自动控制理论第二章1

Chapter 2 mathematical models of systems
La J La f Ra J Ra f La Ra 1 1 ( ) 1 ( 1)1 ua M M CeCm CeCm CeCm CeCm Ce CeCm CeCm
1 1 Te Tm 1 (Te T f Tm ) 1 (T f 1)1 ua (Te Tm M Tm M ) Ce J
Assume the motor idle: Mf = 0, and neglect the friction: f = 0, we have: 2
TeTm d 2 dt 2 d 1 Tm ua dt Ce
T1T2 d y dy 1 T y F 1 2 dt k dt
d 2 uc du T1T2 2 T1 c uc ur dt dt
F k
dy d2y F ky f m 2 dt d t d2y dy d 2 uc duc m 2 f ky F T1T2 2 T1 uc ur dt dt dt dt f m If we make : T1, T2 k f
we have : T1T2 d2y dt 2 T1 dy 1 y F dt k
Compare with example 2.1 and example 2.2:
uc y ;
ur F ua ← Analogous systems
Chapter 2 mathematical models of systems
Example 2.5 :
+
ur R1
R1 R2
A DC-Motor control system
m
y
f
Compare with example 2.1: uc→y, ur→F---analogous systems
Chapter 2 mathematical models of systems
Example 2.3 : An operational amplifier (Op-amp) circuit
Chapter 2 mathematical models of systems
2.1.4 types 1) 2) 3) 4) Differential equations (input-output description) Transfer function Block diagram, signal flow graph State variables ←The Modern Control Theory
Chapter 2 mathematical models of systems
2.1 Introduction 2.2 The input-output description of the physical systems — differential equations 2.3 Linearization of the nonlinear components 2.4 Transfer function 2.5 Transfer function of the typical elements of linear systems 2.6 The block diagram models (dynamic) 2.7 The Signal flow diagram
(2)→(3); (2)→(1); (3)→(1)：
duc dur R2 R3 R2 R3 R4C uc R ( R4 )C ur R R 1 2 3 dt dt
make : R4C T ;
R2 R3 R2 R3 k; ( R4 )C R1 R2 R3
Chapter 2 mathematical models of systems
2.1 Introduction 2.1.1 Why？ Example…the design problem of a temperature control 1) The basis of analyzing or designing the control systems. 2) Easy to discuss the full possible types of the control systems — only in terms of the system’s “mathematical characteristics”. 2.1.2 What is ？ Mathematical models of systems — the mathematical relationships between the system’s variables. 2.1.3 How get？ 1) theoretical approaches 2) experimental approaches 3) discrimination learning
make : RC T1
d 2 uc duc L T2 T1T2 2 T1 uc ur R dt dt
Chapter 2 mathematical models of systems
Example 2.2 : A mechanism
Define: input → F ，output → y. We have:
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. 2.2.1 Examples
La J La f Ra J Ra f La Ra 1 for : 1 ( ) 1 ( 1)1 ua M M CeC m CeCm CeCm CeCm Ce CeC m CeCm
J 2 J3 2 2 ......equivalent moment of inertia 2 i1 i1 i2 f3 f2 here : f f1 2 2 2 ......equivalent friction coefficient i1 i1 i2 Mf M ..........................equivalent torque i1i2 J J1 (can be derived from : 1 i1 2 i1i2 3 )
The input-output description of the DC-Motor control system ：
Tm d 1 1 TeTm 2 Tm (1 k1k2 C ) k1k2 ur (Te M M ) e dt Ce J dt
2.2.2 4 steps to obtain the input-output description (differential equation) of control systems 1) Identify the output and input variables of the control systems. 2) Write the differential equations of each system’s component in terms of the physical laws of the components. * necessary assumption and neglect. * proper approximation. 3) dispel the intermediate(across) variables to get the inputoutput description which only contains the output and input variables.
duc dur we have : T uc k ( ur ) dt dt
Chapter 2 mathematical models of systems
Example 2.4 : A DC motor
Ra La
(J1,f1)
ua
ia
w1
M
(J2,f2)
w2
(J3,f3)
w3
Mf
Rቤተ መጻሕፍቲ ባይዱ R3 C R4
Input →ur
uc R3 i3
output →uc
i2 ur
R1 i1 R1
i3 uc
+
1 ( i3 i2 )dt R4 ( i3 i2 )......(1) C u i2 i1 r ...........................................( 2) R1 1 i3 ( uc R2 i2 ).....................................( 3) R3
Input → ua， output → ω1
dia La Ra ia Ea ua ....(1) dt M C m ia .........................( 2) Ea C e1 .........................( 3) d1 MM J f 1 .....(4) dt
And make: Te
Tm
La ............electric - magnetic time - constant Ra Ra J .......mechanical - electric time - constant CeC m
Ra f Tf ....... friction - electric time - constant CeC m

u f .....................(2) TeTm d dt 2
2
Tm
d 1 1 ua (TeTm M Tm M )......(4) dt Ce J

(2)→(1)→(3)→ (4)，we have：
Chapter 2 mathematical models of systems
R3 R3 DC motor
M
-
uk
trigger
ua
w
load
Uf
rectifier
M
+
techometer
Input → ur, Output →ω; neglect the friction:
uk R2 ( ur u f ) k1( ur u f )........................................(1) R1 ua k 2 uk ......................(3)
Chapter 2 mathematical models of systems
Example 2.1 : A passive circuit
R
L
define: input → ur we have：
output → uc。
ur
i
C
uc
Ri L
du di uc ur i C c dt dt d 2 uc duc LC 2 RC uc ur dt dt
i1 i2
(4)→(2)→(1) and (3)→(1):
La J La f Ra J R f 1 ( ) 1 ( a 1)1 CeC m CeC m CeC m CeCm La Ra 1 ua M M Ce CeC m CeC m

Define:
Chapter 2 mathematical models of systems