线性系统理论与设计

好词好句摘录

词组搭配

1、track desired signals 跟踪所需的信号
2、suppress noise 抑制噪声
3、Necessity and Sufficiency 必要、充分性
4、linear time-varying\time-invariant 线性时变\时不变系统
5、controllability and observability 能控能观性
6、canonical decomposition 规范分解（能控能观分解）
7、without loss of generality\continuity 不失一般性\连续性
8、complex number 复数
9、repeat eigenvalue\simply eigenvalue 重根\单根
10、digital-to-analog converter 数模转换器
11、picewise constant 分段常数
12、Cayley-Hamilton theorem 凯莱 - 汉密尔顿定理
13、one-degree-of-freedom 一自由度
14、disturbance rejection 干扰抑制
15、pole-zero cancelltion 零极点对消
16、positive definite\semidefinite matrix 正定矩阵\正半定矩阵
17、schmidt orthonormalization 施密特正交归一化

好句

1、If a system is not stable, the system may burn out, disintegrate, or saturate when a signal, no matter how small, is applied
如果一个系统是不稳定的，那么无论多么小的信号都可能使得该系统烧坏，瓦解或者饱和。

2、We will introduce the BIBO (bounded-input bounded-output) stability for the zero-state response and marginal and asymptotic stabilities for the zero-input response.
这里将会介绍零状态响应下的有界输入有界输出稳定性和零输入响应下的稳定性及渐进稳定性

3、The system is marginally stable or stable in the sense of Lyapunov if every finite initial state xo excites a bounded response.
如果每个有限初始状态Xo激励得到的是有界响应，则系统在Lyapunov意义下是边缘稳定或稳定的。

4、The equation $\dot x=Ax$ is marginally stable if and only if all eigenvalues of A have zero or negative real parts and those with zero real parts are simple roots of the minimal polynomial of A.
等式$\dot x=Ax$是边缘稳定的当且仅当A的所有特征值都具有零或负实数部分且具有零实数部分的所有特征值是A的最小多项式的单根。

5、All eigenvalues of A have negative real parts if and only if for any given positive definite symmetric matrix N, the Lyapunov equation

$$A'M+MA=-N$$
has a unique symmetric solution M and M is positive definite.
A的所有特征值具有负实部，当且仅当对于任何给定的正定对称矩阵N，Lyapunov方程 $$A'M+MA=-N$$ 具有唯一的对称解M且M是正定的。

6、Marginal and asymptotic stabilities of $\dot x=A(t)x$ are invariant under any Lyapunov transformation.
在任何Lyapunov变换下，$\dot x=A(t)x$的边缘和渐近稳定性是不变的。

7、Controllability deals with whether or not the state of a state-space equation can be controlled from the input,and observability deals with whether or not the initial state can be observed from the output.
能控性表示状态空间方程的状态是否可以从输入控制，而能观性表示是否可以从输出观察初始状态

8、The state equation $\dot x=Ax+Bu$ or the pair (A, B) is said to be controllable if for any initial state $x(0) = x_0$ and any final state $x_1$, there exists an input that transfers $x_0$ to $x_1$ in a finite time. Otherwise $\dot x=Ax+Bu$ or (A, B) is said to be uncontrollable.
如果对于任何初始状态 $x(0) = x_0$ 和任何最终状态$x_1$，存在一个输入能在有限的时间内将状态$x_0$转换到状态$x_1$，则状态方程$\dot x=Ax+Bu$或对（A，B）是可控的。 否则$\dot x=Ax+Bu$或（A，B）被认为是不可控的。

9、Roughly speaking, controllability studies the possibility of steering the state from the input; observability studies the possibility of estimating the state from the output.
大致来说，可控性研究从输入转向状态的可能性;可观测性研究从输出估计状态的可能性。

10、The state equation (6.22) is said to be observable if for any unknown initial state $x(0)$, there exists a finite $t_1 > 0$ such that the knowledge of the input u and the output y over $[0, t_1]$ suffices to determine uniquely the initial state x(0). Otherwise, the equation is said to be unobservable.
如果对于任意一个非已知初始状态$x(0)$,都存在一个有限的时间$t_1 > 0$，在$[0, t_1]$上利用已知的输入u和输出y能够唯一确定初始状态$x(0)$，那么这个系统是能观的。否则，我们称这个系统不能观。

11、The controllability/observability property is invariant under any equivalence transformation.
在任何等价变换下，系统的能观性和能控性是不变的。

12、Where the first $n_1$ columns are any $n_1$ linearly independent columns of C, and the remaining columns can arbitrarily be chosen as long as P is nonsingular.
前$n_1$列是矩阵C的线性独立列，其余列可以任意选取只要使得P非奇异。

13、A state equation (A, b, c, d) is a minimal realization of a proper rational function $\hat g(s)$ if and only if (A, b) is controllable and (A, c) is observable or if and only if

$$dim A = deg \quad\hat g(s)$$
状态空间方程 (A, b, c, d) 是一个合理传递函数的最小实现当且仅当它是能控能观的或者它满足：
$$dim A = deg \quad\hat g(s)$$

14、The concepts of controllability and observability were used in the preceding two chapters to study the internal structure of systems and to establish the relationships between the internal and external descriptions.
在前两章中使用能控性和能观性的概念来研究系统的内部结构并建立内部和外部描述之间的关系。

15、The problem is to design an overall system so that the plant output will follow as closely as possible the reference signal $r(t)$.
问题是设计整个系统，使得设备输出将尽可能接近参考信号$r(t)$。

16、The open-loop control is, in general, not satisfactory if there are plant parameter variations and/or there are noise and disturbance around the system.
如果存在设备参数变化和/或系统周围存在噪声和干扰，则开环控制通常是不令人满意的。

17、A properly designed feedback system, on the other hand,can reduce the effect of parameter variations and suppress noise and disturbance. Therefore feedback control is more widely used in practice.
另一方面，适当设计的反馈系统可以减少参数变化的影响并抑制噪声和干扰。 因此，反馈控制在实践中被更广泛地使用。

18、This is actually a general property of feedback: feedback can shift the poles of a plant but has no effect on the zeros. This can be used to explain hy a state feedback may alter the observability property of a state equation.
这实际上是反馈的一般属性：反馈可以移动设备的极点，但对零点没有影响。 这可以用于解释状态反馈可以改变状态方程的能观性属性。

19、If one or more poles are shifted to coincide with zeros of $\hat g(s)$, then the numerator and denominator of $\hat g(s)$ in (8.17) are not coprime.
如果一个或多个极点偏移与$\hat g(s)$的零点重合，则（8.17）中的$\hat g(s)$的分子和分母不是互质的

20、The larger the distance of the vertical line from the imaginary axis, the faster the response.
垂直线与虚轴的距离越大，响应越快。

21、Therefore a final selection may involve compromises among many conflicting requirements.
因此，最终选择可能涉及许多相冲突的要求之间的折衷。

22、We introduced in the preceding sections state feedback under the implicit assumption that all state variables are available for feedback.
我们在前面的部分中介绍的状态反馈器是基于所有状态变量都可用于反馈的假设。

23、This assumption may not hold in practice either because the state variables are not accessible for direct connection or because sensing devices or transducers are not available or very expensive.
这种假设在实践中可能不成立，因为状态变量对于直接连接是不可访问的，或者因为感测装置或转换器不可用或非常昂贵。

24、Second, and more seriously, if the matrix A has eigenvalues with positive real parts, then even for a very small difference between $x(t_0)$ and $\hat x(t_0)$ for some $t_0$, which may be caused by disturbance or imperfect estimation of the initial state, the difference between $x(t)$ and $\hat x(t)$ will grow with time.
第二，更严重的是，如果矩阵A具有具有正实部的特征值，则即使对于某些$t_0$时刻的$x(t_0)$ 和$\hat x(t_0)$ 之间的非常小的差异，这可能由干扰或是对初始状态不精确估计造成的，导致$x(t)$和$\hat x(t)$之间的差值将随时间增长。

25、 If all eigenvalues of $(A − lc)$ can be assigned arbitrarily, then we can control the rate for $e(t)$ to approach zero or, equivalently, for the estimated state to approach the actual state.
如果可以任意地分配$(A − lc)$的所有特征值，则我们可以控制$e(t)$ 接近零，或者等效地，估计状态接近实际状态。

26、This unique relationship between the excitation and response, input and output, or cause and effect is essential in defining a system.
激励和响应，输入和输出或原因和效果之间的这种独特的关系在定义系统中是至关重要的。

27、 A system with only one input terminal and only one output terminal is called a single-variable system or a single-input single-output (SISO) system.
仅具有一个输入端子和仅一个输出端子的系统被称为单变量系统或单输入单输出（SISO）系统。

28、A system is called a continuous-time system if it accepts continuous-time signals as its input and generates continuous-time signals as its output.
如果系统接受连续时间信号作为其输入并产生连续时间信号作为其输出，则该系统被称为连续时间系统。

29、Thus the response of every linear system can be decomposed into the zero-state response and the zero-input response. Furthermore, the two responses can be studied separately and theirsum yields the complete response.
因此，每个线性系统的响应可以分解为零状态响应和零输入响应。此外，这两个响应可以分别研究，它们的总和是全状态响应。

30、If a system is causal, the output will not appear before an input is applied.
如果系统是因果性的，则在输入激励之前不会出现输出。

31、Improper rational transfer functions will amplify high-frequency noise, which often exists in the real world; therefore improper rational transfer functions rarely arise in practice.
不合理的合理传递函数将放大高频噪声，这在现实世界中经常存在; 因此在实践中很少出现不合理的理性转移函数。

32、In RLC networks, capacitors and inductors can store energy and are associated with state variables.
在RLC网络中，电容器和电感器可以存储能量并且与状态变量相关联。

33、Note that resistors are memoryless elements, and their currents or voltages should not be assigned as state variables.
注意，电阻器是无记忆元件，并且它们的电流或电压不应被分配为状态变量。

34、For most simple RLC networks, once state variables are assigned, their state equations can be developed by applying Kirchhoff’s current and voltage laws, as the next example illustrates.
对于大多数简单的RLC网络，一旦状态变量被分配，它们的状态方程可以通过应用基尔霍夫的电流和电压定律来开发，如下一个例子所示。

35、The procedure is fairly simple: assign state variables and then use branch characteristics and Kirchhoff’s laws to develop state equations.
过程相当简单：分配状态变量，然后使用分支特性和基尔霍夫定律来构造状态方程。

36、The procedure can be stated more systematically by using graph concepts, as we will introduce next.
该过程可以通过使用图形概念更系统地陈述，我们将在下面介绍。

37、Thus a network can be considered to consist of only branches and nodes. A loop is a connection of branches starting from one point and coming back to the same point without passing any point twice.
因此，可以认为网络仅由分支和节点组成。循环是分支的连接,从一个点开始并返回到同一点而不经过任何点两次。

38、The algebraic sum of all voltages along every loop is zero (Kirchhoff’s voltage law).
沿每个环路的所有电压的代数和为零（基尔霍夫的电压定律）。

39、The set of all branches connect to a node is called a cutset. More generally, a cutset of a connected network is any minimal set of branches so that the removal of the set causes the remaining network to be unconnected.
连接到一个节点的所有分支的集合称为小集。 更一般地，连接网络的集合是任何最小分支集合，使得集合的移除导致剩余网络未连接。

40、The linearization discussed for the continuous-time case can also be applied to the discretetime case with only slight modification.
对于连续时间情况讨论的线性化仅稍微修改也可以应用于离散情况。

41、Norms of vectors The concept of norm is a generalization of length or magnitude. Any real-valued function of x, denoted by $|| x ||$, can be defined as a norm if it has the following properties:
向量的范数 范数的概念是长度或幅度的泛化。 如果$|| x ||$表示的x的任何实值函数具有以下属性，则可以将其定义为范数：

42、Even though the data we encounter in practice are all real numbers, complex numbers may arise when we compute eigenvalues and eigenvectors. To deal with this problem, we must extend real linear spaces into complex linear spaces and permit all scalars such as $α_i$ in (3.4) to assume complex numbers.
即使我们在实践中遇到的数据都是实数，当我们计算特征值和特征向量时，可能会出现复数。 为了解决这个问题，我们必须将实线性空间扩展为复杂的线性空间，并允许所有的标量，例如（3.4）中的$α_i$假定复数。

43、If A has repeated eigenvalues, then it may not have a diagonal form representation. However, it has a blockdiagonal and triangular-form representation as we will discuss next.
如果A具有重复的特征值，则它可以不具有对角形式表示。但是，它具有块对角线和三角形形式的表示，我们将在下面讨论。

44、Clearly, if all eigenvalues of A are distinct, then the minimal polynomial equals the characteristic polynomial.
显然，如果A的所有特征值都不同，则最小多项式等于特征多项式。

45、Although computing singular-value decomposition is time consuming, it is very reliable and gives a quantitative measure of the rank.
虽然计算奇异值分解是耗时的，但它是非常可靠的并给出了秩的定量测量。

46、 This discretization is the easiest to carry out but yields the least accurate results for the same T.
这种离散化是最容易进行的，但对于相同的T产生的结果最不精确。

47、Thus equivalent state equations have the same characteristic polynomial and, consequently, the same set of eigenvalues and same transfer matrix.
因此，等效状态方程具有相同的特性多项式，因此，有相同的特征值和相同的传递矩阵。

48、This terminology is justified by the fact that, by using the state equation, we can build an op-amp circuit for the transfer matrix.
这个术语实际上是有道理的，通过使用状态方程，我们可以为传递矩阵构建运算放大器电路。

49、 A matrix $P(t)$ is called a Lyapunov transformation if $P(t)$ is nonsingular, $P(t)$and $\dot P(t)$ are continuous, and $P(t)$ and $P^{-1}(t)$ are bounded for all $t$.Equations(4.69) and (4.70) are said to be Lyapunov equivalent if P(t) is a Lyapunov transformation.
如果矩阵$P(t)$是非奇异的，那么该矩阵被称为李雅普诺夫变换。那么对于任意$t$$P(t)$和$P^{-1}(t)$是连续且有界的。如果$P(t)$是李雅普诺夫变换那么等式(4.69)和(4.70)是等价的。

50、It is clear that if $P(t) = P$ is a constant matrix, then it is a Lyapunov transformation. Thus the (algebraic) transformation in the time-invariant case is a special case of the Lyapunov transformation.
显然，如果$P（t）= P$是常数矩阵，则它是Lyapunov变换。因此，时不变情况下的（代数）变换是Lyapunov变换的特殊情况。

51、The converse problem is to find a state equation from a given impulse response matrix.
相反（转换）的问题是从给定的脉冲响应矩阵中找到状态方程。

52、This LTI state equation is a different realization of the same impulse response.
这个LTI状态方程是相同脉冲响应的不同实现。