STABILITY ANALYSIS OF STATE-SPACE REALIZATIONS FOR MULTIDIMENSIONAL FILTERS WITH OVERFLOW NONLINEARI
linear stability analysis
linear stability analysisLinear stability analysis is a method used in engineering and physics to determine the stability of a system under small perturbations. The basic idea behind linear stability analysis is to linearize the equations governing the system around an equilibrium point and then analyze the eigenvalues and eigenvectors of the resulting linear system.To perform a linear stability analysis, the first step is to determine the equilibrium point of the system, which is the point at which the system is in a steady state. Once the equilibrium point is found, the equations governing the system can be linearized around this point by neglecting higher-order terms in a Taylor series expansion.The resulting linearized equations can be written in the form of a matrix equation Ax = 0, where A is a matrix and x is a vector representing the small perturbations from the equilibrium point. The eigenvalues and eigenvectors of the matrix A then determine the stability of the system.If all eigenvalues of A have negative real parts, then the equilibrium point is stable and any small perturbations will decay over time. If any eigenvalue has a positive real part, then the equilibrium point is unstable and any small perturbation will grow over time. If any eigenvalue has a zero real part, then the stability of the equilibrium point is中性的 and depends on the nature of the eigenvectors corresponding to that eigenvalue.Linear stability analysis is a powerful tool for analyzing the stability of systems, but it has some limitations. It assumes that the system is linear and that the perturbations are small, which may not be accurate in all cases. However, it is often used as a first step in the analysis of more complex systems, and it can provide valuable insights into the stability of a system under different conditions.。
signal and system 英文原版书
signal and system 英文原版书Title: An Overview of the Book "Signal and System"Introduction:The book "Signal and System" is an essential resource for anyone interested in understanding the fundamentals of signal processing and system analysis. It provides a comprehensive and in-depth exploration of the concepts, theories, and applications related to signals and systems. This article aims to provide a detailed overview of the book, highlighting its key points and relevance.I. Fundamental Concepts of Signals and Systems:1.1 Definition and Properties of Signals:- Explanation of signals as time-varying or spatially varying quantities.- Discussion on continuous-time and discrete-time signals.- Description of signal properties such as amplitude, frequency, and phase.1.2 Classification of Signals:- Overview of different types of signals including periodic, aperiodic, deterministic, and random signals.- Explanation of energy and power signals.- Introduction to common signal operations such as time shifting, scaling, and time reversal.1.3 System Classification and Properties:- Definition and classification of systems as linear or nonlinear, time-invariant or time-varying.- Explanation of system properties like causality, stability, and linearity.- Introduction to system representations such as differential equations, transfer functions, and state-space models.II. Time-Domain Analysis of Signals and Systems:2.1 Convolution and Correlation:- Detailed explanation of convolution and its significance in system analysis.- Discussion on correlation as a measure of similarity between signals.- Application of convolution and correlation in practical scenarios.2.2 Fourier Series and Transform:- Introduction to Fourier series and its representation of periodic signals.- Explanation of Fourier transform and its application in analyzing non-periodic signals.- Discussion on the properties of Fourier series and transform.2.3 Laplace Transform:- Overview of Laplace transform and its use in solving differential equations.- Explanation of the relationship between Laplace transform and frequency response of systems.- Application of Laplace transform in system analysis and design.III. Frequency-Domain Analysis of Signals and Systems:3.1 Frequency Response:- Definition and interpretation of frequency response.- Explanation of magnitude and phase response.- Analysis of frequency response using Bode plots.3.2 Filtering and Filtering Techniques:- Introduction to digital and analog filters.- Discussion on different filter types such as low-pass, high-pass, band-pass, and band-stop filters.- Explanation of filter design techniques including Butterworth, Chebyshev, and Elliptic filters.3.3 Sampling and Reconstruction:- Explanation of sampling theorem and its importance in signal processing.- Overview of sampling techniques and their impact on signal reconstruction.- Discussion on anti-aliasing filters and reconstruction methods.IV. System Analysis and Stability:4.1 System Response and Impulse Response:- Explanation of system response to different input signals.- Introduction to impulse response and its relationship with system behavior.- Analysis of system stability based on impulse response.4.2 Transfer Function and Frequency Domain Analysis:- Definition and interpretation of transfer function.- Explanation of frequency domain analysis using transfer function.- Application of transfer function in system design and analysis.4.3 Feedback Systems and Control:- Overview of feedback systems and their role in control theory.- Explanation of stability analysis and design using control theory.- Discussion on PID controllers and their applications.V. Applications of Signal and System Theory:5.1 Communication Systems:- Explanation of modulation techniques and their role in communication systems.- Overview of demodulation techniques and their significance.- Discussion on error control coding and channel equalization.5.2 Digital Signal Processing:- Introduction to digital signal processing and its applications.- Explanation of digital filters and their role in signal processing.- Overview of image and speech processing techniques.5.3 Signal Processing in Biomedical Engineering:- Application of signal processing in biomedical signal analysis.- Discussion on medical imaging techniques such as MRI and CT scans.- Explanation of signal processing methods used in ECG and EEG analysis.Conclusion:The book "Signal and System" provides a comprehensive and detailed exploration of the fundamental concepts, theories, and applications related to signals and systems. It covers a wide range of topics including signal classification, system analysis, frequency-domain analysis, stability, and various applications. By studying this book, readers can gain a solid understanding of signal and system theory, which is essential in various fields such as communication, digital signal processing, and biomedical engineering.。
线性系统英文3
m =1
& &+ kx = 0 x& + fx
（1）
x1 = x & x 1 = x2 Then the system equation & x 2 = − kx1 − fx2
Choose variables as
&= x & x2 = x 1
}
（2）
The spring displacement characterizes the potential energy stored in the spring, and the mass velocity characterizes the kinetic energy stored in the mass. We therefore can express the total energy stored in 1 2 1 2 （3） the system by the function E ( x1 , x2 ) = x2 + kx1 2 2 x = 0 , E (0) = 0 When x ≠ 0 ， E ( x ) > 0 ; On evaluating the energy function along a system trajectory, we can compute the time derivative
d ∂ E d x1 ∂E d x2 & & E ( x1 , x 2 ) = = kx 1 x + 1 + x 2 x 2 = − fx d t ∂x2 d t ∂ x1 d t
2 2
Only when x2 = 0 ，d E /d t = 0 。We conclude that d E /d t < 0 the total energy in the system is strictly decreasing along all other trajectories and converges to zero as time tends to infinity.
特征值wely定理
特征值wely定理The eigenvalue theorem, also known as the Perron-Frobenius theorem, is a fundamental result in linearalgebra that has various applications in different fieldsof science and engineering. This theorem provides insights into the behavior of matrices and their associated eigenvalues. In this response, I will discuss theeigenvalue theorem from multiple perspectives, highlighting its significance, applications, and implications indifferent domains.From a mathematical perspective, the eigenvalue theorem states that for any square matrix, there exists at leastone eigenvalue that is real and non-negative. Furthermore, this eigenvalue is associated with a corresponding eigenvector that has only non-negative entries. This result is particularly important in the study of positive matrices, which are matrices with all non-negative elements. The eigenvalue theorem guarantees the existence of a dominant eigenvalue for positive matrices, which characterizes theirbehavior and provides insights into their long-term dynamics.In the field of dynamical systems, the eigenvalue theorem plays a crucial role in understanding the stability and convergence properties of linear systems. The dominant eigenvalue of a matrix determines the long-term behavior of the associated dynamical system. If the dominant eigenvalue is less than one, the system converges to a stable equilibrium point. On the other hand, if the dominant eigenvalue is greater than one, the system exhibits exponential growth or divergence. This knowledge is vital in various applications, such as control systems, population dynamics, and stability analysis of physical systems.In the context of graph theory, the eigenvalue theorem has significant implications for studying the connectivity and structure of networks. The adjacency matrix of a graph represents its connectivity pattern, and the eigenvalues of this matrix provide valuable information about the graph's properties. The largest eigenvalue of the adjacency matrix,known as the spectral radius, determines the expansion properties of the graph. A larger spectral radius implies a more connected and well-structured network. This concept finds applications in social networks, transportation networks, and communication networks, where understanding the connectivity and robustness of the system is crucial.Another application of the eigenvalue theorem can be found in the field of quantum mechanics. In quantum systems, matrices called Hamiltonians represent the total energy ofa physical system. The eigenvalues of the Hamiltonianmatrix correspond to the possible energy states of the system, while the eigenvectors represent the associated wavefunctions. The eigenvalue theorem ensures that the energy states are real and non-negative, which aligns with the physical interpretation of energy in quantum mechanics. This theorem provides a foundation for understanding the energy spectrum and behavior of quantum systems, enabling predictions and analysis of various physical phenomena.Beyond its mathematical and scientific applications,the eigenvalue theorem has implications in practicaldomains such as data analysis and machine learning. In data analysis, the eigenvalues and eigenvectors of a covariance matrix are used in principal component analysis (PCA). PCA is a dimensionality reduction technique that identifies the most informative features in a dataset. The eigenvalue theorem guarantees that the principal components, which are linear combinations of the original features, capture the maximum variance in the data. This allows for efficient data representation and visualization, facilitating tasks such as clustering, classification, and anomaly detection.In conclusion, the eigenvalue theorem is a powerful result in linear algebra with wide-ranging applications in various fields. From its mathematical significance in positive matrices to its implications in dynamical systems, graph theory, quantum mechanics, and data analysis, the eigenvalue theorem provides valuable insights into the behavior and properties of matrices and their associated eigenvalues. Its applications span from stability analysis and network connectivity to quantum systems and machine learning, making it an essential tool for understanding and solving problems in diverse disciplines.。
边坡稳定分析方法综述
Construction & Decoration148 建筑与装饰2022年3月下 边坡稳定分析方法综述刘琦中铁二十一局集团有限公司 甘肃 兰州 730000摘 要 边坡工程是个十分复杂的系统性工程，其稳定性对于人类的生命财产有重大意义。
如何在复杂地质条件下分析边坡的稳定性历来是学术界和工程界极为关注的课题。
因此，本文从定性分析、定量分析和非确定性三方面进行分析，层次分明地概述了边坡稳定分析方法，综合评述了各自的优缺点，对边坡稳定分析的发展做出展望。
关键词 边坡稳定；定性分析；定量分析；非确定性分析 Overview of Slope Stability Analysis MethodsLiu QiChina Railway 21st Bureau Group Co., Ltd., Lanzhou 730000, Gansu Province, ChinaAbstract Slope engineering is a very complex and systematic engineering, and its stability is of great significance to human life and property. How to analyze slope stability under complex geological conditions has always been a topic of great concern in academic and engineering fields. Therefore, this article analyzes this issue from three aspects: qualitative analysis, quantitative analysis and non-determinancy analysis, clearly outlines the slope stability analysis methods, comprehensively reviews their advantages and disadvantages, and makes a prospect for the development of slope stability analysis.Key words slope stability; qualitative analysis; quantitative analysis; non-determinancy analysis引言我国是一个多山的国家，随着国民经济的发展，工商业及民用建筑剧增，许多工程边坡的分析方法不断产生。
边坡稳定性分析的有限元法与极限平衡法的结合_曾亚武
ZENG Ya-wu，TIAN Wei-ming
(School of Civil Engineering，Wuhan University，Wuhan 430072，China)
Abstract：A method combining the elastoplastic finite element method (FEM) with the limit equilibrium method is proposed to analyze the slope stability problems. The stress，strain and displacement distributions of the slope are calculated by elastoplastic FEM. Then，the stability factor of safety of the slope is calculated with method of slices
+
τ
2 xy
，为第
二应力偏量不变量；θ
=
1 3
cos−1
⎜⎛ ⎜⎝
3
3 2
J3 J 3/2
2
⎟⎞ ⎟⎠
，为
Lode
角； J 3
=
SxSySz
−
S
z
S
2 xy
，为第三应力偏量不变量；
c 为材料的粘聚力；ϕ 为材料的内摩擦角。
在有限元计算中需要计算 ∂F 和 ∂F 的值，而 ∂J 2 ∂J 3
当θ = 0°及θ = 60°时，屈服面出现角点，此时 ∂F ∂J 2
本文应用上述方法进行了一个边坡算例分析， 并将分析结果与各种极限平衡法所得结果进行了比 较。
英语力学词汇英汉对译
力学词汇英汉对译通类名词力学 mechanics牛顿力学 Newtonian mechanics经典力学 classical mechanics静力学 statics运动学 kinematics动力学 dynamics动理学 kinetics宏观力学 macroscopic mechanics,macromechanics细观力学 mesomechanics微观力学 microscopic mechanics,micromechanics一般力学 general mechanics固体力学 solid mechanics流体力学 fluid mechanics理论力学 theoretical mechanics应用力学 applied mechanics工程力学 engineering mechanics实验力学 experimental mechanics计算力学 computational mechanics理性力学 rational mechanics物理力学 physical mechanics地球动力学 geodynamics力 force作用点 point of action作用线 line of action力系 system of forces力系的简化 reduction of force system等效力系 equivalent force system刚体 rigid body力的可传性 transmissibility of force平行四边形定则 parallelogram rule力三角形 force triangle力多边形 force polygon零力系 null-force system平衡 equilibrium力的平衡 equilibrium of forces平衡条件 equilibrium condition平衡位置 equilibrium position平衡态 equilibrium state分析力学 analytical mechanics拉格朗日乘子 Lagrange multiplier拉格朗日[量] Lagrangian拉格朗日括号 Lagrange bracket循环坐标 cyclic coordinate循环积分 cyclic integral哈密顿[量] Hamiltonian哈密顿函数 Hamiltonian function正则方程 canonical equation正则摄动 canonical perturbation正则变换 canonical transformation正则变量 canonical variable哈密顿原理 Hamilton principle作用量积分 action integral哈密顿--雅可比方程 Hamilton-Jacobi equation作用--角度变量 action-angle variables 阿佩尔方程 Appell equation劳斯方程 Routh equation拉格朗日函数 Lagrangian function诺特定理 Noether theorem泊松括号 poisson bracket边界积分法 boundary integral method 并矢 dyad运动稳定性 stability of motion轨道稳定性 orbital stability李雅普诺夫函数 Lyapunov function渐近稳定性 asymptotic stability结构稳定性 structural stability久期不稳定性 secular instability弗洛凯定理 Floquet theorem倾覆力矩 capsizing moment自由振动 free vibration固有振动 natural vibration暂态 transient state环境振动 ambient vibration反共振 anti-resonance衰减 attenuation库仑阻尼 Coulomb damping同相分量 in-phase component非同相分量 out-of -phase component超调量 overshoot参量[激励]振动 parametric vibration 模糊振动 fuzzy vibration临界转速 critical speed of rotation阻尼器 damper半峰宽度 half-peak width集总参量系统 lumped parameter system相平面法 phase plane method相轨迹 phase trajectory等倾线法 isocline method跳跃现象 jump phenomenon负阻尼 negative damping达芬方程 Duffing equation希尔方程 Hill equationKBM方法 KBM method, Krylov-Bogoliu-bov-Mitropol'skii method 马蒂厄方程 Mathieu equation平均法 averaging method组合音调 combination tone解谐 detuning耗散函数 dissipative function硬激励 hard excitation硬弹簧 hard spring, hardening spring谐波平衡法 harmonic balance method久期项 secular term自激振动 self-excited vibration分界线 separatrix亚谐波 subharmonic软弹簧 soft spring ,softening spring软激励 soft excitation邓克利公式 Dunkerley formula瑞利定理 Rayleigh theorem分布参量系统 distributed parameter system优势频率 dominant frequency模态分析 modal analysis固有模态 natural mode of vibration同步 synchronization超谐波 ultraharmonic范德波尔方程 van der pol equation频谱 frequency spectrum基频 fundamental frequencyWKB方法 WKB method, Wentzel-Kramers-Brillouin method 缓冲器 buffer风激振动 aeolian vibration嗡鸣 buzz倒谱 cepstrum颤动 chatter蛇行 hunting阻抗匹配 impedance matching机械导纳 mechanical admittance机械效率 mechanical efficiency机械阻抗 mechanical impedance随机振动 stochastic vibration, random vibration隔振 vibration isolation减振 vibration reduction应力过冲 stress overshoot喘振 surge摆振 shimmy起伏运动 phugoid motion起伏振荡 phugoid oscillation驰振 galloping陀螺动力学 gyrodynamics陀螺摆 gyropendulum陀螺平台 gyroplatform.陀螺力矩 gyroscoopic torque陀螺稳定器 gyrostabilizer陀螺体 gyrostat惯性导航 inertial guidance姿态角 attitude angle方位角 azimuthal angle舒勒周期 Schuler period机器人动力学 robot dynamics多体系统 multibody system多刚体系统 multi-rigid-body system机动性 maneuverability凯恩方法 Kane method转子[系统]动力学 rotor dynamics转子[一支承一基础]系统 rotor-support-foundation system 静平衡 static balancing动平衡 dynamic balancing静不平衡 static unbalance动不平衡 dynamic unbalance现场平衡 field balancing不平衡 unbalance不平衡量 unbalance互耦力 cross force挠性转子 flexible rotor分频进动 fractional frequency precession半频进动 half frequency precession油膜振荡 oil whip转子临界转速 rotor critical speed自动定心 self-alignment亚临界转速 subcritical speed涡动 whirl固体力学弹性力学 elasticity弹性理论 theory of elasticity均匀应力状态 homogeneous state of stress应力不变量 stress invariant应变不变量 strain invariant应变椭球 strain ellipsoid均匀应变状态 homogeneous state of strain应变协调方程 equation of strain compatibility拉梅常量 Lame constants各向同性弹性 isotropic elasticity旋转圆盘 rotating circular disk楔 wedge开尔文问题 Kelvin problem布西内斯克问题 Boussinesq problem艾里应力函数 Airy stress function克罗索夫--穆斯赫利什维利法 Kolosoff-Muskhelishvili method 基尔霍夫假设 Kirchhoff hypothesis板 Plate矩形板 Rectangular plate圆板 Circular plate环板 Annular plate波纹板 Corrugated plate加劲板 Stiffened plate,reinforced Plate 中厚板 Plate of moderate thickness弯[曲]应力函数 Stress function of bending 壳 Shell扁壳 Shallow shell旋转壳 Revolutionary shell球壳 Spherical shell[圆]柱壳 Cylindrical shell锥壳 Conical shell环壳 Toroidal shell封闭壳 Closed shell波纹壳 Corrugated shell扭[转]应力函数 Stress function of torsion 翘曲函数 Warping function半逆解法 semi-inverse method瑞利--里茨法 Rayleigh-Ritz method松弛法 Relaxation method莱维法 Levy method松弛 Relaxation量纲分析 Dimensional analysis自相似[性] self-similarity影响面 Influence surface接触应力 Contact stress赫兹理论 Hertz theory协调接触 Conforming contact滑动接触 Sliding contact滚动接触 Rolling contact压入 Indentation各向异性弹性 Anisotropic elasticity颗粒材料 Granular material散体力学 Mechanics of granular media热弹性 Thermoelasticity超弹性 Hyperelasticity粘弹性 Viscoelasticity对应原理 Correspondence principle褶皱 Wrinkle塑性全量理论 Total theory of plasticity 滑动 Sliding微滑 Microslip粗糙度 Roughness非线性弹性 Nonlinear elasticity大挠度 Large deflection突弹跳变 snap-through有限变形 Finite deformation格林应变 Green strain阿尔曼西应变 Almansi strain弹性动力学 Dynamic elasticity运动方程 Equation of motion准静态的 Quasi-static气动弹性 Aeroelasticity水弹性 Hydroelasticity颤振 Flutter弹性波 Elastic wave简单波 Simple wave柱面波 Cylindrical wave水平剪切波 Horizontal shear wave竖直剪切波 Vertical shear wave体波 body wave无旋波 Irrotational wave畸变波 Distortion wave膨胀波 Dilatation wave瑞利波 Rayleigh wave等容波 Equivoluminal wave勒夫波 Love wave界面波 Interfacial wave边缘效应 edge effect塑性力学 Plasticity可成形性 Formability金属成形 Metal forming耐撞性 Crashworthiness结构抗撞毁性 Structural crashworthiness 拉拔 Drawing破坏机构 Collapse mechanism回弹 Springback挤压 Extrusion冲压 Stamping穿透 Perforation层裂 Spalling塑性理论 Theory of plasticity安定[性]理论 Shake-down theory运动安定定理 kinematic shake-down theorem 静力安定定理 Static shake-down theorem率相关理论 rate dependent theorem载荷因子 load factor加载准则 Loading criterion加载函数 Loading function加载面 Loading surface塑性加载 Plastic loading塑性加载波 Plastic loading wave简单加载 Simple loading比例加载 Proportional loading卸载 Unloading卸载波 Unloading wave冲击载荷 Impulsive load阶跃载荷 step load脉冲载荷 pulse load极限载荷 limit load中性变载 nentral loading拉抻失稳 instability in tension加速度波 acceleration wave本构方程 constitutive equation完全解 complete solution名义应力 nominal stress过应力 over-stress真应力 true stress等效应力 equivalent stress流动应力 flow stress应力间断 stress discontinuity应力空间 stress space主应力空间 principal stress space静水应力状态 hydrostatic state of stress对数应变 logarithmic strain工程应变 engineering strain等效应变 equivalent strain应变局部化 strain localization应变率 strain rate应变率敏感性 strain rate sensitivity应变空间 strain space有限应变 finite strain塑性应变增量 plastic strain increment累积塑性应变 accumulated plastic strain永久变形 permanent deformation内变量 internal variable应变软化 strain-softening理想刚塑性材料 rigid-perfectly plastic Material 刚塑性材料 rigid-plastic material理想塑性材料 perfectl plastic material材料稳定性 stability of material应变偏张量 deviatoric tensor of strain应力偏张量 deviatori tensor of stress应变球张量 spherical tensor of strain应力球张量 spherical tensor of stress路径相关性 path-dependency线性强化 linear strain-hardening应变强化 strain-hardening随动强化 kinematic hardening各向同性强化 isotropic hardening强化模量 strain-hardening modulus幂强化 power hardening塑性极限弯矩 plastic limit bending Moment塑性极限扭矩 plastic limit torque弹塑性弯曲 elastic-plastic bending弹塑性交界面 elastic-plastic interface弹塑性扭转 elastic-plastic torsion粘塑性 Viscoplasticity非弹性 Inelasticity理想弹塑性材料 elastic-perfectly plastic Material 极限分析 limit analysis极限设计 limit design极限面 limit surface上限定理 upper bound theorem上屈服点 upper yield point下限定理 lower bound theorem下屈服点 lower yield point界限定理 bound theorem初始屈服面 initial yield surface后继屈服面 subsequent yield surface屈服面[的]外凸性 convexity of yield surface截面形状因子 shape factor of cross-section沙堆比拟 sand heap analogy屈服 Yield屈服条件 yield condition屈服准则 yield criterion屈服函数 yield function屈服面 yield surface塑性势 plastic potential能量吸收装置 energy absorbing device能量耗散率 energy absorbing device塑性动力学 dynamic plasticity塑性动力屈曲 dynamic plastic buckling塑性动力响应 dynamic plastic response塑性波 plastic wave运动容许场 kinematically admissible Field静力容许场 statically admissible Field流动法则 flow rule速度间断 velocity discontinuity滑移线 slip-lines滑移线场 slip-lines field移行塑性铰 travelling plastic hinge塑性增量理论 incremental theory of Plasticity米泽斯屈服准则 Mises yield criterion普朗特--罗伊斯关系 prandtl- Reuss relation特雷斯卡屈服准则 Tresca yield criterion洛德应力参数 Lode stress parameter莱维--米泽斯关系 Levy-Mises relation亨基应力方程 Hencky stress equation赫艾--韦斯特加德应力空间 Haigh-Westergaard stress space 洛德应变参数 Lode strain parameter德鲁克公设 Drucker postulate盖林格速度方程 Geiringer velocity Equation结构力学 structural mechanics结构分析 structural analysis结构动力学 structural dynamics拱 Arch三铰拱 three-hinged arch抛物线拱 parabolic arch圆拱 circular arch穹顶 Dome空间结构 space structure空间桁架 space truss雪载[荷] snow load风载[荷] wind load土压力 earth pressure地震载荷 earthquake loading弹簧支座 spring support支座位移 support displacement支座沉降 support settlement超静定次数 degree of indeterminacy机动分析 kinematic analysis结点法 method of joints截面法 method of sections结点力 joint forces共轭位移 conjugate displacement影响线 influence line三弯矩方程 three-moment equation单位虚力 unit virtual force刚度系数 stiffness coefficient柔度系数 flexibility coefficient力矩分配 moment distribution力矩分配法 moment distribution method 力矩再分配 moment redistribution分配系数 distribution factor矩阵位移法 matri displacement method单元刚度矩阵 element stiffness matrix单元应变矩阵 element strain matrix总体坐标 global coordinates贝蒂定理 Betti theorem高斯--若尔当消去法 Gauss-Jordan elimination Method 屈曲模态 buckling mode复合材料力学 mechanics of composites复合材料 composite material纤维复合材料 fibrous composite单向复合材料 unidirectional composite泡沫复合材料 foamed composite颗粒复合材料 particulate composite层板 Laminate夹层板 sandwich panel正交层板 cross-ply laminate斜交层板 angle-ply laminate层片 Ply多胞固体 cellular solid膨胀 Expansion压实 Debulk劣化 Degradation脱层 Delamination脱粘 Debond纤维应力 fiber stress层应力 ply stress层应变 ply strain层间应力 interlaminar stress比强度 specific strength强度折减系数 strength reduction factor强度应力比 strength -stress ratio横向剪切模量 transverse shear modulus横观各向同性 transverse isotropy正交各向异 Orthotropy剪滞分析 shear lag analysis短纤维 chopped fiber长纤维 continuous fiber纤维方向 fiber direction纤维断裂 fiber break纤维拔脱 fiber pull-out纤维增强 fiber reinforcement致密化 Densification最小重量设计 optimum weight design网格分析法 netting analysis混合律 rule of mixture失效准则 failure criterion蔡--吴失效准则 Tsai-W u failure criterion达格代尔模型 Dugdale model断裂力学 fracture mechanics概率断裂力学 probabilistic fracture Mechanics格里菲思理论 Griffith theory线弹性断裂力学 linear elastic fracture mechanics, LEFM 弹塑性断裂力学 elastic-plastic fracture mecha-nics, EPFM 断裂 Fracture脆性断裂 brittle fracture解理断裂 cleavage fracture蠕变断裂 creep fracture延性断裂 ductile fracture晶间断裂 inter-granular fracture准解理断裂 quasi-cleavage fracture穿晶断裂 trans-granular fracture裂纹 Crack裂缝 Flaw缺陷 Defect割缝 Slit微裂纹 Microcrack折裂 Kink椭圆裂纹 elliptical crack深埋裂纹 embedded crack[钱]币状裂纹 penny-shape crack预制裂纹 Precrack短裂纹 short crack表面裂纹 surface crack裂纹钝化 crack blunting裂纹分叉 crack branching裂纹闭合 crack closure裂纹前缘 crack front裂纹嘴 crack mouth裂纹张开角 crack opening angle,COA裂纹张开位移 crack opening displacement, COD裂纹阻力 crack resistance裂纹面 crack surface裂纹尖端 crack tip裂尖张角 crack tip opening angle, CTOA裂尖张开位移 crack tip opening displacement, CTOD裂尖奇异场 crack tip singularity Field 裂纹扩展速率 crack growth rate稳定裂纹扩展 stable crack growth定常裂纹扩展 steady crack growth亚临界裂纹扩展 subcritical crack growth 裂纹[扩展]减速 crack retardation止裂 crack arrest止裂韧度 arrest toughness断裂类型 fracture mode滑开型 sliding mode张开型 opening mode撕开型 tearing mode复合型 mixed mode撕裂 Tearing撕裂模量 tearing modulus断裂准则 fracture criterionJ积分 J-integralJ阻力曲线 J-resistance curve断裂韧度 fracture toughness应力强度因子 stress intensity factor HRR场 Hutchinson-Rice-Rosengren Field 守恒积分 conservation integral有效应力张量 effective stress tensor应变能密度 strain energy density能量释放率 energy release rate内聚区 cohesive zone塑性区 plastic zone张拉区 stretched zone热影响区 heat affected zone, HAZ延脆转变温度 brittle-ductile transition temperature剪切带 shear band剪切唇 shear lip无损检测 non-destructive inspection双边缺口试件 double edge notched specimen, DEN specimen单边缺口试件 single edge notched specimen, SEN specimen三点弯曲试件 three point bending specimen, TPB specimen中心裂纹拉伸试件 center cracked tension specimen, CCT specimen 中心裂纹板试件 center cracked panel specimen, CCP specimen紧凑拉伸试件 compact tension specimen, CT specimen大范围屈服 large scale yielding小范围攻屈服 small scale yielding韦布尔分布 Weibull distribution帕里斯公式 paris formula空穴化 Cavitation应力腐蚀 stress corrosion概率风险判定 probabilistic risk assessment, PRA损伤力学 damage mechanics损伤 Damage连续介质损伤力学 continuum damage mechanics细观损伤力学 microscopic damage mechanics累积损伤 accumulated damage脆性损伤 brittle damage延性损伤 ductile damage宏观损伤 macroscopic damage细观损伤 microscopic damage微观损伤 microscopic damage损伤准则 damage criterion损伤演化方程 damage evolution equation 损伤软化 damage softening损伤强化 damage strengthening损伤张量 damage tensor损伤阈值 damage threshold损伤变量 damage variable损伤矢量 damage vector损伤区 damage zone疲劳 Fatigue低周疲劳 low cycle fatigue应力疲劳 stress fatigue随机疲劳 random fatigue蠕变疲劳 creep fatigue腐蚀疲劳 corrosion fatigue疲劳损伤 fatigue damage疲劳失效 fatigue failure疲劳断裂 fatigue fracture疲劳裂纹 fatigue crack疲劳寿命 fatigue life疲劳破坏 fatigue rupture疲劳强度 fatigue strength疲劳辉纹 fatigue striations疲劳阈值 fatigue threshold交变载荷 alternating load交变应力 alternating stress应力幅值 stress amplitude应变疲劳 strain fatigue应力循环 stress cycle应力比 stress ratio安全寿命 safe life过载效应 overloading effect循环硬化 cyclic hardening循环软化 cyclic softening环境效应 environmental effect裂纹片 crack gage裂纹扩展 crack growth, crack Propagation 裂纹萌生 crack initiation循环比 cycle ratio实验应力分析 experimental stress Analysis 工作[应变]片 active[strain] gage基底材料 backing material应力计 stress gage零[点]飘移 zero shift, zero drift应变测量 strain measurement应变计 strain gage应变指示器 strain indicator应变花 strain rosette应变灵敏度 strain sensitivity机械式应变仪 mechanical strain gage直角应变花 rectangular rosette引伸仪 Extensometer应变遥测 telemetering of strain横向灵敏系数 transverse gage factor横向灵敏度 transverse sensitivity焊接式应变计 weldable strain gage平衡电桥 balanced bridge粘贴式应变计 bonded strain gage粘贴箔式应变计 bonded foiled gage粘贴丝式应变计 bonded wire gage桥路平衡 bridge balancing电容应变计 capacitance strain gage补偿片 compensation technique补偿技术 compensation technique基准电桥 reference bridge电阻应变计 resistance strain gage温度自补偿应变计 self-temperature compensating gage 半导体应变计 semiconductor strain Gage集流器 slip ring应变放大镜 strain amplifier疲劳寿命计 fatigue life gage电感应变计 inductance [strain] gage光[测]力学 Photomechanics光弹性 Photoelasticity光塑性 Photoplasticity杨氏条纹 Young fringe双折射效应 birefrigent effect等位移线 contour of equal Displacement暗条纹 dark fringe条纹倍增 fringe multiplication干涉条纹 interference fringe等差线 Isochromatic等倾线 Isoclinic等和线 isopachic应力光学定律 stress- optic law主应力迹线 Isostatic亮条纹 light fringe光程差 optical path difference热光弹性 photo-thermo -elasticity光弹性贴片法 photoelastic coating Method光弹性夹片法 photoelastic sandwich Method动态光弹性 dynamic photo-elasticity空间滤波 spatial filtering空间频率 spatial frequency起偏镜 Polarizer反射式光弹性仪 reflection polariscope残余双折射效应 residual birefringent Effect应变条纹值 strain fringe value应变光学灵敏度 strain-optic sensitivity应力冻结效应 stress freezing effect应力条纹值 stress fringe value应力光图 stress-optic pattern暂时双折射效应 temporary birefringent Effect脉冲全息法 pulsed holography透射式光弹性仪 transmission polariscope实时全息干涉法 real-time holographic interferometry 网格法 grid method全息光弹性法 holo-photoelasticity全息图 Hologram全息照相 Holograph全息干涉法 holographic interferometry全息云纹法 holographic moire technique全息术 Holography全场分析法 whole-field analysis散斑干涉法 speckle interferometry散斑 Speckle错位散斑干涉法 speckle-shearing interferometry, shearography 散斑图 Specklegram白光散斑法 white-light speckle method云纹干涉法 moire interferometry[叠栅]云纹 moire fringe[叠栅]云纹法 moire method云纹图 moire pattern离面云纹法 off-plane moire method参考栅 reference grating试件栅 specimen grating分析栅 analyzer grating面内云纹法 in-plane moire method脆性涂层法 brittle-coating method条带法 strip coating method坐标变换 transformation of Coordinates计算结构力学 computational structural mechanics加权残量法 weighted residual method有限差分法 finite difference method有限[单]元法 finite element method配点法 point collocation里茨法 Ritz method广义变分原理 generalized variational Principle最小二乘法 least square method胡[海昌]一鹫津原理 Hu-Washizu principle赫林格-赖斯纳原理 Hellinger-Reissner Principle 修正变分原理 modified variational Principle约束变分原理 constrained variational Principle 混合法 mixed method杂交法 hybrid method边界解法 boundary solution method有限条法 finite strip method半解析法 semi-analytical method协调元 conforming element非协调元 non-conforming element混合元 mixed element杂交元 hybrid element边界元 boundary element强迫边界条件 forced boundary condition自然边界条件 natural boundary condition离散化 Discretization离散系统 discrete system连续问题 continuous problem广义位移 generalized displacement广义载荷 generalized load广义应变 generalized strain广义应力 generalized stress界面变量 interface variable节点 node, nodal point[单]元 Element角节点 corner node边节点 mid-side node内节点 internal node无节点变量 nodeless variable杆元 bar element桁架杆元 truss element梁元 beam element二维元 two-dimensional element一维元 one-dimensional element三维元 three-dimensional element轴对称元 axisymmetric element板元 plate element壳元 shell element厚板元 thick plate element三角形元 triangular element四边形元 quadrilateral element四面体元 tetrahedral element曲线元 curved element二次元 quadratic element线性元 linear element三次元 cubic element四次元 quartic element等参[数]元 isoparametric element超参数元 super-parametric element亚参数元 sub-parametric element节点数可变元 variable-number-node element 拉格朗日元 Lagrange element拉格朗日族 Lagrange family巧凑边点元 serendipity element巧凑边点族 serendipity family无限元 infinite element单元分析 element analysis单元特性 element characteristics刚度矩阵 stiffness matrix几何矩阵 geometric matrix等效节点力 equivalent nodal force节点位移 nodal displacement节点载荷 nodal load位移矢量 displacement vector载荷矢量 load vector质量矩阵 mass matrix集总质量矩阵 lumped mass matrix相容质量矩阵 consistent mass matrix阻尼矩阵 damping matrix瑞利阻尼 Rayleigh damping刚度矩阵的组集 assembly of stiffness Matrices 载荷矢量的组集 consistent mass matrix质量矩阵的组集 assembly of mass matrices单元的组集 assembly of elements局部坐标系 local coordinate system局部坐标 local coordinate面积坐标 area coordinates体积坐标 volume coordinates曲线坐标 curvilinear coordinates静凝聚 static condensation合同变换 contragradient transformation形状函数 shape function试探函数 trial function检验函数 test function权函数 weight function样条函数 spline function代用函数 substitute function降阶积分 reduced integration零能模式 zero-energy modeP收敛 p-convergenceH收敛 h-convergence掺混插值 blended interpolation等参数映射 isoparametric mapping双线性插值 bilinear interpolation小块检验 patch test非协调模式 incompatible mode节点号 node number单元号 element number带宽 band width带状矩阵 banded matrix变带状矩阵 profile matrix带宽最小化 minimization of band width 波前法 frontal method子空间迭代法 subspace iteration method 行列式搜索法 determinant search method 逐步法 step-by-step method纽马克法 Newmark威尔逊法 Wilson拟牛顿法 quasi-Newton method牛顿-拉弗森法 Newton-Raphson method增量法 incremental method初应变 initial strain初应力 initial stress切线刚度矩阵 tangent stiffness matrix割线刚度矩阵 secant stiffness matrix模态叠加法 mode superposition method平衡迭代 equilibrium iteration子结构 Substructure子结构法 substructure technique超单元 super-element网格生成 mesh generation结构分析程序 structural analysis program前处理 pre-processing后处理 post-processing网格细化 mesh refinement应力光顺 stress smoothing组合结构 composite structure流体力学流体动力学 fluid dynamics连续介质力学 mechanics of continuous media 介质 medium流体质点 fluid particle无粘性流体 nonviscous fluid, inviscid fluid 连续介质假设 continuous medium hypothesis 流体运动学 fluid kinematics水静力学 hydrostatics液体静力学 hydrostatics支配方程 governing equation伯努利方程 Bernoulli equation伯努利定理 Bernonlli theorem毕奥-萨伐尔定律 Biot-Savart law欧拉方程 Euler equation亥姆霍兹定理 Helmholtz theorem开尔文定理 Kelvin theorem涡片 vortex sheet库塔-茹可夫斯基条件 Kutta-Zhoukowski condition 布拉休斯解 Blasius solution达朗贝尔佯廖 d'Alembert paradox雷诺数 Reynolds number施特鲁哈尔数 Strouhal number随体导数 material derivative不可压缩流体 incompressible fluid质量守恒 conservation of mass动量守恒 conservation of momentum能量守恒 conservation of energy动量方程 momentum equation能量方程 energy equation控制体积 control volume液体静压 hydrostatic pressure涡量拟能 enstrophy压差 differential pressure流[动] flow流线 stream line流面 stream surface流管 stream tube迹线 path, path line流场 flow field流态 flow regime流动参量 flow parameter流量 flow rate, flow discharge 涡旋 vortex涡量 vorticity涡丝 vortex filament涡线 vortex line涡面 vortex surface涡层 vortex layer涡环 vortex ring涡对 vortex pair涡管 vortex tube涡街 vortex street卡门涡街 Karman vortex street 马蹄涡 horseshoe vortex对流涡胞 convective cell卷筒涡胞 roll cell涡 eddy涡粘性 eddy viscosity环流 circulation环量 circulation速度环量 velocity circulation 偶极子 doublet, dipole驻点 stagnation point总压[力] total pressure总压头 total head静压头 static head总焓 total enthalpy能量输运 energy transport速度剖面 velocity profile库埃特流 Couette flow单相流 single phase flow单组份流 single-component flow均匀流 uniform. flow非均匀流 nonuniform. flow二维流 two-dimensional flow三维流 three-dimensional flow准定常流 quasi-steady flow非定常流 unsteady flow, non-steady flow 暂态流 transient flow周期流 periodic flow振荡流 oscillatory flow分层流 stratified flow无旋流 irrotational flow有旋流 rotational flow轴对称流 axisymmetric flow不可压缩性 incompressibility不可压缩流[动] incompressible flow浮体 floating body定倾中心 metacenter阻力 drag, resistance减阻 drag reduction表面力 surface force表面张力 surface tension毛细[管]作用 capillarity来流 incoming flow自由流 free stream自由流线 free stream line外流 external flow进口 entrance, inlet出口 exit, outlet扰动 disturbance, perturbation分布 distribution传播 propagation色散 dispersion弥散 dispersion附加质量 added mass ,associated mass 收缩 contraction镜象法 image method无量纲参数 dimensionless parameter 几何相似 geometric similarity运动相似 kinematic similarity动力相似[性] dynamic similarity平面流 plane flow势 potential势流 potential flow速度势 velocity potential复势 complex potential复速度 complex velocity流函数 stream function源 source汇 sink速度[水]头 velocity head拐角流 corner flow空泡流 cavity flow超空泡 supercavity超空泡流 supercavity flow空气动力学 aerodynamics低速空气动力学 low-speed aerodynamics 高速空气动力学 high-speed aerodynamics 气动热力学 aerothermodynamics亚声速流[动] subsonic flow跨声速流[动] transonic flow超声速流[动] supersonic flow锥形流 conical flow楔流 wedge flow叶栅流 cascade flow非平衡流[动] non-equilibrium flow细长体 slender body细长度 slenderness钝头体 bluff body钝体 blunt body翼型 airfoil翼弦 chord薄翼理论 thin-airfoil theory构型 configuration后缘 trailing edge迎角 angle of attack失速 stall脱体激波 detached shock wave波阻 wave drag诱导阻力 induced drag诱导速度 induced velocity临界雷诺数 critical Reynolds number前缘涡 leading edge vortex附着涡 bound vortex约束涡 confined vortex综合类：广义连续统力学 generalized continuum mechanics简单物质 simple material纯力学物质 purely mechanical material微分型物质 material of differential type积分型物质 material of integral type混合物组份 constituents of a mixture非协调理论 incompatibility theory微极理论 micropolar theory决定性原理 principle of determinism等存在原理 principle of equipresence局部作用原理 principle of objectivity客观性原理 principle of objectivity电磁连续统理论 theory of electromagnetic continuum 内时理论 endochronic theory非局部理论 nonlocal theory混合物理论 theory of mixtures里夫林-矣里克森张量 Rivlin-Ericksen tensor声张量 acoustic tensor半向同性张量 hemitropic tensor各向同性张量 isotropic tensor应变张量 strain tensor伸缩张量 stretch tensor连续旋错 continuous dislination连续位错 continuous dislocation动量矩平衡 angular momentum balance余本构关系 complementary constitutive relations共旋导数 co-rotational derivative, Jaumann derivative 非完整分量 anholonomic component爬升效应 climbing effect协调条件 compatibility condition错综度 complexity当时构形 current configuration能量平衡 energy balance变形梯度 deformation gradient有限弹性 finite elasticity熵增 entropy production标架无差异性 frame. indifference弹性势 elastic potential熵不等式 entropy inequality极分解 polar decomposition低弹性 hypoelasticity参考构形 reference configuration响应泛函 response functional动量平衡 momentum balance奇异面 singular surface贮能函数 stored-energy function内部约束 internal constraint物理分量 physical components本原元 primitive element普适变形 universal deformation速度梯度 velocity gradient测粘流动 viscometric flow当地导数 local derivative岩石力学 rock mechanics原始岩体应力 virgin rock stress构造应力 tectonic stress三轴压缩试验 three-axial compression test 三轴拉伸试验 three-axial tensile test三轴试验 triaxial test岩层静态应力 lithostatic stress吕荣 lugeon地压强 geostatic pressure水力劈裂 hydraulic fracture咬合[作用] interlocking内禀抗剪强度 intrinsic shear strength循环抗剪强度 cyclic shear strength残余抗剪强度 residual shear strength土力学 soil mechanics孔隙比 void ratio内磨擦角 angle of internal friction休止角 angle of repose孔隙率 porosity围压 ambient pressure渗透系数 coefficient of permeability [抗]剪切角 angle of shear resistance渗流力 seepage force表观粘聚力 apparent cohesion粘聚力 cohesion稠度 consistency固结 consolidation主固结 primary consolidation次固结 secondary consolidation固结仪 consolidometer浮升力 uplift扩容 dilatancy有效应力 effective stress絮凝[作用] flocculation主动土压力 active earth pressure 被动土压力 passive earth pressure 土动力学 soil dynamics应力解除 stress relief次时间效应 secondary time effect 贯入阻力 penetration resistance 沙土液化 liquefaction of sand泥流 mud flow多相流 multiphase flow马格努斯效应 Magnus effect韦伯数 Weber number环状流 annular flow泡状流 bubble flow层状流 stratified flow平衡流 equilibrium flow二组份流 two-component flow冻结流 frozen flow均质流 homogeneous flow二相流 two-phase flow。
自动控制原理与设计 第5版 英文
自动控制原理与设计第5版英文Automatic Control Principles and Design 5th EditionIntroduction:Automatic control principles and design play a crucial role in various engineering fields, enabling the efficient operation of systems and processes. In this article, we delve into the key concepts and applications of automatic control, as outlined in the 5th edition of the book "Automatic Control Principles and Design."Chapter 1: Introduction to Automatic ControlAutomatic control is the use of control systems to regulate processes or machines without human intervention. It encompasses a wide range of applications, from simple domestic appliances to complex industrial systems. The chapter provides an overview of the basic principles and benefits of automatic control.Chapter 2: Modeling of Dynamic SystemsThe accurate modeling of dynamic systems is essential for effective control design. This chapter explores different techniques for modeling linear and nonlinear systems, including differential equations, transfer functions, and state-space representations. Real-world examples are used to illustrate the modeling process.Chapter 3: Time-Domain Analysis of Control SystemsTime-domain analysis allows us to examine the transient and steady-state responses of control systems. This chapter covers the analysis of first-and second-order systems, as well as higher-order systems. It also introduces the concept of system stability and the influence of system parameters on stability.Chapter 4: Frequency-Domain Analysis of Control SystemsFrequency-domain analysis provides insights into the behavior of control systems in the frequency spectrum. The chapter discusses transfer function analysis, Bode plots, Nyquist plots, and the relationship between the time- and frequency-domain representations of systems. Practical examples are included to enhance understanding.Chapter 5: Control System Design by Root Locus TechniqueThe root locus technique is a graphical method that aids in control system design. This chapter explores the construction of the root locus plot and its use in determining system stability, transient response, and controller design. Design guidelines and examples are presented to illustrate the application of this powerful tool.Chapter 6: Control System Design by Frequency Response TechniqueFrequency response techniques provide an alternative approach to control system design. This chapter discusses the design of compensators and filters using frequency response methods such as gain and phase margins and loop shaping. The advantages and limitations of this design approach are highlighted.Chapter 7: State-Space Analysis and DesignState-space analysis offers a modern and comprehensive framework for control system design. This chapter presents the concepts of state variables, state equations, observability, and controllability. The methods for state feedback control and observer design are also covered, along with their applications.Chapter 8: Digital Control SystemsThe design and implementation of control systems in the digital domain are covered in this chapter. It discusses the advantages of digital control, discretization of continuous-time systems, sampling and quantization, and various digital control algorithms. Practical considerations and implementation issues are addressed.Chapter 9: Introduction to Nonlinear Systems and ControlNonlinear systems present unique challenges in control design. This chapter introduces the basics of nonlinear systems and control techniques. It covers phase-plane analysis, describing functions, feedback linearization, and sliding mode control. Real-world examples demonstrate the application of these methods.Conclusion:The 5th edition of "Automatic Control Principles and Design" provides a comprehensive and up-to-date resource for understanding the principles and design techniques in automatic control. Through the exploration of different chapters, readers gain the necessary knowledge and skills to design effective control systems for a wide range of applications. This book serves as aninvaluable guide for students, researchers, and professionals in the field of automatic control.。
离散时间信号处理(英文版)chap7-第5讲
差分方程用于描述离散时间信号的处理过程 ，例如数字滤波器、离散控制系统等。
state equation
定义
01
状态方程是描述离散时间系统动态行为的另一种数学
模型，它给出了系统内部状态变量的变化规律。
形式
02
状态方程的一般形式为 x[n] = f(x[n-1], x[n-2], ...,
u[n])
分类
根据系统对初始条件的敏感度，可以分为渐近稳定、指数稳定和不稳定。
stability criterion
劳斯-赫尔维茨准则
对于线性时不变离散系统，如果系统 的特征方程的所有根都位于复平面的 左半部分，则该系统是稳定的。
尼可拉斯-贝塔朗准则
对于线性时不变离散系统，如果系统 的所有状态方程的系数矩阵都是稳定 的，则该系统是稳定的。
Sequences are the most basic representation, where each element in the sequence corresponds to a point in time.
Arrays and matrices are more complex representations that can encode additional information about the relationships between different points in time.
The discrete time signal processing has its roots in the development of digital computers in the 1940s. With the advent of digital technology, it became possible to process signals in a discrete manner, rather than continuously. This led to the development of various algorithms and techniques for discrete time signal processing.
边坡毕业设计
目录摘要 (I)Abstract .................................................................................................................................................. I I 第一章绪论 . (1)1.1课题研究的目的和意义 (1)1.2 国内外的研究及发展现状 (1)1.3边坡稳定分析方法 (2)1.3.1 (2)1.3.2 (2)1.3.4 (2)1.4 治理措施 (3)1.4.1 放坡： (3)1.4.2 支挡设计（ (3)1.4.2.2 抗滑桩 (3)1.4.3.3 排水设计 (3)1.5设计内容： (3)1.6 拟解决的主要问题 (4)1.7技术方案： (4)1.8 技术路线 (5)第二章工程概况 (6)2.1 场地工程地质与水文地质条件 (6)2.2 地层岩性 (6)2.3 岩体分类 (7)2.5水文地质条件 (7)2.3 岩土物理力学特征分析与评价 (8)2.4 边坡工程治理措施分析 (8)2.5 结论与建议 (8)第三章稳定性计算 (9)3.1 滑坡推力计算 (9)3.1.1传力系数法计算公式 (9)3.2.1 1—1断面处的计算 (13)3.2.1 2—2 断面处的计算（圆弧法） (20)3.2.3 2—2 断面处的计算（平面滑动法） (27)第四章支挡结构设计 (34)4.1抗滑桩和锚杆设计 (34)4.2 剩余下滑力计算（断面2最危险滑面为第三潜在滑动面） (34)4.3抗滑桩设计 (34)4.4 各截面桩侧应力、剪力及弯矩计算 (38)4.5桩的结构设计 (42)4.6 锚杆设计 (45)4.7 边坡排水设计 (47)4.8坡地排水沟设计 (48)第五章施工技术 (49)5.1 施工准备及平面布置 (49)5.3施工平面布置说明 (50)5.4 锚杆施工 (50)5.5抗滑桩施工 (51)第六章结论和建议 (54)6.1 结论 (54)致谢 (55)参考文献 (57)摘要伴随着现代工程技术的迅速发展，在道路工程和建筑工程中，也出现了大量的边坡问题，有些地方甚至出现了高大边坡，防治起来问题重重。
离散时间迭代的LCC-S型无线电能传输系统建模及稳定性分析
LCC-S 型 WPT 系统的 EDF 模型、GSSA 模型和离散
法
[4 - 5]
averaging,GSSA) 法
[6 - 7]
以及扩展描述函数( extend-
ed describing function,EDF) 法
[8]
。 这 3 种建模方法
时间迭代模型进行临界稳定预测值的对比。 最后,
仿真和 实 验 验 证 所 建 立 模 型 和 稳 定 性 分 析 的 正
2
lished models and the correctness of stability analysis.
Keywords:wireless power transfer system; LCC-S compensated; discrete time iterative; delay time; stability analysis; oscillation period
主要由全桥逆变器、谐振补偿网络、全桥整流器以及
间迭代模型和样本数据模型。 离散时间迭代模型以
MOSFET 的导通电阻之和;R1 表示 L1 和 C1 等效串联
数提高了一倍。 文献[9] 和文献[10] 提出了离散时
其高精度而闻名。
在 WPT 系统中,整个控制过程主要包括 3 个步
骤:采样、计算和脉宽调制 [11 - 12] ( pulse width modu-
sated WPT system. Finally, simulation and experimental verification are conducted to validate the estab
上,进一步推导电压型比例积分控制 LCC-S 型 WPT
弹性力学专业英语词汇_elasticity
弹性力学elasticity弹性理论theory of elasticity均匀应力状态homogeneous state of stress应力不变量stress invariant应变不变量strain invariant应变椭球strain ellipsoid均匀应变状态homogeneous state of strain应变协调方程equation of strain compatibility拉梅常量Lame constants各向同性弹性isotropic elasticity旋转圆盘rotating circular disk楔wedge开尔文问题Kelvin problem布西内斯克问题Boussinesq problem艾里应力函数Airy stress function克罗索夫-穆斯赫利什维利法Kolosoff-Muskhelishvili method 基尔霍夫假设Kirchhoff hypothesis板Plate矩形板Rectangular plate圆板Circular plate环板Annular plate波纹板Corrugated plate加劲板Stiffened plate,reinforced Plate中厚板Plate of moderate thickness弯[曲]应力函数Stress function of bending 壳Shell扁壳Shallow shell旋转壳Revolutionary shell球壳Spherical shell[圆]柱壳Cylindrical shell锥壳Conical shell环壳Toroidal shell封闭壳Closed shell波纹壳Corrugated shell扭[转]应力函数Stress function of torsion 翘曲函数Warping function半逆解法semi-inverse method瑞利——里茨法Rayleigh-Ritz method松弛法Relaxation method莱维法Levy method松弛Relaxation量纲分析Dimensional analysis自相似[性] self-similarity影响面Influence surface接触应力Contact stress赫兹理论Hertz theory协调接触Conforming contact滑动接触Sliding contact滚动接触Rolling contact压入Indentation各向异性弹性Anisotropic elasticity颗粒材料Granular material散体力学Mechanics of granular media 热弹性Thermoelasticity超弹性Hyperelasticity粘弹性Viscoelasticity对应原理Correspondence principle褶皱Wrinkle塑性全量理论Total theory of plasticity 滑动Sliding微滑Microslip粗糙度Roughness非线性弹性Nonlinear elasticity大挠度Large deflection突弹跳变snap-through有限变形Finite deformation格林应变Green strain阿尔曼西应变Almansi strain弹性动力学Dynamic elasticity运动方程Equation of motion准静态的Quasi-static气动弹性Aeroelasticity水弹性Hydroelasticity颤振Flutter弹性波Elastic wave简单波Simple wave柱面波Cylindrical wave水平剪切波Horizontal shear wave 竖直剪切波Vertical shear wave 体波body wave无旋波Irrotational wave畸变波Distortion wave膨胀波Dilatation wave瑞利波Rayleigh wave等容波Equivoluminal wave勒夫波Love wave界面波Interfacial wave边缘效应edge effect塑性力学Plasticity可成形性Formability金属成形Metal forming耐撞性Crashworthiness结构抗撞毁性Structural crashworthiness拉拔Drawing破坏机构Collapse mechanism回弹Springback挤压Extrusion冲压Stamping穿透Perforation层裂Spalling塑性理论Theory of plasticity安定[性]理论Shake-down theory运动安定定理kinematic shake-down theorem 静力安定定理Static shake-down theorem率相关理论rate dependent theorem载荷因子load factor加载准则Loading criterion加载函数Loading function加载面Loading surface塑性加载Plastic loading塑性加载波Plastic loading wave 简单加载Simple loading比例加载Proportional loading 卸载Unloading卸载波Unloading wave冲击载荷Impulsive load阶跃载荷step load脉冲载荷pulse load极限载荷limit load中性变载nentral loading拉抻失稳instability in tension加速度波acceleration wave本构方程constitutive equation 完全解complete solution名义应力nominal stress过应力over-stress真应力true stress等效应力equivalent stress流动应力flow stress应力间断stress discontinuity应力空间stress space主应力空间principal stress space静水应力状态hydrostatic state of stress对数应变logarithmic strain工程应变engineering strain等效应变equivalent strain应变局部化strain localization应变率strain rate应变率敏感性strain rate sensitivity应变空间strain space有限应变finite strain塑性应变增量plastic strain increment累积塑性应变accumulated plastic strain永久变形permanent deformation内变量internal variable应变软化strain-softening理想刚塑性材料rigid-perfectly plastic Material 刚塑性材料rigid-plastic material理想塑性材料perfectl plastic material材料稳定性stability of material应变偏张量deviatoric tensor of strain应力偏张量deviatori tensor of stress应变球张量spherical tensor of strain应力球张量spherical tensor of stress路径相关性path-dependency线性强化linear strain-hardening应变强化strain-hardening随动强化kinematic hardening各向同性强化isotropic hardening强化模量strain-hardening modulus幂强化power hardening塑性极限弯矩plastic limit bending Moment塑性极限扭矩plastic limit torque弹塑性弯曲elastic-plastic bending弹塑性交界面elastic-plastic interface弹塑性扭转elastic-plastic torsion粘塑性Viscoplasticity非弹性Inelasticity理想弹塑性材料elastic-perfectly plastic Material 极限分析limit analysis极限设计limit design极限面limit surface上限定理upper bound theorem上屈服点upper yield point下限定理lower bound theorem下屈服点lower yield point界限定理bound theorem初始屈服面initial yield surface后继屈服面subsequent yield surface屈服面[的]外凸性convexity of yield surface 截面形状因子shape factor of cross-section 沙堆比拟sand heap analogy屈服Yield屈服条件yield condition屈服准则yield criterion屈服函数yield function屈服面yield surface塑性势plastic potential能量吸收装置energy absorbing device能量耗散率energy absorbing device塑性动力学dynamic plasticity塑性动力屈曲dynamic plastic buckling塑性动力响应dynamic plastic response塑性波plastic wave运动容许场kinematically admissible Field 静力容许场statically admissible Field流动法则flow rule速度间断velocity discontinuity滑移线slip-lines滑移线场slip-lines field移行塑性铰travelling plastic hinge塑性增量理论incremental theory of Plasticity米泽斯屈服准则Mises yield criterion普朗特——罗伊斯关系prandtl- Reuss relation特雷斯卡屈服准则Tresca yield criterion洛德应力参数Lode stress parameter莱维——米泽斯关系Levy-Mises relation亨基应力方程Hencky stress equation赫艾——韦斯特加德应力空间Haigh-Westergaard stress space 洛德应变参数Lode strain parameter德鲁克公设Drucker postulate盖林格速度方程Geiringer velocity Equation结构力学structural mechanics结构分析structural analysis结构动力学structural dynamics拱Arch三铰拱three-hinged arch抛物线拱parabolic arch圆拱circular arch穹顶Dome空间结构space structure空间桁架space truss雪载[荷] snow load风载[荷] wind load土压力earth pressure地震载荷earthquake loading弹簧支座spring support支座位移support displacement支座沉降support settlement超静定次数degree of indeterminacy机动分析kinematic analysis结点法method of joints截面法method of sections结点力joint forces共轭位移conjugate displacement影响线influence line三弯矩方程three-moment equation单位虚力unit virtual force刚度系数stiffness coefficient柔度系数flexibility coefficient力矩分配moment distribution力矩分配法moment distribution method力矩再分配moment redistribution分配系数distribution factor矩阵位移法matri displacement method单元刚度矩阵element stiffness matrix单元应变矩阵element strain matrix总体坐标global coordinates贝蒂定理Betti theorem高斯——若尔当消去法Gauss-Jordan elimination Method 屈曲模态buckling mode复合材料力学mechanics of composites复合材料composite material纤维复合材料fibrous composite单向复合材料unidirectional composite泡沫复合材料foamed composite颗粒复合材料particulate composite层板Laminate夹层板sandwich panel正交层板cross-ply laminate斜交层板angle-ply laminate层片Ply多胞固体cellular solid膨胀Expansion压实Debulk劣化Degradation脱层Delamination脱粘Debond纤维应力fiber stress层应力ply stress层应变ply strain层间应力interlaminar stress比强度specific strength强度折减系数strength reduction factor 强度应力比strength -stress ratio横向剪切模量transverse shear modulus 横观各向同性transverse isotropy正交各向异Orthotropy剪滞分析shear lag analysis短纤维chopped fiber长纤维continuous fiber纤维方向fiber direction纤维断裂fiber break纤维拔脱fiber pull-out纤维增强fiber reinforcement致密化Densification最小重量设计optimum weight design网格分析法netting analysis混合律rule of mixture失效准则failure criterion蔡——吴失效准则Tsai-W u failure criterion达格代尔模型Dugdale model断裂力学fracture mechanics概率断裂力学probabilistic fracture Mechanics格里菲思理论Griffith theory线弹性断裂力学linear elastic fracture mechanics, LEFM 弹塑性断裂力学elastic-plastic fracture mecha-nics, EPFM 断裂Fracture脆性断裂brittle fracture解理断裂cleavage fracture蠕变断裂creep fracture延性断裂ductile fracture晶间断裂inter-granular fracture准解理断裂quasi-cleavage fracture穿晶断裂trans-granular fracture裂纹Crack裂缝Flaw缺陷Defect割缝Slit微裂纹Microcrack折裂Kink椭圆裂纹elliptical crack深埋裂纹embedded crack[钱]币状裂纹penny-shape crack预制裂纹Precrack短裂纹short crack表面裂纹surface crack裂纹钝化crack blunting裂纹分叉crack branching裂纹闭合crack closure裂纹前缘crack front裂纹嘴crack mouth裂纹张开角crack opening angle,COA裂纹张开位移crack opening displacement, COD裂纹阻力crack resistance裂纹面crack surface裂纹尖端crack tip裂尖张角crack tip opening angle, CTOA裂尖张开位移crack tip opening displacement, CTOD 裂尖奇异场crack tip singularity Field裂纹扩展速率crack growth rate稳定裂纹扩展stable crack growth定常裂纹扩展steady crack growth亚临界裂纹扩展subcritical crack growth 裂纹[扩展]减速crack retardation止裂crack arrest止裂韧度arrest toughness断裂类型fracture mode滑开型sliding mode张开型opening mode撕开型tearing mode复合型mixed mode撕裂Tearing撕裂模量tearing modulus断裂准则fracture criterionJ积分J-integralJ阻力曲线J-resistance curve断裂韧度fracture toughness应力强度因子stress intensity factor HRR场Hutchinson-Rice-Rosengren Field 守恒积分conservation integral有效应力张量effective stress tensor应变能密度strain energy density能量释放率energy release rate内聚区cohesive zone塑性区plastic zone张拉区stretched zone热影响区heat affected zone, HAZ延脆转变温度brittle-ductile transition temperature剪切带shear band剪切唇shear lip无损检测non-destructive inspection双边缺口试件double edge notched specimen, DEN specimen单边缺口试件single edge notched specimen, SEN specimen三点弯曲试件three point bending specimen, TPB specimen中心裂纹拉伸试件center cracked tension specimen, CCT specimen 中心裂纹板试件center cracked panel specimen, CCP specimen紧凑拉伸试件compact tension specimen, CT specimen大范围屈服large scale yielding小范围攻屈服small scale yielding韦布尔分布Weibull distribution帕里斯公式paris formula空穴化Cavitation应力腐蚀stress corrosion概率风险判定probabilistic risk assessment, PRA 损伤力学damage mechanics损伤Damage连续介质损伤力学continuum damage mechanics 细观损伤力学microscopic damage mechanics 累积损伤accumulated damage脆性损伤brittle damage延性损伤ductile damage宏观损伤macroscopic damage细观损伤microscopic damage微观损伤microscopic damage损伤准则damage criterion损伤演化方程damage evolution equation损伤软化damage softening损伤强化damage strengthening损伤张量damage tensor损伤阈值damage threshold损伤变量damage variable损伤矢量damage vector损伤区damage zone疲劳Fatigue低周疲劳low cycle fatigue应力疲劳stress fatigue随机疲劳random fatigue 蠕变疲劳creep fatigue腐蚀疲劳corrosion fatigue 疲劳损伤fatigue damage 疲劳失效fatigue failure疲劳断裂fatigue fracture 疲劳裂纹fatigue crack疲劳寿命fatigue life疲劳破坏fatigue rupture疲劳强度fatigue strength 疲劳辉纹fatigue striations 疲劳阈值fatigue threshold 交变载荷alternating load 交变应力alternating stress 应力幅值stress amplitude 应变疲劳strain fatigue应力循环stress cycle应力比stress ratio安全寿命safe life过载效应overloading effect 循环硬化cyclic hardening循环软化cyclic softening环境效应environmental effect裂纹片crack gage裂纹扩展crack growth, crack Propagation 裂纹萌生crack initiation循环比cycle ratio实验应力分析experimental stress Analysis 工作[应变]片active[strain] gage基底材料backing material应力计stress gage零[点]飘移zero shift, zero drift应变测量strain measurement应变计strain gage应变指示器strain indicator应变花strain rosette应变灵敏度strain sensitivity机械式应变仪mechanical strain gage直角应变花rectangular rosette引伸仪Extensometer应变遥测telemetering of strain横向灵敏系数transverse gage factor横向灵敏度transverse sensitivity焊接式应变计weldable strain gage平衡电桥balanced bridge粘贴式应变计bonded strain gage粘贴箔式应变计bonded foiled gage粘贴丝式应变计bonded wire gage桥路平衡bridge balancing电容应变计capacitance strain gage补偿片compensation technique补偿技术compensation technique基准电桥reference bridge电阻应变计resistance strain gage温度自补偿应变计self-temperature compensating gage 半导体应变计semiconductor strain Gage集流器slip ring应变放大镜strain amplifier疲劳寿命计fatigue life gage电感应变计inductance [strain] gage光[测]力学Photomechanics光弹性Photoelasticity光塑性Photoplasticity杨氏条纹Young fringe双折射效应birefrigent effect等位移线contour of equal Displacement暗条纹dark fringe条纹倍增fringe multiplication干涉条纹interference fringe等差线Isochromatic等倾线Isoclinic等和线isopachic应力光学定律stress- optic law主应力迹线Isostatic亮条纹light fringe光程差optical path difference热光弹性photo-thermo -elasticity光弹性贴片法photoelastic coating Method 光弹性夹片法photoelastic sandwich Method 动态光弹性dynamic photo-elasticity空间滤波spatial filtering空间频率spatial frequency起偏镜Polarizer反射式光弹性仪reflection polariscope残余双折射效应residual birefringent Effect 应变条纹值strain fringe value应变光学灵敏度strain-optic sensitivity应力冻结效应stress freezing effect应力条纹值stress fringe value应力光图stress-optic pattern暂时双折射效应temporary birefringent Effect脉冲全息法pulsed holography透射式光弹性仪transmission polariscope实时全息干涉法real-time holographic interferometry网格法grid method全息光弹性法holo-photoelasticity全息图Hologram全息照相Holograph全息干涉法holographic interferometry全息云纹法holographic moire technique全息术Holography全场分析法whole-field analysis散斑干涉法speckle interferometry散斑Speckle错位散斑干涉法speckle-shearing interferometry, shearography 散斑图Specklegram白光散斑法white-light speckle method云纹干涉法moire interferometry[叠栅]云纹moire fringe[叠栅]云纹法moire method云纹图moire pattern离面云纹法off-plane moire method参考栅reference grating试件栅specimen grating分析栅analyzer grating面内云纹法in-plane moire method脆性涂层法brittle-coating method条带法strip coating method坐标变换transformation of Coordinates计算结构力学computational structural mechanics 加权残量法weighted residual method有限差分法finite difference method有限[单]元法finite element method配点法point collocation里茨法Ritz method广义变分原理generalized variational Principle最小二乘法least square method胡[海昌]一鹫津原理Hu-Washizu principle赫林格-赖斯纳原理Hellinger-Reissner Principle修正变分原理modified variational Principle约束变分原理constrained variational Principle混合法mixed method杂交法hybrid method边界解法boundary solution method有限条法finite strip method半解析法semi-analytical method协调元conforming element非协调元non-conforming element混合元mixed element杂交元hybrid element边界元boundary element强迫边界条件forced boundary condition 自然边界条件natural boundary condition 离散化Discretization离散系统discrete system连续问题continuous problem广义位移generalized displacement广义载荷generalized load广义应变generalized strain广义应力generalized stress界面变量interface variable节点node, nodal point[单]元Element角节点corner node边节点mid-side node内节点internal node无节点变量nodeless variable杆元bar element桁架杆元truss element梁元beam element二维元two-dimensional element 一维元one-dimensional element 三维元three-dimensional element 轴对称元axisymmetric element 板元plate element壳元shell element厚板元thick plate element三角形元triangular element四边形元quadrilateral element四面体元tetrahedral element曲线元curved element二次元quadratic element线性元linear element三次元cubic element四次元quartic element等参[数]元isoparametric element超参数元super-parametric element亚参数元sub-parametric element节点数可变元variable-number-node element 拉格朗日元Lagrange element拉格朗日族Lagrange family巧凑边点元serendipity element巧凑边点族serendipity family无限元infinite element单元分析element analysis单元特性element characteristics刚度矩阵stiffness matrix几何矩阵geometric matrix等效节点力equivalent nodal force节点位移nodal displacement节点载荷nodal load位移矢量displacement vector载荷矢量load vector质量矩阵mass matrix集总质量矩阵lumped mass matrix相容质量矩阵consistent mass matrix阻尼矩阵damping matrix瑞利阻尼Rayleigh damping刚度矩阵的组集assembly of stiffness Matrices 载荷矢量的组集consistent mass matrix质量矩阵的组集assembly of mass matrices单元的组集assembly of elements局部坐标系local coordinate system局部坐标local coordinate面积坐标area coordinates体积坐标volume coordinates曲线坐标curvilinear coordinates静凝聚static condensation合同变换contragradient transformation形状函数shape function试探函数trial function检验函数test function权函数weight function样条函数spline function代用函数substitute function降阶积分reduced integration零能模式zero-energy modeP收敛p-convergenceH收敛h-convergence掺混插值blended interpolation等参数映射isoparametric mapping双线性插值bilinear interpolation小块检验patch test非协调模式incompatible mode节点号node number单元号element number带宽band width带状矩阵banded matrix变带状矩阵profile matrix带宽最小化minimization of band width 波前法frontal method子空间迭代法subspace iteration method 行列式搜索法determinant search method 逐步法step-by-step method纽马克法Newmark威尔逊法Wilson拟牛顿法quasi-Newton method牛顿-拉弗森法Newton-Raphson method 增量法incremental method初应变initial strain初应力initial stress切线刚度矩阵tangent stiffness matrix割线刚度矩阵secant stiffness matrix模态叠加法mode superposition method平衡迭代equilibrium iteration子结构Substructure子结构法substructure technique超单元super-element网格生成mesh generation结构分析程序structural analysis program 前处理pre-processing后处理post-processing网格细化mesh refinement应力光顺stress smoothing组合结构composite structure流体动力学fluid dynamics连续介质力学mechanics of continuous media 介质medium流体质点fluid particle无粘性流体nonviscous fluid, inviscid fluid连续介质假设continuous medium hypothesis 流体运动学fluid kinematics水静力学hydrostatics液体静力学hydrostatics支配方程governing equation伯努利方程Bernoulli equation伯努利定理Bernonlli theorem毕奥-萨伐尔定律Biot-Savart law欧拉方程Euler equation亥姆霍兹定理Helmholtz theorem开尔文定理Kelvin theorem涡片vortex sheet库塔-茹可夫斯基条件Kutta-Zhoukowski condition 布拉休斯解Blasius solution达朗贝尔佯廖d&;am #39;Alembert paradox雷诺数Reynolds number施特鲁哈尔数Strouhal number随体导数material derivative不可压缩流体incompre ible fluid质量守恒co ervation of ma动量守恒co ervation of momentum能量守恒co ervation of energy动量方程momentum equation能量方程energy equation控制体积control volume液体静压hydrostatic pre ure 涡量拟能e trophy压差differential pre ure流[动] flow流线stream line流面stream surface流管stream tube迹线path, path line流场flow field流态flow regime流动参量flow parameter流量flow rate, flow discharge 涡旋vortex涡量vorticity涡丝vortex filament涡线vortex line涡面vortex surface涡层vortex layer涡环vortex ring涡对vortex pair涡管vortex tube涡街vortex street卡门涡街Karman vortex street 马蹄涡horseshoe vortex对流涡胞convective cell卷筒涡胞roll cell涡eddy涡粘性eddy viscosity环流circulation环量circulation速度环量velocity circulation偶极子doublet, dipole驻点stagnation point总压[力] total pre ure总压头total head静压头static head总焓total enthalpy能量输运energy tra ort速度剖面velocity profile库埃特流Couette flow单相流single phase flow单组份流single-component flow 均匀流uniform flow非均匀流nonuniform flow二维流two-dime ional flow三维流three-dime ional flow准定常流quasi-steady flow非定常流u teady flow, non-steady flow 暂态流tra ient flow周期流periodic flow振荡流oscillatory flow分层流stratified flow无旋流irrotational flow有旋流rotational flow轴对称流axisymmetric flow不可压缩性incompre ibility不可压缩流[动] incompre ible flow浮体floating body定倾中心metacenter阻力drag, resistance减阻drag reduction表面力surface force表面张力surface te ion毛细[管]作用capillarity来流incoming flow自由流free stream自由流线free stream line外流external flow进口entrance, inlet出口exit, outlet扰动disturbance, perturbation分布distribution传播propagation色散di ersion弥散di ersion附加质量added ma ,a ociated ma 收缩contraction镜象法image method无量纲参数dime ionle parameter 几何相似geometric similarity运动相似kinematic similarity动力相似[性] dynamic similarity平面流plane flow势potential势流potential flow速度势velocity potential复势complex potential复速度complex velocity流函数stream function源source汇sink速度[水]头velocity head拐角流corner flow空泡流cavity flow超空泡supercavity超空泡流supercavity flow空气动力学aerodynamics低速空气动力学low- eed aerodynamics 高速空气动力学high- eed aerodynamics 气动热力学aerothermodynamics亚声速流[动] su onic flow跨声速流[动] tra onic flow超声速流[动] supersonic flow锥形流conical flow楔流wedge flow叶栅流cascade flow非平衡流[动] non-equilibrium flow细长体slender body细长度slenderne钝头体bluff body钝体blunt body翼型airfoil翼弦chord薄翼理论thin-airfoil theory构型configuration后缘trailing edge迎角angle of attack失速stall脱体激波detached shock wave波阻wave drag诱导阻力induced drag诱导速度induced velocity临界雷诺数critical Reynolds number 前缘涡leading edge vortex附着涡bound vortex约束涡confined vortex气动中心aerodynamic center气动力aerodynamic force气动噪声aerodynamic noise气动加热aerodynamic heating离解di ociation地面效应ground effect气体动力学gas dynamics稀疏波rarefaction wave热状态方程thermal equation of state喷管Nozzle普朗特-迈耶流Prandtl-Meyer flow瑞利流Rayleigh flow可压缩流[动] compre ible flow可压缩流体compre ible fluid绝热流adiabatic flow非绝热流diabatic flow未扰动流undisturbed flow等熵流isentropic flow匀熵流homoentropic flow兰金-于戈尼奥条件Rankine-Hugoniot condition 状态方程equation of state量热状态方程caloric equation of state完全气体perfect gas拉瓦尔喷管Laval nozzle马赫角Mach angle马赫锥Mach cone马赫线Mach line马赫数Mach number马赫波Mach wave当地马赫数local Mach number 冲击波shock wave激波shock wave正激波normal shock wave斜激波oblique shock wave头波bow wave附体激波attached shock wave 激波阵面shock front激波层shock layer压缩波compre ion wave反射reflection折射refraction散射scattering衍射diffraction绕射diffraction出口压力exit pre ure超压[强] over pre ure反压back pre ure爆炸explosion爆轰detonation缓燃deflagration水动力学hydrodynamics液体动力学hydrodynamics 泰勒不稳定性T aylor i tability 盖斯特纳波Gerstner wave斯托克斯波Stokes wave瑞利数Rayleigh number自由面free surface波速wave eed, wave velocity 波高wave height波列wave train波群wave group波能wave energy表面波surface wave表面张力波capillary wave规则波regular wave不规则波irregular wave浅水波shallow water wave 深水波deep water wave重力波gravity wave椭圆余弦波cnoidal wave潮波tidal wave涌波surge wave破碎波breaking wave船波ship wave非线性波nonlinear wave孤立子soliton水动[力]噪声hydrodynamic noise水击water hammer空化cavitation空化数cavitation number空蚀cavitation damage超空化流supercavitating flow水翼hydrofoil水力学hydraulics洪水波flood wave涟漪ri le消能energy di ipation海洋水动力学marine hydrodynamics 谢齐公式Chezy formula欧拉数Euler number弗劳德数Froude number水力半径hydraulic radius水力坡度hvdraulic slope高度水头elevating head水头损失head lo水位water level水跃hydraulic jump含水层aquifer排水drainage排放量discharge壅水曲线back water curve压[强水]头pre ure head过水断面flow cro -section明槽流open cha el flow孔流orifice flow无压流free surface flow有压流pre ure flow缓流subcritical flow急流supercritical flow渐变流gradually varied flow急变流rapidly varied flow临界流critical flow异重流de ity current, gravity flow 堰流weir flow掺气流aerated flow含沙流sediment-laden stream降水曲线dropdown curve沉积物sediment, deposit沉[降堆]积sedimentation, deposition沉降速度settling velocity流动稳定性flow stability不稳定性i tability奥尔-索末菲方程Orr-Sommerfeld equation 涡量方程vorticity equation泊肃叶流Poiseuille flow奥辛流Oseen flow剪切流shear flow粘性流[动] viscous flow层流laminar flow分离流separated flow二次流secondary flow近场流near field flow远场流far field flow滞止流stagnation flow尾流wake [flow]回流back flow反流reverse flow射流jet自由射流free jet管流pipe flow, tube flow内流internal flow拟序结构coherent structure 猝发过程bursting proce表观粘度a arent viscosity运动粘性kinematic viscosity 动力粘性dynamic viscosity 泊poise厘泊centipoise厘沱centistoke剪切层shear layer次层sublayer流动分离flow separation层流分离laminar separation 湍流分离turbulent separation 分离点separation point附着点attachment point再附reattachment再层流化relaminarization起动涡starting vortex驻涡standing vortex涡旋破碎vortex breakdown涡旋脱落vortex shedding压[力]降pre ure drop压差阻力pre ure drag压力能pre ure energy型阻profile drag滑移速度slip velocity无滑移条件non-slip condition壁剪应力skin friction, frictional drag壁剪切速度friction velocity磨擦损失friction lo磨擦因子friction factor耗散di ipation滞后lag相似性解similar solution局域相似local similarity气体润滑gas lubrication液体动力润滑hydrodynamic lubrication 浆体slurry泰勒数Taylor number纳维-斯托克斯方程Navier-Stokes equation 牛顿流体Newtonian fluid边界层理论boundary later theory边界层方程boundary layer equation 边界层boundary layer附面层boundary layer层流边界层laminar boundary layer 湍流边界层turbulent boundary layer 温度边界层thermal boundary layer 边界层转捩boundary layer tra ition 边界层分离boundary layer separation 边界层厚度boundary layer thickne位移厚度di lacement thickne动量厚度momentum thickne能量厚度energy thickne焓厚度enthalpy thickne注入injection吸出suction泰勒涡Taylor vortex速度亏损律velocity defect law形状因子shape factor测速法anemometry粘度测定法visco[si] metry流动显示flow visualization油烟显示oil smoke visualization孔板流量计orifice meter频率响应frequency re o e油膜显示oil film visualization阴影法shadow method纹影法schlieren method烟丝法smoke wire method丝线法tuft method氢泡法nydrogen bu le method相似理论similarity theory相似律similarity law部分相似partial similarity定理pi theorem, Buckingham theorem 静[态]校准static calibration动态校准dynamic calibration风洞wind tu el激波管shock tube激波管风洞shock tube wind tu el水洞water tu el拖曳水池towing tank旋臂水池rotating arm basin扩散段diffuser测压孔pre ure tap皮托管pitot tube普雷斯顿管preston tube斯坦顿管Stanton tube文丘里管Venturi tubeU形管U-tube压强计manometer微压计micromanometer多管压强计multiple manometer静压管static [pre ure]tube流速计anemometer风速管Pitot- static tube激光多普勒测速计laser Do ler anemometer, laser Do ler velocimeter 热线流速计hot-wire anemometer热膜流速计hot- film anemometer流量计flow meter粘度计visco[si] meter涡量计vorticity meter传感器tra ducer, se or压强传感器pre ure tra ducer热敏电阻thermistor示踪物tracer时间线time line脉线streak line尺度效应scale effect壁效应wall effect堵塞blockage堵寒效应blockage effect动态响应dynamic re o e响应频率re o e frequency底压base pre ure菲克定律Fick law巴塞特力Ba et force埃克特数Eckert number格拉斯霍夫数Grashof number努塞特数Nu elt number普朗特数prandtl number雷诺比拟Reynolds analogy施密特数schmidt number斯坦顿数Stanton number对流convection自由对流natural convection, free convec-tion 强迫对流forced convection热对流heat convection质量传递ma tra fer传质系数ma tra fer coefficient 热量传递heat tra fer传热系数heat tra fer coefficient 对流传热convective heat tra fer 辐射传热radiative heat tra fer 动量交换momentum tra fer能量传递energy tra fer传导conduction热传导conductive heat tra fer 热交换heat exchange临界热通量critical heat flux浓度concentration扩散diffusion扩散性diffusivity扩散率diffusivity扩散速度diffusion velocity分子扩散molecular diffusion 沸腾boiling蒸发evaporation气化gasification凝结conde ation。
Input to state stability Basic concepts and results
Input to State Stability
3
the convergent-input convergent-state (CICS) and the bounded-input bounded-state (BIBS) properties. We should remark that, for simplicity of exposition, we concentrate here solely on stability notions relative to globally attractive steady states. However, the general theory allows consideration of more arbitrary attractors (so that norms get replaced by, for example, distances to certain compact sets), and one may also consider local versions, as well as robust and/or adaptive concepts associated to the ones that we will define. 2.2 Merging two different views of stability Broadly speaking, there are two main competing approaches to system stability: the state-space approach usually associated with the name of Lyapunov, and the operator approach, of which George Zames was one of the main proponents and developers and which was the subject of major contributions by Sandberg, Willems, Safonov, and others. Our objective is in a sense (Fig. 3) that of merging these “Lyapunov” and
水轮机电液调节系统及装置技术规程(英文版)
水轮机电液调节系统及装置技术规程(英文版)以下是为您生成的二十个关于水轮机电液调节系统及装置技术规程相关的英语释义、短语、单词、用法及双语例句：---1. **“水轮机电液调节系统”**：Hydroelectric turbine electro-hydraulic regulating system- 释义：用于控制水轮机运行的电液结合的调节系统- 短语：optimize the hydroelectric turbine electro-hydraulic regulating system（优化水轮机电液调节系统）- 单词：hydroelectric（水电的）、turbine（涡轮机、水轮机）、electro-hydraulic（电液的）、regulating（调节）- 用法：This paper focuses on the performance of the hydroelectric turbine electro-hydraulic regulating system.（这篇论文关注水轮机电液调节系统的性能。
）- 双语例句：The stability of the hydroelectric turbine electro-hydraulic regulating system is crucial for efficient power generation.（水轮机电液调节系统的稳定性对于高效发电至关重要。
）2. **“装置”**：Device / Installation- 释义：设备、仪器；安装、设置- 短语：testing device（测试装置）、installation procedure（装置安装程序）- 单词：test（测试）、procedure（程序、步骤）- 用法：The new device has improved the efficiency of the system.（新装置提高了系统的效率。
三阶离散扩张状态观测器的稳定性分析及其综合_邵立伟
第37卷第2期 2008年4月信息与控制Infor m ation and Contro lV o.l 37,N o .2A pr .,2008文章编号:1002-0411(2008)02-0135-05三阶离散扩张状态观测器的稳定性分析及其综合邵立伟1,廖晓钟1,夏元清1,韩京清2(1.北京理工大学,北京 100081; 2.中国科学院数学与系统科学研究院,北京 100080)收稿日期:2006-11-02摘 要:在连续扩张状态观测器(Ex tended S tate Observer ,ES O )结构的基础上,提出了一种离散形式的三阶ESO,并给出了稳定性分析.当系统扰动变化不是很快的时候,所提出的ES O 能够将状态估计误差和扰动估计误差限制在一个很小的范围内.仿真结果证明了理论分析的有效性.关键词:扩张状态观测器;自抗扰控制器;离散;稳定性中图分类号:T P13 文献标识码:AStability Analysis and Synthesis of Third O rder Discrete Extended State O bserverS HAO L-i w ei 1,2,LIAO X iao -zhong 1,2,XIA Yuan -qing 1,HAN Jing -q i n g3(1.B eiji ng In stit u t e of T ec hnolo gy,B eiji ng 100081,Ch ina;2.A c ade my ofM a t he ma tics and Syst e m S cie nces ,Chinese A c ade my of Sciences ,B eiji ng 100080,Ch ina )Ab stract :A d i scre te t h ird o rder ESO (Ex tended State O bserve r)is propo sed which is based on the frame o f t he continuous ESO,and the analysis of its stab ility is presented .W hen the dist urbances do no t vary too much ,the pro -posed ES O can render the esti m ati on errors of syste m states and dist urbances to be constra i ned i n a s m a ll bounded reg i on .Si m ulati on results sho w t he effecti veness o f the theo re ti ca l analysis .K ey words :ex tended state observe r ;acti ve auto dist urbance rejecti on contro ll er ;d iscrete ;stability1 引言(Introduction)扩张状态观测器[1](Extended Sta te Observer ,ESO)是自抗扰控制器[2](Acti v e D istur bance Re j e c -ti o n Contro ller ,ADRC )的核心单元,它不仅能得到不确定对象的状态,还能获得对象模型中的内扰和外扰的实时作用量,可以通过ESO 消除闭环系统的静差.闭环系统能够自动补偿对象模型的内扰和外扰,使其变为线性系统的标准形:积分器串联型,从而实现动态系统的动态反馈线性化.ESO 除了可以作为ADRC 的单元外,本身还有很多其他用途,如作为状态观测器可以观测系统状态,也可以用来滤波,还可以作为扰动观测器.在国内,ESO 的稳定性的研究一直为众多学者所关注[3],同时国外有关学者也在这方面进行了探索[4],但没有系统地提出ADRC 的思想.文[3]使用分段光滑的李亚普诺夫技术来分析和证明了二阶ESO 的稳态误差和稳定性.而自稳定域法[5~7]也已用于二阶ESO 的稳定性分析,并已经推广到三阶ESO 的稳定性与收敛性分析中,但是对于离散ESO 的形式与稳定性却没有讨论.本文根据连续ESO 的形式,提出了一种离散的线性ESO 结构,并进行了稳定性分析.理论分析证明,当扰动变化不是很快的时候,所提出的算法能将状态估计误差和扰动估计误差限制在很小的范围内,即误差的阶为O (T 2),其中T 为系统的采样周期.仿真实验表明,只要采样时间足够小,状态估计误差和扰动估计误差就能限制在足够小的范围内.2 问题描述(P roble m description)考虑如下S I SO 二阶系统x ##=f(x #,x,w (t))+bu y =x(1)其中f (x #,x,w (t))已经包含了系统内部结构的模型扰动和系统外扰,即系统的总扰动.令x 1=x,x 2=x #,d (t)=f (x #,x,w (t)),整理成状态空间方程为136信息与控制37卷x#1=x2x#2=d(t)+bu(2)令x(t)=x1x2,H=01,D=b,F=1,C=[10],得系统方程为x#(t)=H x(t)+D u+F d(t)y(t)=Cx(t)(3)对式(3)进行离散化,假设采样时间为T,采用zero-order-ho l d(ZOH)来离散对象模型[4],记x(k)=x(kT),y(k)=y(k T),u(k)=u(kT),d(k)=d(k T),离散化后得x(k+1)=Ax(k)+B u(k)+E d(k)+O(T2)y(k)=Cx(k)(4)其中A=exp(H T)=1T01,B=Q T0exp(H S)D d S=Tb,E=Q T0exp(H S)F d S=0T.而且未知扰动也转化成式(5)d(k)=Q T0e H S d((k+1)T-S)d S(5)为了设计离散的ESO,做如下假设.假设:采样时间足够小,以至于扰动f(x#,x,w(t))在两个相邻采样周期内变化不是很大,在这个前提下,式(6)成立.d(k+1)-d(k)I O(T2)(6)而且,如下关系成立O(T n)+O(T n+1)U O(T n)P n I N(7)3离散ESO稳定性分析(Stability analysisof discrete ES O)采用ZOH方法对连续线性ESO进行离散化,得到下面的离散形式z1(k+1)=z1(k)+T(z2(k)-B c01(z1(k)-x1(k)))z2(k+1)=z2(k)+T(q(k)-B c02(z1(k)-x1(k))+bu(k))q(k+1)=q(k)-T(B c03(z1(k)-x1(k)))(8)式中B c01、B c02、B c03分别为ESO的三个可调参数,z1是对x1的状态估计,z2是对x2的状态估计,q(k)是d(k)的估计值,这正体现了ESO的精髓,即将未知扰动作为系统一个扩张状态进行估计.令z(k)=z1(k)z2(k),y^(k)=Cz(k),B01=T B c01,B02=T B c02,B03=T B c03,L1=B01B02,L2=[B03],结合系统输出方程y(k)=x1(k),y^(k)=z1(k),整理式(8)得z(k+1)=Az(k)+B u(k)+E q(k)-L1(y^(k)-y(k))q(k+1)=q(k)-L2(y^(k)-y(k))y^(k)=Cz(k)(9)式(9)即得到的离散的ESO形式,其中L1、L2为ESO的可调参数,z为对系统状态的估计,q为对扰动的估计.对离散系统可以采用式(9)进行状态和扰动估计.定义状态误差e(k)=x(k)-x^(k),则输出估计误差为y~(k)=y(k)-y^(k)=Ce(k).从式(9)和式(4)得到误差方程为e(k+1)=(A-L1C)e(k)+E(d(k)-q(k))(10)令G(k)=d(k)-q(k),得G(k+1)=d(k+1)-q(k+1)=G(k)-(q(k+1)-q(k))+(d(k+1)-d(k))=G(k)-L2Ce(t)+(d(k+1)-d(k))(11)e(k+1)G(k+1)=A-L1C E-L2C Ie(k)G(k)+O(T2)d(k+1)-d(k)=(M-LG)e(k)G(k)+O(T2)(12)y~(k)=[C0]e(k)G(k)=Ge(k)G(k)(13)其中M=A E0I=1T001T001,L=L1L2=B01B02B03,G=[C0]=[100].式(12)中,O(T2)因为是T2的同阶无穷小,所以当采样周期T比较小的时,这项可以忽略,得到广义误差方程为e(k+1)G(k+1)=(M-LG)e(k)G(k)y~(k)(14)2期邵立伟等:三阶离散扩张状态观测器的稳定性分析及其综合137从式(14)可知,如果(M ,G )可观测,则(M -LG )能稳定,也就是说,如果(M ,G )可观测,则存在L 使(M -LG )是稳定的.因为GGM G M2=101T 012TT2,显然只要采样周期T 不等于0,则(M ,G )可观测,即存在L 使观测器稳定.下面对L 参数的具体要求进行分析.若要求所设计的ESO 稳定,需要(M ,G )可观测,则要求M -LG 的特征根均分布在单位圆内. M -LG 的特征多项式为t z I -(M -LG )t =z-1+B 01-T 0B 02z-1-TB 03z-1=z 3+(B 01-3)z 2+(T B 02-2B 01+3)z +(T 2B 03-T B 02+B 01-1)(15)令z =(w +1)/(w -1),代入式(15)中进行双线性变换,化简后得w 域特征方程T 2B 03w 3+(2T B 02-3T 2B 03)w 2+(4B 01-4T B 02+3T 2B 03)w +(8-4B 01+2T B 02-T 2B 03)=0(16)由Routh 判据可知M -LG 稳定的条件为T 2B 03>02T B 02-3T 2B 03>04B 01-4T B 02+3T 2B 03>08-4B 01+2T B 02-T 2B 03>0(2T B 02-3T 2B 03)(4B 01-4T B 02+3T 2B 03)>T 2B 03(8-4B 01+2T B 02-T 2B 03)(17)式(17)为离散ESO 稳定的条件,这可以为ESO 参数调节提供参考价值.由此可见,ESO 的稳定性与系统采样周期以及ESO 的系数都有关系.对以上的分析进行归纳可知,如果ESO 的参数能保证M -LG 是稳定的,则从式(14)中可知,当k y ]时,e (k )y 0且G (k )y 0,即 当k y ]时,x^(k)y x (k ),q (k )y d (k)(18)这样,尽管设计的离散ESO 不能提供百分之百的精确估计,但是它能保证估计误差在O (T 2)范围内,只要能够加快采样频率,那估计误差就能更小.所以,ESO 的稳定性与采样周期有关,其估计误差也与采样周期有关,采样周期越小,ESO 的估计误差就越小.自抗扰控制器(ADRC)利用跟踪微分器(Trac -k i n g D ifferentiator ,TD)来安排过渡过程并提取其微分信号,结合扩张状态观测器ESO 估计对象状态和不确定扰动作用,最后利用安排的过渡过程与状态估计之间误差的非线性组合(N on -L i n ear Co m b i n a -tion ,NLC )和扰动估计量的补偿来生成控制信号.对于时变、强非线性、强耦合、大时滞等广泛的被控对象,ADRC 均能给出很好的控制效果,系统的控制结构图如图1所示.图1 ADRC 的控制框图F i g.1 The con tro l frame o f ADRC4 仿真结果(Si m ulation results)仿真的对象为式(1)所示的二阶系统,x ##=f (x ,x #,t)+w (t)+uy =x其中,f (x,x #,t)=-t x t x #-0.2(x +cos x #)为模型的参数内扰,w (t)=0.5sgn(cos (t /2))为外部的扰动.由于仿真对象是一个非线性系统,采用常规PI 控制器很难对其进行控制,其阶跃响应控制效果如图2所示.可见,PI 控制器很难控制好阶跃信号,更别说跟踪方波等其它信号了.图2 P I 控制的阶跃响应曲线F i g.2 Step response curve w it h P I con tro l138 信 息 与 控 制37卷采用离散ESO 进行状态和扰动估计,结合AD-RC 进行控制.输入为方波信号,系统实验波形如图3~图7.图3 系统的输入信号F i g .3 T he i npu t si gna l o f system图4 ESO 的输出估计误差F i g .4 T he output esti m ate error o f ESO图5 ESO 的扰动估计误差F i g .5 T he d i sturbance esti m a te erro r o f ESO图6 系统的输出波形F i g.6 T he ou t put wave o f syste m图7 系统的跟踪误差F ig .7 The traci ng error of syste m图3~图7中,系统的采样时间为0.0001s ,图3为系统的给定信号,ESO 的输出(状态)估计误差如图4所示,扰动估计误差如图5所示,可见大部分时间,ESO 的估计精度很高.结合ADRC 控制后的输出波形为图6,图7为跟踪误差,可见,输出能很好地跟踪输入,跟踪的动态精度和稳态精度都很好,这正是得益于ESO 对状态和扰动的准确及时的估计,才使ADRC 能很好地控制这个非线性对象. 理论分析表明,ESO 的状态和扰动估计误差与系统的采样时间有直接关系.为了证明这点,保持ESO 的参数不变,将系统的采样时间变为0.005s ,得到的系统波形为图8~图11.可见,采样时间变大以后,ESO 的状态估计误差和扰动估计误差都变大了,以至于ADRC 不能通过ESO 对状态和扰动的准确估计来对系统进行很好的控制,此时,系统的输出波形如图8所示.这说明,ESO 的估计准确性与采样时间有很大关系,采样越快,ESO 的估计误差就越小.2期邵立伟等:三阶离散扩张状态观测器的稳定性分析及其综合139图8 采样周期变大后ESO 的输出观测误差F i g.8 T he ou t put esti m a te erro r o f ES O w hen samp li ngti m e is larg er图9 采样周期变大后ESO 的扰动估计误差F i g .9 T he disturbance esti m ate error of ESO whensamp li ng ti m e is larger图10 变采样时间后的系统跟踪曲线F ig .10 T he syste m trac i ng curve when sa m pli ng ti m e i s l arger5 结论(Conclusion)本文根据连续ESO 的形式,提出了一种离散形图11 变采样时间后的系统跟踪误差F i g .11 The sy stem trac i ng erro r when sa m pli ng ti m e is larger式的ESO 观测器,并且从理论上分析了ESO 的稳定性,得出ESO 的状态估计误差和扰动估计误差与系统的采样时间有直接关系,采样时间越小,ESO 的估计准确度就越高,从而利用ADRC 的闭环控制效果就越好.本文还给出ESO 参数稳定时应该满足的条件,对ESO 的调节有着一定的指导意义.仿真结果证明了理论分析的有效性.参 考 文 献(R eferences)[1] 韩京清.一类不确定对象的扩张状态观测器[J].控制与决策,1995,10(1):85~88.[2] 韩京清.自抗扰控制器及其应用[J].控制与决策,1998,13(1):19~23.[3] 韩京清,张荣.二阶扩张状态观测器的误差分析[J].系统科学与数学,1999,19(4):465~471.[4]C hang J L.App l ying d i screte -ti m e proportional integral observers f or state and d i s t urb ance es ti m ations [J].I EEE T ran s acti on s onAuto m atic C ontro,l 2006,51(5):814~818.[5] 夏元清,黄焕袍,韩京清.不确定时滞系统ADRC 控制[J ].中南工业大学学报,2003,34(4):383~385.[6] 黄一,韩京清.非线性连续二阶扩张状态观测器的分析与设计[J ].科学通报,2000,45(13):1373~1379.[7] 要晓梅,王庆林,韩京清.大纯滞后纯积分对象的二阶自抗扰控制[J].控制工程,2002,9(6):7~9,17.作者简介邵立伟(1979-),男,博士,助理研究员.研究领域为非线性控制,电机控制.廖晓钟(1962-),女,博士,教授,博士生导师.研究领域为电力电子,电气传动.。
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[8], [li], [12], [24], [25], [31], [33], [51], [611, [67], [90], [91], [92], [981, [102], [103], [1061,
[107], [108], [114], [120], and see also the discussion in Chapter 6). However, a great deal of work which addresses qualitative issues concerning 2-D digital filters endowed with overflow nonlinearities remains to be accomplished. We consider the state-space quarter plane model of 2-D digital filters described by
acteristics (cf. Chapter 6). These nonlinearities constitute a generalization of the usual types
of overflow arithmetic employed in practice, including saturation, zeroing, two's complement and triangular overflow arithmetic. In our approach, we do not characterize these nonlinearities by sector conditions; instead, we characterize them by the range of the nonlinear function representing the overflow arithmetic. In Section 7.2, we prove several results for the global asymptotic stability of the null solution of 2-D digital filters described by (7.1) (Theorem 7.1, Corollary 7.1, and Proposition 7.1). In Section 7.3, we establish our main result for the present chapter (Theorem 7.2). We generalize our results to multidimensional cases in Section 7.4 (Theorem 7.3 and Corollary 7.2). We demonstrate the applicability of the present results by means of specific examples in Section 7.5. We conclude this chapter in Section 7.6.
Q = G - ATGA
(7.3)
is positive definite, then the null solution of the 2-D digital filter (7.1) is globally asymptotically stable, where
A =
An :: A12]
(7.4)
7.1
Introduction
As we discussed in Chapter 5, in the implementation of linear digital filters, signals are usuMly represented and processed in a finite wordlength format. Therefore, such implementations will naturally give rise to several kinds of nonlinear effects, such as overflow and quantization. The stability analysis of 2-D (two-dimensional) as well as multidimensional digital filters subject to such nonlinearities has been of increasing interest in recent years (cf. [1], [2], [7], [8], [64], [69], [1131). Since finite wordlength realizations of digital filters result in systems which inherently are nonlinear, the asymptotic stability of such filters (under zero-input), as in many other nonlinear systems, is of great interest in practice. The global asymptotic stability of the null solution guarantees the non-existence of limit cycles (overflow oscillations) in the realized digital fitters. In this chapter, we establish new results for the global asymptotic stability of zero-input 2-D state-space digital filters with overflow nonlinearities. We do not consider quantization effects in the present chapter. The stability properties of I - D (one dimensional) digital filters subject to overflow nonlinearities have been investigated extensively during the last two decades (el. [6], [7],
f zh(k,O) = 0 for k _> K, x'(k,O) = 0 for k > K,
xh(0,1) = 0 for I > L, x~(0, l) = 0 for I > L. For the asymptotic stability of the 2-D digital filter (7.1) with initial conditions (7.2), a well known result states that (cf. [32], also see [1]) if there exists a diagonal positive definite matrix G such that (7.2)
Condition (7.3) can not be applied to the case where the absolute values of some of the diagonal elements of matrix A axe greater than or equal to 1. In this chapter, we utilize Lyapunov's Second Method to establish new results for the global asymptotic stability of the null solution of the 2-D system (7.1). One of our results generalizes condition (7.3) and shows that the matrix G in (7.3) can be relaxed to certain classes of positive definite matrices. We also provide necessary and sufficient conditions under which positive definite matrices can be used to construct for the 2-D system (7.1) quadratic form Lyapunov functions with the desired property that the overflow nonlinearity never increases the values of these functions. Another result for testing the global asymptotic stability of the null solution for 2-D fixed-point digital filters is given by Bauer and Jury [8] (which is established for the shiftvariant case). For shift-invariant 2-D digital filters described by (7.1), the results obtained
79 in [8] require that
p(IAI) < 1,
where p(.) denotes the spectral radius (see definition on page 42) and
(7.5)
IAI =
[la,jI]. Condition