2013/3/10
Principles of Automatic Control
By Chunyue Song (宋春跃)
Qinmin Yang (杨秦敏)
浙江大学控制科学与工程学系
1
Course Objective / Requirement
 Where to find (or contact) me:
– Office：Control Building New-507 (Yuquan)
– Office hour MW 14:00-15:00 (or by appointment)
– E-mail：qmyang@iipc.zju.edu.cn
– Mobile: 15168218448 (588448)
 Where to find (or contact) Prof. Song: 宋春跃
– Office：Control Building Old-110 (Yuquan)
– E-mail：cysong@iipc.zju.edu.cn
– Mobile: 15857189809
 Teaching Assistant: 吴俏
– Office：Control Building Old-113 (Yuquan)
– E-mail：wendyxk@126.com
– Mobile: 15158114423
Any questions？
2
1
2013/3/10
Course Objective / Requirement
 Objectives:
cybernetics
– To introduce the fundamental principles of control theory
– To develop the basic understanding of control systems
– To learn how to analyze and how to design automatic
control systems
– Bonus: English?
– More to say ...
 Requirements:
– Preparation, attendance and review
– Let me know if you must miss a class
– Finish homework assignments on schedule
– Penalty for late delivery (80% full credit)
 Grading policy:
– Attendance, homework, quiz: 30%
– Midterm: 10-20% (April 17th)
– Final Exam: 50-60%
– Final Project (PAC II)
3
Course Objective / Requirement
 Where to find course materials
– Courses center: http://www.cse.zju.edu.cn/eclass/
 控制系主页－本科教学平台－本科网络课程平台
 核心课程：自动控制原理 I
– 我的主页：http://mypage.zju.edu.cn/en/qmyang
 Prerequisites
– Calculus, Ordinary Differential Equation, Linear Algebra,
Complex Function Theory, Circuit
4
2
2013/3/10
Course Objective / Requirement
 Textbooks ＆ Refs:
– English:
1. John J.D’Azzo, Constantine H.Houpis, 2001, 2006
Linear Control System Analysis and Design (Fourth or Fifth Edition)
2. Richard C. Dorf, Robert H. Bishop, 2008
Modern Control Systems（Eleventh Edition)
3. Karl Johan Astrom & Richard M. Murray, 2008
Feedback Systems---An Introduction for Scientists and Engineers
5
Course Objective / Requirement
 Textbooks ＆ Refs:
– Chinese:
1. 孙优贤、王慧主编，《自动控制原理》，化学工业出版社，2012
2. 胡寿松主编. 《自动控制原理》（第四版）；2001，科学出版社
3. 周春晖主编. 《化工过程控制原理》（第二版）；1998，化工出版社
4. 《自动控制系统》 (第八版) ；汪小帆，李翔译，2004，高等教育出版社
5. 《现代控制系统》（第八版），谢红卫等译，2001，2006，高教出版社
6. 《动态系统的反馈控制》 (第四版) ；朱齐丹等译, 电子工业 出版社
6
3
2013/3/10
CHAPTER 1
Introduction to Control Systems
控制科学与工程学系
Outline
 Course Outline
 Introduction to Control Systems
 Definitions
 Historical Background
 Mathematical Background
 Engineering Control Problem
 Basic Form of Block Diagram
8
4
2013/3/10
Introduction to Control Systems
 "The philosophers have only interpreted
the world, in various ways; the point is,
however, to change it.“
- in ‘Theses on Feuerbach’, Karl Marx 1845
The capability of effective
control is the unique
characteristic distinguishing
human beings and animals.
9
Introduction: What Can Control Achieve?

Make a system behave as desired

Keep output constant or varying as wanted (most
processes, aircraft)

Stabilize unstable system (inverted pendulum)

Reduce disturbance effects (quadrotor )

New freedom and tools for designers
Y(s)＝G(s)X(s)
X(s)
G(s)
Block diagram
10
5
2013/3/10
Key Words
 Required knowledge
– Model
 Mathematic, diagram, response, rule, etc…
– Control theory
 Linear, nonlinear …
– System theory
 Optimal design, robustness …
 Assumptions to make
11
Key Words
 System
 (Automatic) Control System
 Input, Output
 Open-loop Control system
 Closed-loop Control system
(Negative Feedback-control systems)
 Dynamic, Static
12
6
2013/3/10
Basic Concepts – Systems
 System
– A collection of components which are coordinated
together to perform a specific function.
 Static System
– I/O relationship is not time dependent
– the output is determined by the current input only
 Dynamic System
– A system with a memory.
– For example, the input value at time t will influence
the output at future instant.
 A system interact with their environment through a
controlled boundary.
13
Basic Concepts – Interaction
 The interaction is defined in terms of variables.
i. System input
ii. System output
iii. Environmental disturbances
iv. State variables (internal variables) and Subsystems
14
7
2013/3/10
Basic Concepts – System Variables
 The system’s boundary depends upon the defined
objective function of the system.
 The system’s function is expressed in terms of
measured output variables (CV).
 The system’s operation is manipulated through
control input variables (MV).
 The system’s operation is also affected in an
uncontrolled manner through disturbance input
variables (DV).
– measured variable = CV = controlled variable 被控变量
– control variable = MV = manipulated variable 控制变量
15
Introduction to Control Systems
 Control: is the process of causing a system
variable (e.g, temperature, position….) to conform
to a desired value or trajectory, called reference
value or trajectory, by means of some sort of
efforts.
 自动控制系统：在无人直接参与下可使生产过程或其
他过程按期望规律或预定程序进行的控制系统。
 A control system consisting of interconnected
components is designed to achieve a desired
purpose.
16
8
2013/3/10
Manual vs. Automatic Control
 Example: driving a car implies controlling the vehicle to follow
the desired path and arrive safely at a planned destination.
 If you drive the car yourself, you are performing a manual
control of the car. If you design a machine to do it then you
build an automatic control system.
17
Closed-Loop Control: -- Ex. A manual control system





Objective: To control direction and speed of car
Outputs: Actual direction and speed of car
Control inputs: throttle pedal, brake, steering wheel
Disturbances: Road surface and grade, wind, obstacles
Possible subsystems: The car in mechanics, the
engine system, the power steering system, the braking
system, suspension system, ...
18
9
2013/3/10
Closed-Loop Control: -- Ex. A manual control system
(a) Automobile
steering control system.
视觉与触觉
(b) The driver uses the
difference between the
actual and the desired
direction of travel to
generate a controlled
adjustment of the
steering wheel.
(c) Typical directionof-travel response.
Adaptive Cruise Control----ACC
19
Unmanned Ground Vehicles
 – A teleoperated UGV is a vehicle that is controlled by a
human operator at a remote location via a communications
link.
 – An autonomous UGV is essentially an autonomous robot
but is specifically a vehicle that operates on the surface of
the ground.
20
10
2013/3/10
Open Loop Control: Ex. Microwave
Input: Set time
Output: Temperature
If the temperature of
the stuff to be warmed
is not satisfactory, it
cannot automatically
alter the time.
Timer
21
Open Loop Control: Ex. DC shunt motor
(See P1-2, Fig. 1.1(b)直流并励电动机）
Input: armature voltage
(电枢电压）
Output: speed of the
shaft（轴）
If a variation of the
speed from the desired
value appears, due to a
change of mechanical
load on the shaft, it is
no way to change the
value of input quantity
automatically (e.g.
electric fan).
dynamic part
并激磁场
22
11
2013/3/10
Open-Loop System: Functional block diagram
(See P2, Fig. 1.1(c))
Reference
Input
Reference
selector
Control Dynamic
Output
unit
Input
Figure 1.1 (c) Functional block diagram of a open-loop control system
Temperature
time
Microwave
voltage
motor
Timer
Speed of the motor
Desired output
Setpoint/Reference
Actuating device
Temperature
speed
Process/Plant
output
The actual output has no effect upon the control input is called
open-loop control system.
23
From Open-loop to Closed-Loop Control System
(See P3, Fig. 1.2)
Reference Input
Command
Input
Reference
selector
+
error
－
Feedback signal
Actuating signal
Forward
elements
System dynamics
Output
Feedback
element
Figure 1.2 Functional block diagram of a closed-loop control system
• The closed-loop control system implies that the action results
from the comparison between the output and input quantities in
order to maintain the output at the desired value.
•
The output is controlled in order to achieve the desired value.
24
12
2013/3/10
Colsed-Loop Control: Ex.-- Household Furnace
(See P4, Fig. 1.3)（家用取暖炉）
Desired room
temperature setting
Actual room
temperature
Thermostat
Hot-air
vents
Furnace
Relay
Fig. 1.3 Home heating control system
bimetallic strip: difference in
thermal expansion in the two
metals leads to a much larger
sideways displacement of the strip
bimetallic coil
 If temperature inside the house is below (or above) the desired value,
furnace turns on (turns off) until the temperature inside the house is
slightly higher (lower) than the desired temperature.
25
Colsed-Loop Control: Ex.-- Household Furnace
(See P3-4, Fig. 1.3)（家用取暖炉）
Error
signal
Desired
temp.
Relay switch
signal
gain
+
heat
furnace
temp.
house
temperature
sensor
controller
dynamic part
26
13
2013/3/10
Closed-Loop Control System: Ex.-- A manual control system
Dynamic
h
Objective: To maintain
the level in the tank a
desired value
If h<H, close valve;
If h>H, open valve;
Controlled Variable
fluid Level Height H
If h=H, maintain unchanged
27
Closed-Loop Control System: Ex.-- A manual control system
Actuating
Desired
level.
signal
+
(Error)
control
rules
valve
signal
fluid
valve
dynamic
level
tank
-
Man’s brain
level sensor:
man’s eyes
The system components
28
14
2013/3/10
Closed-Loop Control: Ex. -- Automatic control system
FT
FC
Feedforward control
Set point
给定值H
Measurement Device
Actual Value h
HC
HT
测量值
Control Variable
控制量u
Actuating Signal
Feedback control
Actuator
带控制点的工艺流程图
29
Open-Loop/Closed-Loop Systems
A control system is an interconnection of components.
Each component is represented by a block in a diagram.
Open Loop
Desired
output
Control
signal
Controller
Actuator
Plant
Plant
output
Closed-Loop
Actuating
Desired
output
signal
Controller
+
Control
signal
Actuator
Plant
Plant
output
(Error)
Sensor
30
15
2013/3/10
Colsed-Loop Control: Ex. --Elevator
(See P4, Fig. 1.4 Automatic elevator)
Chicago Sears
Tower
(more than
2400 steps,
103 floors and
443m high)
• The express elevator in the Sears Tower is
designed so that it ascends or descends in 55
sec., with maximum passenger comfort.
31
Definitions (See P6-7)
System – An interconnection of elements and devices for a desired purpose.
Control System – An interconnection of components forming a system
configuration that will provide a desired response.
Forward element (process; system
dynamics) –The unit that reacts to an
actuating signal to produce a desired
output, i.e. the device, plant, or
system under control. The input and
output relationship represents the
cause-and-effect relationship of the
process.
input
output
process
Process to be controlled
32
16
2013/3/10
Definitions (See P6-7)
Command input (desired output) – The motivating input signal to the
system, which is independent of the output of the system..
Reference selector (reference input element) – The unit that establishes
the value of the reference input (in terms of the desired output).
Reference input – Produced by the reference selector, it is the actual
signal input to the control system.
Disturbance input – The signal to the system that has an unwanted effect
on the system output.
Output (controlled variable) – The quantity that must be maintained at a
prescribed value, i.e, it must follow the command input without
responding to disturbance inputs.
33
Definitions (See P6-7)
Actuating signal (error signal) – It is the input to the control unit that
causes the output to have the desired value. Usually, it is the difference
between the reference input and the feedback signal.
Control variable (manipulated variable) – The signal that is the output of
the control unit, and as an input act directly on the process (system
dynamic). The device that performs the action called actuating device.
Actuating signal
Control
unit
Control variable
34
17
2013/3/10
Definitions (See P6-7)
Open-Loop Control Systems
utilize a controller or
control actuator to obtain
the desired response.
Disturbance input
Feedback Element－The unit that provides the means for feeding back the output quantity.
Closed-Loop Control Systems
utilizes feedback to compare
the actual output to the
desired output response.
SISO (single-input single output) Control System
MIMO (multi-input multi-output)
Multivariable Control System
35
Examples of Modern Control Systems
36
18
2013/3/10
Definitions (See P6-7)
干扰输入
前馈控
制器
给定值
输入
比较器
反馈控制器 过程
反馈通道
被控对象
输出
传感器
• Feedback Control and Feedforward Control
－ hybrid control
37
Examples of Modern Control Systems
38
19
2013/3/10
Examples of Modern Control Systems
39
Categorizing（不同分类）
• Open-loop and Closed-loop control System
• Feedback Control and Feedforward Control and
Feedback-Feedforward Control
• SISO Control System and MIMO Control System
• Set-point Control System and Servo Control System
(定值控制系统）
（随动或称伺服控制系统）
• Analog (continuous) Control System and Sampleddata (discrete) Control System
40
20
2013/3/10
Outline
 Course Outline
 Introduction to Control Systems
 Definitions
 Historical Background
 Mathematical Background
 Engineering Control Problem
 Basic Form of Block Diagram
41
Historical Background
http://www.asc-cybernetics.org/foundations/timeline.htm
 First automatic feedback
system
invented in modern Europe
(Holland) by Cornelis Drebbel
(1572-1633) for temperature
regulation of a furnace used to
heat an incubator (培养的器具,
孵化小鸡的培育器)

Intake of heat from the valve above

Temperature sensor – alcohol & mercury & stick

Principle – thermal expansion and contraction of alcohol
42
21
2013/3/10
Historical Background
 First automatic feedback controller used in an
industrial process invented by James Watt in 1769 –
the flyball governor – used to mantain the rotating
speed of a shaft in a steam engine
 J.C.Maxwell (1868)
formulated the first
mathematical study of the
stability of feedback
control applied to a
governor.
Watt’s Flyball Governor
(18th century)
飞球式（离心式）调速器
43
Historical Background
 The first historical feedback system,
claimed by Russia, is the water-level
float regulator said to have been
invented by I. Polzunov in 1765.
The float detects the water
level and controls the valve
that covers the water inlet in
the boiler.
44
22
2013/3/10
Historical Background
 E.J.Routh determined a criteria for stability analysis in 1877
 Lyapunov (1893) studied stability of motion (ODEs)
 The Wright Brothers achieved controlled flight (1903)
 Sperry (1910) worked on gyroscopes（陀螺仪） and autopilots
 Black proposed feedback electronic amplifier in 1927（AT&T,
Bell)
 Nyquist derived a frequency domain stability criterion in 1932
45
Historical Background
 Bode developed frequency response methods in 1938
 Ziegler-Nichols developed a method for PID tuning and Wiener
develops optimal filter design in 1942
 Evans developed the Root Locus in 1948
 Pontryagin formulated the Maximum Principle in 1956
 Bellman developed Dynamic Programming in 1957
 Kalman formulated Optimal Estimation in 1960
Modern control
Milestone
 Hoff invented the Microprocessor in 1969, then digital control is used
extensively
 State variable methods and optimal control were further developed in
1970-1980
 Robust and Nonlinear Control developed in 1980-1990
 Multivariable Predictive Control developed and employed in process
industry in 1980
46
23
2013/3/10
Historical Background
 Switched control, hybrid control, and control using convex
optimization methods is developed in 1990-2000
 Feedback control is widely used in automobiles and reliable,
robust systems are heavily demanded in manufacturing starting
in 1994
 First ever autonomous rover vehicle, known as Sojourner
explores successfully the Martian surface in 1997. Second lands
successfully in January 3 2004.
 Control abstractions and control of piecewise-affine systems
are developed starting in 1998-1999 and are still research topics
as well as hybrid control
47
Cybernetics
“自动控制理论” VS “控制论”
 The term cybernetics stems from the Greek
κυβερνήτης (kybernētēs, steersman, governor,
pilot, or rudder — the same root as
government)
 The essential goal of cybernetics is to
understand and define the functions and
processes of systems that have goals and that
participate in circular, causal chains that move
from action to sensing to comparison with
desired goal, and again to action.
48
24
2013/3/10
Cybernetics
 Some descriptions:
“a science concerned with the study of systems of any
nature which are capable of receiving, storing, and
processing information so as to use it for control”
- A.N. Kolmogorov 苏联最伟大的数学家之一
“the science of control and communication in
the animal and the machine”-Norbert Wiener,
应用数学家、控制专家 《控制论-关于在动物
或机器中控制或通讯的科学》，1948
维纳抓住了一切通信和控制系统的共同特点，
即它们都包含着一个信息传输和信息处理的过
程：双向信息传输、调整、适应、随机。。。
49
Cybernetics
 “控制论”是一门“横断科学”
控制论的研究对象不是物质的特定结构或运动的特定形态，而是
普遍的结构和行为方式
 关键概念：目的性、负反馈、稳定性
 关键概念：控制、通讯、反馈、信息、适应、复杂性、网络、自
组织、自主性
“控制就是通讯”(维纳)
生命体和机器的统一 人机关系（服务、合作、竞争）
 控制论的影响：控制理论、计算机科学、运筹学、优化理论、信
息理论、人工智能、神经网络、认知科学、动力系统。。。
50
25
2013/3/10
IFAC (International Federation of Automatic Control)
http://www.ifac.org/;
http://www.ifac-control.org/
Education
Organization
Application
Journals
Industrialization
Conferences
51
CAA (Chinese Association of Automation
52
26
2013/3/10
Outline
 Couse Outline
 Introduction to Control Systems
 Definitions
 Historical Background
 Mathematical Background
 Engineering Control Problem
 Basic Form of Block Diagram
53
Mathematical background
• Classical control system:
• the solution of differential equations
• Laplace transform
• steady-state frequency-response
• root-locus
• Modern control system:
• state variable methods (linear algebra)
• Computer-aided analyze & design program
• MATLAB……….
54
27
2013/3/10
Outline
 Course Outline
 Introduction to Control Systems
 Definitions
 Historical Background
 Mathematical background
 Engineering Control Problem
 Basic Form of Block Diagram
55
Engineering Control Problem
1. Establish control objectives
• Qualitative: Ex. do not use too much fuel
• Quantitative: Ex. step response overshoot < 20%
2. Establish system configuration and select sensors
and actuators
3. Obtain a model of the plant, the actuator and the
sensor
• Analytic
• From measured data (system identification)
56
28
2013/3/10
Engineering Control Problem
4.
5.
Design a controller
• Select techniques
• Choose parameters
Analyze closed loop performance – meet desired
specifications?
• Yes – stop, you are done.
• No – iterate going back to step 4 by changing
parameters and/or technique
57
Engineering Control Problem
reasoning
abstract
Physical problem
Theory
Mathematics model
application
Implementation
Modeling
analytical
system IDs
Dynamic model
Controller
Design
Control algorithm
Root-Locus PI
Control
Satisfy?
Requirement
Analysis
Performance Specifications
58
29
2013/3/10
Applications
Energy generation
Entertainment
Energy transmission
Instrumentation
Process control
Mechanics
Discrete manufacturing
Materials
Communication
Physics
Transportation
Biology
Buildings
Economics
59
Outline
 Introduction to Control Systems
 Definitions
 Historical Background
 Mathematical background
 Engineering Control Problem
 Course Outline
 Conclusions
 Basic Form of Block Diagram
60
30
2013/3/10
Outline of This Course (see P16-18)
 Mathematical foundation for modeling physical
systems and obtaining time solutions
 First undergraduate course in control and
control system design
 Classical Control Theory
 Modern Control Theory
 Advanced undergraduate or graduate topics
61
Course arrangement
 There are three short semesters to finish the
course: spring, summer and fall semester.
 We will mainly learn two foundation parts in control
theory: classical (traditional) and modern control
theory.
 Both analog and sampled-data single-input singleoutput (SISO) feedback-control systems are
covered in details with respect to the textbook
chapter 1 to 15 (16).
62
31
2013/3/10
Outline & Schedule of This Course
 Introductions to the control system (Chpt.1)
 Foundations of Classical Control
 System modeling (writing system equations)
Spring
(Chpt.2-5)
 Control-system characteristics (Chpt.2,3,4,6)
 Root locus (Chpt.7,10)
 Frequency response (Chpt. 8,11)
Summer
 Foundations of Modern Control Theory（Fall)
 Digital control system
 State variable feedback
63
Outline
 Introduction to Control Systems
 Definitions
 Historical Background
 Mathematical background
 Engineering Control Problem
 Course Outline
 Conclusions
 Basic Form of Block Diagram
64
32
2013/3/10
Conclusions
 An exciting field
 Use of feedback often revolutionary
 Rapid growth of applications
 Interveaved with other subjects
 Many unsolved problems
65
References:
 http://www.ifac.org/
 http://www.ifac-control.org/
 http://www.cds.caltech.edu/~murray/amwiki/Main_Page
 A magazine: IEEE Control System Magazine
 （美）理查德 M.穆拉里编著，陈虹、马彦译，信息爆
炸时代的控制，北京：科学出版社

………….
66
33
2013/3/10
例1. 开环的药物注射控制系统是医学领域最常见的应用实例。
它运用的所用药物的剂量与疗效之间的关系（数学模型）。由于
微型葡萄糖传感器还不成熟，胰岛素注射控制系统也采用开环系
统。现在要求设计一个能调节糖尿病人血糖浓度的系统。也即控
制系统要向病人注射剂量适中的胰岛素。
健康人士的血糖和胰岛素浓度
67
Example 1. 胰岛素注射控制系统也采用开环系统。现在要求设
计一个能调节糖尿病人血糖浓度的系统。也即控制系统要向病
人注射剂量适中的胰岛素。
可编程信
号发生器
电机泵与阀门
胰岛素注射速率
放大器
人体、血液与胰腺
实际血糖浓度
实际血糖浓度
期望的血糖浓度
葡萄糖
68
34
2013/3/10
Example 2. Consider a disk drive system：
（1）Control objectives？（2）Controlled variable？
（3）What are the performance indices?
（4）How to define the whole control system?
臂的转动
转轴
激励电机
磁头滑片
69
Example 2. Consider a disk drive system：
（1）Control objectives？（2）Controlled variable？
（3）What are the performance indices?
（4）How to define the whole control system?
A：(1) Position the head to a desired point quickly and
accurately, in order to read the data on the disk.
(2) Position of the head.
(3) How quickly？（1800－7200 r/m）How accurately? 1um
(4)
激励电机与读臂
70
35
2013/3/10
New Demands for Control
 任何重要的工程技术领域，都离不开控制技术
– 传统的：化工、石化、钢铁等制造业；汽车、家电…
– 新兴的：能源、新材料、高铁、航空航天、交通
 新的需求
– 节能减排的需求
– 快速适应市场变化的需求
– 非传统、高端对象的控制需求：超大型、高速、强非
线性、高度耦合
 新的控制技术
– 传统的PID 多变量控制与优化
– 系统控制和系统工程
71
Outline
 Introduction to Control Systems
 Definitions
 Historical Background
 Mathematical background
 Engineering Control Problem
 Course Outline
 Conclusions
 Basic Form of Block Diagram
72
36
2013/3/10
Basic form of block diagram
 The representation of physical components by blocks is shown
in earlier. For each block the transfer function provides the
dynamic mathematical relationship between the input and
output including feedback (See Chapter 1, Fig. 1.2).
 For control systems with several components, the transfer
function of each component is usually placed in a box.
Y(s)＝G(s)U(s)
U(s)
G(s)
Block diagram
73
Basic form of block diagram
 方块图是控制系统或对象中每个环节（元件）的功能和
信号流向的图解表示。每一个方块填写环节（元件）的传
递函数，指向方块的箭头表示该环节的输入信号，离开方
块的箭头表示该环节的输出信号，它是输入信号与方块内
的传递函数运算后的结果。注意箭头方还标明了相应的信
号符号（有时“＋”会省略）。
 根据方块图与传递函数的定义，可以直接由系统各个环
节之间的关系用图解的方式描述系统的信息传递－－也是
一种建模方法。
74
37
2013/3/10
Determination of the overall transfer function
Block diagram: cascade or in series
u  u1
H1 ( s)
y1  u2
H 2 (s)
y  y2
Y ( s )  Y2 ( s )  H 2 ( s)U 2 ( s)  H 2 ( s )Y1 ( s)  H 2 ( s ) H1 ( s )U ( s )
G(s) 
Y (s)
 H 2 ( s ) H1 ( s )
U (s)
N blocks are in series
75
Determination of the overall transfer function
Block diagram: parallel
u1
H1 ( s)
y1
u

u2
H 2 ( s)
y
G(s) 
Y (s)
 H1 ( s )  H 2 ( s )
U (s)
y2
Y ( s )  Y1 ( s )  Y2 ( s )  H1 ( s )U1 ( s)  H 2 ( s )U 2 ( s)  H1 ( s)  H 2 ( s ) U ( s )
N blocks are in parallel
76
38
2013/3/10
Determination of the overall transfer function
Block diagram: Feedback (P136)
r

u1
G (s)
y2
H (s )
y1
c
u2
C ( s )  G ( s )U 1 ( s )  G ( s )R( s )  Y2 ( s )  G ( s )R( s )  H ( s )C ( s )
C ( s )1  G ( s ) H ( s )  G ( s ) R( s )
( s ) 
C(s)
G( s )

R( s ) 1  G ( s ) H ( s )
positive feedback
( s ) 
C(s)
G( s )

R( s ) 1  G ( s ) H ( s )
negative feedback
Note!
Note!
77
Determination of the overall transfer function
Block diagram: Feedback
u1
u
±
y2
H1 ( s)
H 2 ( s)
y1
y
U(s)
H1 ( s)
1  H1 ( s) H 2 (s)
Y(s)
u2
Y ( s )  H1 ( s )U1 ( s )  H1 ( s )U ( s )  Y2 ( s )  H1 ( s )U ( s )  H 2 ( s )Y ( s )
Y ( s )1  H1 ( s ) H 2 ( s)  H1 ( s )U ( s )
( s ) 
Y (s)
H1 ( s )

U ( s ) 1 H1 ( s ) H 2 ( s )
where “+” means positive feedback and “－”
implied negative feedback.
78
39
2013/3/10
Basic form of block diagram
Ex.---- In the figure, given the differential equations of G(s)
and H(s) : dc (t )
db(t )
20
 5b(t )  10c(t )
 10c(t )  20e(t )
6
dt
dt
R
C
M E
Solve the transfer
10
G(s)
function with zero
B
initial conditions
H(s)
C ( s) B( s)
E (s) C ( s)
B( s)
E (s)
C (s)
R( s)
E (s)
R( s)
C(s)
20
G (s) 
A：6 sC ( s )  10C ( s )  20 E ( s )  E(s) 
6 s  10
20sB(s)  5B(s)  10C (s)  B(s) 
C(s)
H (s) 
10
20 s  5
79
Basic form of block diagram
Ex.---- In the figure, given the differential equations of G(s)
and H(s): dc (t )
db(t )
20
 5b(t )  10c(t )
6
 10c(t )  20e(t )
dt
dt
Solve the transfer
R
C
M E
10
G(s)
function with zero
initial conditions
B
H(s)
E
(
s
)
B( s )
C (s)
R( s )
E ( s)
R( s)
Open-loop TF：Gopen 
B(s)
20
10
200
 G (s)  H (s) 


E (s)
6 s  10 20 s  5 (6 s  10)(20 s  5)
Closed-loop TF：Gclosed   B 
C ( s)
?
R( s)
Error TF：
Ge   e 
E ( s)
?
R( s )
80
40
2013/3/10
Basic form of block diagram
Ex.---- In the figure, given the differential equations of G(s)
R
C
M E
and H(s):
10
Gclosed   B 
Closed-loop TF：
Ge   e 
Error TF：
G (s) 
Gclosed   B 
20
6 s  10
C (s)
?
R( s)
Characteristic
Equation
G(s)
C ( s)
?
R( s)
B
H(s)
E ( s)
?
R( s )
H (s) 
10
20 s  5
Gopen 
200
(6 s  10)(20 s  5)
20
6 s  10
20
10
(6 s  10) (20 s  5)
200(20 s  5)
400 s  100


(6 s  10)(20 s  5)  200 12 s 2  23s  25
G(s)
 B  10 

1  G (s) H ( s) 1 
10 
81
Basic form of block diagram
Ex.---- In the figure, given the differential equations of G(s)
and H(s): dc (t )
db(t )
20
 5b(t )  10c(t )
6
 10c(t )  20e(t )
dt
dt
Solve the transfer
R
C
M E
10
G(s)
equation with zero E ( s)
R( s )
initial conditions
B
H(s)
Ge   e 
Characteristic
Equation
E (s)
?
R( s)
 e  10 
1
1  G (s)H (s)
1 0 (1 2 s 2  2 3 s  5 )

12s2  23s  25
82
41
2013/3/10
Industrial Examples
Atomic Force Microscope
83
Some pictures are cited from Internet with
copy rights still reserved by their original
owners.
Enjoy this course and good luck!
qmyang@iipc.zju.edu.cn
Qinmin Yang: 15168218448 (588448)
控制科学与工程学系
42