技术

为快速进行模型的降阶,结合平衡截断（Balanced Truncation,BT）方法和特征正交分解（Proper Orthogonal Decomposition,POD）方法提出一种模型降阶方法.该方法采用频域POD快照矩阵低阶逼近系统的可控、可观Gram矩阵;通过奇异值分解（Singular Value Decomposition,SVD）提取BT＋POD模态,对低能量模态截断形成降阶子空间,并将其映射到全阶系统,从而形成基于状态空间的降阶模型（Reduced Order Model,ROM）;该模型就成为全阶模型（Full Order Model,FOM）的ROM.通过对阶数n=406的LTI SISO系统和阶数n=9的2区间电力系统进行的验证表明,在保留BT方法输入输出平衡特性的基础上,该方法效率高于BT方法.     

Data-driven model reduction-based nonlinear MPC for large-scale distributed parameter systems

Model predictive control (MPC) has been effectively applied in process industries since the 1990s. Models in the form of closed equation sets are normally needed for MPC, but it is often difficult to obtain such formulations for large nonlinear systems. To extend nonlinear MPC (NMPC) application to nonlinear distributed parameter systems (DPS) with unknown dynamics, a data-driven model reduction-based approach is followed. The proper orthogonal decomposition (POD) method is first applied off-line to compute a set of basis functions. Then a series of artificial neural networks (ANNs) are trained to effectively compute POD time coefficients. NMPC, using sequential quadratic programming is then applied. The novelty of our methodology lies in the application of POD's highly efficient linear decomposition for the consequent conversion of any distributed multi-dimensional space-state model to a reduced 1-dimensional model, dependent only on time, which can be handled effectively as a black-box through ANNs. Hence we construct a paradigm, which allows the application of NMPC to complex nonlinear high-dimensional systems, even input/output systems, handled by black-box solvers, with significant computational efficiency. This paradigm combines elements of gain scheduling, NMPC, model reduction and ANN for effective control of nonlinear DPS. The stabilization/destabilization of a tubular reactor with recycle is used as an illustrative example to demonstrate the efficiency of our methodology. Case studies with inequality constraints are also presented.     

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.     

Passivity‐based sliding mode controller/observer for second‐order nonlinear systems

A passivity‐based sliding mode control for a class of second‐order nonlinear systems with matched disturbances is proposed in this paper. Firstly, a nonlinear sliding surface is designed using feedback passification, in which the passivity is employed to guarantee the closed‐loop system's stability. The passivity‐based controller comprising a discontinuous term guarantees globally asymptotical convergence to the sliding surface. A sliding mode‐based control law that satisfies the reaching and sliding condition is also developed. Moreover, the passivity‐based sliding mode observer is also developed to effectively estimate the system states. Compared with conventional sliding mode control, the proposed control scheme has a shorter reaching time; and hence, the system performance is less affected by disturbances, thus eliminating the need to increase the control input gain. Finally, simulation results demonstrate the validity of the proposed method.     

Linear Parameter-varying Approach for Modeling and Control of Rapid Thermal Processes

In the present paper, a new approach is presented to model and control single wafer rapid thermal processing (RTP) systems. In the past decade, RTP has achieved acceptance as the mainstream technology for semiconductor manufacturing. Thermal processing is one of the most efficient ways to control the phase-structure properties. Moreover, the time duration of RTP systems reduces the so-called thermal budget significantly compared to the traditional methods. RTP implementation is based on the use of light from heating lamps to provide a heat flux. This process is highly nonlinear due to the radiative heat transfer and material properties. By invoking the first principles-based models, we develop in this paper a linear parameter-varying (LPV) model to directly account for the nonlinearities within the system. The model is then discretized into a high-order LPV model; thereafter, principal component analysis (PCA) method is utilized to reduce the number of LPV model’s scheduling variables, followed by the use of proper orthogonal decomposition (POD) for model order reduction. Using the reduced order model, we then design a gain-scheduled controller to satisfy an induced L₂ gain performance for tracking of a temperature profile and show improvement over other controller design methods suggested in the literature.     

Nonlinear system identification using two-dimensional wavelet-based state-dependent parameter models

This article presents a nonlinear system identification approach that uses a two-dimensional (2-D) wavelet-based state-dependent parameter (SDP) model. In this method, differing from our previous approach, the SDP is a function with respect to two different state variables, which is realised by the use of a 2-D wavelet series expansion. Here, an optimised model structure selection is accomplished using a PRESS-based procedure in conjunction with orthogonal decomposition (OD) to avoid any ill-conditioning problems associated with the parameter estimation. Two simulation examples are provided to demonstrate the merits of the proposed approach.     

Hybrid‐impulsive second‐order sliding mode control: Lyapunov approach

Dynamic system of relative degree two controlled by discontinuous‐hybrid‐impulsive feedback in the presence of bounded perturbations is considered. The state feedback impulsive‐twisting control exhibits a uniform exact finite time convergence to the second‐order sliding mode with zero convergence time. The output feedback discontinuous control augmented by a simplified hybrid‐impulsive functions provides uniform exact convergence with zero convergence time of the system's states to a real second‐order sliding mode in the presence of bounded perturbations. Only ‘snap’ knowledge of the output derivative, that is, the knowledge of the output derivative in isolated time instants, is required. The output feedback hybrid‐impulsive control with practically implemented impulsive actions asymptotically drives the system's states to the origin. The Lyapunov analysis of the considered hybrid‐impulsive‐discontinuous system proves the system's stability. The efficacy of the proposed control technique is illustrated via computer simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Synchronization of A Class of Uncertain Chaotic Systems with Lipschitz Nonlinearities Using State‐Feedback Control Design: A Matrix Inequality Approach

This paper proposes a new state‐feedback stabilization control technique for a class of uncertain chaotic systems with Lipschitz nonlinearity conditions. Based on Lyapunov stabilization theory and the linear matrix inequality (LMI) scheme, a new sufficient condition formulated in the form of LMIs is created for the chaos synchronization of chaotic systems with parametric uncertainties and external disturbances on the slave system. Using Barbalat's lemma, the suggested approach guarantees that the slave system synchronizes to the master system at an asymptotical convergence rate. Meanwhile, a criterion to find the proper feedback gain vector F is also provided. A new continuous‐bounded nonlinear function is introduced to cope with the disturbances and uncertainties and obtain a desired control performance, i.e. small steady‐state error and fast settling time. Several criteria are derived to guarantee the asymptotic and robust stability of the uncertain master–slave systems. Furthermore, the proposed controller is independent of the order of the system's model. Numerical simulation results are displayed with an expected satisfactory performance compared to the available methods.     

A robust multi-model predictive controller for distributed parameter systems

In this work a robust nonlinear model predictive controller for nonlinear convection-diffusion-reaction systems is presented. The controller makes use of a collection of reduced order approximations of the plant (models) reconstructed on-line by projection methods on proper orthogonal decomposition (POD) basis functions. The model selection and model update step is based on a sufficient condition that determines the maximum allowable process-model mismatch to guarantee stable control performance despite process uncertainty and disturbances. Proofs on the existence of a sequence of feasible approximations and control stability are given.Since plant approximations are built on-line based on actual measurements, the proposed controller can be interpreted as a multi-model nonlinear predictive control (MMPC). The performance of the MMPC strategy is illustrated by simulation experiments on a problem that involves reactant concentration control of a tubular reactor with recycle.     

Nonlinear controls for a class of discrete‐time bilinear systems

For a discrete‐time neutrally stable bilinear system, a nonlinear state feedback control based on the passivity design has been proposed to stabilize the system globally and asymptotically. This paper shows that the decay rate resulting from the passivity control is not exponential, and the system's response speed becomes very sluggish asymptotically. A ‘normalized’ nonlinear control is therefore proposed to achieve exponential stability. The new exponentially stabilizing control not only improves the system's response speed, but also enhances the system's robustness against small parametric perturbations. Copyright © 2003 John Wiley & Sons, Ltd.     

A trajectory‐based method for constructing null controllable regions

The problem of determining a system's set of stabilizable states, the null controllable region (NCR), is intricately related to the problem of determining control Lyapunov functions. In this paper, we address the problem of construction of the NCR for control‐affine nonlinear systems with input constraints. To this end, we explain how the boundary of the NCR is covered by time‐optimal trajectories. To construct the NCR, we employ an algorithm based on Pontryagin's minimum principle, which integrates optimal trajectories of a special smooth system in reverse time from the boundary of an initial NCR estimate. We illustrate this algorithm with linear and nonlinear system examples.     

A Receding Horizon Sliding Controller for Automotive Engine Coldstart: Design and Hardware‐in‐the‐Loop Testing With an Echo State Network High‐Fidelity Model

The aim of the current study is to probe the potential of receding horizon sliding control (RHSC) technique for reducing the coldstart hydrocarbon (HC) emissions of automotive spark‐ignited (SI) engines. The RHSC approach incorporates the potentials of sliding control (SC) and nonlinear model predictive control (NMPC) to employ the future information of the considered engine to keep the system's trajectories close to a stable manifold. To calculate the control commands, the authors adopt an efficient optimization technique, known as the multivariate quadratic fit sectioning algorithm (MQFSA), and also, define three different objective functions, based on l₁, l₂, and l_∞ norms. To demonstrate the efficacy of RHSC controller, its performance is compared with two other well‐known controllers extracted from the literature, namely NMPC and Pontryagin's minimum principle (PMP)‐based controllers. Through numerical simulations for three distinctive operating conditions, it is demonstrated that the RHSC controller is very effective for reducing the total tailpipe HC emissions over the coldstart period of the considered engine system. Moreover, by conducting a hardware‐in‐the‐loop (HIL) test using an echo state network high‐fidelity model, it is indicated that the computational speed of calculating control commands is fast enough to enable RHSC to be used for real‐time implementations in practice.     

Enhanced POD projection basis with application to shape optimization of car engine intake port

In this paper we present a rigorous method for the construction of enhanced Proper Orthogonal Decomposition (POD) projection bases for the development of efficient Reduced Order Models (ROM). The resulting ROMs are seen to exactly interpolate global quantities by design, such as the objective function(s) and nonlinear constraints involved in the optimization problem, thus narrowing the search space by limiting the number of constraints that need to be explicitly included in the statement of the optimization problem. We decompose the basis into two subsets of orthogonal vectors, one for the representation of constraints and the other one, in a complementary space, for the minimization of the projection errors. An explicit algorithm is presented for the case of linear objective functions. The proposed method is then implemented within a bi-level ROM and is illustrated with an application to the multi-objective shape optimization of a car engine intake port for two competing objectives: CO₂ emissions and engine power. We show that optimization using the proposed method produces Pareto dominant and realistic solutions for the flow fields within the combustion chamber, providing further insight into the flow properties.     

Control‐oriented modeling and deployment of tensegrity–membrane systems

This study addresses control‐oriented modeling and control design of tensegrity–membrane systems. Lagrange's method is used to develop a control‐oriented model for a generic system. The equations of motion are expressed as a set of differential‐algebraic equations (DAEs). For control design, the DAEs are converted into second‐order ordinary differential equations (ODEs) based on coordinate partitioning and coordinate mapping. Because the number of inputs is less than the number of state variables, the system belongs to the class of underactuated nonlinear systems. A nonlinear adaptive controller based on the collocated partial feedback linearization (PFL) technique is designed for system deployment. The stability of the closed‐loop system for the actuated coordinates is studied using the Lyapunov stability theory. Because of system complexity, numerical tests are used to conduct stability analysis for the dynamics of the underactuated coordinates, which represents the system's zero dynamics. For the tensegrity–membrane systems studied in this work, analytical proof of zero dynamics stability remains an open theoretical problem. An H_∞ controller is implemented for rapid stabilization of the system at the final deployed configuration. Simulations are conducted to test the performance of the two controllers. The simulation results are presented and discussed in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

Model reduction and controller design for a nonlinear heat conduction problem using finite element method

The mathematical models for dynamic distributed parameter systems are given by systems of partial differential equations. With nonlinear material properties, the corresponding finite element (FE) models are large systems of nonlinear ordinary differential equations. However, in most cases, the actual dynamics of interest involve only a few of the variables, for which model reduction strategies based on system theoretical concepts can be immensely useful. This paper considers the problem of controlling a three dimensional profile on nontrivial geometries. Dynamic model is obtained by discretizing the domain using FE method. A nonlinear control law is proposed which transfers any arbitrary initial temperature profile to another arbitrary desired one. The large dynamic model is reduced using proper orthogonal decomposition (POD). Finally, the stability of the control law is proved through Lyapunov analysis. Results of numerical implementation are presented and possible further extensions are identified.     

Efficient computation of higher order frequency response functions for nonlinear systems with,and without,a constant term

The behaviour of a nonlinear system can be profoundly affected by the presence of a constant or dc term in the system governing equation. These changes are reflected in the nonlinear frequency response characteristics of the system which provide a powerful insight into the system's dynamics. In this article, a new and efficient algorithm is presented for computing the higher order Volterra frequency response functions from nonlinear time-domain models that may contain a constant term. A comparison with previous methods is included to demonstrate the significant gains in computational efficiency that are achieved using the new method. The algorithm is applicable to systems modelled by nonlinear differential, or difference, equations and is easily automated. Several examples are used to illustrate the method, and to highlight the importance of dc terms in nonlinear system analysis.     

油藏注水开发低阶模型及最优控制仿真研究

油藏注水开发最优控制问题计算规模大、控制变量与计算网格多,且控制变量与目标函数之间的关系为一组非线性偏微分方程控制,若直接进行数值求解,对于目前的计算机计算速度和存储空间是个巨大负担.本文采用最佳正交分解(proper orthogonal decomposition,POD)方法提出了基于低阶模型的油藏注水开发最优控制问题,这样,控制变量与目标函数之间的复杂关系被转变为解析函数,仅以少量的POD系数作为优化变量且只需采用非线性规划方法即可求解,大幅度地降低了原问题的求解复杂度与计算量.以二维五点井网的一个井组为应用实例进行仿真研究,结果表明:基于低阶模型的最优控制问题所求解的最大生产净现值与经典的伴随梯度法相比仅有不超过2.5%的误差,且计算速度优势极为明显,当网格数为40×40时,计算速度可提高30倍以上,网格数越多,计算速度优势越明显,当网格数为70×70时,可提速60倍以上.     

A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization

Predicting the transient response of structures by high-fidelity simulation models within design optimization and uncertainty quantification often leads to unacceptable computational cost. This paper presents a reduced-order modeling (ROM) framework for approximating the transient response of linear elastic structures over a range of design and random parameters. The full-order response is projected onto a lower-dimensional basis spanned by modes computed from a proper orthogonal decomposition (POD) of full-order model simulation results at multiple calibration points. The basis is further enriched by gradients of the POD modes with respect to the design/random parameters. A truncation strategy is proposed to compensate for the increase in basis vectors due to the proposed enrichment strategies. The accuracy, efficiency and robustness of the proposed framework are studied with a two-dimensional model problem. The numerical results suggest that the proposed ROM approach is well suited for large parameter changes and that the number of basis vectors needs to be increased only linearly with the number of design and random parameters to maintain a particular ROM performance. The application of the proposed ROM approach to robust shape optimization demonstrates significant savings in computational cost over using full-order models. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.