MATLAB控制系统仿真作业1

一、控制系统的模型与转换

1．请将下面的传递函数模型输入到matlab 环境。

]52)1)[(2(24)(322

33++++++=s s s s s s s G )

99.02.0)(1(568

.0)(22+--+=z z z z z H ，T=0.1s >> s=tf('s');

G=(s^3+4*s+2)/(s^3*(s^2+2)*((s^2+1)^3+2*s+5)); G

Transfer function:

s^3 + 4 s + 2 ------------------------------------------------------ s^11 + 5 s^9 + 9 s^7 + 2 s^6 + 12 s^5 + 4 s^4 + 12 s^3

>> num=[1 0 0.56];

den=conv([1 -1],[1 -0.2 0.99]); H=tf(num,den,'Ts',0.1)

Transfer function: z^2 + 0.56 ----------------------------- z^3 - 1.2 z^2 + 1.19 z - 0.99

2．请将下面的零极点模型输入到matlab 环境。请求出上述模型的零极点，并绘制其位置。

)1)(6)(5()1)(1(8)(22+++-+++=s s s s j s j s s G )

2.8()

6.2)(2.3()(1

511-++=----z z z z z H ，T=0.05s

>>z=[-1-j -1+j]; p=[0 0 -5 -6 -j j]; G=zpk(z,p,8)

Zero/pole/gain: 8 (s^2 + 2s + 2) -------------------------- s^2 (s+5) (s+6) (s^2 + 1)

>>pzmap(G)

Zero/pole/gain:

z^5 (z+0.3125) (z+0.3846) ------------------------- (z-0.122)

Sampling time: 0.05

>>pzmap （H ）

二、线性系统分析

1. 请分析下面传递函数模型的稳定性。

21)(23+++=

s s s s G 1

3)50600300(1

3)(22+++++=

s s s s s s G

>> num=[1];

den=[1 2 1 2]; G=tf(num,den); eig(G)' ans =

-2.0000 0.0000 - 1.0000i 0.0000 + 1.0000i

可见，系统有两个特征根在虚轴上，一个特征根在虚轴左侧，所以系统是临界稳定的。

G=tf(num,den); eig(G)' ans =

-1.9152 -0.1414

0.0283 - 0.1073i 0.0283 + 0.1073i

可见，有两个特征根在虚轴右侧，所以系统是不稳定的。

2. 请判定下面离散系统的稳定性。

)

05.025.02.0(2

3)(23+--+-=

z z z z z H

)

34039.804.10215.20368.791

.1576.1112.2)(1

2345

12-++--++=-------z z z z z z z z H

>> num=[-3 2];

den=[1 -0.2 -0.25 0.05]; H=tf(num,den,'Ts',0.1); [eig(H) abs(eig(H))] ans =

-0.5000 0.5000 0.5000 0.5000

0.2000 0.2000

可以看出，由于各个特征根的模均小于1，所以可以判定闭环系统是稳定的。

[eig(H) abs(eig(H))] ans =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4.1724 4.1724 0.3755 + 0.1814i 0.4170 0.3755 - 0.1814i 0.4170 -0.5292 0.5292 -0.2716 0.2716

0.1193 0.1193

可以看出，由于4.1724这个特征根的模大于1，所以可以判定闭环系统是不稳定的。

3. 设描述系统的传递函数为

40320

10958411812467284224494536546364032018576022208812266436380598251418)(2345678

234567+++++++++++++++=s s s s s s s s s s s s s s s s G ，假定系统

具有零初始状态，请求出单位阶跃响应曲线和单位脉冲响应曲线。

>> num=[18 514 5982 36380 122664 22088 185760 40320]; den=[1 36 546 4536 22449 67284 118124 109584 40320]; G=tf(num,den)

Transfer function:

18 s^7 + 514 s^6 + 5982 s^5 + 36380 s^4 + 122664 s^3 + 22088 s^2 + 185760 s + 40320 -----------------------------------------------------------------------------------------

s^8 + 36 s^7 + 546 s^6 + 4536 s^5 + 22449 s^4 + 67284 s^3 + 118124 s^2 + 109584 s + 40320

>> step(G ,10) >> impulse(G ,10)

单位脉冲响应：

1. 请分析下面传递函数模型阶跃响应。

21)(2

+++=

s s s s G

利用Simulink 建模，建立系统仿真模型如下：

单击启动仿真按钮，双击示波器得到系统的阶跃响应如下：

2. 请分析下面离散系统的脉冲响应。

13)50600300(1

43)(2

223++++++=s s s s s s s G

利用Simulink 建模，建立系统仿真模型如下：

单击启动仿真按钮，双击示波器得到系统的脉冲响应如下：

3. 对离散采样系统进行分析，并求出其阶跃响应。

G(s)

零阶保持器u(t)

y(t)

其中：2

373

)(2

3++++=

s s s s s G

利用Simulink 建模，建立系统仿真模型如下：

单击启动仿真按钮，双击示波器得到系统的阶跃响应如下：

4. 设计控制器，使得下列系统稳定。

)2.1)(2)(3()3.2)(1()(+-+++=

s s s s s s G

利用Simulink 建模，未连入控制器时，仿真模型和响应如下：

利用Simulink 建模，设计控制器：

从响应输出图形可以看出，连入控制器后系统稳定，性能明显提高。

四、基于MATLAB 的PID 控制器设计

设计题目：

1. 应用Ziegler — Nichols 算法设计PID 控制器，实现系统的闭环稳定，并比较对各个系统的控制效果。

)1s (1G (s)+=

未连入PID 控制器时的系统仿真及其性能指标如下：

可见，未调节时的系统性能有待提高，需设计PID控制器连入。

输入：

>> num=1;

den=conv( [1,1],conv( [1,1],[1,1] ));

Step(num,den);

K=dcgain (num,den)

得出：

K =1

根据图形，得出：

L=1.86 T=4.4

利用自定义的Ziegler_std函数求出Kp、Ti、Td

输入：

>> K=1;

L=1.86;

T=4.4;

[num,den,Kp,Ti,Td]=Ziegler_std (3,[K,L,T])

得出：

num =

2.6400 2.8387 1.5262

den =

根据得出的Kp 、Ti 、Td 值，设计PID 控制器，并利用利用Simulink 仿真建模。仿真模型及其响应如下：

可见，加入PID 控制器调节后，系统性能明显改善。

)1s (1G (s)+=

未连入PID 控制器时的系统仿真及其性能指标如下：

输入：

>> num=1;

den=conv([1,1],conv([1,1],…;

conv( [1,1],conv( [1,1],[1,1] ))));

Step(num,den);

K=dcgain (num,den)

得出：

K = 1

根据图形，得出：

L=3.4 T=6.8

利用自定义的Ziegler_std函数求出Kp、Ti、Td

输入：

>> K=1;

L=3.4;

T=6.8;

[num,den,Kp,Ti,Td]=Ziegler_std (3,[K,L,T])

得出:

num =

4.0800 2.4000 0.7059

den =

1 0

Kp =

2.4000

Ti =

6.8000

Td =

1.7000

根据得出的Kp、Ti、Td值，设计PID控制器，并利用利用Simulink仿真建模。仿真模型及其响应如下：

可见，加入PID控制器调节后，系统性能明显改善。

3)1 s(1

-1.5s G(s)

利用Simulink建模，未连入控制器时，仿真模型和响应如下：

输入:

>>num=[-1.5 1];

den=conv( [1,1],conv( [1,1],[1,1] ));

Step(num,den);

K=dcgain (num,den)

得出:

K =1

根据图形，得出：

L=1.8 T=5.7

利用自定义的Ziegler_std函数求出Kp、

Ti、Td

输入:

>> K=1;

L=1.8

T=5.7;

[num,den,Kp,Ti,Td]=Ziegler_std (3,[K,L,T])

得出:

num =

3.4200 3.8000 2.1111

den =

Kp =

3.8000

Ti =

3.6000

Td =

0.9000

根据得出的Kp、Ti、Td值，设计PID控制器，并利用利用Simulink仿真建模。仿真模型及其响应如下：

可见，加入PID控制器调节后，系统性能明显改善。

五、模糊控制器设计

设计任务：试设计一个模糊控制器，实现对室内温度的控制的模拟。

参考输入：

（1）温度18-40℃范围内分为七个论域，NB NM NS ZE PS PM PB；隶属度函数满足高斯分布；

（2）温度变化率-2 ~2℃范围内分为七个论域，NB NM NS ZE PS PM PB；隶属度函数满足高斯分布；

参考输出：

变频空调输出的控制信号。在一定范围内分为七个论域，NB NM NS ZE PS PM PB，隶属度函数为常数1。

模糊推理过程，output=输入隶属度函数值*输出论域的中心值。

注：本模糊程序采用PAM控制方式的压缩机，则其输出的转速范围为：0~10500转/分。

控制规则：

% % if input is NB and errorinput is PS, then output is NM; % % if input is NB and errorinput is PM, then output is NM; % % if input is NB and errorinput is PB, then output is NS; % % if input is NM and errorinput is NB, then output is NB; % % if input is NM and errorinput is NM, then output is NM; % % if input is NM and errorinput is NS, then output is NM; % % if input is NM and errorinput is ZE, then output is NM; % % if input is NM and errorinput is PS, then output is NM; % % if input is NM and errorinput is PM, then output is NS; % % if input is NM and errorinput is PB, then output is NS; % % if input is NS and errorinput is NB, then output is NM; % % if input is NS and errorinput is NM, then output is NS; % % if input is NS and errorinput is NS, then output is NS; % % if input is NS and errorinput is ZE, then output is NS; % % if input is NS and errorinput is PS, then output is NS; % % if input is NS and errorinput is PM, then output is ZE; % % if input is NS and errorinput is PB, then output is ZE; % % if input is ZE and errorinput is NB, then output is NS; % % if input is ZE and errorinput is NM, then output is ZE; % % if input is ZE and errorinput is NS, then output is ZE; % % if input is ZE and errorinput is ZE, then output is ZE; % % if input is ZE and errorinput is PS, then output is ZE;

% % if input is ZE and errorinput is PM, then output is PS; % % if input is ZE and errorinput is PB, then output is PS;

% % if input is PS and errorinput is NB, then output is ZE; % % if input is PS and errorinput is NM, then output is PS; % % if input is PS and errorinput is NS, then output is PS;

% % if input is PS and errorinput is ZE, then output is PS;

% % if input is PS and errorinput is PS, then output is PS;

% % if input is PS and errorinput is PM, then output is PM; % % if input is PS and errorinput is PB, then output is PM; % % if input is PM and errorinput is NB, then output is PS; % % if input is PM and errorinput is NM, then output is PS; % % if input is PM and errorinput is NS, then output is PM; % % if input is PM and errorinput is ZE, then output is PM; % % if input is PM and errorinput is PS, then output is PM; % % if input is PM and errorinput is PM, then output is PM; % % if input is PM and errorinput is PB, then output is PB; % % if input is PB and errorinput is NB, then output is PS; % % if input is PB and errorinput is NM, then output is PM; % % if input is PB and errorinput is NS, then output is PM; % % if input is PB and errorinput is ZE, then output is PM; % % if input is PB and errorinput is PS, then output is PB;

>> x1 = (18:0.1:40)';

y0 = gaussmf(x1, [1 18]);

y1 = gaussmf(x1, [1 21]);

y2 = gaussmf(x1, [1 25]);

y3 = gaussmf(x1, [1 29]);

y4 = gaussmf(x1, [1 33]);

y5 = gaussmf(x1, [1 37]);

y6 = gaussmf(x1, [1 40]);

plot(x1,[y0 y1 y2 y3 y4 y5 y6])

2.误差图：

程序为：

>> x1 = (-2:0.1:2)';

y0 = gaussmf(x1, [0.5 -2]);

y1 = gaussmf(x1, [0.5 -1.3]); y2 = gaussmf(x1, [0.5 -0.7]); y3 = gaussmf(x1, [0.5 0]);

y4 = gaussmf(x1, [0.5 0.7]);

y5 = gaussmf(x1, [0.5 1.3]);

y6 = gaussmf(x1, [0.5 2]);

plot(x1,[y0 y1 y2 y3 y4 y5 y6])

3.程序为；

x=35;

ex=-0.8;

% define input type in fuzzy zone

y0 = gaussmf(x, [1 18]);

y1 = gaussmf(x, [1 21]);

y2 = gaussmf(x, [1 25]);

y3 = gaussmf(x, [1 29]);

y4 = gaussmf(x, [1 33]);

y5 = gaussmf(x, [1 37]);

y6 = gaussmf(x, [1 40]);

a=[y0 y1 y2 y3 y4 y5 y6];

b=max(a);

% caculate input in fuzzy zone,get input_type and input_authorityvalue if x<=40 & x>=18

if b==a(1)

type='NB';

authorityvalue=y0;

elseif b==a(2)

type='NM';

authorityvalue=y1;

elseif b==a(3)

type='NS';

authorityvalue=y2;

elseif b==a(4)

type='ZE';

authorityvalue=y3;

elseif b==a(5)

type='PS';

authorityvalue=y4;

elseif b==a(6)

type='PM';

authorityvalue=y5;

elseif b==a(7)

type='PB';

authorityvalue=y6;

end

else

if x>40

type='PB';

authorityvalue=1;

elseif x<18

type='NB';

authorityvalue

%error calculate.

ey0 = gaussmf(x, [0.5 -2]);

ey1 = gaussmf(x, [0.5 -1.3]);

ey2 = gaussmf(x, [0.5 -0.7]);

ey3 = gaussmf(x, [0.5 0]);

ey4 = gaussmf(x, [0.5 0.7]);

ey5 = gaussmf(x, [0.5 1.3]);

ey6 = gaussmf(x, [0.5 2]);

a=[ey0 ey1 ey2 ey3 ey4 ey5 ey6];

b=max(a);

% caculate input in fuzzy zone,get input_type and input_authorityvalue if x<=2 & x>=-2

if b==a(1)

etype='NB';

eauthorityvalue=y0;

elseif b==a(2)

etype='NM';

eauthorityvalue=y1;

elseif b==a(3)

etype='NS';

eauthorityvalue=y2;

elseif b==a(4)

etype='ZE';

eauthorityvalue=y3;

elseif b==a(5)

etype='PS';

eauthorityvalue=y4;

elseif b==a(6)

etype='PM';

eauthorityvalue=y5;

elseif b==a(7)

etype='PB';

eauthorityvalue=y6;

end

else

if x>2

etype='PB';

eauthorityvalue=1;

elseif x<-2

etype='NB';

eauthorityvalue=1;

end

if type=='NB'& etype=='NB'

out=authorityvalue*0.001;

% if input is NB and errorinput is NB, then output is NB;

elseif type=='NB'& etype=='NM'

out=authorityvalue*0.001;

% if input is NB and errorinput is NM, then output is NB;

elseif type=='NB'& etype=='NS'

out=authorityvalue*0.001;

% if input is NB and errorinput is NS, then output is NB;

elseif type=='NB'& etype=='ZE'

out=authorityvalue*1750;

% if input is NB and errorinput is ZE, then output is NM;

elseif type=='NB'& etype=='PS'

out=authorityvalue*1750;

% if input is NB and errorinput is PS, then output is NM;

elseif type=='NB'& etype=='PM'

out=authorityvalue*1750;

% if input is NB and errorinput is PM, then output is NM;

elseif type=='NB'& etype=='PB'

out=authorityvalue*3500;

% if input is NB and errorinput is PB, then output is NS;

elseif type=='NM'& etype=='NB'

out=authorityvalue*0.001;

% if input is NM and errorinput is NB, then output is NB;

elseif type=='NM'& etype=='NM'

out=authorityvalue*1750;

% if input is NM and errorinput is NM, then output is NM;

elseif type=='NM'& etype=='NS'

out=authorityvalue*1750;

% if input is NM and errorinput is NS, then output is NM;

elseif type=='NM'& etype=='ZE'

out=authorityvalue*1750;

% if input is NM and errorinput is ZE, then output is NM;

elseif type=='NM'& etype=='PS'

out=authorityvalue*1750;

% if input is NM and errorinput is PS, then output is NM;

elseif type=='NM'& etype=='PM'

out=authorityvalue*3500;

% if input is NM and errorinput is PM, then output is NS;

elseif type=='NM'& etype=='PB'

out=authorityvalue*3500;

% if input is NM and errorinput is PB, then output is NS;

elseif type=='NS'& etype=='NB'