时域投影模型降阶方法的小样本误差估计

投影方法作为一类重要的模型降阶方法,其计算过程稳定,易于实现,但在理论上鲜有良好的时域误差估计结果．本文提出一种基于小样本估计过程的时域投影模型降阶误差估计方法．该方法首先将降阶过程中产生的误差分解为两部分,然后对各部分利用小样本估计方法进行估计．文中分别对线性和非线性输入输出系统进行小样本误差估计分析．此外,该方法能对线性系统的扰动问题进行分析,进一步的数值算例验证了该方法的有效性．     

Stabilization of a solid propellant rocket instability by state feedback

In this paper a globally stabilizing feedback boundary control law for an arbitrarily fine discretization of a one‐dimensional nonlinear PDE model of unstable burning in solid propellant rockets is presented. The PDE has a destabilizing boundary condition imposed on one part of the boundary. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs, properly modified to accommodate the imposed destabilizing nonlinear boundary condition at the burning end, we transform the original system into a target system that is asymptotically stable in l²‐norm with the same type of boundary condition at the burning end, and homogeneous Dirichlet boundary condition at the control end. Copyright © 2003 John Wiley & Sons, Ltd.     

Reduced-order modeling of weakly nonlinear MEMS devices with Taylor-series expansion and Arnoldi approach

In this paper, we present a new technique by combining the Taylor series expansion with the Arnoldi method to automatically develop reduced-order models for coupled energy domain nonlinear microelectromechanical devices. An electrostatically actuated fixed-fixed beam structure with squeeze-film damping effect is examined to illustrate the model-order reduction method. Simulation results show that the reduced-order nonlinear models can accurately capture the device dynamic behavior over a much larger range of device deformation than the conventional linearized model. Compared with the fully meshed finite-difference method, the model reduction method provides accurate models using orders of magnitude less computation. The reduced MEMS device models are represented by a small number of differential and algebraic equations and thus can be conveniently inserted into a circuit simulator for fast and efficient system-level simulation.     

Gain-scheduled controller synthesis for a nonlinear PDE

Linear parameter-varying (LPV) modelling and control of a nonlinear partial differential equation (PDE) is considered in this article. The one-dimensional viscous Burgers' equation is discretised using a finite difference scheme; the boundary conditions are taken as control inputs and the velocities at two grid points are assumed to be measurable. A nonlinear high-order state space model is generated and proper orthogonal decomposition is used for model order reduction. After assessing the accuracy of the reduced model, a low-order functional observer is designed to estimate the reduced states which are linear combinations of the velocities at all grid points. A discrete-time quasi-LPV model that is affine in scheduling parameters is derived based on the reduced model. A polytopic LPV controller is synthesised based on a generalised plant containing the LPV model and the functional observer. More generally, the proposed method can be used to design an LPV controller for a quasi-LPV system with non-measurable scheduling parameters. Simulation results demonstrate the high tracking performance and disturbance and measurement noise rejection capabilities of the designed LPV controller compared with a linear quadratic Gaussian (LQG) controller based on a linearised model.     

An uncertainty vector adjustment for process modelling error compensation

A new uncertainty vector adjustment (UVA) was developed to compensate for the adverse effects of modelling errors and, thereby, to improve the robustness of two nonlinear control strategies: input output linearization; and Su-Hunt-Meyer transformation. The UVA provides an appropriate adjustment to the transformation relationships obtained from a nonlinear control strategy according to the overall effect of modelling errors. The effectiveness of the approach was verified by introducing parametric errors into the simulation of an evaporative stage of the liquor burning unit of the Bayer process for the production of alumina. Significant improvements in responses were observed based on the integral of the time-weighted absolute error performance index.     

直拉硅单晶非均匀相变温度场最优控制

针对直拉硅单晶固液界面相变温度场的非均匀性导致晶体直径不均匀问题,提出一种基于偏微分方程(PDE)模型的温度场最优控制策略.考虑生长速率波动的影响,建立了一种改进的提拉动力学模型,确定了域边界演化动力学关系.研究基于抛物型PDE的时变空间域对流扩散过程的温度模型,描述了域运动在对流扩散系统上的单向耦合.针对无限维分布参数系统建模控制难问题,采用谱方法进行系统近似,选取整个空间域的全局和正交的空间基函数,通过Galerkin方法对无限维系统进行降维,获得了该系统的近似模型.采用线性二次型方法控制晶体生长温度,通过仿真实验对相变温度场模型进行验证.结果表明,优化后的模型能够获得较为平稳的晶体生长速率,减小了生长直径的波动,使得固液界面径向温度分布更加均匀,验证了该方法的有效性.     

A nonlinear sliding mode control design approach based on neural network modelling

A complete nonlinear framework for the modelling and robust control of nonlinear systems is proposed. The use of neural networks for continuous time modelling to obtain a certain nonlinear canonical form is investigated. The model obtained is used with recently proposed dynamic sliding mode controller design methods. The robustness bounds needed for controller design are determined from modelling errors. A modified version of the backpropagation theorem is also introduced. Copyright © 1999 John Wiley & Sons, Ltd.     

An improved noise removal model based on nonlinear fourth-order partial differential equations

In this paper, we propose an improved noise removal model based on a nonlinear fourth-order partial differential equation (PDE). It associates with the minimization of a certain energy subject to spatially varying constraints involving local variance measures. We discuss the existence and uniqueness of the solutions for the proposed model. The main advantage of the proposed method over the related methods is that it can not only preserve textures but also avoid the staircase effect in smooth regions in the process of denoising. Experimental results illustrate advantages of our proposed method in visual improvement as well as an increase in the signal-to-noise ratio over related PDE methods.     

Modified Smith predictor with a robust disturbance reduction scheme for linear systems with small time delays

This paper presents a robust disturbance reduction scheme using an artificial neural network (ANN) for linear systems with small time delays. It is assumed that the nominal linear systems are stable, minimum phase and relative degree one systems. The proposed structure is an integration of a modified Smith predictor and an ANN‐based disturbance reduction scheme. Unlike other disturbance rejection methods, the proposed approach does not require information about unknown load disturbance frequencies. An ANN is used to approximate the unknown load disturbances and to enhance the robustness of the proposed disturbance reduction scheme against modelling errors in the estimated time delay and the process model. Connective weights of the ANN are trained on‐line using a back‐propagation algorithm until uncertainties resulting from unknown load disturbances and modelling errors are minimized. The simulation results show the effectiveness of the presented disturbance reduction scheme for controlling linear delay systems subjected to step or periodic unknown load disturbances.     

Robust identification method for nonlinear model structures and its application to high-performance aircraft

A common assumption is that the model structure is known for modelling high performance aircraft. In practice, this is not the case. Actually, structure identification plays the most important role in the processing of nonlinear system modelling. The integration of mode structure identification and parameter estimation is an efficient method to construct the model for high performance aircraft, which is nonlinear and also contains uncertainties. This article presents an efficient method for identifying nonlinear model structure and estimating parameters for high-performance aircraft model, which contains uncertainties. The parameters associated with nonlinear terms are considered one after the other if they should be included in the nonlinear model until a stopping criterion is met, which is based on Akaike's information criterion. A numerically efficient U-D factorisation is presented to avoid complex computation of high-order matrices. The proposed method is applied to flight test data of a high-performance aircraft. The results demonstrate that the proposed method could obtain the good aircraft model with a reasonably good fidelity based on the comparison with flight test data.     

Improving prediction of exchange rates using Differential EMD

Volatility is a key parameter when measuring the size of errors made in modelling returns and other financial variables such as exchanged rates. The autoregressive moving-average (ARMA) model is a linear process in time series; whilst in the nonlinear system, the generalised autoregressive conditional heteroskedasticity (GARCH) and Markov switching GARCH (MS-GARCH) have been widely applied. In statistical learning theory, support vector regression (SVR) plays an important role in predicting nonlinear and nonstationary time series variables. In this paper, we propose a new algorithm, differential Empirical Mode Decomposition (EMD) for improving prediction of exchange rates under support vector regression (SVR). The new algorithm of Differential EMD has the capability of smoothing and reducing the noise, whereas the SVR model with the filtered dataset improves predicting the exchange rates. Simulations results consisting of the Differential EMD and SVR model show that our model outperforms simulations by a state-of-the-art MS-GARCH and Markov switching regression (MSR) models.     

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.     

Cooperative information space distributed macromodels

The paper introduces a new approach for the dynamic distributed modelling using the variation principle, applied to a functional on trajectories of random process, and its connection to the process's information functional. The model equations in partial derivatives (PDE) are found by the solution of the variation problem for this functional. This allows us to build a two-level information model with a random process at the microlevel and dynamic process at the macrolevel. The informational macromodels are connected to the equations of irreversible thermodynamics and kinetics. The paper focuses on the problem of a space-time consolidation, which is a new area in the PDE theory, directly connected to modelling complex systems. The synthesized cooperative distributed macromodel is formed during the optimal time-spaced movement, directed toward the equalization and collectivization of the model operator eigenvectors. The mathematical formalism has been applied for the constructive solution to the problems of the object identification, combined with optimal control's synthesis and process’ consolidation. The procedure, demonstrated on this paper's examples and cooperative macromodels, leads to the creation of a dynamic information hierarchical network. The developed computer-based methodology and software were practically used for systems modelling, identification, and control of a diversity of information interactions in some physical (technological) and non-physical (economical and information) objects.     

Robust end-point optimizing feedback for nonlinear dynamic processes

The paper suggests two novel approaches to the synthesis of robust end-point optimizing feedback for nonlinear dynamic processes. Classically, end-point optimization is performed only for the nominal process model using optimal control methods, and the question of performance robustness to disturbances and model-plant mismatch remains unaddressed. The present contribution addresses the end-point optimization problem for nonlinear affine systems with fixed final time through robust optimal feedback methods. In the first approach, a nonlinear state feedback is derived that robustly optimizes the final process state. This solution is obtained through series expansion of the Hamilton-Jacobi-Bellman PDE with an active opponent disturbance. As reliable measurements or estimates of all states may not always be available, the second approach also robustly optimizes the process end-point, but uses output rather than state information. This direct use of measurement information is preferred since the choice of a state estimator for robust state feedback is non-trivial even when the observability issue is addressed. A linear time-variant output corrector is obtained by feedback parametrization and numerical optimization of a nonlinear H _∞ cost functional. A number of possible variations and alternatives to both approaches are also discussed. As model-plant mismatch is particularly common with chemical batch processes, the suitability of the robust optimizing feedback is demonstrated on a semi-batch reactor simulation example, where robustness to several realistic mismatches is investigated and the results are compared against those for the optimal open-loop policy and the optimal feedback designed for the nominal model.     

Fast character modeling with sketch-based PDE surfaces

Virtual characters are 3D geometric models of characters. They have a lot of applications in multimedia. In this paper, we propose a new physics-based deformation method and efficient character modelling framework for creation of detailed 3D virtual character models. Our proposed physics-based deformation method uses PDE surfaces. Here PDE is the abbreviation of Partial Differential Equation, and PDE surfaces are defined as sculpting force-driven shape representations of interpolation surfaces. Interpolation surfaces are obtained by interpolating key cross-section profile curves and the sculpting force-driven shape representation uses an analytical solution to a vector-valued partial differential equation involving sculpting forces to quickly obtain deformed shapes. Our proposed character modelling framework consists of global modeling and local modeling. The global modeling is also called model building, which is a process of creating a whole character model quickly with sketch-guided and template-based modeling techniques. The local modeling produces local details efficiently to improve the realism of the created character model with four shape manipulation techniques. The sketch-guided global modeling generates a character model from three different levels of sketched profile curves called primary, secondary and key cross-section curves in three orthographic views. The template-based global modeling obtains a new character model by deforming a template model to match the three different levels of profile curves. Four shape manipulation techniques for local modeling are investigated and integrated into the new modelling framework. They include: partial differential equation-based shape manipulation, generalized elliptic curve-driven shape manipulation, sketch assisted shape manipulation, and template-based shape manipulation. These new local modeling techniques have both global and local shape control functions and are efficient in local shape manipulation. The final character models are represented with a collection of surfaces, which are modeled with two types of geometric entities: generalized elliptic curves (GECs) and partial differential equation-based surfaces. Our experiments indicate that the proposed modeling approach can build detailed and realistic character models easily and quickly.

     

Constrained nonlinear multivariable control of a fluid catalytic cracking process

In this paper, a nonlinear constrained optimization strategy is proposed and applied to the reactor-regenerator section of a fluid catalytic cracking (FCC) unit. A nonlinear dynamic model of the fluid catalytic cracking process was used for the dynamic analysis of the plant and nonlinear multivariable control system. The model realistically simulates the riser-reactor and the one stage regenerator by assembling the mass and energy balances on the system of reactions. The model results were tested in a real-time application and the results were used to provide the initial values for the nonlinear control system design. A dynamic parameter update algorithm was used to reduce the effect of large modelling errors by regularly updating the model parameters. The constrained nonlinear optimization algorithm and strategies were tested in real-time on the fluid catalytic cracking reactor-regenerator. The results compared favourably to those from a linear multivariable controller.     

A Nonlinear Structure Tensor with the Diffusivity Matrix Composed of the Image Gradient

We propose a nonlinear partial differential equation (PDE) for regularizing a tensor which contains the first derivative information of an image such as strength of edges and a direction of the gradient of the image. Unlike a typical diffusivity matrix which consists of derivatives of a tensor data, we propose a diffusivity matrix which consists of the tensor data itself, i.e., derivatives of an image. This allows directional smoothing for the tensor along edges which are not in the tensor but in the image. That is, a tensor in the proposed PDE is diffused fast along edges of an image but slowly across them. Since we have a regularized tensor which properly represents the first derivative information of an image, the tensor is useful to improve the quality of image denoising, image enhancement, corner detection, and ramp preserving denoising. We also prove the uniqueness and existence of solution to the proposed PDE.     

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.