matlab传函转zpk形式,实验一：状态方程、传函为模型的系统的仿真

一、实验目的

1.掌握各数学模型之间的转换与数学模型的参数获取，掌握相关MATLAB命令

2.掌握欧拉法和RK法的递推公式

3.掌握欧拉法和RK法的MATLAB算法实现

二、实验内容

1.分别用欧拉法和四阶龙格-库塔法求解如下系统：

(1)R=4KΩ，C=1µF，L=1H， 仿真时间：0.03s

(2)R=280Ω，C=8µF，L=2H，仿真时间：0.1s

两组参数下电路的单位阶跃响应。 (初始条件均为零)。

2. 熟悉并理解掌握MATLAB模型转换以及参数提取指令(针对上述系统写出它们的运行结果)

G=tf(num,den)　　　　G1=tf(G)

G=zpk(z,p,k)　　　 　G1=zpk(G)

G=ss(A,B,C,D)　　　　G3=ss(G)

[num,den]=tfdata(G)

[Z,P,K]=zpkdata(G)

[A,B,C,D]=ssdata(G)

三、实验要求

1. 预习和完成实验内容

2. 完成实验报告，内容至少包括：理论分析，实验程序，程序运行结果以及结果分析。

四、实验内容

1.建模

列方程：

状态空间表达式(时域)：

写成矩阵形式：

拉氏变换后得传递函数(复域)：

2. 实验代码：

%********** 参数提取 *************

R = 4000;

C = 0.000001;

L = 1;

num = [1];

den = [L*C, R*C, 1];

G = tf(num, den)

G1 = tf(G)

G2 = zpk(G)

G3 = ss(G)

[Num, Den] = tfdata(G)

[Z, P, K] = zpkdata(G)

[A, B, C, D] = ssdata(G)

问题(1)：

clear;

clc;

%******************************************%%

% Question 1

%******************************************%%

R1 = 4000;

C1 = 0.000001;

L1 = 1;

A1 = [0, 1/C1;-1/L1,-R1/L1];

B1 = [0;1/L1];

h1 = 0.00003; % step size

y1 = [0;0];

y2 = [0;0];

% ******** Euler method **********%

for i1 = 0:1:1000

t1 = i1*h1; % simulation time: 0.03s

x1(i1+1) = t1;

k1 = A1 * y1 + B1;

k2 = A1 * (y1+h1*k1) + B1;

y1 = y1 + (k1 +k2)*h1/2;

result_for_Euler_1(i1+1) = y1(1,1);

end

% ******** 4 order Runge-Kutta method **********%

for i2 = 0:1:1000

t2 = i2*h1; % simulation time: 0.03s

x2(i2+1) = t2;

k1 = A1 * y2 + B1;

k2 = A1 * (y2+h1*k1/2) + B1;

k3 = A1 * (y2+h1*k2/2) + B1;

k4 = A1 * (y2+h1*k3) + B1;

y2 = y2 + (k1 +k2)*h1/2;

result_for_RK_1(i2+1) = y2(1,1);

end

subplot(2,2,1);plot(x1,result_for_Euler_1,'b');

xlabel(' time /s ');ylabel(' Uc /V ');title('Result for Euler method');legend('Euler');

subplot(2,2,2);plot(x2,result_for_RK_1,'r');

xlabel(' time /s ');ylabel(' Uc /V ');title('Result for Runge-Kutta method');legend('Runge-Kutta');

subplot(2,1,2);plot(x1,result_for_Euler_1,'b', x2,result_for_RK_1,'r');

xlabel(' time /s ');ylabel(' Uc /V ');

title('Comparision of results between Euler and RK');legend('Euler','Runge-Kutta');

问题(2)：

clear;

clc;

%******************************************%%

% Question 2

%******************************************%%

R1 = 280;

C1 = 0.000008;

L1 = 2;

A1 = [0, 1/C1;-1/L1,-R1/L1];

B1 = [0;1/L1];

h1 = 0.0002; % step size

y1 = [0;0];

y2 = [0;0];

% ******** Euler method **********%

for i1 = 0:1:500

t1 = i1*h1; % simulation time: 0.1s

x1(i1+1) = t1;

k1 = A1 * y1 + B1;

k2 = A1 * (y1+h1*k1) + B1;

y1 = y1 + (k1 +k2)*h1/2;

result_for_Euler_1(i1+1) = y1(1,1);

end

% ******** 4 order Runge-Kutta method **********%

for i2 = 0:1:500

t2 = i2*h1; % simulation time: 0.1s

x2(i2+1) = t2;

k1 = A1 * y2 + B1;

k2 = A1 * (y2+h1*k1/2) + B1;

k3 = A1 * (y2+h1*k2/2) + B1;

k4 = A1 * (y2+h1*k3) + B1;

y2 = y2 + (k1 +k2)*h1/2;

result_for_RK_1(i2+1) = y2(1,1);

end

subplot(2,2,1);plot(x1,result_for_Euler_1,'b');

xlabel(' time /s ');ylabel(' Uc /V ');title('Result for Euler method');legend('Euler');

subplot(2,2,2);plot(x2,result_for_RK_1,'r');

xlabel(' time /s ');ylabel(' Uc /V ');title('Result for Runge-Kutta method');legend('Runge-Kutta');

subplot(2,1,2);plot(x1,result_for_Euler_1,'b', x2,result_for_RK_1,'r');

xlabel(' time /s ');ylabel(' Uc /V ');

title('Comparision of results between Euler and RK');legend('Euler','Runge-Kutta');

3. 运行结果

问题(1)：

问题(2)：