控制系统计算机辅助设计第四章习题15-21

第四章习题 第15题. 程序： clc

[nP,dP]=paderm(2,0,3);%将延时环节进行pade 降阶近似 s=tf('s');%定义laplace 算子 G=(s-1)/(s+1)^5;

rlocus(G*tf(nP,dP))%画出函数的根轨迹 结果：

由图可知，当0

-6

-4

-2

02

4

-6

-4

-2

02

4

60.28

0.420.56

0.7

0.820.91

0.975

0.14

0.280.42

0.56

0.70.82

0.910.97512345671234

5

67System: untitled1Gain: 4.05

P ole: 0.0389 + 0.872i Damping: -0.0445Overshoot (%): 115Frequency (rad/sec): 0.873System: untitled1

Gain: 2.71P ole: -0.0112 - 0.851i Damping: 0.0131Overshoot (%): 96

Frequency (rad/sec): 0.851System: untitled1Gain: 8.58e+004P ole: 0.0204 - 4.81i

Damping: -0.00425Overshoot (%): 101

Frequency (rad/sec): 4.81System: untitled1Gain: 7.11e+004

P ole: -0.00783 + 4.69i

Damping: 0.00167Overshoot (%): 99.5

Frequency (rad/sec): 4.69

0.14Root Locus

Real Axis

I m a g i n a r y A x i s

第16题 程序：

clc

s=tf('s');%定义laplace 算子

G=1/(s*(s+1)*(s+20)*(s+40));%输入传递函数 rlocus(G),grid %画出根轨迹图形

结果：

如图所示：能使闭环主导极点在阻尼大约为0.707时，K=367

Root Locus

Real Axis

I m a g i n a r y A x i s

-2.5

-2

-1.5

-1

-0.50

0.5

1

1.5

-2.5

-2-1.5-1-0.50

0.511.5

20.64

0.760.86

0.940.985

0.160.340.50.640.760.86

0.940.9850.5

11.522.5System: G Gain: 367

P ole: -0.481 + 0.476i Damping: 0.711

Overshoot (%): 4.17

Frequency (rad/sec): 0.676

0.16

0.34

0.5

第17题 程序： clc

z=tf('z');%定义z 算子

H=1/((z+0.8)*(z-0.8)*(z-0.99)*(z-0.368));%输入H 函数 rlocus(H);%画出函数的根轨迹 K=0.223;%增益K

step(feedback(H*K,1))%画出增益K=0.223时的系统阶跃响应曲线 结果：

如图所示，可以看出，当0

-5

-4-3-2-1012345

-5-4-3-2-1012

345System: H Gain: 0.223P ole: 0.0118Damping:

Overshoot (%):

Frequency (rad/sec):

Root Locus

Real Axis

I m a g i n a r y A x i s

取K=0.223，画出阶跃响应曲线如下：

020406080100120140160

-1

-0.5

0.5

1

1.5

2

x 10

6

Step Response

Time (sec)

A m p l i t u d e

第18题 程序： clc

z=tf('z',0.1);%定义z 算子

H1=1/((z+0.8)*(z-0.8)*(z-0.99)*(z-0.368));%输入H 函数 H=d2c(H1); H2=z^(-8);

[n,d]=paderm(8,0,3); Q=tf(n,d); R=H*Q;

rlocus(R);%画出函数的根轨迹 step(feedback(G*0.01,1))

运行结果：

-6

-4

-2

02

4

6

-6-4

-2

2

4

System: R Gain: 0.0119

P ole: 0.0444 + 0.236i Damping: -0.185Overshoot (%): 181

Frequency (rad/sec): 0.24

Root Locus

Real Axis

I m a g i n a r y A x i s

0102030405060

-2

-1.5-1-0.500.51

1.52

2.5x 10

5

Step Response

Time (sec)

A m p l i t u d e

19题 程序：

clc

s=tf('s');

G=(s^2*(s^2+3*s+4.32))/(s^5+3*s^4+4.63*s^3+1.23*s^2+1.629*s+1.638); rlocus(G) figure

step(feedback(G*2.68,1))

运行结果：从图上时看出K=2.68时系统稳定

-3

-2.5-2-1.5

-1-0.500.5

-2-1.5

-1

-0.5

0.5

1

1.5

2

System: G Gain: 2.68

P ole: 0.00299 + 0.368i Damping: -0.00814Overshoot (%): 103Frequency (rad/sec): 0.368System: G

Gain: 2.68

P ole: 0.00299 - 0.368i Damping: -0.00814Overshoot (%): 103

Frequency (rad/sec): 0.368

Root Locus

Real Axis

I m a g i n a r y A x i s

0500100015002000250030003500400045005000

-2

-1.5

-1

-0.5

0.5

1

1.5

2

x 10

6

Step Response

Time (sec)

A m p l i t u d e

第20题 （1） 程序： clc

s=tf('s');%定义laplace 算子

G=8*(s+1)/(s^2*(s+5)*(s^2+6*s+10));%输入开环传递函数模型

bode(G)%画出bode 图 figure;

nyquist(G),grid %画出nyquist 图 figure;

nichols(G)%画出nichols 图

[Gm,r,wcg,wcp]=margin(G),grid %求系统的幅值裕度和相位裕度 Step(feedback(G,1)) 运行结果： Gm =4.8750 r =3.5676 wcg =1.0000 wcp =0.4125

Gm>1,r>0 所以系统稳定

-150-100

-50

50

M a g n i t u d e (d B

)10

-1

10

10

1

10

2

-360

-315-270-225-180

-135P h a s e (d e g )

Bode Diagram

Frequency (rad/sec)

-8

-7-6-5-4-3-2-101

-0.25

-0.2-0.15-0.1-0.0500.050.1

0.150.20.25

0 dB -20 dB

-10 dB -6 dB

-4 dB -2 dB 20 dB

10 dB 6 dB 4 dB 2 dB Nyquist Diagram

Real Axis

I m a g i n a r y A x i s

-360

-315

-270

-225

-180

-135

-90

-45

-160-140-120-100-80-60-40-20

02040 6 dB

3 dB 1 dB 0.5 dB 0.25 dB 0 dB

-1 dB -3 dB -6 dB -12 dB -20 dB -40 dB -60 dB -80 dB -100 dB -120 dB -140 dB -160 dB

Nichols Chart

Open-Loop P hase (deg)

O p e n -L o o p G a i n (d B )

050100150200250300350400

0.20.40.60.811.2

1.41.61.8

2Step Response

Time (sec)

A m p l i t u d e

（2） 程序：

clc

s=tf('s');%定义laplace 算子

G=(4*(s/3+1))/(s*(0.02*s+1)*(0.05*s+1)*(0.1*s+1));%输入开环传递函数模型

bode(G)%画出bode 图

figure;

nyquist(G),% figure

nichols(G)

[Gm,r,wcg,wcp]=margin(G),grid Step(feedback(G,1))

运行结果： Gm =8.1405 r =88.8362 wcg =38.4943 wcp =8.2231

-100-50

50

M a g n i t u d e (d B

)10

-1

10

10

1

10

2

10

3

-270

-225-180-135-90

-45P h a s e (d e g )

Bode Diagram

Frequency (rad/sec)

-1

-0.8-0.6-0.4-0.200.20.40.60.8

-15-10

-5

5

10

15

Nyquist Diagram

Real Axis

I m a g i n a r y A x i s

00.51

1.52

2.5

0.10.20.30.40.50.6

0.70.80.9

1Step Response

Time (sec)

A m p l i t u d e

(3)

程序：clc

A=[0 2 1;-3 -2 0;1 3 4]; B=[4;3;2]; C=[1 2 3]; D=[];

G=ss(A,B,C,D); G1=tf(G);

bode(G1);%画出bode 图 figure;

nyquist(G);%画出nyquist 图 figure;

nichols(G);%画出nichols 图

[Gm,r,wcg,wcp]=margin(G)%求系统的幅值裕度和相位裕度 Step(feedback(G,1)) 运行结果：

-20-10

10

20

M a g n i t u d e (d B )10

-2

10

-1

10

10

1

10

2

-180

-135-90-450P h a s e (d e g )

Bode Diagram

Frequency (rad/sec)

-1

-0.500.51 1.52

-5-4-3-2-1012

345Nyquist Diagram

Real Axis

I m a g i n a r y A x i s

-180

-135-90

-450

-20-15

-10

-5

5

10

15

Nichols Chart

Open-Loop P hase (deg)

O p e n -L o o p G a i n (d B )

024********

1618

123456

789x 10

6

Step Response

Time (sec)

A m p l i t u d e

（4） 程序： clc

z=tf('z',0.1);

H=(0.45*(z+1.31)*(z+0.054)*(z-0.957))/z*(z-1)*(z-0.368)*(z-0.99);

bode(H)%画出bode 图

figure;

nyquist(H)%画出nyquist 图 figure;

nichols(H)%画出nichols 图

[Gm,r,wcg,wcp]=margin(H),grid %求系统的幅值裕度和相位裕度 figure

Step(feedback(G,1))

Gm = 0.4904 r = -62.7190 wcg =15.2109 wcp = 11.3909 系统不稳定

-150-100

-50

50

M a g n i t u d e (d B )10

-2

10

-1

10

10

1

10

2

180360540720900P h a s e (d e g )

Bode Diagram

Frequency (rad/sec)

-3

-2-10123

-4-3

-2

-1

1

2

3

4

Nyquist Diagram

Real Axis

I m a g i n a r y A x i s

090180270360450540630720810900

-160

-140-120-100-80-60-40-20

02040 6 dB

3 dB 1 dB 0.5 dB 0.25 dB 0 dB -1 dB -3 dB -6 dB -12 dB -20 dB -40 dB -60 dB -80 dB -100 dB -120 dB -140 dB -160 dB

Nichols Chart

Open-Loop P hase (deg)

O p e n -L o o p G a i n (d B )

024681012141618

123456

789x 10

6

Step Response

Time (sec)

A m p l i t u d e

（5）

clc

s=tf('s');

G=(6*(-s+4))/s^2*(0.5*s+1)*(0.1*s+1); bode(G)%画出bode 图 figure;

nyquist(G)%画出nyquist 图 figure;

nichols(G)%画出nichols 图

[Gm,r,wcg,wcp]=margin(G)%求系统的幅值裕度和相位裕度 figure

Step(feedback(G,1))

程序：

Gm = 0 r = Inf wcg = 0 wcp =NaN 系统不稳定

102030405060

70M a g n i t u d e (d B

)10

-1

10

10

1

10

2

10

3

180

225

270

P h a s e (d e g )

Bode Diagram

Frequency (rad/sec)

180

210

240270

1020

30

40

50

60

70

80

Nichols Chart

Open-Loop P hase (deg)

O p e n -L o o p G a i n (d B )

-1000

-900-800-700-600-500-400-300-200-1000

-60-40

-20

20

40

60

Nyquist Diagram

Real Axis

I m a g i n a r y A x i

s

00.51 1.5

2 2.5

3 3.5

0.5

1

1.5

2

2.5

3

3.5

x 10

6

Step Response

Time (sec)

A m p l i t u d e

（6） 程序： clc

G=(-10*s^3-60*s^2+110*s+60)/(s^4+17*s^3+82*s^2+130*s+100);

bode(G)%画出bode 图 figure;

nyquist(G)%画出nyquist 图 figure;

nichols(G)%画出nichols 图

[Gm,r,wcg,wcp]=margin(G)%求系统的幅值裕度和相位裕度 figure

Step(feedback(G,1))

Gm =0.8392 r = -18.6130 wcg = 5.0824 wcp =7.0277 系统不稳定、

-40-30-20-100

10M a g n i t u d e (d

B )10

-2

10

-1

10

10

1

10

2

10

3

90

180270360450P h a s e (d e g )

Bode Diagram

Frequency (rad/sec)

-1.5

-1-0.500.51 1.5

-2-1.5

-1

-0.5

0.5

1

1.5

2

Nyquist Diagram

Real Axis

I m a g i n a r y A x i s

90

135180225270315360405

-35-30

-25

-20

-15

-10

-5

5

Nichols Chart

Open-Loop P hase (deg)

O p e n -L o o p G a i n (d B )