线性系统的频率响应分析实验报告_动态系统的建模与分析

阅读量：

参考：DR_CAN

1.介绍

解决一个控制系统的问题：

对研究对象进行分析
控制器设计
测试

分析被控对象的物理特性及动态表现，在这个基础上建立数学模型，数学模型可以是动力学模型、热力学模型、流体力学模型和经济学模型等，然后在数学模型的基础上进行控制器的设计，为满足不同的要求就要应用不同的控制方法（传统控制控制、PID控制、非线性控制、自适应控制和优化控制等）,紧接着选择测试平台，可以是仿真平台、实验室模型样机和真实设备等。最后不断将实验结果与模型比较，对数学模型不断的验证和更新。

涉及的内容： 动态系统建模：

电力，KCL，KVL
流体
热力学
机械系统

拉普拉斯+微分方程 时域分析频域分析

2.电路系统建模

基础元件：

流速：
$equation?tex=i%3D%5Cfrac%7Bdq%7D%7Bdt%7D%5C%5C$

电阻电压：
equation?tex=e_R%3DiR%5C%5C

电量:
equation?tex=q%3DCe_c%5C%5C
$equation?tex=e_c%3D%5Cfrac%7B1%7D%7BC%7Dq%3D%5Cfrac%7B1%7D%7BC%7D%5Cint_0%5Etidt%5C%5C$

电感：
$equation?tex=e_L%3DL%5Cfrac%7Bdi%7D%7Bdt%7D%3DLi%5E%7B%5Cprime%7D%5C%5C$

基尔霍夫定律 KCL ：所有进入某节点的电流的总和等于所有离开这个节点的电流的总和
equation?tex=i_1%2Bi_2-i_3-i_4%3D0%5C%5C

KVL ：沿着闭合回路所有元件两端的电压的代数和为零
equation?tex=e_R-e%3D0%5C%5C

KVL
equation?tex=e_L%2Be_C%2Be_R-e%3D0%5C%5C
$equation?tex=Li%5E%7B%5Cprime%7D%2B%5Cfrac%7B1%7D%7BC%7D%5Cint_0%5Etidt%2BiR%3De%5C%5C$

两边求导
$equation?tex=Li%5E%7B%5Cprime%5Cprime%7D%2Bi%5E%7B%5Cprime%7DR%2B%5Cfrac%7B1%7D%7BC%7Di%3De%5E%7B%5Cprime%7D%5C%5C$

equation?tex=L%3D2H+
$equation?tex=C%3D%5Cfrac%7B1%7D%7B4%7DF$
equation?tex=R_1%3D1%5COmega
equation?tex=R_2%3D3%5COmega

loop 1 :
equation?tex=e_L%2Be_C-e_i%3D0%5C%5C

loop 2 :
equation?tex=e_%7BR1%7D%2Be_%7BR2%7D-e_C%3D0%5C%5C

合并：
equation?tex=e_L%2Be_%7BR1%7D%2Be_%7BR2%7D-e_i%3D0%5C%5C

这是一个大圈，因此在用KVL时，不一定都用小圈，也可用大圈。
equation?tex=e_L%3DLi_1%5E%7B%5Cprime%7D%3D2i_1%5E%7B%5Cprime%7D
$equation?tex=e_C%3D%5Cfrac%7B1%7D%7BC%7D%5Cint_0%5Et+%28i_1-i_2%29+dt%3D4%5Cint_0%5Et+%28i_1-i_2%29+dt$
equation?tex=e_%7BR1%7D%3Di_2R_1%3Di_2

loop 1 :
equation?tex=2Li_1%5E%7B%5Cprime%7D%2B4%5Cint_0%5Et+%28i_1-i_2%29+dt-e_i%3D0%5Ctag%7B1%7D%5C%5C

loop 2 :
equation?tex=4i_2-4%5Cint_0%5Et+%28i_1-i_2%29+dt%3D0%5Ctag%7B2%7D%5C%5C

由（1）（2）式得
equation?tex=2i_1%5E%7B%5Cprime%7D%2B4i_2-e_i%3D0%5Ctag%7B3%7D%5C%5C

由（2）得
equation?tex=i_2%5E%7B%5Cprime%7D%3Di_1-i_2%5C%5C+i_2%5E%7B%5Cprime%5Cprime%7D%3Di_1%5E%7B%5Cprime%7D-i_2%5E%7B%5Cprime%7D+%5Ctag%7B4%7D%5C%5C

由（3）（4）式得
equation?tex=2%28i_2%5E%7B%5Cprime%5Cprime%7D%2Bi_2%5E%7B%5Cprime%7D%29%2B4i_2%3De_i%5Ctag%7B5%7D%5C%5C

求
equation?tex=e_o 和
equation?tex=e_i 的关系
equation?tex=e_o%3De_R%3D3i_2%5Ctag%7B6%7D%5C%5C

由（5）（6）式得
equation?tex=2%28e_o%5E%7B%5Cprime%5Cprime%7D%2Be_o%5E%7B%5Cprime%7D%29%2B4e_o%3D3e_i%5C%5C

小结： KVL列方程，然后消掉自己定义的电流

loop 1:
$equation?tex=i_1R1%2B%28i_1-i_2%29R2-e_i%3D0%5C%5C+%5Cfrac%7B16%7D%7B3%7Di_1-4i_2-e_i%3D0%5Ctag%7B1%7D%5C%5C$

loop 2:
equation?tex=%28i_2-i_3%29R3%2Bi_2R4-%28i_1-i_2%29R2%3D0%5C%5C+-4i_1%2B9i_2-3i_3%3D0%5Ctag%7B2%7D%5C%5C

loop 3:
$equation?tex=%5Cfrac%7B1%7D%7BC%7D%5Cint_0%5Et+i_3+dt-%28i_2-i_3%29R3%3D0%5C%5C+i_3%3D3C%28i_2%5E%7B%5Cprime%7D-i_3%5E%7B%5Cprime%7D%29%5Ctag%7B3%7D%5C%5C$

我们的目的是找到
equation?tex=e_i 和
equation?tex=e_o 的关系,而
equation?tex=e_o%3D2i_2 ,因此想先消去
equation?tex=i_1 和
equation?tex=i_3 ，再消去
equation?tex=i_2

由（1）（2）式得
$equation?tex=-4%28%5Cfrac%7B3%7D%7B4%7Di_2%2B%5Cfrac%7B3%7D%7B16%7De_i%29%2B9i_2-3i_3%3D0%5C%5C+2i_2-i_3-%5Cfrac%7B1%7D%7B4%7De_i%3D0%5Ctag%7B4%7D%5C%5C$

由（3）（4）式得
$equation?tex=2i_2-3C%28i_2%5E%7B%5Cprime%7D-i_3%5E%7B%5Cprime%7D%29-%5Cfrac%7B1%7D%7B4%7De_i%3D0%5Ctag%7B5%7D%5C%5C$

（5）式还有
equation?tex=i_3%5E%7B%5Cprime%7D 没消去，为了不引入新的变量，对（4）式求导
$equation?tex=2i_2%5E%7B%5Cprime%7D-i_3%5E%7B%5Cprime%7D-%5Cfrac%7B1%7D%7B4%7De_i%5E%7B%5Cprime%7D%3D0%5Ctag%7B6%7D%5C%5C$

由（5）（6）式得
$equation?tex=2i_2-3C%28i_2%5E%7B%5Cprime%7D-%282i_2%5E%7B%5Cprime%7D-%5Cfrac%7B1%7D%7B4%7De_i%5E%7B%5Cprime%7D%29%29-%5Cfrac%7B1%7D%7B4%7De_i%3D0%5C%5C+2i_2%2B3Ci_2%5E%7B%5Cprime%7D-%5Cfrac%7B3%7D%7B4%7DCe_i%5E%7B%5Cprime%7D-%5Cfrac%7B1%7D%7B4%7De_i%3D0%5Ctag%7B7%7D%5C%5C$

只有电流
equation?tex=i_2 ，这样就可以引入
equation?tex=e_o 了
$equation?tex=e_o%2B%5Cfrac%7B3%7D%7B2%7DCe_o%5E%7B%5Cprime%7D%3D%5Cfrac%7B1%7D%7B4%7De_i%2B%5Cfrac%7B3%7D%7B4%7DCe_i%5E%7B%5Cprime%7D%5Ctag%7B8%7D%5C%5C$

3.流体系统建模

流体系统的几个基本元素： 此处默认为不可压缩的均质流体
equation?tex=%5Cbegin%7Barray%7D%5Bb%5D%7B%7Cc%7Cc%7C%7D+%5Chline++%E5%AF%86%E5%BA%A6+%26+%5Crho+%26+kg%2Fm%5E3+%5C%5C+%5Chline+%E6%B5%81%E9%87%8Fflow+rate%26++q++%26+m%5E3%2Fs%5C%5C+%5Chline%E4%BD%93%E7%A7%AF%26v%26+m%5E3%5C%5C+%5Chline%E9%AB%98%E5%BA%A6hight%26h%26m+%5C%5C+%5Chline%E5%8E%8B%E5%BC%BApressure%26P%26+N%2Fm%5E2%5C%5C+%5Cend%7Barray%7D%5C%5C

压强有三个概念，比如说对于容器的液体来说，它的高度是
equation?tex=h ，横截面积是
equation?tex=A ，由流体重力产生的压强称之为静压（Hydrostatic Pressure）
$equation?tex=%5Cbegin%7Baligned%7D+P_%7BHydro%7D+%26+%3D+%5Cfrac%7BF_%7BHydro%7D%7D%7BA%7D++%3D+%5Cfrac%7Bmg%7D%7BA%7D++%3D+%5Cfrac%7B%5Crho+V+g%7D%7BA%7D%5C%5C++%26+%3D+%5Cfrac%7B%5Crho+Ah+g%7D%7BA%7D++%3D+%5Cfrac%7B%5Crho++g+h%7D%7BA%7D+%3D+%5Crho++g+h+%5Cend%7Baligned%7D%5C%5C$

除了液体的压强以外还有大气压强，绝对压强（Absolute Pressure）
equation?tex=P_%7Babs%7D%3DP_a%2BP_%7BHydro%7D%3DP_a%2B+%5Crho++g+h%5C%5C

测量出来的压力称为表压（Gauge Pressure）
equation?tex=P_%7Bgauge%7D%3DP_%7Babs%7D-P_a%3D+%5Crho++g+h%5C%5C

流阻Fluid Resistance 产生流阻的原因是流体在流动的过程中，通过一些管道连接等，这些都会阻碍流体的流动，因此会产生压差，压差和流量相关
equation?tex=P_1-P_2%3D%5Crho+q+R%5C%5C
equation?tex=%5Crho+q%3Dkg%2Fm%5E3%5Ccdot+m%5E3%2Fs%3Dkg%2Fs :每秒钟通过横截面的流体的质量，两边的压力差越大，每秒钟流过的流体的越多。

流阻和电阻的概念非常相似
equation?tex=e_1-e_2%3Di+R%5C%5C

理想压源
equation?tex=P_2%3DP_1%2BP_s%5C%5C

基本法则-质量守恒定律Conseration of Mass 有了基本元素，还需要基本法则把它们联系在一起，就像电路当中有基尔霍夫定律，在力学当中有牛顿定律一样，这里面我们用到的是质量守恒定律，容器内流体质量的变化
$equation?tex=%5Cfrac%7B%5Cmathrm%7Bd%7D+m%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D%5Cdot%7Bm%7D_%7Bin%7D-%5Cdot%7Bm%7D_%7Bout%7D%5C%5C$

式子两边除以
equation?tex=%5Crho
$equation?tex=%5Cfrac%7B%5Cmathrm%7Bd%7D+V%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D%5Cdot%7Bq%7D_%7Bin%7D-%5Cdot%7Bq%7D_%7Bout%7D%5C%5C$
$equation?tex=%5CRightarrow+A%5Cfrac%7B%5Cmathrm%7Bd%7D+h%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D%5Cdot%7Bq%7D_%7Bin%7D-%5Cdot%7Bq%7D_%7Bout%7D%5C%5C$
$equation?tex=%5CRightarrow+%5Cfrac%7B%5Cmathrm%7Bd%7D+h%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D%5Cfrac%7B1%7D%7BA%7D%28%5Cdot%7Bq%7D_%7Bin%7D-%5Cdot%7Bq%7D_%7Bout%7D%29%5C%5C$

容器底部受到的压力
equation?tex=P%3DP_a%2B%5Crho+gh%5C%5C

其动态方程为
$equation?tex=%5Cbegin%7Baligned%7D++%5Cfrac%7B%5Cmathrm%7Bd%7D+P%7D%7B%5Cmathrm%7Bd%7D+t%7D+++%26%3D+%5Cfrac%7B%5Cmathrm%7Bd%7D+%7D%7B%5Cmathrm%7Bd%7D+t%7D%28P_a%2B%5Crho+gh%29%5C%5C++%26+%3D+%5Crho+g%5Cfrac%7B%5Cmathrm%7Bd%7D+h%7D%7B%5Cmathrm%7Bd%7D+t%7D++%5C%5C%26+%3D+%5Cfrac%7B+%5Crho+g%7D%7BA%7D%28%5Cdot%7Bq%7D_%7Bin%7D-%5Cdot%7Bq%7D_%7Bout%7D%29+%5Cend%7Baligned%7D%5C%5C$

进口处为
equation?tex=q_%7Bin%7D ,出口处
equation?tex=q_%7Bout%7D ，容器得横截面积为
equation?tex=A ，出口流阻为u
equation?tex=R ，求液面高度的动态方程
$equation?tex=%5Cfrac%7Bdh%7D%7Bdt%7D$ .

由质量守恒定律
$equation?tex=%5Cfrac%7B%5Cmathrm%7Bd%7D+V%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D%5Cdot%7Bq%7D_%7Bin%7D-%5Cdot%7Bq%7D_%7Bout%7D%5C%5C$
$equation?tex=%5CRightarrow+%5Cfrac%7B%5Cmathrm%7Bd%7D+h%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D%5Cfrac%7B1%7D%7BA%7D%28%5Cdot%7Bq%7D_%7Bin%7D-%5Cdot%7Bq%7D_%7Bout%7D%29%5C%5C$

流阻压差
equation?tex=P_1-P_a%3D%5Crho+q_%7Bout%7DR%5C%5C
$equation?tex=+%5Cbegin%7Baligned%7D+q_%7Bout%7D+%26%3D%5Cfrac%7BP_1-P_a%7D%7B%5Crho+R%7D%5C%5C+%26%3D%5Cfrac%7BP_a-%5Crho+gh-P_a%7D%7B%5Crho+R%7D%5C%5C+%26%3D%5Cfrac%7Bgh%7D%7B+R%7D+%5Cend%7Baligned%7D%5C%5C$
$equation?tex=%5CRightarrow+%5Cfrac%7B%5Cmathrm%7Bd%7D+h%7D%7B%5Cmathrm%7Bd%7D+t%7D+%3D%5Cfrac%7B%5Cdot%7Bq%7D_%7Bin%7D%7D%7BA%7D-%5Cfrac%7Bgh%7D%7B+AR%7D%5C%5C$

4.拉普拉斯变换

拉普拉斯变换是控制理论的基础，它广泛的应用于工程分析当中，它可以把时域(
equation?tex=t )上的函数变换到复数域(
equation?tex=s%3D%5Csigma%2Bjw )上，从而大大简化系统分析的难度和复杂程度。
equation?tex=f%28t%29+%5Cto+F%28s%29%5C%5C

先从一个简单的电路系统开始，它的动态方程
$equation?tex=e%5E%7B%27%7D%3DLi%5E%7B%27%27%7D%2BRi%5E%7B%27%7D%2B%5Cfrac%7B1%7D%7BC%7Di%5C%5C$

定义系统的输入为
equation?tex=e ，输出为
equation?tex=i ，分析电流的变化。本质上就是求解微分方程的过程，假设
equation?tex=g%28t%29 就是变化过程，
隐含了系统的特征，就是微分方程表现出来的内容，三者的关系其实是一个卷积的过程。因此分析这样一个系统，它涉及到了卷积和微分方程，分析和计算起来都非常麻烦，而且不是很直观。拉普拉斯变换可以帮助我们解决这些问题，通过拉普拉斯变换，微分方程变成了代数方程，卷积运算变成了乘法运算。

对时域函数
equation?tex=f%28t%29 作拉普拉斯变换：
equation?tex=%5Cmathcal%7BL%5Bf%28t%29%5D%7D%3DF%28s%29%3D%5Cint_0%5E%5Cinfty+f%28t%29e%5E%7B-st%7Ddt%5C%5C
是一个平面图形，经过拉普拉斯变换后三维的复数域。当
equation?tex=%5Csigma%3D0 时，从箭头的方向看过去，就是傅里叶变换，可以看到拉普拉斯变换和傅里叶变换的关系。
equation?tex=F%28s%29%3DF%28w%29%3D%5Cint_0%5E%5Cinfty+f%28t%29e%5E%7B-jwt%7Ddt%5C%5C

从上向下看就是复平面，做工程的往往会关注系统的极点和零点在复平面上的位置.

指数函数
equation?tex=f%28t%29%3De%5E%7B-at%7D

的拉普拉斯变换 ：
$equation?tex=%5Cbegin%7Baligned%7D+%5Cmathcal%7BL%7D%5Bf%28t%29%5D++%26+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D+e%5E%7B-st%7Ddt%5C%5C++%26+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-%28a%2Bs%29t%7Ddt%5C%5C++%26+%3D+-%5Cfrac%7B1%7D%7Ba%2Bs%7D+%5Cleft.+e%5E%7B-%28a%2Bs%29t%7D%5Cright%7C_0%5E%5Cinfty%5C%5C++%26+%3D+%5Clim_%7Bt+%5Cto+0%7D-%5Cfrac%7B1%7D%7Ba%2Bs%7De%5E%7B-%28a%2Bs%29t%7D-%28-%5Cfrac%7B1%7D%7Ba%2Bs%7D%29%5C%5C++%26+%3D+%5Cfrac%7B1%7D%7Ba%2Bs%7D+%5Cend%7Baligned%7D%5C%5C$

拉普拉斯变换的重要性质：符合线性变换，线性变换符合叠加原理
equation?tex=%5Cmathcal%7BL%7D%5Baf%28t%29%2Bbg%28t%29%5D%3DaF%28s%29%2BbG%28s%29+%5C%5C

正弦
equation?tex=%5Csin+at

的拉普拉斯变换 ：根据欧拉公式转化为复指数
equation?tex=e%5E%7Bi%5Ctheta%7D%3D%5Ccos+%5Ctheta%2Bi%5Csin+%5Ctheta%5C%5C
equation?tex=e%5E%7B-i%5Ctheta%7D%3D%5Ccos+%5Ctheta-i%5Csin+%5Ctheta%5C%5C

两式相减：
$equation?tex=e%5E%7Bi%5Ctheta%7D-e%5E%7B-i%5Ctheta%7D%3D2i%5Csin+%5Ctheta%5C%5C+%5Csin%5Ctheta%3D%5Cfrac%7Be%5E%7Bi%5Ctheta%7D-e%5E%7B-i%5Ctheta%7D%7D%7B2i%7D%5C%5C$
$equation?tex=%5Csin+at%3D%5Cfrac%7Be%5E%7Biat%7D-e%5E%7B-iat%7D%7D%7B2i%7D%5C%5C$

因为拉普拉斯变换是一个线性变换：
$equation?tex=%5Cbegin%7Baligned%7D+%5Cmathcal%7BL%7D%5Cleft+%5B+%5Cfrac%7Be%5E%7Biat%7D%7D%7B2i%7D+%5Cright+%5D+-%5Cmathcal%7BL%7D%5Cleft+%5B+%5Cfrac%7Be%5E%7B-iat%7D%7D%7B2i%7D+%5Cright+%5D++%26+%3D++%5Cfrac%7B1%7D%7B2i%7D+%5Cleft+%28+%5Cmathcal%7BL%7D%5Cleft+%5B+e%5E%7Biat%7D+%5Cright+%5D+-%5Cmathcal%7BL%7D%5Cleft+%5B+e%5E%7B-iat%7D+%5Cright+%5D++%5Cright+%29+%5C%5C%3Cbr+%2F%3E%26+%3D++%5Cfrac%7B1%7D%7B2i%7D+%5Cleft+%28+%5Cfrac%7B1%7D%7Bs-ai%7D-%5Cfrac%7B1%7D%7Bs%2Bai%7D+%5Cright+%29+%5C%5C++%26+%3D+%5Cfrac%7B1%7D%7B2i%7D+%5Cleft+%28+%5Cfrac%7B2ai%7D%7B%28s-ai%29%28s%2Bai%29%7D+%5Cright+%29%5C%5C%3Cbr+%2F%3E%26+%3D++%5Cfrac%7Ba%7D%7Bs%5E2%2Ba%5E2%7D++%5Cend%7Baligned%7D%5C%5C$

导数的拉普拉斯变换：
equation?tex=%5Cmathcal%7BL%7D%5Cleft+%5B+f%5E%7B%27%7D%28t%29+%5Cright+%5D+%3D%5Cint_0%5E%7B%2B%5Cinfty%7Df%5E%7B%27%7D%28t%29e%5E%7B-st%7Ddt%5C%5C

复合函数求积分，用到分部积分：
equation?tex=%5Cint_0%5E%7B%2B%5Cinfty%7Df%5E%7B%27%7D%28t%29g%28t%29dt%3Df%28t%29g%28t%29-%5Cint_0%5E%7B%2B%5Cinfty%7Df%28t%29g%5E%7B%27%7D%28t%29dt%5C%5C
equation?tex=g%28t%29%3De%5E%7B-st%7D%2Cg%5E%7B%27%7D%28t%29%3D-se%5E%7B-st%7D

equation?tex=F%28s%29 为拉普拉斯变换，很多时候都把初始条件设置为
equation?tex=f%280%29%3D0 。

同理可得
equation?tex=%5Cmathcal%7BL%7D%5Cleft+%5B+f%5E%7B%27%27%7D%28t%29+%5Cright+%5D+%3D++s%5E2F%28s%29-sf%280%29-f%5E%7B%27%7D%280%29%5C%5C
$equation?tex=%5Cmathcal%7BL%7D%5Cleft+%5B+%5Cint_0%5E%5Cinfty+f%28t%29dt+%5Cright+%5D+%3D++%5Cfrac%7B1%7D%7Bs%7D+F%28s%29%5C%5C$

卷积的拉普拉斯变换 能够将卷积运算变成乘积运算，大大简化运算和分析的复杂程度。
equation?tex=%5Cmathcal%7BL%7D%5Cleft+%5B+f%28t%29%2Ag%28t%29+%5Cright+%5D%3DF%28s%29G%28s%29+%5C%5C

回到最初的电路的动态方程：
$equation?tex=e%5E%7B%27%7D%3DLi%5E%7B%27%27%7D%2BRi%5E%7B%27%7D%2B%5Cfrac%7B1%7D%7BC%7Di%5C%5C$

两端作拉普拉斯变换：
$equation?tex=sE%28s%29%3DLs%5E2I%28s%29%2BsRI%28s%29%2B%5Cfrac%7B1%7D%7BC%7DI%28s%29%5C%5C$
$equation?tex=I%28s%29%3D%5Cfrac%7Bs%7D%7BLs%5E2%2BsR%2B%5Cfrac%7B1%7D%7BC%7D%7DE%28s%29+%5C%5C$

可以看到，经过拉普拉斯变换把微分方程变换为代数方程，它只有加减乘除，非常的简单。下图方框称之为传递函数。

5.拉普拉斯变换的收敛域（ROC）与逆变换（ILT）

指数函数的拉普拉斯变换 ：
$equation?tex=%5Cbegin%7Baligned%7D+%5Cmathcal%7BL%7D%5Bf%28t%29%5D++%26+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D+e%5E%7B-st%7Ddt%5C%5C++%26+%3D+%5Cint_0%5E%5Cinfty+e%5E%7B-%28a%2Bs%29t%7Ddt%5C%5C++%26+%3D+-%5Cfrac%7B1%7D%7Ba%2Bs%7D+%5Cleft.+e%5E%7B-%28a%2Bs%29t%7D%5Cright%7C_0%5E%5Cinfty%5C%5C++%5Cend%7Baligned%7D%5C%5C$

如果
equation?tex=s%3D-2a%EF%BC%8C%5Cmathcal%7BL%7D%5Bf%28t%29%5D 是发散的
equation?tex=+%5Cint_0%5E%5Cinfty+e%5E%7B-%28a%2Bs%29t%7Ddt%3D+%5Cint_0%5E%5Cinfty+e%5E%7Bat%7Ddt%5C%5C

加上限制条件，收敛域ROC（Region of lonvergence）,把
equation?tex=s%3D%5Csigma%2Bjw%E4%BB%A3%E5%85%A5
equation?tex=%5Cint_0%5E%5Cinfty+e%5E%7B-at%7D+e%5E%7B-%EF%BC%88%5Csigma%2Bjw%EF%BC%89t%7Ddt%3D%5Cint_0%5E%5Cinfty+e%5E%7B-%28a%2B%5Csigma%29t%7De%5E%7B-jwt%7Ddt%5C%5C

根据欧拉公式：
equation?tex=e%5E%7B-jwt%7D%3D%5Ccos+wt-i%5Csin+wt%5C%5C
equation?tex=%5Cleft+%7C+e%5E%7B-jwt%7D+%5Cright+%7C+%3D%5Ccos%5E2+wt%2B%5Csin%5E2+wt%3D1%5C%5C
这一项仅仅带来的是振动，并不会对系统的收敛产生影响。因此收敛域为
equation?tex=-%28a%2B%5Csigma%29%3C0%5C+%5CRightarrow+%5Csigma+%3E-a%5C+%5CRightarrow+Re%28s%29+%3E-a%5C

前面我们已经知道，拉普拉斯能简化运算和分析，为什么还需要微分方程？因为微分方程能够描述动态世界的数学手段。

在经典控制理论和现代控制理论当中，研究对象一般是常系数微分方程，对应的系统就是线性时不变系统，如果是非线性系统的话，一般会在平衡点附近作线性化处理，或者直接采用非线性分析手段。

用拉普拉斯变换求解微分方程的三个步骤：

时域转化到复频域
,这里用到拉普拉斯变换
求解代数方程
把结果从复频域转回时域，用到拉普拉斯逆变换

拉普拉斯逆变换

例子
$equation?tex=%5Cbegin%7Baligned%7D+F%28s%29++%26+%3D+%5Cfrac%7B-s%2B5%7D%7Bs%5E2%2B5s%2B4%7D%5C%5C++%26+%3D+%5Cfrac%7B-s%2B5%7D%7B%28s%2B4%29%28s%2B1%29%7D%5C%5C++%26+%3D+%5Cfrac%7BA%7D%7Bs%2B4%7D%2B%5Cfrac%7BB%7D%7Bs%2B1%7D%5C%5C++%26+%3D+%5Cfrac%7BA%28s%2B1%29%2BB%28s%2B4%29%7D%7B%28s%2B4%29%28s%2B1%29%7D%5C%5C+%5Cend%7Baligned%7D%5C%5C$
equation?tex=%5CRightarrow+A%28s%2B1%29%2BB%28s%2B4%29%3D-s%2B5%5C%5C
equation?tex=s%3D-1%3A%5Cquad+B%28-1%2B4%29%3D1%2B5+%5CRightarrow+B%3D2
equation?tex=s%3D-4%3A%5Cquad+-3A%3D9+%5CRightarrow+A%3D-3
$equation?tex=F%28s%29++%3D++%5Cfrac%7B-3%7D%7Bs%2B4%7D%2B%5Cfrac%7B2%7D%7Bs%2B1%7D%5C%5C$

两端拉普拉斯逆变换：
$equation?tex=%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft+%5B+F%28s%29+%5Cright+%5D+++%3D+%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft+%5B+%5Cfrac%7B-3%7D%7Bs%2B4%7D%2B%5Cfrac%7B2%7D%7Bs%2B1%7D+%5Cright+%5D+%5C%5C+%3D-3e%5E%7B-4t%7D%2B2e%5E%7B-t%7D%5C%5C$
equation?tex=s%3D-1%2Cs%3D-4 称为极点Pole

$equation?tex=%5Cbegin%7Baligned%7D+F%28s%29++%26+%3D+%5Cfrac%7B4s%2B8%7D%7Bs%5E2%2B2s%2B5%7D%5C%5C++%26+%3D+%5Cfrac%7B4s%2B8%7D%7B%28s%2B1%2B2i%29%28s%2B1-2i%29%7D%5C%5C++%26+%3D+%5Cfrac%7BA%7D%7Bs%2B1%2B2i%7D%2B%5Cfrac%7BB%7D%7Bs%2B1-2i%7D%5C%5C++%5Cend%7Baligned%7D%5C%5C$
equation?tex=%5CRightarrow+A%3Di%2B2%2CB%3D-i%2B2
$equation?tex=F%28s%29++%3D+%5Cfrac%7Bi%2B2%7D%7Bs%2B1%2B2i%7D%2B%5Cfrac%7B-i%2B2%7D%7Bs%2B1-2i%7D%5C%5C$

其中，根据欧拉公式有
equation?tex=e%5E%7Bi%5Ctheta%7D%3D%5Ccos+%5Ctheta%2Bi%5Csin+%5Ctheta%5Ctag%7B1%7D%5C%5C
equation?tex=e%5E%7B-i%5Ctheta%7D%3D%5Ccos+%5Ctheta-i%5Csin+%5Ctheta%5Ctag%7B2%7D%5C%5C

(2)-(1)
$equation?tex=%5Csin+2t%3D-%5Cfrac%7Be%5E%7B-2it%7D-e%5E%7B2it%7D%7D%7B2i%7D%5C%5C$
$equation?tex=%5Ccos+2t%3D%5Cfrac%7Be%5E%7B-2it%7D%2Be%5E%7B2it%7D%7D%7B2%7D%5C%5C$

6.拉&传&微的关系

重点讲解传递函数

这部分内容非常重要，对经典控制理论、根轨迹、伯德图、信号处理等学习都有很大的帮助，因为都是从这里伸展出去的。

流体系统

$equation?tex=%5Cfrac%7B%5Cmathrm%7Bd%7D+h%7D%7B%5Cmathrm%7Bd%7D+t%7D%2B+%5Cfrac%7Bg%7D%7B+AR%7Dh%3D%5Cfrac%7B%5Cdot%7Bq%7D_%7Bin%7D%7D%7BA%7D%5C%5C$

令A=1
equation?tex=x%3Dh 输出
equation?tex=u%3Dq_%7Bin%7D 输入
$equation?tex=%5Cdot+x%28t%29%2B+%5Cfrac%7Bg%7D%7B+R%7Dx%28t%29%3Du%28t%29%5C%5C$

两端作拉普拉斯变换：
$equation?tex=sX%28s%29%2B%5Cfrac%7Bg%7D%7B+R%7DX%28s%29%3DU%28s%29%2C%5Cquad+x%280%29%3D0%5C%5C$
$equation?tex=s%2B%5Cfrac%7Bg%7D%7B+R%7D%5Cfrac%7BX%28s%29%7D%7BU%28s%29%7D%3D%5Cfrac%7B1%7D%7Bs%2B%5Cfrac%7Bg%7D%7B+R%7D%7D%3DG%28s%29%5C%5C$
equation?tex=G%28s%29 称为传递函数

假设系统的输入为常数，对常数作拉普拉斯变换
equation?tex=u%28t%29%3DC%3DCe%5E0
$equation?tex=%5Cmathcal%7BL%7D%5Bu%28t%29%5D%3DC%5Cfrac%7B1%7D%7Bs%2B0%7D+%3DC%5Cfrac%7B1%7D%7Bs%7D+%5C%5C$
$equation?tex=%5Cbegin%7Baligned%7D+X%28s%29++%26+%3D+U%28s%29G%28s%29++%3D+C%5Cfrac%7B1%7D%7Bs%7D+%5Ccdot+%5Cfrac%7B1%7D%7Bs%2B%5Cfrac%7Bg%7D%7B+R%7D%7D%5C%5C+%26+%3D+C%5Cleft+%28+%5Cfrac%7BA%7D%7Bs%7D%2B+%5Cfrac%7BB%7D%7Bs%2B%5Cfrac%7Bg%7D%7B+R%7D%7D+%5Cright+%29+%5C%5C+%26+%3D+C%5Cfrac%7BA%28s%2B%5Cfrac%7Bg%7D%7B+R%7D%29%2BBs%7D%7Bs%28s%2B%5Cfrac%7Bg%7D%7B+R%7D%29%7D++%5Cend%7Baligned%7D%5C%5C$
$equation?tex=s%3D0%3AA%3D%5Cfrac%7BR%7D%7Bg%7D$
$equation?tex=s%3D-%5Cfrac%7Bg%7D%7B+R%7D%3AB%3D-%5Cfrac%7BR%7D%7Bg%7D$
$equation?tex=%5Cbegin%7Baligned%7D+X%28s%29++%3D+C%5Cleft+%28+%5Cfrac%7BR%7D%7Bg%7D%5Cfrac%7B1%7D%7Bs%7D-%5Cfrac%7BR%7D%7Bg%7D+%5Cfrac%7B1%7D%7Bs%2B%5Cfrac%7Bg%7D%7B+R%7D%7D+%5Cright+%29++%5Cend%7Baligned%7D%3D%5Cfrac%7BCR%7D%7Bg%7D%5Cleft++%28+%5Cfrac%7B1%7D%7Bs%7D-%5Cfrac%7B1%7D%7Bs%2B%5Cfrac%7Bg%7D%7B+R%7D%7D+%5Cright+%29%5C%5C$
$equation?tex=%5Cbegin%7Baligned%7D+x%28t%29+%26+%3D+%5Cfrac%7BCR%7D%7Bg%7D%5Cleft++%28+e%5E%7B0t%7D-e%5E%7B-%5Cfrac%7Bg%7D%7B+R%7Dt%7D+%5Cright+%29%5C%5C++%26+%3D+%5Cfrac%7BCR%7D%7Bg%7D%5Cleft++%28+1-e%5E%7B-%5Cfrac%7Bg%7D%7B+R%7Dt%7D+%5Cright+%29+%5Cend%7Baligned%7D%5C%5C$

当时间
equation?tex=t+%5Cto+%5Cinfty ，系统收敛到
$equation?tex=%5Cfrac%7BCR%7D%7Bg%7D$ 。系统的关键点在指数部分，
equation?tex=0t 不变，
$equation?tex=%5Cfrac%7BCR%7D%7Bg%7Dt$ 随着时间不断的衰减，所以系统是稳定的。

7.一阶系统的单位阶跃响应

流体系统

动态方程：
$equation?tex=%5Cdot+x%28t%29%2B+%5Cfrac%7Bg%7D%7B+R%7Dx%28t%29%3Du%28t%29%5C%5C$

输出是一阶，输入是单位阶跃，称为一阶系统的单位阶跃响应 Unit Step Response.

一般形式：

equation?tex=u%28t%29%3D+%5Cbegin%7Bcases%7D+0%5Cquad+%2Ct%3D0+%5C%5C+1%5Cquad%2Ct%3E0+%5Cend%7Bcases%7D+
$equation?tex=%5Cmathcal%7BL%7D%5Bu%28t%29%5D%3D%5Cfrac%7B1%7D%7Bs%2B0%7D+%3D%5Cfrac%7B1%7D%7Bs%7D+%5C%5C$
$equation?tex=X%28s%29%3D%5Cfrac%7B1%7D%7Bs%7D%5Ccdot%5Cfrac%7Ba%7D%7Bs%2Ba%7D%3D%5Cfrac%7BA%7D%7Bs%7D%2B%5Cfrac%7BB%7D%7Bs%2Ba%7D%3D%5Cfrac%7BA%28s%2Ba%29%2BBs%7D%7Bs%28s%2Ba%29%7D%5C%5C$
equation?tex=A%28s%2Ba%29%2BBs%3Da
equation?tex=s%3D-a%3A B=-1
equation?tex=s%3D0%3A A=1

两边作拉普拉斯逆变换：
$equation?tex=%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft+%5B+X%28s%29+%5Cright+%5D+%3D%5Cmathcal%7BL%7D%5E%7B-1%7D+%5Cleft+%5B+%5Cfrac%7B1%7D%7Bs%7D-%5Cfrac%7B1%7D%7Bs%2Ba%7D++%5Cright+%5D%5C%5C$
equation?tex=x%28t%29%3D1-e%5E%7B-at%7D%5C%5C

a越大收敛越快。

时间常数 time constant
$equation?tex=t%3D%5Ctau+%3D%5Cfrac%7B1%7D%7Ba%7D%5C%5C$
$equation?tex=x%28%5Ctau%29%3D1-e%5E%7B-a%5Cfrac%7B1%7D%7Ba%7D%7D%3D1-e%5E%7B-1%7D%3D0.63%5C%5C$

即
$equation?tex=%5Ctau+%3D%5Cfrac%7B1%7D%7Ba%7D$ 的时候达到最终状态的63%。

有时候还会引入另一个概念-稳定时间(Steady State)（整定时间）Setting time
equation?tex=T_%7Bss%7D%3D4%5Ctau%5C%5C
$equation?tex=x%28T_%7Bss%7D%29%3Dx%28%5Cfrac%7B4%7D%7Ba%7D%29%3D1-e%5E%7B-a%5Cfrac%7B4%7D%7Ba%7D%7D%3D1-e%5E%7B-4%7D%3D0.98%5C%5C$

对于一阶线性系统来说，时间常数是特有的，因此可以用时间常数作系统识别。

根据上一节有：
$equation?tex=x%28t%29+++%3D+%5Cfrac%7BCR%7D%7Bg%7D%5Cleft++%28+1-e%5E%7B-%5Cfrac%7Bg%7D%7B+R%7Dt%7D+%5Cright+%29%5C%5C$

4秒钟达到稳定时间:
equation?tex=T_%7Bss%7D%3D4%5C%5C
equation?tex=%5CRightarrow+%5Ctau%3D1%5C%5C

系统的传递函数：
$equation?tex=G%28s%29%3D%5Cfrac%7B1%7D%7Bs%2B%5Cfrac%7Bg%7D%7BR%7D+%7D+%5C%5C$
$equation?tex=%5CRightarrow+%5Ctau%3D%5Cfrac%7BR%7D%7Bg%7D%3D1%5C%5C$
$equation?tex=+%5Cfrac%7BCR%7D%7Bg%7D%3D5+%5CRightarrow+C%3D5%5C%5C$

一阶系统与信号处理

一阶系统是一个低通滤波器，低通滤波器只反映了低频变化，高频变化则被过滤了。对于流体系统来说，容器内的液体就起到了抵抗高速变化的作用，是因为它有积累，所以说有积累的都是低通滤波器，它对高速变化不敏感。最典型的积累就是积分，如：
$equation?tex=%5Cint+%5Ccos+x+%2B%5Ccos+100x+dx%3D%5Csin+x+%2B%5Cfrac%7B1%7D%7B100%7D+%5Csin+100x+%2BC%5C%5C$

高频变化被缩放100倍，相当于被过滤掉了。所以说大家平时多做积累，有了容量以后面对高速变化 的世界才可以做到处乱不惊。

另一个角度

一阶线性时不变系统1st order LTI：
equation?tex=%5Cdot+x%2Bax%3Dau%2C%5Cquad+t%5Cge+0%2Cu%3D1%5C%5C
equation?tex=%5Cdot+x%3Da%281-x%29%5C%5C

equation?tex=x%28t%29 单调增，
逐渐减小，
增加速度减缓，最后为零，可以得到一样的图。

其他情况，
equation?tex=a%3C0%2Cx%280%29+%5Cne+0 等

Phase-Portrait

8.频率响应与滤波器

信号通过线性时不变系统后频率不变
equation?tex=M_i+%5Csin%28wt%2B%5Cphi_i%29%5Cto+M_o+%5Csin%28wt%2B%5Cphi_o%29%5C%5C

振幅响应 Magnitude Response：
$equation?tex=%5Cfrac%7BM_o%7D%7BM_i%7D%3DM+%5C%5C$

辐角响应 Phase Response：
equation?tex=%5Cphi_o-%5Cphi_i%3D%5Cphi%5C%5C

一般形式：
$equation?tex=%5Cbegin%7Baligned%7D+u%28t%29++%26+%3D+A%5Csin+wt+%2BB%5Ccos+wt%5C%5C++%26+%3D+%5Csqrt%7BA%5E2%2BB%5E2%7D%5Cleft+%28+%5Cfrac%7BA%7D%7B%5Csqrt%7BA%5E2%2BB%5E2%7D%7D+%5Csin+wt+%2B%5Cfrac%7BB%7D%7B%5Csqrt%7BA%5E2%2BB%5E2%7D%7D+%5Ccos+wt+%5Cright+%29+%5C%5C+%26+%3D+%5Csqrt%7BA%5E2%2BB%5E2%7D%5Cleft+%28+%5Ccos+%5Cphi_i+%5Csin+wt+%2B+%5Csin+%5Cphi_i+%5Ccos+wt+%5Cright+%29+%5C%5C+%26+%3D+%5Csqrt%7BA%5E2%2BB%5E2%7D+%5Csin+%28wt+%2B%5Cphi_i+%29%5C%5C+%26%3DM_i+%5Csin%28wt%2B%5Cphi_i%29+%5Cend%7Baligned%7D%5C%5C$

两边作拉普拉斯变换：
$equation?tex=%5Cbegin%7Baligned%7D+U%28s%29++%26+%3D+%5Cfrac%7BAw%7D%7Bs%5E2%2Bw%5E2%7D+%2B%5Cfrac%7BBs%7D%7Bs%5E2%2Bw%5E2%7D+%5C%5C++%26+%3D+%5Cfrac%7BAw%2BBs%7D%7Bs%5E2%2Bw%5E2%7D+%5C%5C++%26+%3D+%5Cfrac%7BAw%2BBs%7D%7B%28s%2Bjw%29%28s-jw%29%7D+%5Cend%7Baligned%7D+%5C%5C$

其中，
equation?tex=j%3D%5Csqrt%7B-1%7D
$equation?tex=G%28s%29%3D%5Cfrac%7BD%28s%29%7D%7BN%28s%29%7D+%3D%5Cfrac%7BD%28s%29%7D%7B%28s-p_1%29%28s-p_2%29%5Ccdots+%28s-p_n%29%7D+%5C%5C$
equation?tex=p_1%2Cp_2%2C%5Ccdots+%2Cp_n :极点Poles
$equation?tex=%5Cbegin%7Baligned%7D+X%28s%29++%26+%3D+U%28s%29G%28s%29%5C%5C++%26+%3D+%5Cfrac%7BAw%2BBs%7D%7B%28s%2Bjw%29%28s-jw%29%7D%5Ccdot++%5Cfrac%7BD%28s%29%7D%7B%28s-p_1%29%28s-p_2%29%5Ccdots+%28s-p_n%29%7D%5C%5C+%26+%3D+%5Cfrac%7BK_1%7D%7Bs%2Bjw%7D%2B++%5Cfrac%7BK_2%7D%7Bs-jw%7D%2B%5Cfrac%7BC_1%7D%7Bs-p_1%7D%2B%5Cfrac%7BC_2%7D%7Bs-p_2%7D%2B%5Ccdots%2B%5Cfrac%7BC_n%7D%7Bs-p_1%7D%5C%5C+%26+%3D+%5Cfrac%7BK_1%28s-jw%29N%28s%29%2BK_2%28s%2Bjw%29N%28s%29%2BC_1%28s%2Bjw%29%28s-jw%29%2BC_2%28s%2Bjw%29%28s-jw%29%28s-p_2%29%5Ccdots+%28s-p_n%29%2B%5Ccdots+%7D%7B%28s%2Bjw%29%28s-jw%29%28s-p_1%29%28s-p_2%29%5Ccdots+%28s-p_n%29%7D%5C%5C+%26+%3D+%5Cfrac%7BAw%2BBs%7D%7B%28s%2Bjw%29%28s-jw%29%28s-p_1%29%28s-p_2%29%5Ccdots+%28s-p_n%29%7D+%5Cend%7Baligned%7D+%5C%5C$

拉普拉斯逆变换：
equation?tex=x%28t%29%3DK_1e%5E%7B-jwt%7D%2BK_2e%5E%7Bjwt%7D%2BC_1e%5E%7Bp_1t%7D%2BC_2e%5E%7Bp_2t%7D%2B%5Ccdots+%2BC_ne%5E%7Bp_nt%7D%5C%5C

对于稳定系统，
equation?tex=p_1%2Cp_2%2C%5Ccdots+%2Cp_n 的实部小于0，有
equation?tex=X_%7Bss%7D%28t%29%3DK_1e%5E%7B-jwt%7D%2BK_2e%5E%7Bjwt%7D%5C%5C

ss:Steady State 稳态，由上式可以看出频率响应 就是稳态响应 。求
equation?tex=K_1%2CK_2 :
equation?tex=K_1%28s-jw%29N%28s%29%2BK_2%28s%2Bjw%29N%28s%29%2BC_1%28s%2Bjw%29%28s-jw%29%5C%5C%2BC_2%28s%2Bjw%29%28s-jw%29%28s-p_2%29%5Ccdots+%28s-p_n%29%2B%5Ccdots%3D%28Aw%2BBs%29D%28s%29%5C%5C
equation?tex=s%3D-jw
equation?tex=K_1%28-jw-jw%29N%28-jw%29%2B0%3D%28Aw-Bjw%29D%28-jw%29%5C%5C
$equation?tex=K_1%3D%5Cfrac%7BAw-Bjw%7D%7B-2jw%7D%5Ccdot+%5Cfrac%7BD%28-jw%29%7D%7BN%28-jw%29%7D%3D%5Cfrac%7BB%2BAj%7D%7B2%7DG%28-jw%29%5C%5C$
equation?tex=s%3Djw
$equation?tex=K_2%3D%5Cfrac%7BB-Aj%7D%7B2%7DG%28jw%29%5C%5C$

复数表达：
equation?tex=G%28jw%29%3D%5Cleft+%7C+G%28jw%29+%5Cright+%7C+e%5E%7Bj+%5Cphi_%7BG%7D%7D%5C%5C
$equation?tex=%5Cbegin%7Baligned%7D+X_%7Bss%7D%28t%29++%26+%3D+%5Cfrac%7BB%2BAj%7D%7B2%7D++%5Cleft+%7C+G%28jw%29+%5Cright+%7C+e%5E%7B-j+%5Cphi_%7BG%7D%7D+e%5E%7B-jwt%7D%5C%5C+%26+%2B+%5Cfrac%7BB-Aj%7D%7B2%7D++%5Cleft+%7C+G%28jw%29+%5Cright+%7C+e%5E%7Bj+%5Cphi_%7BG%7D%7D+e%5E%7Bjwt%7D%5C%5C+%26+%3D+%5Cfrac%7B1%7D%7B2%7D+%5Cleft+%7C+G%28jw%29+%5Cright+%7C%5Cleft+%28+%28B%2BAj%29e%5E%7B-%28%5Cphi_G+%2Bwt%29j%7D+%2B%28B-Aj%29e%5E%7B%28%5Cphi_G+%2Bwt%29j%7D%5Cright+%29+%5C%5C+%5Cend%7Baligned%7D%5C%5C$

欧拉公式：
equation?tex=%5Cbegin%7Baligned%7D+e%5E%7B-%28%5Cphi_G+%2Bwt%29j%7D++%26+%3D+%5Ccos+%28-%28%5Cphi_G+%2Bwt%29%29%2Bj+%5Csin+%28-%28%5Cphi_G+%2Bwt%29%29%5C%5C++%26+%3D+%5Ccos+%28%5Cphi_G+%2Bwt%29-j+%5Csin+%28%5Cphi_G+%2Bwt%29%5C%5C+e%5E%7B%28%5Cphi_G+%2Bwt%29j%7D++%26+%3D+%5Ccos+%28%5Cphi_G+%2Bwt%29%2Bj+%5Csin+%28%5Cphi_G+%2Bwt%29+%5Cend%7Baligned%7D%5C%5C
$equation?tex=%5Cbegin%7Baligned%7D+X_%7Bss%7D%28t%29+%3D+%5Cfrac%7B1%7D%7B2%7D+%5Cleft+%7C+G%28jw%29+%5Cright+%7C+%28+%26+B%5Ccos+%28%5Cphi_G+%2Bwt%29-Bj+%5Csin+%28%5Cphi_G+%2Bwt%29+%5C%5C+%2B%26+Aj+%5Ccos+%28%5Cphi_G+%2Bwt%29%2BA+%5Csin+%28%5Cphi_G+%2Bwt%29%5C%5C+%2B%26B%5Ccos+%28%5Cphi_G+%2Bwt%29%2BBj+%5Csin+%28%5Cphi_G+%2Bwt%29+%5C%5C-++%26+Aj+%5Ccos+%28%5Cphi_G+%2Bwt%29%2BA+%5Csin+%28%5Cphi_G+%2Bwt%29%29%5C%5C+%5Cend%7Baligned%7D%5C%5C$
$equation?tex=%5Cbegin%7Baligned%7D+X_%7Bss%7D%28t%29++%26++%3D+%5Cfrac%7B1%7D%7B2%7D+%5Cleft+%7C+G%28jw%29+%5Cright+%7C+%282B%5Ccos+%28%5Cphi_G+%2Bwt%29%2B2A+%5Csin+%28%5Cphi_G+%2Bwt%29%29%5C%5C+%26%3D%5Cleft+%7C+G%28jw%29+%5Cright+%7C%5Csqrt%7BA%5E2%2BB%5E2%7D%5Cleft+%28+%5Cfrac%7BB%7D%7B%5Csqrt%7BA%5E2%2BB%5E2%7D%7D+%5Ccos+%28%5Cphi_G+%2B+wt%29%2B+%5Cfrac%7BA%7D%7B%5Csqrt%7BA%5E2%2BB%5E2%7D%7D+%5Csin+%28%5Cphi_G+%2B+wt%29++%5Cright+%29+%5C%5C+%26+%3D+%5Cleft+%7C+G%28jw%29+%5Cright+%7C+M_i+%5Csin%28wt%2B%5Cphi_i+%2B%5Cphi_G%29%5C%5C+%26+%3D+M_G+M_i+%5Csin%28wt%2B%5Cphi_i+%2B%5Cphi_G%29+%5Cend%7Baligned%7D%5C%5C$

非常非常的重要:
equation?tex=M_G%3D+%5Cleft+%7C+G%28jw%29+%5Cright+%7C%5C%5C
equation?tex=%5Cphi_G%3D%5Cangle+G%28jw%29%5C%5C

积分
equation?tex=%5Cint+u%28t%29dt
$equation?tex=G%28s%29%3D%5Cfrac%7B1%7D%7Bs%7D$
$equation?tex=G%28jw%29%3D%5Cfrac%7B1%7D%7Bjw%7D%3D-%5Cfrac%7B1%7D%7Bjw%7Dj%5E2%3D-%5Cfrac%7B1%7D%7Bw%7Dj%5C%5C$
$equation?tex=%5Cleft+%7C+G%28jw%29+%5Cright+%7C+%3D%5Cfrac%7B1%7D%7Bw%7D%5C%5C$
equation?tex=w 越大，频率越高，频率响应越小，因此频率响应是一个低通滤波器

例子：
$equation?tex=A+%5Csin+wt+%5Cto+%5Cfrac%7B1%7D%7Bs%7D+%5Cto+%5Cfrac%7B1%7D%7Bw%7DA+%5Csin+%28wt-%5Cfrac%7B%5Cpi%7D%7B2%7D%29%5C%5C$
$equation?tex=+%5Csin+2t+%5Cto+%5Cfrac%7B1%7D%7Bs%7D+%5Cto+%5Cfrac%7B1%7D%7B2%7D+%5Csin+%282t-%5Cfrac%7B%5Cpi%7D%7B2%7D%29%3D-%5Cfrac%7B1%7D%7B2%7D+%5Ccos+2t%5C%5C$

9.一阶系统的频率响应

一阶系统：
$equation?tex=G%28s%29%3D%5Cfrac%7Ba%7D%7Bs%2Ba%7D%5C%5C$
equation?tex=s%3Djw
$equation?tex=G%28jw%29%3D%5Cfrac%7Ba%7D%7Ba%2Bjw%7D%3D%5Cfrac%7Ba%28a-jw%29%7D%7Ba%5E2%2Bw%5E2%7D%5C%5C+%3D%5Cfrac%7Ba%5E2%7D%7Ba%5E2%2Bw%5E2%7D-%5Cfrac%7Baw%7D%7Ba%5E2%2Bw%5E2%7Dj%5C%5C$

$equation?tex=%5Cbegin%7Baligned%7D+%5Cleft+%7C+G%28jw%29+%5Cright+%7C++%26+%3D+%5Csqrt%7B%5Cleft+%28+%5Cfrac%7Ba%5E2%7D%7Ba%5E2%2Bw%5E2%7D+%5Cright+%29%5E2+%2B%5Cleft+%28+%5Cfrac%7Baw%7D%7Ba%5E2%2Bw%5E2%7D+%5Cright+%29+%5E2%7D%5C%5C++%26+%3D+%5Csqrt%7B+%5Cfrac%7Ba%5E2%28a%5E2%2Bw%5E2%29%7D%7B%28a%5E2%2Bw%5E2%29%5E2%7D+%7D%5C%5C++%26+%3D+%5Csqrt%7B%5Cfrac%7B1%7D%7B1%2B%5Cleft+%28+%5Cfrac%7Bw%7D%7Ba%7D++%5Cright+%29+%5E2%7D%7D+%5Cend%7Baligned%7D%5C%5C$

当
equation?tex=w+%5Cll+a%2C%5Cleft+%7C+G%28jw%29+%5Cright+%7C+%5Cto+1

当
$equation?tex=w+%3D+a%2C%5Cleft+%7C+G%28jw%29+%5Cright+%7C+%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B2%7D+%7D%3D0.707$

当
equation?tex=w+%5Cgg+a%2C%5Cleft+%7C+G%28jw%29+%5Cright+%7C+%5Cto+0

所以一阶系统的频率响应是一个低通滤波

总结： 无论是室内空调系统、流体系统还是含电容器的电路系统，容器就是一个缓冲器，其本质是抑制高速变化。缓冲也会带来延迟。

Matlab 仿真

积分前后的对比

滤波信号后与原函数的对比滤波信号延迟45°，振幅变为0.707左右

$equation?tex=%5Cleft+%7C+G%28jw%29+%5Cright+%7C++%3D+%5Csqrt%7B%5Cfrac%7B1%7D%7B1%2B%5Cleft+%28+%5Cfrac%7Bw%7D%7Ba%7D++%5Cright+%29+%5E2%7D%7D%5C%5C$
equation?tex=w 和
equation?tex=a 互换，就是高通滤波器：
$equation?tex=%5Cleft+%7C+G%28jw%29+%5Cright+%7C++%3D+%5Csqrt%7B%5Cfrac%7B1%7D%7B1%2B%5Cleft+%28+%5Cfrac%7Ba%7D%7Bw%7D++%5Cright+%29+%5E2%7D%7D%5C%5C$
$equation?tex=G%28s%29%3D%5Cfrac%7Bs%7D%7Bs%2Ba%7D%5C%5C$

把纵轴改为
equation?tex=-20%5Clog+%5Cleft+%7C+G%28jw%29+%5Cright+%7C ,就得到伯德图了。

10.二阶系统对初始条件的动态响应

二阶系统无处不在,运动现象普遍是二阶系统，如牛顿第二定律
equation?tex=F%3Dma%3Dm+%5Cddot+x%5C%5C

质量弹簧阻尼系统

阻尼和速度成正比，牛顿第二定律：
$equation?tex=%5Cddot+x%2B%5Cfrac%7BB%7D%7Bm%7D%5Cdot+x%2B%5Cfrac%7Bk%7D%7Bm%7Dx%3DF%5C%5C$
$equation?tex=w_n%3D%5Csqrt%7B%5Cfrac%7Bk%7D%7Bm%7D%7D$ :固有频率Natural Frequency
$equation?tex=%5Czeta++%3D%5Cfrac%7BB%7D%7B2%5Csqrt%7Bkm%7D%7D$ :阻尼比Damping Ratio

研究零初始条件，无外力的情况下：
equation?tex=F%3D0%2Cx%280%29%3Dx_0%2C%5Cdot+%7Bx%7D%280%29%3D%5Cdot+x_0

将条件代入：
equation?tex=%5Cddot+x%2B2%5Czeta+w_n+%5Cdot+x%2Bw_n%5E2x%3D0%5C%5C

simulink
equation?tex=%5Cddot+x%3D-2%5Czeta+w_n+x-w_n%5E2x
equation?tex=x%280%29%3D5
equation?tex=%5Cdot+x%280%29%3D0

位置为5，速度为0

equation?tex=x%28t%29%3De%5E%7B%5Clambda+t%7D%2C%5Cdot+x%28t%29%3D%5Clambda+e%5E%7B%5Clambda+t%7D%2C%5Cddot+x%28t%29%3D%5Clambda%5E2+e%5E%7B%5Clambda+t%7D
equation?tex=%5Clambda%5E2+e%5E%7B%5Clambda+t%7D%2B2%5Czeta+w_n+%5Clambda+e%5E%7B%5Clambda+t%7D%2Bw_n%5E2+e%5E%7B%5Clambda+t%7D%3D0%5C%5C

特征方程 Characteristic Equation:
equation?tex=%5Clambda%5E2+%2B2%5Czeta+w_n+%5Clambda+%2Bw_n%5E2+%3D0%5C%5C
$equation?tex=%5Clambda+%3D%5Cfrac%7B-2%5Czeta+w_n+%5Cpm+%5Csqrt%7B4%5Czeta%5E2+w_n%5E2-4w_n%5E2%7D%7D%7B2%7D%5C%5C$
equation?tex=%5Clambda_1+%3D-%5Czeta+w_n+%2B+w_n+%5Csqrt%7B%5Czeta%5E2+-1%7D%5C%5C+%5Clambda_2+%3D-%5Czeta+w_n+-+w_n+%5Csqrt%7B%5Czeta%5E2+-1%7D%5C%5C

equation?tex=%5Czeta%3E1 过阻尼Over damped
equation?tex=%5Clambda_1+%3C+-%5Czeta+w_n+%2B+w_n+%5Czeta%3D0%5C%5C
equation?tex=%5Clambda_2+%3C+-%5Czeta+w_n+-+w_n+%5Czeta%3C0%5C%5C
equation?tex=%5Clambda_1+%3E%5Clambda_2%EF%BC%8Ce%5E%7B%5Clambda_2+t%7D 快
equation?tex=x%28t%29%3DC_1e%5E%7B%5Clambda_1t%7D%2BC_2e%5E%7B%5Clambda_2t%7D%5C%5C
equation?tex=%5Czeta+%3D2

临界阻尼 Critial damped
equation?tex=%5Clambda_1%3D%5Clambda_2+%3D+-+w_n
equation?tex=x%28t%29%3D%28C_1%2BC_2%29e%5E%7B%5Clambda+t%7D%5C%5C

比过阻尼收敛速度快一些

欠阻尼 Under damped
equation?tex=%5Clambda_1+%3D-%5Czeta+w_n+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%5C%5C+%5Clambda_2+%3D-%5Czeta+w_n+-+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%5C%5C
equation?tex=%5Cbegin%7Baligned%7D+x%28t%29++%26+%3D+e%5E%7B-%5Czeta+w_n+t%7D%5Cleft+%28+C_1+%5Ccos+w_n+%5Csqrt%7B1-%5Czeta%5E2%7Dt%2BC_2+%5Csin+w_n+%5Csqrt%7B1-%5Czeta%5E2%7Dt+%5Cright+%29%5C%5C++%26+%3D+e%5E%7B-%5Czeta+w_n+t%7D+%5Csqrt%7BC_1%2BC_2%7D%5Csin+%28w_dt%2B%5Cphi%29+%5Cend%7Baligned%7D%5C%5C

其中
equation?tex=w_d%3Dw_n+%5Csqrt%7B1-%5Czeta%5E2%7D 为阻尼固有频率，
$equation?tex=%5Cphi%3D%5Carctan+%5Cfrac%7BC_1%7D%7BC_2%7D$

从
equation?tex=x%28t%29 的表达式可以看出是震荡衰减的
equation?tex=%5Czeta+%3D0.2

equation?tex=%5Czeta+%3D0
equation?tex=%5Cbegin%7Baligned%7D+x%28t%29++%26+%3D+e%5E%7B-%5Czeta+w_n+t%7D%5Cleft+%28+C_1+%5Ccos+w_n+%5Csqrt%7B1-%5Czeta%5E2%7Dt%2BC_2+%5Csin+w_n+%5Csqrt%7B1-%5Czeta%5E2%7Dt+%5Cright+%29%5C%5C++%26+%3D+e%5E0%5Cleft+%28+C_1+%5Ccos+w_nt%2BC_2+%5Csin+w_n+t+%5Cright+%29%5C%5C+%26+%3D++%5Csqrt%7BC_1%2BC_2%7D%5Csin+%28w_nt%2B%5Cphi%29+%5Cend%7Baligned%7D%5C%5C

这是正弦函数，没有衰减

equation?tex=-1%3C+%5Czeta+%3C0
equation?tex=%5Cbegin%7Baligned%7D+x%28t%29++%26+%3D+e%5E%7B-%5Czeta+w_n+t%7D%5Cleft+%28+C_1+%5Ccos+w_n+%5Csqrt%7B1-%5Czeta%5E2%7Dt%2BC_2+%5Csin+w_n+%5Csqrt%7B1-%5Czeta%5E2%7Dt+%5Cright+%29%5C%5C++%26+%3D+e%5E%7B-%5Czeta+w_n+t%7D+%5Csqrt%7BC_1%2BC_2%7D%5Csin+%28w_dt%2B%5Cphi%29+%5Cend%7Baligned%7D%5C%5C
equation?tex=-%5Czeta+w_n+t+%3E0

11.二阶系统的单位阶跃响应

弹簧质量阻尼系统

equation?tex=%5Cddot+x%2B2%5Czeta+w_n+%5Cdot+x%2Bw_n%5E2x%3DF%5C%5C

输入：
$equation?tex=u%28t%29%3D%5Cfrac%7BF%7D%7Bw_n%5E2%7D%2Cw_n%5E2$ 为单位化输出：
equation?tex=x%28t%29
equation?tex=%5Cddot+x%2B2%5Czeta+w_n+%5Cdot+x%2Bw_n%5E2x%3Dw_n%5E2u%28t%29%5C%5C

上一节用的是微分方程的通解和特解，这小节用拉普拉斯变换：
equation?tex=s%5E2X%28s%29%2B2%5Czeta+w_nsX%28s%29%2Bw_n%5E2X%28s%29%3Dw_n%5E2U%28s%29%5C%5C

传递函数：
$equation?tex=H%28s%29%3D%5Cfrac%7BX%28s%29%7D%7BU%28s%29%7D%3D%5Cfrac%7Bw_n%5E2%7D%7Bs%5E2%2B2%5Czeta+w_ns%2Bw_n%5E2%7D%5C%5C$

单位阶跃
$equation?tex=X%28s%29%3DU%28s%29H%28s%29%5C%5C+%3D%5Cfrac%7B1%7D%7Bs%7D+%5Ccdot+%5Cfrac%7Bw_n%5E2%7D%7Bs%5E2%2B2%5Czeta+w_ns%2Bw_n%5E2%7D%5C%5C$

极点
equation?tex=s%3Dp_1%3D0
equation?tex=s%5E2%2B2%5Czeta+w_ns%2Bw_n%5E2%3D0%5C%5C
$equation?tex=%5Cbegin%7Baligned%7D+s++%26+%3D+%5Cfrac%7B-2%5Czeta+w_n+%5Cpm+%5Csqrt%7B4%5Czeta%5E2+w_n%5E2-4w_n%5E2%7D%7D%7B2%7D%5C%5C++%26+%3D+-%5Czeta+w_n+%5Cpm+w_n+%5Csqrt%7B%5Czeta%5E2-1%7D+%5Cend%7Baligned%7D%5C%5C$

欠阻尼
equation?tex=0%3C%5Czeta+%3C1
equation?tex=s+%3D-%5Czeta+w_n+%5Cpm+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%5C%5C
equation?tex=p_2%3D-%5Czeta+w_n+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D
equation?tex=p_3%3D-%5Czeta+w_n+-+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D
$equation?tex=%5Cbegin%7Baligned%7D+X%28s%29++%26+%3D+%5Cfrac%7BA%7D%7Bs-p_1%7D+%2B%5Cfrac%7BB%7D%7Bs-p_2%7D+%2B%5Cfrac%7BC%7D%7Bs-p_3%7D+%5C%5C++%26+%3D+%5Cfrac%7BA%28s-p_2%29%28s-p_3%29%2BB%28s-p_1%29%28s-p_3%29%2BC%28s-p_1%29%28s-p_2%29%7D%7B%28s-p_1%29%28s-p_2%29%7D++%5Cend%7Baligned%7D%5C%5C$

equation?tex=s%3Dp_1%3D0 时：
equation?tex=A%28-p_2%29%28-p_3%29%3Dw_n%5E2%5C%5C
equation?tex=A%28%5Czeta+w_n+-+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%29%28%5Czeta+w_n+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%29%3Dw_n%5E2%5C%5C
equation?tex=A%28%5Czeta%5E2+w_n%5E2+%2B+w_n%5E2+%281-%5Czeta%5E2+%29%29%3Dw_n%5E2%5C%5C
equation?tex=A++%3D1%5C%5C

equation?tex=s%3Dp_2%3D-%5Czeta+w_n+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D 时:
equation?tex=B%28-%5Czeta+w_n+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D-0%29%28-%5Czeta+w_n%5C%5C+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%2B%5Czeta+w_n+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%29%3Dw_n%5E2%5C%5C

equation?tex=B%28-2i%5Czeta+w_n%5E2%5Csqrt%7B1-%5Czeta%5E2+%7D+-2w_n%5E2%281-%5Czeta%5E2+%29%29%3Dw_n%5E2%5C%5C
$equation?tex=%5Cbegin%7Baligned%7D+B++%26+%3D+-%5Cfrac%7B1%7D%7B2%7D+%5Ccdot+%5Cfrac%7B1%7D%7B%281-%5Czeta%5E2+%29%2B+i%5Czeta+%5Csqrt%7B1-%5Czeta%5E2+%7D%7D%5C%5C+%26+%3D+-%5Cfrac%7B1%7D%7B2%7D+%5Ccdot+%5Cfrac%7B%281-%5Czeta%5E2+%29-i%5Czeta+%5Csqrt%7B1-%5Czeta%5E2+%7D%7D%7B%281-%5Czeta%5E2+%29%5E2%2B%5Czeta%5E2+%281-%5Czeta%5E2+%29%7D%5C%5C+%26+%3D+-%5Cfrac%7B1%7D%7B2%7D+%5Ccdot+%5Cfrac%7B1-%5Czeta%5E2+-i%5Czeta+%5Csqrt%7B1-%5Czeta%5E2+%7D%7D%7B1-%5Czeta%5E2%7D%5C%5C+%26+%3D+-%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+-+%5Cfrac%7B%5Czeta+%5Csqrt%7B1-%5Czeta%5E2+%7D%7D%7B1-%5Czeta%5E2%7Di++%5Cright+%5D+%5C%5C+%26+%3D+-%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+-+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5D+%5C%5C+%5Cend%7Baligned%7D%5C%5C$

equation?tex=s%3Dp_3 :
equation?tex=C%3DB%5E%2A

$equation?tex=%5Cbegin%7Baligned%7D+X%28s%29+%26+%3D+%5Cfrac%7B1%7D%7Bs-p_1%7D+%5C%5C%26-+%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+-+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5D%5Cfrac%7B1%7D%7Bs-p_2%7D%5C%5C+%26-+%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+%2B+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5D+%5Cfrac%7B1%7D%7Bs-p_3%7D+%5C%5C%5Cend%7Baligned%7D%5C%5C$
$equation?tex=%5Cbegin%7Baligned%7D+x%28t%29+%26+%3D1-%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+-+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5De%5E%7Bp_2t%7D%5C%5C+%26+-+%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+%2B+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5D+e%5E%7Bp_3t%7D%5C%5C+%26+%3D1-%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+-+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5De%5E%7B%28-%5Czeta+w_n+%2B+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%29t%7D%5C%5C+%26+-+%5Cfrac%7B1%7D%7B2%7D+%5Ccdot%5Cleft+%5B+1+%2B+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5D+e%5E%7B%28-%5Czeta+w_n+-+iw_n+%5Csqrt%7B1-%5Czeta%5E2+%7D%29t%7D%5C%5C+%26+%3D1-%5Cfrac%7B1%7D%7B2%7De%5E%7B-%5Czeta+w_n+t%7D+%5Ccdot+%5Cleft+%28+%5Cleft+%5B+1+-+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5De%5E%7Biw_dt%7D-+%5Cleft+%5B+1+%2B+%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di++%5Cright+%5D+e%5E%7B-iw_dt%7D+%5Cright+%29+%5C%5C+%26+%3D1-e%5E%7B-%5Czeta+w_n+t%7D+%5Ccdot+%5Cleft+%28%5Cfrac%7B1%7D%7B2%7D+%5Cleft+%28+e%5E%7Biw_dt%7D+%2Be%5E%7B-iw_dt%7D+%5Cright+%29%2B+%5Cfrac%7B1%7D%7B2%7D%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7Di+%5Cleft+%28+-e%5E%7Biw_dt%7D+%2Be%5E%7B-iw_dt%7D+%5Cright+%29%5Cright+%29+%5C%5C+%26+%3D1-e%5E%7B-%5Czeta+w_n+t%7D+%5Ccdot+%5Cleft+%28%5Ccos+w_dt%2B%5Cfrac%7B%5Czeta+%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7D+%5Csin+w_dt+%5Cright+%29+%5C%5C+%26+%3D1-e%5E%7B-%5Czeta+w_n+t%7D+%5Csqrt%7B1%2B%5Cfrac%7B%5Czeta%5E2+%7D%7B1-%5Czeta%5E2%7D%7D++%5Csin+%28w_dt+%2B%5Cphi+%29%5C%5C+%26+%3D1-e%5E%7B-%5Czeta+w_n+t%7D+%5Csqrt%7B%5Cfrac%7B1+%7D%7B1-%5Czeta%5E2%7D%7D++%5Csin+%28w_dt+%2B%5Cphi+%29+%5Cend%7Baligned%7D%5C%5C$

因此是震荡衰减的。

Matlab 仿真

12.二阶系统的性能分析与比较

如何衡量系统的性能？

欠阻尼动态响应：
$equation?tex=x%28t%29%3D1-e%5E%7B-%5Czeta+w_n+t%7D+%5Csqrt%7B%5Cfrac%7B1+%7D%7B1-%5Czeta%5E2%7D%7D++%5Csin+%28w_dt+%2B%5Cphi+%29%5C%5C$
equation?tex=T_d 延迟时间Delay time：系统达到稳态50%所需的时间
equation?tex=T_r 上升时间Rise time：100%
equation?tex=M_p 最大超调量 Max Overshoot
equation?tex=%28x_p-1%29%5Ctimes+100+%5C%25%5C%5C
equation?tex=x_p%3Dx%28t_p%29
equation?tex=T_%7Bss%7D 稳态时间或调节时间Setting time

:
$equation?tex=1%3D1-e%5E%7B-%5Czeta+w_n+t_r%7D+%5Csqrt%7B%5Cfrac%7B1+%7D%7B1-%5Czeta%5E2%7D%7D++%5Csin+%28w_dt_r+%2B%5Cphi+%29%5C%5C$
equation?tex=%5Csin+%28w_dt_r+%2B%5Cphi+%29%3D0%5C%5C
equation?tex=w_dt_r+%2B%5Cphi%3D%5Cpi%5C%5C
$equation?tex=t_r+%3D%5Cfrac%7B%5Cpi-%5Cphi%7D%7Bw_d%7D%5C%5C$
equation?tex=M_p :
equation?tex=%5Cdot+x%3D0%2C%5Cquad+1st
$equation?tex=t_p%3D%5Cfrac%7B%5Cpi-%7D%7Bw_d%7D%5C%5C$
$equation?tex=M_p%3De%5E%7B-%5Cfrac%7B%5Czeta+%5Cpi%7D%7B%5Csqrt%7B1-%5Czeta%5E2%7D%7D%7D+%5Ctimes+100+%5C%25%5C%5C$
equation?tex=T_%7Bss%7D : 2%
$equation?tex=T_%7Bss%7D%3D%5Cfrac%7B4%7D%7B%5Czeta+w_n%7D%5C%5C$

5%
$equation?tex=T_%7Bss%7D%3D%5Cfrac%7B3%7D%7B%5Czeta+w_n%7D%5C%5C$

分析手段和方法

计分规则：1分，2分，3分
越小越好
越小越好
越小越好

雷达图

13.二阶系统频率响应分析

不同阻尼比的频率响应
equation?tex=M_i+%5Csin%28wt%2B%5Cphi_i%29%5Cto+G%28s%29+%5Cto+M_o+%5Csin%28wt%2B%5Cphi_o%29%5C%5C

振幅响应：
$equation?tex=%5Cfrac%7BM_o%7D%7BM_i%7D%3D%5Cleft+%7C+G%28jw%29+%5Cright+%7C+%5C%5C$

辐角响应：
equation?tex=%5Cphi_o+-%5Cphi_i+%3D+%5Cangle+G%28jw%29%5C%5C

用这个结论分析二阶系统

传递函数：
$equation?tex=G%28s%29%3D%5Cfrac%7Bw_n%5E2%7D%7Bs%5E2%2B2%5Czeta+w_ns%2Bw_n%5E2%7D%5C%5C$
equation?tex=w_n :固有频率
equation?tex=%5Czeta :阻尼比
$equation?tex=%5Cbegin%7Baligned%7D+G%28jw%29++%26+%3D+%5Cfrac%7Bw_n%5E2%7D%7B-w%5E2%2B2%5Czeta+w_nwj%2Bw_n%5E2%7D%5C%5C++%26+%3D+%5Cfrac%7B1%7D%7B-%5Cfrac%7Bw%5E2%7D%7Bw_n%5E2%7D%2B2%5Czeta+%5Cfrac%7Bw%7D%7Bw_n%7Dj%2B1%7D%5C%5C+%26+%3D+%5Cfrac%7B1%7D%7B-%5COmega%5E2%2B2%5Czeta+%5COmega+j%2B1%7D%5C%5C+%26+%3D+%5Cfrac%7B1-%5COmega%5E2%2B2%5Czeta+%5COmega+j%7D%7B%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2+%7D%5C%5C+%26+%3D+%5Cfrac%7B1-%5COmega%5E2%7D%7B%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2+%7D-+%5Cfrac%7B2%5Czeta+%5COmega+%7D%7B%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2+%7Dj%5C%5C+%5Cend%7Baligned%7D%5C%5C$

其中
$equation?tex=%5COmega%3D%5Cfrac%7Bw%7D%7Bw_n%7D$ ,输入频率比上固有频率

振幅响应：
$equation?tex=%5Cbegin%7Baligned%7D+%5Cleft+%7C+G%28jw%29++%5Cright+%7C++%26+%3D+%5Csqrt%7B+Real%5Cleft+%28+G%28jw%29+%5Cright+%29%5E2+%2B+I_m+%5Cleft+%28+G%28jw%29+%5Cright+%29%5E2%7D+%5C%5C+%26+%3D%5Csqrt%7B+%5Cfrac%7B%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2%7D%7B%5Cleft+%28+%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2+%5Cright+%29+%5E2%7D%7D%5C%5C+%26+%3D%5Csqrt%7B+%5Cfrac%7B1%7D%7B%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2+%7D%7D+%5Cend%7Baligned%7D%5C%5C$

分析

equation?tex=%5COmega%3D0%2C%5Comega%3D0
equation?tex=%5Cleft+%7C+G%28jw%29++%5Cright+%7C+%3D1%5C%5C

equation?tex=%5COmega+%5Cto+%5Cinfty%2C%5Comega+%5Cgg+%5Comega_n
equation?tex=%5Cleft+%7C+G%28jw%29++%5Cright+%7C+%5Cto+1%5C%5C

equation?tex=%5COmega+%3D1%2C%5Comega+%3D%5Comega_n ，输入频率等于固有频率
$equation?tex=%5Cleft+%7C+G%28jw%29++%5Cright+%7C+%3D+%5Cfrac%7B1%7D%7B2%5Czeta%7D%5C%5C$
equation?tex=%5Czeta+%3C+0.5

equation?tex=%5Czeta+%3E+0.5

因此在
equation?tex=%5COmega%3D1 一定存在极值求极值，令
equation?tex=f%28%5COmega%29%3D%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2%5C%5C
$equation?tex=%5Cbegin%7Baligned%7D+%5Cfrac%7B%5Cmathrm%7Bd%7D+f%28%5COmega%29%7D%7B%5Cmathrm%7Bd%7D+%5COmega%7D++%26+%3D+%281-%5COmega%5E2%29%5E2%2B4%5Czeta%5E2+%5COmega%5E2%5C%5C+%26+%3D+2%281-%5COmega%5E2%29%28-2%5COmega%29%2B8%5Czeta%5E2+%5COmega%5C%5C+%26+%3D+%5COmega%28-1%2B%5COmega%5E2%2B2%5Czeta%5E2+%29%3D0%5C%5C+%26%5CRightarrow+-1%2B%5COmega%5E2%2B2%5Czeta%5E2+%3D+0%5C%5C+%26%5CRightarrow+%5COmega%3D%5Csqrt%7B1-2%5Czeta%5E2+%7D+%5Cend%7Baligned%7D%5C%5C$

当
equation?tex=1-2%5Czeta%5E2+%3E0 时存在极值
equation?tex=%5Comega%3D%5Comega_n%5Csqrt%7B1-2%5Czeta%5E2+%7D%5C%5C

这个频率称为系统的谐振振频率，
equation?tex=%5Czeta+%5Cto+0 ，谐振频率和固有频率非常接近
equation?tex=%5Comega+%5Capprox+%5Comega_n 。

当输入频率等于谐振频率时：
$equation?tex=%5Cleft+%7C+G%28jw%29++%5Cright+%7C+_%7B%5Comega+%3D+%5Comega_n%5Csqrt%7B1-2%5Czeta%5E2+%7D%7D%3D%5Cfrac%7B1%7D%7B2%5Czeta+%5Csqrt%7B1-%5Czeta%5E2+%7D%7D+%5C%5C$

equation?tex=%5Czeta+%3D1%2C%5Comega%3D%5Comega_n 时：
$equation?tex=%5Cleft+%7C+G%28jw%29++%5Cright+%7C+%3D+%5Cfrac%7B1%7D%7B2%7D+%5C%5C$

equation?tex=%5Czeta+%3D0.5%2C%5Comega%3D%5Comega_n%EF%BC%8C%5Comega+%3D+%5Comega_n%5Csqrt%7B1-2%5Czeta%5E2+%7D 时：
equation?tex=%5Cleft+%7C+G%28jw%29++%5Cright+%7C+%3D+1+%5C%5C

equation?tex=%5Czeta+%3D0%2C%5Comega%3D%5Comega_n 时：
equation?tex=%5Cleft+%7C+G%28jw%29++%5Cright+%7C+%5Cto+%5Cinfty+%5C%5C

对于阻尼比比较小的系统来说，如果外力的频率在谐振频率（极值）附近，那么系统就会表现出强烈的振幅响应，不同的系统有不同的谐振频率，对外界刺激响应也就不同。

不同阻尼比的频率响应

14.伯德图

伯德图是表示频率响应的图示方法，频率响纵坐标改为
equation?tex=-20+%5Clog+M ，和辐角响应合称伯德图。

对于传递函数：
$equation?tex=G%28s%29%3D%5Cfrac%7Bs%2B2%7D%7Bs%2B4%7D%5C%5C$

直接在命令窗口输入：

复制代码

    >> bode([1 2],[1 4])

dB decibel 分贝 dec 指十分之一，bel人名，分贝表示的是电话、电报的信号损失
$equation?tex=dB%3D10%5Clog_%7B10%7D+%5Cfrac%7BP_M%7D%7BP_R%7D%5C%5C$
equation?tex=P_M : 测量功率
equation?tex=P_R : 参考功率

加对数是为了把较大的数值降低，便于记录，如
equation?tex=10%5Clog_%7B10%7D+10%5E%7B20%7D%3D120+dB%5C%5C

振幅和功率为平方关系
equation?tex=P%3Df%28M%5E2%29%5C%5C
$equation?tex=%5Cbegin%7Baligned%7D+dB++%26+%3D+10%5Clog_%7B10%7D+%5Cfrac%7BP_M%7D%7BP_R%7D+%5C%5C+%26+%3D+10%5Clog_%7B10%7D+%5Cleft+%28+%5Cfrac%7BM_o%7D%7BM_i%7D+%5Cright+%29%5E2+%5C%5C+%26+%3D+20%5Clog_%7B10%7D+%5Cfrac%7BM_o%7D%7BM_i%7D%5C%5C+%26+%3D+20%5Clog_%7B10%7D+M+%5Cend%7Baligned%7D+%5C%5C$

积分
$equation?tex=G%28j%5Comega%29%3D%5Cfrac%7B1%7D%7Bj+%5Comega%7D%3D-%5Cfrac%7B1%7D%7B+%5Comega%7Dj$
$equation?tex=%5Cleft+%7C+G%28j%5Comega%29+%5Cright+%7C+%3D%5Cfrac%7B1%7D%7B+%5Comega%7D%5C%5C$
$equation?tex=20%5Clog_%7B10%7D+%5Cfrac%7B1%7D%7B+%5Comega%7D+%3D+-20%5Clog_%7B10%7D+%5Comega%5C%5C$

$equation?tex=%5Cleft+%7C+G%28j+%5Comega%29+%5Cright+%7C+%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B1%2B%5Cleft+%28+%5Cfrac%7B%5Comega+%7D%7Ba%7D++%5Cright+%29%5E2%7D%7D%5C%5C$
$equation?tex=%5Cangle+G%28j+%5Comega%29%3D-%5Carctan+%5Cleft+%28+%5Cfrac%7B%5Comega+%7D%7Ba%7D++%5Cright+%29+%5C%5C$

低频
equation?tex=%5Comega+%5Cll+a
equation?tex=%5Cleft+%7C+G%28j+%5Comega%29+%5Cright+%7C+%3D1%5C%5C
equation?tex=20%5Clog_%7B10%7D+%5Cleft+%7C+G%28j+%5Comega%29+%5Cright+%7C+%3D+0%5C%5C

截至频率
equation?tex=%5Comega+%3Da
$equation?tex=%5Cleft+%7C+G%28j+%5Comega%29+%5Cright+%7C+%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B2%7D%7D%5C%5C$
equation?tex=20%5Clog_%7B10%7D+%5Cleft+%7C+G%28j+%5Comega%29+%5Cright+%7C+%3D+-20%5Clog_%7B10%7D+%5Csqrt%7B2%7D+%3D+-3dB%5C%5C
equation?tex=%5Cangle+G%28j+%5Comega%29%3D-%5Carctan+1%3D-45%C2%B0%5C%5C

高频
equation?tex=%5Comega+%5Cgg+a
$equation?tex=%5Cleft+%7C+G%28j%5Comega%29+%5Cright+%7C+%3D%5Cfrac%7B1%7D%7B+%5Comega%7D%5C%5C$
$equation?tex=20%5Clog_%7B10%7D+%5Cfrac%7B1%7D%7B+%5Comega%7D+%3D+-20%5Clog_%7B10%7D+%5Comega%5C%5C$
equation?tex=%5Cangle+G%28j+%5Comega%29%3D-%5Carctan+1%3D-90%C2%B0%5C%5C

equation?tex=G%28j%5Comega%29%3DG_1%28j%5Comega%29+%5Ccdot+G_2%28j%5Comega%29%3Dr_1e%5E%7Bj+%5Ctheta_1%7D%5Ccdot+r_2e%5E%7Bj+%5Ctheta_2%7D%3Dr_1r_2e%5E%7Bj+%28%5Ctheta_1%2B%5Ctheta_2%29%7D%5C%5C
equation?tex=%5Cleft+%7C+G%28j%5Comega%29+%5Cright+%7C+%3D20+%5Clg+%5Cleft+%7C+G_1%28j%5Comega%29+%5Cright+%7C+%2B20+%5Clg+%5Cleft+%7C+G_2%28j%5Comega%29+%5Cright+%7C+%5C%5C

例：
$equation?tex=G%28s%29%3D%5Cfrac%7B+s%2B4%7D%7Bs%2B8%7D%3D%5Cfrac%7B4%28%5Cfrac%7B1%7D%7B4%7Ds%2B1+%29%7D%7B8%28%5Cfrac%7B1%7D%7B8%7Ds%2B1+%29%7D%3D%5Cfrac%7B1%7D%7B2%7D++%5Ccdot+%28%5Cfrac%7B1%7D%7B4%7Ds%2B1+%29+%5Ccdot+%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B8%7Ds%2B1%7D+%5C%5C$

拆分
equation?tex=%5Cto 叠加

复制代码

    >> bode([0 1],[0 2])

复制代码

    >> bode([1/4 1],[0 1])

复制代码

    >> bode([0 1],[1/8 1])

复制代码

    >> bode([1 4],[1 8])

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1.介绍

2.电路系统建模

3.流体系统建模

4.拉普拉斯变换

5.拉普拉斯变换的收敛域（ROC）与逆变换（ILT）

6.拉&传&微的关系

7.一阶系统的单位阶跃响应

8.频率响应与滤波器

9.一阶系统的频率响应

10.二阶系统对初始条件的动态响应

11.二阶系统的单位阶跃响应

12.二阶系统的性能分析与比较

13.二阶系统频率响应分析

14.伯德图

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