线性系统理论-中英文对照

线性系统理论学习指南英文回答：Linear system theory is a fundamental topic in thefield of control engineering. It provides a mathematical framework for analyzing and designing systems that can be described by linear equations. In this learning guide, I will explain the key concepts and techniques in linear system theory.Firstly, let's start with the definition of a linear system. A linear system is a mathematical model that can be represented by a set of linear equations. These equations describe the relationship between the inputs and outputs of the system. The behavior of a linear system can be analyzed using techniques such as transfer functions, state-space representations, and frequency domain analysis.One important concept in linear system theory is the transfer function. The transfer function is a mathematicalrepresentation of the input-output relationship of a linear system. It is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero. The transfer function provides valuable information about the system's stability, dynamics, and frequency response.To illustrate this, let's consider an example of a simple linear system. Imagine a spring-mass-damper system, where a mass is attached to a spring and a damper. Theinput to the system is the force applied to the mass, andthe output is the displacement of the mass. By applying Newton's second law and solving the resulting differential equation, we can derive the transfer function of the system. This transfer function will tell us how the displacement of the mass is related to the applied force.Another important concept in linear system theory isthe state-space representation. The state-space representation provides a more comprehensive description of a linear system by considering both the inputs and outputs as well as the internal states of the system. It consistsof a set of first-order differential equations that describe the evolution of the system's states over time. The state-space representation is particularly useful for analyzing the system's controllability, observability, and stability.To further illustrate this, let's consider a control system that aims to regulate the temperature of a room. The inputs to the system are the heater power and the outside temperature, and the output is the room temperature. By formulating the system dynamics and the output equation in state-space form, we can analyze the controllability of the system and design a suitable controller to achieve the desired temperature regulation.In addition to transfer functions and state-space representations, frequency domain analysis is another important tool in linear system theory. Frequency domain analysis allows us to study the system's behavior in the frequency domain, which is particularly useful for analyzing the system's stability and frequency response. Techniques such as Bode plots, Nyquist plots, and rootlocus plots are commonly used in frequency domain analysis.To illustrate this, let's consider a feedback control system that aims to stabilize an unstable plant. By analyzing the system's transfer function in the frequency domain using Bode plots, we can determine the stability margins and design a suitable controller to stabilize the system.中文回答：线性系统理论是控制工程领域的基础知识。

电子工程中的线性系统理论线性系统理论是电子工程中非常重要的一部分内容。

其涉及到信号处理、控制系统、通信系统等多个领域。

本文将对线性系统理论的定义、特征、基本理论等方面进行简要介绍。

一、线性系统的定义线性系统是指其输入和输出具有线性关系的系统。

简单地说，就是许多输入信号叠加组成的输出信号，与单独输入信号的输出信号相加之和完全相同。

其中输入信号可以是电压、电流、功率等，输出信号也可以是同样的类型。

例如，如果一个系统的输入信号为 $x_1$ 和 $x_2$，对应的输出信号为 $y_1$ 和 $y_2$，则该系统是线性的，当且仅当：$$y_1 = ax_1 + bx_2 \\y_2 = cx_1 + dx_2$$其中 $a,b,c,d$ 均为常量。

二、线性系统的特征1. 叠加性：线性系统具有叠加性，即当系统中输入信号为$x_1$ 和 $x_2$ 时，对应的输出信号分别为 $y_1$ 和 $y_2$，则系统中同时输入 $x_1+x_2$ 时，输出信号为 $y_1+y_2$。

2. 抑制性：线性系统具有抑制性，即输入信号越大，输出信号越小。

如果输入信号的某一部分被视为噪声，则线性系统可以减小噪声的影响，同时保持信号的大部分原始信息。

3. 延时特性：线性系统具有延时特性，即在特定的时间段内输入信号可以得到响应。

例如，音频系统在接收到输入信号后需要一定时间来处理信号，并绘制出相应的声音波形。

三、线性系统的基本理论1. 系统函数和频率响应系统函数是将输入信号转换为输出信号的函数，通常用$H(s)$ 或 $H(jw)$ 表示，其中 $s$ 是连续时间变量，$jw$ 是离散时间变量，表示系统的频率响应。

频率响应是指系统在不同频率下的输出功率和输入功率之比，通常用 $H(jw)$ 表示。

2. 系统的稳定性稳定性是指系统在输入端输入有限信号时输出端不会产生无限响应的性质。

在线性系统中，通常采用相对稳定性来描述系统的稳定性，这意味着系统相对于任意有限的输入信号都稳定。

[Linear system theory and design] Absolutely integrable 绝对可积Adder 加法器Additivity 可加性Adjoint 伴随Aeronautical航空的Arbitrary 任意的Asymptotic stability渐近稳定Asymptotic tracking 渐近跟踪Balanced realization 平衡实现Basis 基BIBO stability 有界输入有界输出稳定Black box 黑箱Blocking zero 阻塞零点Canonical decomposition 规范分解Canonical规范Capacitor 电容Causality 因果性Cayley-Hamilton theorem 凯莱-哈密顿定理Characteristic polynominal 特征多项式Circumflex 卷积Coefficient 系数Cofactor 余因子Column degree 列次数Column-degree-coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Common Divisor公共因式Companion-form matrix 规范型矩阵Compensator 调节器，补偿器Compensator equation补偿器方程Control configuration 控制构型Controllability 能控性Convolution 卷积Conventional常规的Coprimeness互质Corollary推论Cyclic matrix 循环矩阵Dead beat design 有限拍设计Decoupling 解耦Degree of rational function有理矩阵的次数Description of system系统描述Derivative 导数Determinant 行列式Diagonal对角型Discretization 离散化Disturbance rejectionDivisor 因式Diverge分叉Duality 对偶Eigenvalue特征值Eigenvextor 特征向量Empirical 经验Equivalence 等价Equivalence transformation等价变换Exhaustive 详细的Exponential function 指数函数Extensively广泛地Filter 滤波Finite time有限时间Finite 有限的Fraction 分式Polynomial fraction 多项式分式Fundamental cutest 基本割集Fundamental loop 基本回路Fundamental matrix 基本解阵Gramian 格拉姆Geometric几何的Hankel matrix Hankel 矩阵Hankel singular valueHomogeneity奇次性Hurwitz polynomial Hurwitz 多项式Implementable transfer function 可实现的传递函数Impulse response 脉冲响应Impulse response matrix 脉冲响应矩阵Impulse response sequence 脉冲响应序列Index指数Inductance 自感系数Inductor 电感Integrator 积分器Internal model principle 内模原理Intuitively 直观的Inverse 逆Jacobian 雅可比Jordan block约当块Jordan form matrix 约当型矩阵Lapalace expansion 拉普拉斯展开Lapalace transform 拉普拉斯变换Left multiple 左倍式Linear algebraic equation线性代数方程Linear space 线性空间Linearity 线性性Linearization 线性化Linearly dependent 线性相关Linearly independent 线性无关Lumpedness 集中（参数）性Lyapunov equation Lyapunov方程Lyapunov theorem Lyapunov定理Lyapunov transform Lyapunov变换Markov parameter 马尔科夫参数Magnitude 模Manuscript 手稿，原稿Marginal stability 稳定Matrix 矩阵Minimal polynomial 最小多项式Minimal realization 最小实现Minimum energy control 最小能量控制Minor 子行列式Model matching 模型匹配Model reduction 模型降维Monic 首1的Monic polynomial首1的多项式Multiple 倍式Multiplier 乘法器Nilpotent matrix 幂为零的矩阵Nominal equation 标称（名义）方程Norm 范数Nonsingular非奇异Null space 零空间Nullity 零度（零空间的维数）Observability 能观性Observer 观测器Op-amp circuit 运算发大器Orthogonal 正交的Orthogonal matrix正交矩阵Orthogonality of vectors 向量的正交性Orthonormal set 规范正交集Orthonormalization 规范正交化Oscillator振荡器Parenthesis圆括号Parameterization 参数化Pendulum钟摆Periodic 周期的Periodic state system 周期状态系统Pertinent 相关的Plant被控对象Pole 极点Pole placement 极点配置Pole-zero cancellation 零极点对消Pole-zero excess inequalityPolynomial多项式Polynomial matrix 多项式矩阵Column degree 列次数Column degree 列次数Column-degree-coefficient matrix 列次数系数矩阵Column echelon form 列梯形Column indices 列指数集Column reduced 列既约Echelon formLeft divisor 左因式Left multiple 左倍式Right divisor 右因式Right multiple 右倍式Row degree 行次数Row-degree-coefficient matrix 行次数系数矩阵Row reduced行既约Unimodular 幺模Positive正的Positive definite matrix正定矩阵Positive semidefinite matrix正定矩阵Power series 幂级数Primary dependent column 主相关列Primary dependent column 主相关行Prime素数的Principal minor 主子行列式Pseudoinverse 伪逆Pseudostate 伪状态QR decomposition QR 分解Quadratic二次的Quadratic form二次型Range space 值域空间Rank秩Rational function 有理函数Biproper双真Improper 不真Proper 真Strictly proper 严格真Rational matrix 有理矩阵Biproper双真Improper 不真Proper 真Strictly proper 严格真Reachability 可达性Realization 状态空间可实现性Balanced 平衡的（实现）Companion form 规范型（实现）Controllable form能控型（实现）Input-normal 输入规范（实现）Minimal 最小（实现）Observable-form能控型（实现）Output-normal 输出规范（实现）Time-varying 时变（实现）Reduced 既约的Regulator problem 调节器问题Relaxedness 松弛Response 响应Impulse 脉冲Zero-input 零输入Zero-state 零状态Remainder 余数Resistor 电阻Resultant 结式Generalized Resultant 广义结式Sylvester Resultant Sylvester（西尔维斯特）结式Right divisor右因子Robust design 鲁棒设计Robust tracking 鲁棒跟踪Row indices 行指数集Saturate 使饱和Scalar 标量Schmidt orthonormalizationSemidefinite 半正定的Similar matrices 相似矩阵Separation property 分离原理Servomechanism 司服机制Similar transformation 相似变换Singular value 奇异值Singular value decomposition奇异值分解Spectrum 谱Stability 稳定Asymptotic stability 渐近稳定BIBO stability 有界输入有界输出稳定Stability in the sense of Lyapunov 李雅普诺夫意义下的稳定Marginal stability 临界稳定Total stability 整体稳定Stabilization镇定（稳定化）State 状态State variable 状态变量State equation 状态方程Discrete-time State equation 离散时间状态方程Discretization State equation 离散化状态方程Equivalent State equation 等价状态方程Periodic State equation 周期状态方程Reduction State equation 状态方程Solution of State equation 状态方程的解Time-varying State equation 时变状态方程State estimator 状态估计器Asymptotic State estimator 渐近状态估计器Closed-loop State estimator 闭环状态估计器Full-dimentional State estimator 全维状态估计器Open-loop State estimator 开环状态估计器Reduced-dimentional State estimator 降维状态估计器State feedback 状态反馈State-space equation 状态空间模型State transition matrix 状态转移矩阵Superposition property 叠加性Sylvester resultant （西尔维斯特）结式Sylvester’s inequality （西尔维斯特）不等式System 系统Superposition叠加Terminology 术语Total stability 整体稳定Trace of matrix 矩阵的迹Tracking 跟踪Transfer function 传递函数Discrete Transfer function离散传递函数Pole of Transfer function传递函数的极点Zero of Transfer function传递函数的零点Transfer matrix 传递函数矩阵Blocking zero of matrix 传递函数矩阵的阻塞零点Pole of Transfer function传递函数矩阵的极点Transmission Zero of Transfer function传递函数矩阵的传输零点Transistor 晶体管Transmission zero 传输零点Tree 树Truncation operator 截断算子Trajectory 轨迹Transpose 转置Triangular 三角形的Unimodular matrix 么模阵Unit-time-delay element 单位时间延迟单元Unity-feedback 单位反馈Vandermonde matrix Vandermonde矩阵Vector 向量well posed 适定的well-posedness 适定性Yield 等于，得出Zero 零点Blocking zero 阻塞零点Minimum-phase zero 最小相位零点nonminimum-phase zero 非最小相位零点transmission zero 传输零点zero-input response 零输入响应zero-pole-gain form 零极点增益形式zero-state equivalence零状态等价zero-state response零状态响应z-transform z-变换。