书籍：Matlab实用教程
工具：Matlab2021a
在线工具：https://www.cainiaojc.com/tool/octave/
缺少一些包
在线工具：https://octave-online.net/
比较健全。
电脑信息：Intel® Xeon® CPU E5-2603 v3 @ 1.60GHz
系统类型：64位操作系统，基于X64的处理器 windows10 专业版
第6章 线性控制系统分析与设计
6.1. 线性系统的描述
6.1.1 状态空间描述法
wn=1; A=[0 1;-wn^2 -2*zeta*wn]; B=[0;wn^2]; C=[1 0]; D=0; G=ss(A,B,C,D)
6.1.2 传递函数描述法
num=1; den=[1 1.414 1]; G=tf(num,den)
6.1.3 零-极点描述法
num=1; den=[1 1.414 1]; z=roots(num) p=roots(den) zpk(z,p,1)num=1; den=[1 1.414 1]; z=roots(num) p=roots(den) [r,p,k]=residue(num,den)
6.1.4 离散系统的数学描述
1、状态空间描述法
a=[-1.5 -0.5;1 0]; b=[1;0]; c=[0 0.5]; d=0; G=ss(a,b,c,d,0.1)
2、脉冲传递函数描述法
num1=[0.5 0]; den=[1 -1.5 0.5]; G1=tf(num1,den,-1)num2=[0 0.5]; den=[1 -1.5 0.5]; G2=filt(num2,den)
3、零-极点增益描述法
G3=zpk([0],[0.5 1],0.5,-1)
6.2 线性系统模型之间的转换
6.2.1 连续系统模型之间的转换
1、系统模型的转换
A、状态空间模型的获得
num=[0 3 2;1 2 3]; den=[3 5 2 1]; G11=tf(num(1,:),den) G12=tf(num(2,:),den) G=ss([G11;G12])
B、传递函数的获得：
num=[0 3 2;1 2 3]; den=[3 5 2 1]; G11=tf(num(1,:),den) G12=tf(num(2,:),den) G=ss([G11;G12]) G1=tf(G)
C、零-极点模型的获得
num=[0 3 2;1 2 3]; den=[3 5 2 1]; G11=tf(num(1,:),den) G12=tf(num(2,:),den) G=ss([G11;G12]) G1=tf(G) G2=zpk(G) G3=zpk(G1)
2、模型参数的获取
num=[0 3 2;1 2 3]; den=[3 5 2 1]; G11=tf(num(1,:),den) G12=tf(num(2,:),den) G=ss([G11;G12]) G1=tf(G) G2=zpk(G) G3=zpk(G1) [a,b,c,d]=ssdata(G1) [num,den]=tfdata(G2) [z,p,k]=zpkdata(G)
3、模型类型的检验
继续上面的实例：
class(G)ans = ssisa(G,'tf')ans = 06.2.2 连续系统与离散系统之间的转换
1、c2d命令
a=[0 1;-1 -1.414]; b=[0;1]; c=[1 0]; d=0; G=ss(a,b,c,d); Gd=c2d(G,0.1)
2、d2c命令
G=d2c(Gd)
3、d2d命令
Gd2=d2d(Gd,0.3)
6.2.3 模型对象的属性
1、模型对象的属性
2、get命令和set命令
num=1; den=[1 1.414 1]; G=tf(num,den); get(G)num: p-by-m cell array of row vectors (m = number of inputs) den: p-by-m cell array of row vectors (p = number of outputs) tfvar: string (usually s or z) inv: logical (true for negative powers of TF variable) tsam: scalar (sample time in seconds) inname: m-by-1 cell vector of strings outname: p-by-1 cell vector of strings ingroup: struct with indices as fields outgroup: struct with indices as fields name: string notes: string or cell of strings userdata: any data typeset(G,'den',[1 2 1]) G
3、直接获取和修改属性
G.den=[1 1.414 1]; G
6.3 结构框图的模型表示
1、串联结构
2、并联结构
3、反馈结构
G1=tf(1,[1 2 1]) G2=tf(1,[1 1]); G3=tf(1,[2 1]); G4=tf(1,[1 0]); G12=G1+G2 G34=G3-G4 G=feedback(G12,G34,-1)Transfer function 'G1' from input 'u1' to output ... 1 y1: ------------- s^2 + 2 s + 1 Continuous-time model. Transfer function 'G12' from input 'u1' to output ... s^2 + 3 s + 2 y1: --------------------- s^3 + 3 s^2 + 3 s + 1 Continuous-time model. Transfer function 'G34' from input 'u1' to output ... -s - 1 y1: --------- 2 s^2 + s Continuous-time model. Transfer function 'G' from input 'u1' to output ... 2 s^4 + 7 s^3 + 7 s^2 + 2 s y1: ------------------------------------- 2 s^5 + 7 s^4 + 8 s^3 + s^2 - 4 s - 2 Continuous-time model.G1=ss(tf(1,[1 2 1])); G2=tf(1,[1 1]); G12=G1+G2G12.a = x1 x2 x3 x1 0 -1 0 x2 1 -2 0 x3 0 0 -1 G12.b = u1 x1 -1 x2 0 x3 1 G12.c = x1 x2 x3 y1 0 -1 1 G12.d = u1 y1 0 Continuous-time model.4、复杂的结构框图
G1=tf(1,[1 0]); G2=tf(1,[1 1 0]); G3=tf(1,[1 1 0]); G4=tf(-2,1); G5=tf(-1,1); G6=tf(1,[1 0]); G7=tf(-1,[1 1]); Sys=append(G1,G2,G3,G4,G5,G6,G7)Transfer function 'Sys' from input 'u1' to output ... 1 y1: - s y2: 0 y3: 0 y4: 0 y5: 0 y6: 0 y7: 0 Transfer function 'Sys' from input 'u2' to output ... y1: 0 1 y2: ------- s^2 + s y3: 0 y4: 0 y5: 0 y6: 0 y7: 0 Transfer function 'Sys' from input 'u3' to output ... y1: 0 y2: 0 1 y3: ------- s^2 + s y4: 0 y5: 0 y6: 0 y7: 0 Transfer function 'Sys' from input 'u4' to output ... y1: 0 y2: 0 y3: 0 y4: -2 y5: 0 y6: 0 y7: 0 Transfer function 'Sys' from input 'u5' to output ... y1: 0 y2: 0 y3: 0 y4: 0 y5: -1 y6: 0 y7: 0 Transfer function 'Sys' from input 'u6' to output ... y1: 0 y2: 0 y3: 0 y4: 0 y5: 0 1 y6: - s y7: 0 Transfer function 'Sys' from input 'u7' to output ... y1: 0 y2: 0 y3: 0 y4: 0 y5: 0 y6: 0 -1 y7: ----- s + 1 Continuous-time model.Q=[1 6 5; 2 1 7; 3 2 0; 4 3 0; 5 4 0; 6 2 0; 7 3 0;]; INPUTS=1; OUTPUTS=4; G=connect(Sys,Q,INPUTS,OUTPUTS)warning: tf: converting to minimal state-space for MIMO TF interconnections Transfer function 'G' from input 'u1' to output ... -2 s - 2 y1: --------------------------------------------- s^6 + 3 s^5 + 3 s^4 + 1 s^3 - 1 s^2 - 3 s - 3 Continuous-time model.nblocks=7; n1=1;d1=[1 0]; n2=1;d2=[1 1 0]; n3=1;d3=[1 1 0]; n4=-2;d4=1; n5=-1;d5=1; n6=1;d6=[1 0]; n7=-1;d7=[1 1]; blkbuild Q=[1 6 5; 2 1 7; 3 2 0; 4 3 0; 5 4 0; 6 2 0; 7 3 0;]; INPUTS=1; OUTPUTS=4; [A,B,C,D]=connect(a,b,c,d,Q,INPUTS,OUTPUTS)error: 'blkbuild' undefined near line 1, column 16.4 线性系统的时域分析
6.4.1 零输入响应分析
1、连续系统的零输入响应
G1=tf(12,[1 4]); H=tf(1,[1 3]); GG=feedback(G1,H) G=ss(GG) initial(G,[1 2])Transfer function 'GG' from input 'u1' to output ... 12 s + 36 y1: -------------- s^2 + 7 s + 24 Continuous-time model. G.a = x1 x2 x1 8.882e-16 2.4 x2 -10 -7 G.b = u1 x1 -3.6 x2 12 G.c = x1 x2 y1 0 1 G.d = u1 y1 0 Continuous-time model.
2、离散系统的脉冲响应
6.4.2 脉冲响应分析
1、连续系统的脉冲响应
impulse(G)t=0:0.1:10; y=impulse(G,t)y = 1.2000e+01 7.5497e+00 4.0623e+00 1.6432e+00 1.6386e-01 -5.9848e-01 -8.7579e-01 -8.6531e-01 -7.1369e-01 -5.1764e-01 -3.3270e-01 -1.8457e-01 -7.9778e-02 -1.4242e-02 2.0712e-02 3.4565e-02 3.5596e-02 3.0085e-02 2.2258e-02 1.4605e-02 8.3333e-03 3.8090e-03 9.1778e-04 -6.7324e-04 -1.3494e-03 -1.4569e-03 -1.2637e-03 -9.5397e-04 -6.3875e-04 -3.7414e-04 -1.7943e-04 -5.2379e-05 1.9574e-05 5.1993e-05 5.9295e-05 5.2887e-05 4.0757e-05 2.7837e-05 1.6711e-05 8.3587e-06 2.7966e-06 -4.3865e-07 -1.9710e-06 -2.3984e-06 -2.2049e-06 -1.7357e-06 -1.2090e-06 -7.4291e-07 -3.8574e-07 -1.4311e-07 1.5897e-09 7.3178e-08 9.6345e-08 9.1548e-08 7.3676e-08 5.2335e-08 3.2882e-08 1.7658e-08 7.1104e-09 6.6947e-10 -2.6423e-09 -3.8397e-09 -3.7847e-09 -3.1170e-09 -2.2580e-09 -1.4494e-09 -8.0260e-10 -3.4562e-10 -6.0204e-11 9.1714e-11 1.5164e-10 1.5574e-10 1.3142e-10 9.7109e-11 6.3639e-11 3.6251e-11 1.6517e-11 3.9221e-12 -2.9957e-12 -5.9242e-12 -6.3760e-12 -5.5215e-12 -4.1630e-12 -2.7839e-12 -1.6281e-12 -7.7863e-13 -2.2505e-13 8.7922e-14 2.2846e-13 2.5960e-13 2.3114e-13 1.7789e-13 1.2135e-13 7.2742e-14 3.6295e-14 1.2055e-14 -2.0222e-15 -8.6705e-15 -1.0505e-14 -9.6384e-15 -7.5773e-152、离散系统的脉冲响应
a=[-2 0;0 -3]; b=[1;1]; c=[1 -4]; d=1; dimpulse(a,b,c,d,1,10)error: 'dimpulse' undefined near line 1, column 16.4.3 阶跃响应分析
1、连续阶跃响应
G1=tf(12,[1 4]); H=tf(1,[1 3]); G=feedback(G1,H) step(G)Transfer function 'G' from input 'u1' to output ... 12 s + 36 y1: -------------- s^2 + 7 s + 24 Continuous-time model.t1=0:0.1:5; y1=step(G,t1); plot(t1,y1)t2=0:0.5:5; y2=step(G,t2); plot(t2,y2)2、离散系统的阶跃响应
6.4.4 任意输入的响应
1、连续系统的任意输入响应
t1=0:0.1:5; u=sin(t); G1=tf(1,[1 1.414 1]) G2=tf(1,[1 0.6 1]) lsim(G1,'r',G2,'bo',u,t)Transfer function 'G1' from input 'u1' to output ... 1 y1: ----------------- s^2 + 1.414 s + 1 Continuous-time model. Transfer function 'G2' from input 'u1' to output ... 1 y1: --------------- s^2 + 0.6 s + 1 Continuous-time model.2、离散系统的任意输入响应
num=[2 5 1]; den=[1 2 3]; t=0:0.1:5; u=sin(t); y=dlsim(num,den,u)error: 'dlsim' undefined near line 1, column 36.4.5 系统的结构参数
1、极点和零点
num=[5 100]; den=[1 8 32 80 100]; G=tf(num,den); p=pole(G) [z,gain]=tzero(G) [p,z]=pzmap(G)p = -1 + 3i -1 - 3i -3 + 1i -3 - 1i error: 'tzero' undefined near line 1, column 10 The 'tzero' function belongs to the control package from Octave Forge but has not yet been implemented. p = -1 + 3i -1 - 3i -3 + 1i -3 - 1i z = -202、闭环系统的阻尼系数和固有频率
[wn,zeta]=damp(G)wn = 3.1623 3.1623 3.1623 3.1623 zeta = 0.9487 0.9487 0.3162 0.31623、时域响应的稳态增益
[wn,zeta]=damp(G)wn = 3.1623 3.1623 3.1623 3.1623 zeta = 0.9487 0.9487 0.3162 0.3162
