Performance bounds and suboptimal policies for linear stochastic control via LMIs

In a recent paper, the authors showed how to compute performance bounds for infinite‐horizon stochastic control problems with linear system dynamics and arbitrary constraints, objective, and noise distribution. In this paper, we extend these results to the finite‐horizon case, with asymmetric costs and constraint sets. In addition, we derive our bounds using a new method, where we relax the Bellman equation to an inequality. The method is based on bounding the objective with a general quadratic function, and using linear matrix inequalities (LMIs) and semidefinite programming (SDP) to optimize the bound. The resulting LMIs are more complicated than in the previous paper (which only used quadratic forms) but this extension allows us to obtain good bounds for problems with substantial asymmetry, such as supply chain problems. The method also yields very good suboptimal control policies, using control‐Lyapunov methods. Copyright © 2010 John Wiley & Sons, Ltd.     

基于混杂系统方法的一类采样数据系统鲁棒故障检测

针对具有连续时间过程噪声和离散时间测量噪声的采样数据系统, 提出了一种新的鲁棒故障检测直接设计方法. 首先利用具有有限跳变的线性系统作为残差产生器, 采样数据系统的鲁棒故障检测设计问题被描述成采样数据滤波问题, 然后给出有限跳变线性系统有界实引理的线性矩阵不等式(LMI)表达形式, 基于此, 推导出采样数据系统鲁棒故障检测滤波器的存在条件及设计参数, 并将所提方法推广到具有结构不确定性的采样数据系统上. 所设计的滤波器能够保证残差与故障之间误差最小, 并对过程噪声、测量噪声、结构不确定性等因素鲁棒. 最后, 通过数值仿真对所提方法的可行性进行了验证.     

A priori error estimation of a four-node Reissner–Mindlin plate element for elasto-dynamics

The explicit finite element method for transient dynamics of linear elasticity by Reissner–Mindlin plate model is introduced. For clamped rectangular plate, the a priori error estimates are derived for the four-node Bathe–Dvorkin element. For fixed thickness, the convergence rates of deflection, rotation, and their velocities, measured both in H¹-norm and L²-norm, can possibly all be optimal under certain conditions. In some cases, the numerical examples show that the convergence rate stays optimal for a certain range of thickness. In other cases, however, the deterioration in rate of convergence and even locking may occur to the velocity terms.     

Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

Identification in the presence of classes of unmodeled dynamics andnoise

Identification involves obtaining a model from an a priori chosen model class(es) using finite corrupted data. The corruption may be due to several reasons ranging from noise to unmodeled dynamics, since the real system may not belong to the model class. Two popular approaches-probabilistic and set-membership identification-deal with this problem by imposing temporal constraints on the noise sample paths. We differentiate between the two sources of error by imposing different types of constraints on the corruption. If the source of corruption is noise, we model it by imposing temporal constraints on the possible realizations of noise. On the other hand, if it results from unmodeled dynamics informational constraints are imposed. Contrary to probabilistic identification where the parameters of the identified model converge to the true parameters in the presence of noise, current results in set-membership identification do not have this convergence property. Our approach leads to bridging this gap between probabilistic and set-membership identification when the source of corruption is noise. For the case when both unmodeled dynamics and noise are present, we derive consistency results for the case when the unmodeled dynamics can be described either by a linear time-invariant system or by a static nonlinearity     

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grünwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grünwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects. An erratum to this article can be found at     

Control of nonlinear, continuous, dynamic systems via finite elements, sensitivity analysis, and optimization

A general methodology to design open loop controllers for nonlinear, dynamic, continuous systems is presented and applied to control a single flexible link (SFL). In this application, the partial differential equations that describe the beam system are first analyzed via the finite element method (FEM) and Newmark integration method. Two open loop control inputs to achieve specified system performance criteria are then computed by posing and solving inverse dynamics problems. These analyses use nonlinear programming (NLP) algorithms and analytical gradients that are computed by the direct sensitivity method. The open loop control is verified experimentally. Closed loop controller synthesis for linear time invariant (LTI) and linear time varying systems (LTV) is relatively well understood. To apply this knowledge base to the control of the SFL, the nonlinear finite element plant model is linearized and recast in standard state space form.     

The new numerical method for solving the system of two-dimensional Burgers’ equations

In this paper, the system of two-dimensional Burgers’ equations are solved by local discontinuous Galerkin (LDG) finite element method. The new method is based on the two-dimensional Hopf–Cole transformations, which transform the system of two-dimensional Burgers’ equations into a linear heat equation. Then the linear heat equation is solved by the LDG finite element method. The numerical solution of the heat equation is used to derive the numerical solutions of Burgers’ equations directly. Such a LDG method can also be used to find the numerical solution of the two-dimensional Burgers’ equation by rewriting Burgers’ equation as a system of the two-dimensional Burgers’ equations. Three numerical examples are used to demonstrate the efficiency and accuracy of the method.     

Linear and unit-resulting refutations for Horn theories

We present a new transformation method by which a given Horn theory is transformed in such a way that resolution derivations can be carried out which are both linear (in the sense of Prologs SLD-resolution) and unit-resulting (i.e the resolvents are unit clauses). This is not trivial since although both strategies alone are complete, their naïve combination is not. Completeness is recovered by our method through a completion procedure in the spirit of Knuth-Bendix completion, however with different ordering criteria. A powerful redundancy criterion helps to find a finite system quite often. The transformed theory can be used in combination with linear calculi such as e.g. (theory) model elimination to yield sound, complete and efficient calculi for full first order clause logic over the given Horn theory. As an example application, our method discovers a generalization of the well-known linear paramodulation calculus for the combined theory of equality and strict orderings. The method has been implemented and has been tested in conjunction with a model elimination theorem prover.     

一种带有色量测噪声的非线性系统辨识方法

利用最大似然判据, 本文提出了一种带有色量测噪声的非线性系统辨识方法. 首先, 利用量测差分方法将有色量测噪声白色化, 获得新的量测方程, 从而将带有色量测噪声的非线性系统辨识问题转化成带白色量测噪声和一步延迟状态的非线性系统辨识问题. 其次, 利用期望最大化(Expectation maximization, EM)算法提出了一种新的基于最大似然估计的非线性系统辨识方法, 该算法由期望步骤(Expectation step, E-step)和最大化步骤(Maximization step, M-step)两部分组成. 在期望步骤中, 基于当前估计的参数并利用带有色量测噪声的高斯近似滤波器和平滑器, 近似计算完整的对数似然函数的期望. 在最大化步骤中, 近似计算的似然函数期望值被最大化, 并且通过解析更新获得噪声参数估计, 通过Newton更新方法获得模型参数的估计. 最后, 数值仿真验证了本文提出算法的有效性.     

Extended state space recursive least squares

A new extended state space recursive least squares (ESSRLS) algorithm is proposed for state estimation of nonlinear systems. It is based on state space recursive least squares (SSRLS) approach and uses first order linearization of the system. It inherits the capability of obtaining state estimate without knowledge of process and measurement noise covariance matrices (Q and R respectively). The proposed approach is considered to provide new design option for scenarios where noise statistics and system dynamics vary. ESSRLS is initialized using delayed recursion method and a forgetting factor λ is employed to optimize the performance. The selection of λ can be problem specific as shown through experimental validations. However a value closer to and less than unity is generally recommended. Theoretical bases are validated by applying this algorithm to problems of tracking a non-conservative oscillator, a damped system with amplitude death and a signal modeled by mixture of Gaussian kernels. Simulation results show an MSE performance gain of 20 dB and 23 dB over extended Kalman filter (EKF) and unscented Kalman filter (UKF) while tracking van der Pol oscillator without knowledge about noise variances. The computational complexity of ESSRLS falls within that of EKF and UKF.     

A robust adaptive control method for Wiener nonlinear systems

This paper proposes a new robust adaptive control method for Wiener nonlinear systems with uncertain parameters. The considered Wiener systems are different from the previous ones in the sense that we consider nonlinear block approximation error, process noise, and measurement noise. The parameterization model is obtained based on the inverse of the nonlinear function block. The adaptive control method is derived from a modified criterion function that can overcome non‐minimum phase property of the linear subsystem. The parameter adaptation is performed by using a robust recursive least squares algorithm with a deadzone weighted factor. The control law compensates the model error by incorporating the unmodeled dynamics estimation. Theoretical analysis indicates that the closed‐loop system stability can be guaranteed under mild conditions. Numerical examples including an industrial problem are studied to validate the results. Copyright © 2016 John Wiley & Sons, Ltd.     

电容耦合电阻层析成像并行电阻测量原理研究

传统的电阻层析成像(ERT)系统的电极与被测介质直接接触,容易产生电极极化、电化学腐蚀等问题,近年来出现的电容耦合非接触电导测量技术与ERT技术相结合的电容耦合电阻层析成像系统提出了一种新的非接触测量方法来解决此问题。但是目前该系统的电阻测量原理决定了其测量模式是串行的,成为限制系统数据采集速度的一个重要因素。针对该问题,本文引入了新的测量模型,提出了并行电阻测量原理,实现了数据采集系统,大大提高了系统速度。本文主要研究内容包括:分析了新型传感器的测量模型,提出了新的电阻求解方法;利用有限元法获得了新测量模型下管道内灵敏场分布;研制了相应的12电极样机系统。初步实验测试结果表明,该系统在具有较好测量精度的同时,其数据采集速度可达120帧/s,有效提高了电容耦合电阻层析成像系统的实时性。     

Linear scaling first-principles molecular dynamics with controlled accuracy

In our quest for accurate linear scaling first-principles molecular dynamics methods for pseudopotential DFT calculations, we investigate the accuracy of real-space grid approaches, with finite differences and spherical localization regions. We examine how the positions of the localization centers affect accuracy and the convergence rate in the optimization process. In particular we investigate the accuracy of the atomic forces computation compared to the standard O(N³) approach. We show the exponential decay of the error on the energy and forces with the size of the localization regions for a variety of realistic physical systems. We propose a new algorithm to automatically adapt the localization centers during the ground state computation which allows for molecular dynamics simulations with diffusion processes. The combination of algorithms proposed lead to a genuine linear scaling First-Principles Molecular Dynamics method with controlled accuracy. We illustrate our approach with examples of microcanonical molecular dynamics with localized orbitals.     

Modalities for model checking: branching time logic strikes back

We consider automatic verification of finite state concurrent programs. The global state graph of such a program can be viewed as a finite (Kripke) structure, and amodel checking algorithm can be given for determining if a given structure is a model of a specification expressed in a propositional temporal logic. In this paper, we present a unified approach for efficient model checking under a broad class of generalized fairness constraints in a branching time framework extending that of Clarke et al. (1983). Our method applies to any type of fairness expressed in a certain canonical form. Almost all ‘practical’ types of fairness from the literature, including the fundamental notions of impartiality, weak fairness, and strong fairness, can be succintly written in our canonical form. Moreover, our branching time approach can easily be adapted to handle types of fairness (such as fair reachability of a predicate) which cannot even be expressed in a linear temporal logic. We go on to argue that branching time logic is always better than linear time logic for model checking. We show that given any model checking algorithm for any system of linear time logic (in particular, for the usual system of linear time logic) there is a model checking algorithm of the same order of complexity (in both the structure and formula size) for the corresponding full branching time logic which trivially subsumes the linear time logic in expressive power (in particular, for the system of full branching time logic CTL^*). We also consider an application of our work to the theory of finite automata on infinite strings.     

基于多体系统动力学和有限元法的联合仿真在车桥耦合振动研究中的应用*

为了实现车桥耦合振动精细化仿真研究,利用多体系统动力学软件SIMPACK建立完整的车辆空间模型,采用空间杆系和板壳混合单元有限元方法建立桥梁的动力分析模型;然后将车辆和桥梁两个子系统在轮轨接触面离散的信息点上进行数据交换,实现车桥耦合振动联合仿真分析。以高速铁路上的简支梁桥为研究对象,采用基于多体系统动力学和有限元法结合的联合仿真技术,计算了弹性轮轨接触时动车组列车以不同车速通过桥梁的空间耦合振动响应,证明了该研究方法的可行性。     

Discrete‐Valued Model Predictive Control Using Sum‐of‐Absolute‐Values Optimization

In this paper, we propose a new design method of discrete‐valued model predictive control for continuous‐time linear time‐invariant systems based on sum‐of‐absolute‐values (SOAV) optimization. The finite‐horizon discrete‐valued control design is formulated as an SOAV optimal control, which is an expansion of L¹ optimal control. It is known that under the normality assumption, the SOAV optimal control exists and takes values in a fixed finite alphabet set if the initial state lies in a subset of the reachable set. In this paper, we analyze the existence and discreteness property for systems that do not necessarily satisfy the normality assumption. Then, we extend the finite‐horizon SOAV optimal control to infinite‐horizon model predictive control (MPC). We give sufficient conditions for the recursive feasibility and the stability of the MPC‐based feedback system in the presence of bounded noise. Simulation results show the effectiveness of the proposed method.     

Remeshing-free Graph-based Finite Element Method for Fracture Simulation

Fracture produces new mesh fragments that introduce additional degrees of freedom in the system dynamics. Existing finite element method (FEM) based solutions suffer from increasing computational cost as the system matrix size increases. We solve this problem by presenting a graph-based FEM model for fracture simulation that is remeshing-free and easily scales to high-resolution meshes. Our algorithm models fracture on the graph induced in a volumetric mesh with tetrahedral elements. We relabel the edges of the graph using a computed damage variable to initialize and propagate fracture. We prove that non-linear, hyper-elastic strain energy density is expressible entirely in terms of the edge lengths of the induced graph. This allows us to reformulate the system dynamics for the relabelled graph without changing the size of the system dynamics matrix and thus prevents the computational cost from blowing up. The fractured surface has to be reconstructed explicitly only for visualization purposes. We simulate standard laboratory experiments from structural mechanics and compare the results with corresponding real-world experiments. We fracture objects made of a variety of brittle and ductile materials, and show that our technique offers stability and speed that is unmatched in current literature.     

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.