西安理工大学自动控制理论双语chap

2.2 The input-output description of the physical systems — differential equations The input-output description—description of the mathematical relationship between the output variable and the input variable of physical systems.
Ea Ce1.........................(3)
M
M
J
d1
dt
f 1 .....(4)
(4)→(2)→(1) and (3)→(1):
La J CeCm
••
1
(
La f CeCm
Ra J CeCm
•
)1
(
Ra f CeCm
1)1
1 Ce
ua
La CeCm
•
M
Ra CeCm
M
(i3 i2 )dt R4(i3 i2 )......(1)
uc
i2
i1
ur R1
...........................................(2)
i3
1 R3
(uc
R2i2 ).....................................(3)
(2)→(3);
Example 2.4 : A DC motor
Ra
ua
ia
La
(J1,f1)
w1
M
(Jw22,f2) (J3w,3f3)
Mf
Input → ua， output → ω1
i1 i2
La
dia dt
Ra ia
Ea
ua ....(1)
M Cmia .........................(2)
M Mf
..........................equivalent torque
i1i2
(can be derived from :1 i12 i1i23 )
Make:
Te
La Ra
............electric - magnetic
time - constant
Tm
Ra J CeCm
2) The basis of analyzing or designing the control systems. 2.1.2 What is ？
Mathematical models of systems — the mathematical relationships between the system’s variables. 2.1.3 How get？
(2)→(1);
(3)→(1)：
R4C
duc dt
uc
R2 R3 R1
(
R2R3 R2 R3
R4 )C
dur dt
பைடு நூலகம்ur
make : R4C T;
R2 R3 k; R1
(
R2 R3 R2 R3
R4 )C
we have :
T
duc dt
uc
k(
dur dt
ur )
Chapter 2 mathematical models of systems
2.2.1 Examples
Chapter 2 mathematical models of systems
Example 2.1 : A passive circuit
RL
define: input → ur output → uc。 we have：
ur
i
C uc
Ri
L
di dt
uc
ur
i C duc dt
F
ky
f
dy dt
m
d2y d 2t
m
d2y dt 2
f
dy dt
ky
F
If we make :
f k
T1,
m f
T2
we have :
T1T2
d2y dt 2
T1
dy dt
y
1 k
F
Compare with example 2.1: uc→y, ur→F---analogous systems
Chapter 2 mathematical models of systems
2.1 Introduction 2.1.1 Why？
1) Easy to discuss the full possible types of the control systems —only in terms of the system’s “mathematical characteristics”.
Chapter 2 mathematical models of systems
Example 2.3 : An operational amplifier (Op-amp) circuit
R2
R3
Input →ur output →uc
i2
ur R1 i1
-
+
R1
C R4
i3
uc
R3i3
1 C
Chapter 2 mathematical models of systems
J
J1
J2 i12
J3 i12i22
......equivalent
moment
of
inertia
here :
f
f1
f2 i12
f3 i12i22
......equivalent
friction
coefficient
1) theoretical approaches 2) experimental approaches 3) discrimination learning
Chapter 2 mathematical models of systems
2.1.4 types 1) Differential equations 2) Transfer function 3) Block diagram、signal flow graph 4) State variables
LC
d 2uc dt 2
RC
duc dt
uc
ur
make :
RC T1
L T2 R
T1T2
d 2uc dt 2
T1
duc dt
uc
ur
Chapter 2 mathematical models of systems
Example 2.2 : A mechanism
Fk
m
y
f
Define: input → F ，output → y. We have:
.......mechanical
- electric
time
-
constant
Tf
Ra f ....... friction- electric CeCm