### Mathematical Models of Systems

```‫بسم ا‪ ...‬الرحمن الرحيم‬
‫سیستمهای کنترل خطی‬
‫پاییز ‪1389‬‬
‫دکتر حسین بلندي‪ -‬دکتر سید مجید اسما عیل زاده‬
‫مرور‬
‫‪2‬‬
‫‪ )1‬استخراج معادالت ديفرانسيل از مدل فيزيكي سيستم‪.‬‬
‫‪ )2‬استخراج مدل رياض ي سيستم و خالصه کردن نتيجه بصورت يك بلوك دياگرام‪.‬‬
‫‪ )3‬نتيجه خالصه شدن يك سيگنال فلوگراف‪.‬‬
‫مثال ‪:‬‬
‫)‪N (s‬‬
‫‪T (s) ‬‬
‫)‪D(s‬‬
‫تحليل پاسخ سيستم‬
‫اعمال‬
‫وروديهای‬
‫تست‬
‫پايداري مطلق و نسبي‬
‫طراحی ‪ :‬تنظیم پارامترها‬
‫جبران‬
‫سازها‬
‫‪ )1‬تك ورودي ‪ -‬تك خروجي‬
‫‪ )2‬تابع تبديل در حوزة ‪s‬‬
‫‪ )3‬روشهاي فركانس ي‬
‫‪ )4‬رسم مكان هندس ي ريشهها‬
‫حصول اهداف‬
‫کنترلی‬
‫طراحی‬
‫جبرانسازها‬
OBJECTIVES
 On completion of this course, the student will be able to do the
following:
 Define the basic terminologies used in controls systems
 Explain advantages and drawbacks of open-loop and closed loop
control systems
 Obtain models of simple dynamic systems in ordinary differential
equation, transfer function, state space, or block diagram form
 Obtain overall transfer function of a system using either block
diagram algebra, or signal flow graphs, or Matlab tools
 Compute and present in graphical form the output response of
control systems to typical test input signals
5
 Explain the relationship between system output response and transfer
function characteristics or pole/zero locations
 Determine the stability of a closed-loop control systems using the RouthHurwitz criteria
 Analyze the closed loop stability and performance of control systems based
on open-loop transfer functions using the Root Locus technique
 Design PID or lead-lag compensator to improve the closed loop system
stability and performance using the Root Locus technique
 Analyze the closed loop stability and performance of control systems based
on open-loop transfer functions using the frequency response techniques
 Design PID or lead-lag compensator to improve
6
Topics Covered
 Modeling of control systems using ode, block diagrams, and transfer
functions
 Block diagrams and signal flow graphs
 Modeling and analysis of control systems using state space methods
 Analysis of dynamic response of control systems, including transient
response, steady state response, and tracking performance.
 Closed-loop stability analysis using the Routh-Hurwitz criteria
 Stability and performance analysis using the Root Locus techniques
 Control system design using the Root Locus techniques
 Stability and performance analysis using the frequency response
techniques
 Control system design using the frequency response techniques
7
1.
2.
R.C. Dorf and R.H. Bishop, Modern Control Systems,
10th Edition, Prentice Hall, 2008,
Golnaraghi and Kuo, Automatic Control Systems,,
ninth edition, Wiley, 2009
8
•
•
•
•
Midterm
Final
Quiz
H.W.
40%
40%
10%
10%
9
Mathematical Models of Systems
Objectives
• We use quantitative mathematical models of
physical systems to design and analyze
control systems.
• The dynamic behavior is generally described
by ordinary differential equations.
11
A wide range of physical Systems, including:
• mechanical,
• hydraulic, and
• electrical
could be considered.
13
Introduction
• To understand and control complex
systems we must obtain quantitative
mathematical models of system.
16
• A model is a representation of the
process or a system existing in reality
or planned for realization which
expresses the essential attributes of a
process or a system in a useful form.
Norbert Wiener, 1945
17
18
Approach to dynamic systems
• 1.
• 2.
• 3.
• 4.
• 5.
• 6.
Define the system and its components.
Formulate the MM and list the necessary
assumptions.
Write the differential equations describing the
model.
Solve the equations for the desired output
variables.
Examine the solutions and the assumptions.
If necessary, reanalyze or redesign the
system.
19
Differential Equations of Physical
Systems
The differential equations describing the
dynamic performance of a physical
system are obtained by utilizing the
physical laws of the process.
A differential equation is any algebraic
equality which involves either differentials
or derivatives.
20
This approach applies equally well to;
•
Mechanical,
•
Electrical,
•
Fluid,
•
Thermodynamic systems.
21
Physical laws
The physical laws define relationships
between fundamental quantities and are
usually represented by equations.
22
‫جمعبندی‬
‫اولیه‪:‬‬
‫بطور كلي دو ديدگاه جهت مدلسازي وجود دارد ‪:‬‬
‫الف‪ :‬تقسيم نمودن سيستم به اجزاء تشكيل دهنده و مدلسازي آن توسط روابط رياض ي‪.‬‬
‫ب ‪ :‬شناسايي پارامتري سيستم ‪ :‬در اين حالت آزمايشهايي سيستم انجام ميپذيرد و با‬
‫بررس ي نتايج حاصله يك مدل رياض ي براي سيستم تعيين ميشود‪.‬‬
‫‪ ‬در راستاي پايهگذاري و تبيين سيستم‪ ،‬مدل بدست آمده بايد مبين پارامترهای زير‬
‫باشد‪:‬‬
‫ـ ارتباط ديناميكي بين پارامترهاي دستگاه‬
‫ـ ورودي كارانداز‬
‫ـ خروجي قابل اندازهگيري باشد‪.‬‬
‫نکاتی که در مدلسازی سيستمها بايد در نظر داشت‬
‫‪ ‬مدلسازي دربرگيرنده اطالعات دروني سيستم بوده و همچنين ارتباط بين ‪effect ,‬‬
‫‪ cause‬متغيرهاي سيستم ميباشد‪.‬‬
‫‪ ‬پايه و اساس اصلي جهت انجام كار استفاده از قوانين فيزيكي حاكم بر سيستم‬
‫ميباشد‪.‬‬
‫‪ ‬انتخاب متغيرهاي حالت در روش متغيرهاي فيزيكي براساس عناصر موجود‬
‫نگهدارنده انرژي سيستم بنا ميشود‪.‬‬
‫‪ ‬متغير فيزيكي در معادلة انرژي براي هر عنصر نگهدارنده انرژي ميتواند بعنوان‬
‫متغير حالت سيستم انتخاب شود‪ .‬الزم به يادآوری است که متغيرهاي فيزيكي بايد‬
‫بگونه ای انتخاب شوند كه ناوابسته باشند‪.‬‬
‫عناصر نگهدارندة‬
‫انرژي‬
Common Physical Laws
In GeneraL:
• Circuit: KCL: S(i into a node) = 0
KVL: S(v along a loop) = 0
RLC: v=Ri, i=Cdv/dt, v=Ldi/dt
• Linear motion: Newton: ma = SF
Hooke’s law: Fs = KDx
damping: Fd = CDx_dot
• Angular motion: Euler: Ja = St
t  KDq
t  CDq_dot
Symbols and units
28
Voltage-current, voltage-charge, and impedance
relationships for capacitors, resistors, and inductors
Ex.1; Find ODE eqn.
RLC network
L
di (t )
dt
Ri(t )
1
i
(
t
)
dt

vc

C
di (t )
1
L
 Ri (t )   i (t )dt  v(t )
dt
C
1
i (t )dt

C
.‫ را بنویسید‬ode ‫معادالت‬
di1

e
(
t
)

R
i
(
t
)

L
 VC (t )  
1 1
1

dt

di 2

V
(
t
)

L
 R2 i2  
 C
2
dt

ic  i1  i2  C dvc

dt
: 2 ‫مثال‬
‫سيستمهای مکانيکی‬
‫(‪ )1‬انتقالي ‪ :‬مجموعة نيروها برابر است با حاصلضرب شتاب در جرم (‪)N‬‬
‫‪F  ma‬‬
‫(‪ )2‬دوراني ‪ :‬مجموعة گشتاورها برابر است با حاصلضرب ممان اينرس ي در شتاب‬
‫زاويهاي‬
‫‪t  I .‬‬
‫اجزای اصلی سيستمهای مکانيکی‬
: 3 ‫مثال‬
mx  Bx  Kx  f (t )
x  
B
K
1
x  x  f (t )
m
m
m
:
 x1  x2

 
B
K
1
x2   x2  x1  f (t )

m
m
m

: 4 ‫مثال‬
N 2Y2   K 2Y2  B2Y2  B1Y1  B1Y2  K1Y1  K1Y2
N1Y1   B1 ( x1  x2 )  k 1 ( y1  y2 )  f (t )
MECHANICAL ROTATIONAL SYSTEMS
36
Transforms
• The term transform refers to a mathematical
operation that takes a given function and returns a
new function.
• The transformation is often done by means of an
integral formula.
• Commonly used transforms are named after Laplace
and Fourier.
37
• Transforms are frequently used to change
a complicated problem into a simpler one.
• The simpler problem is then solved,
usually using elementary algebraic means.
• The solution to the simpler problem is
taken over to the original problem using
the inverse transform.
38
39
Laplace transform
• Laplace transform can significantly reduce the
effort required to solve linear differential equations.
• A major benefit is that this transformation convert
differential equations to algebraic equations, which
can simplify the mathematical manipulations
required to obtain a solution.
40
41
General solution procedure:
• Step 1. Take the Laplace transform of both sides of the
differential equation.
• Step 2. Solve for Y(s)
If the expression for Y(s) does not appear in Laplace Transform
Table
• Step 3a. Factor the characteristic equation polynomial.
• Step 3b. Perform the partial fraction expansion.
• Step 4. Use the inverse Laplace transform relations to find y(t).
42
Example 5
d2y
dy
 4  3 y (t )  2r (t )
2
dt
dt
dy
y(0) 1, (0)  0, r(t) 1, t  0
dt
[s 2Y (s)  sy(0)]  4[sY (s)  y(0)]  3Y (s)  2R(s)
43
Example 5
( s  4)
2
Y ( s)  2

( s  4s  3) s( s 2  4s  3)
q(s)  s 2  4s  3  (s  1)(s  3)  0
 3 / 2  1/ 2    1
1/ 3  2 / 3
Y ( s)  




 Y1 (s)  Y2 (s)  Y3 (s)


 (s  1) (s  3)   (s  1) (s  3)  s
44
Example 5
1 3t  2
 3 t  1 3t  
t
y(t )   e 
e    1e  e  
2
3
2
 
 3
2
limt  y(t) 
3

45
• The solution of the differential equation involves use of
Laplace transforms as an intermediate step.
• Any change in the initial conditions or in the forcing
function requires that the complete solution be redeliver.
46
The transfer function - a modified
approach.
• The transfer function is an algebraic expression for
the dynamic relation between input and output of the
process model.
• It is defined so as to be independent of initial
conditions and of the particular choice of forcing
function.
47
G(s)
• To obtain the transfer function G(s) of the
LTI system, we take the Laplace transform
on both sides of the equation, and assume
zero initial conditions.
48
Properties of the G(s)
•
•
•
•
The G(s) is defined only for a LTI system.
All initial conditions of the system are set to zero.
The G(s) is independent of the input of the system.
The G(s) of a continuous-data system is expressed
only as a function of the complex variable s.
• For discrete-data systems modeled by difference
equations, the transfer function is a function of z
when the z-transform is used.
49
• A transfer function can be derived only
for a LTI differential equation model.
50
A transfer function
• A transfer function of the LTI system is
defined as a ratio of the Laplace transform
of the output variable to the Laplace
transform of the input variable, with all
initial conditions assumed to be zero.
51
EX. 6:
An automobile shock absorber
Spring-mass-damper
Free-body diagram
52
The automobile shock absorber
d 2 y (t )
dy(t )
M
b
 ky(t )  r (t )
2
dt
dt
Output
Y(s)
1
 G(s) 

Input
R(s) Ms 2  bs  k

53
Transfer function of the RC
network
EX.7:
V1(s) = (R + 1/Cs) I(s)
V2(s) = I(s) 1/Cs
G(s) = V2(s)/V1(s) = 1/(RC s + 1) = 1/T/(s + 1/T)
54
The transfer function of the RC network is obtained by writing
the Kirchhoff voltage equation.
The circuit is a voltage divider, where
V2(s)/V1(s) = Z2(s)/(Z1(s) + Z2(s)),
where Z1(s)= R and Z2 = 1/Cs
The single pole s = -1/T
55
Ex.8; Find G(S)?
RLC network
L
di (t )
dt
Ri(t )
1
i (t )dt

C
di (t )
1
L
 Ri (t )   i (t )dt  v(t )
dt
C
di (t )
1
L
 Ri (t )   i (t )dt  v(t )
dt
C
as q(t)   i(t)dt
d 2 q (t )
dq(t ) 1
L
R
 q(t )  v(t )
2
dt
dt
C
output vC , q(t)  CvC (t )
2
d vC (t )
dv C (t )
LC
 RC
 vC (t )  v(t )
2
dt
dt
LCs 2 VC ( s )  RCsVC ( s )  VC ( s )  V ( s )
1
VC ( s)
1
LC
G(s) 


V ( s ) LCs 2  RCs  1 s 2  R s  1
L
LC
Ex. 9:
Mesh analysis
Mesh 1
Mesh 2
Write equations around the meshes
m esh 1
R1 I1  LsI1  LsI 2  V ( s )
m esh 2
1
LsI 2  R2 I 2 
I 2  LsI1  0
Cs
Sum of impedance
around mesh 1
Sum of applied voltages
around the mesh
( R1  Ls) I1  LsI 2  V ( s )
1 

 LsI1   Ls  R2 
I 2  0
Cs 

Sum of impedance common
to two meshes
Sum of impedance
around mesh 2
 Ls
  I  V ( s )
 R1  Ls
1


1  


R

Ls
Ls

I
0
 2  

2

Cs 

Cram er' s Rule
 R1  Ls V ( s )

  Ls
0
LsV ( s )



I2 
D
D
Determinant
1 

2
D  R1  Ls  Ls  R2    Ls 
Cs 


R1  Ls  LCs 2  R2Cs  1  L2Cs 3

Cs
LC ( R1  R2 ) s 2  R1 R2C  L s  R1

Cs
2
LCs V ( s )
I 2 
LC ( R1  R2 ) s 2  R1 R2C  L s  R1


• A transfer function of LTI system is
defined as the Laplace transform of the
impulse response, with initial conditions
set to zero.
65
Input-Output description
• A transfer function is an input-output
description of the behavior of a system.
• Thus the transfer function description
does not include any information
concerning the internal structure of the
system.
66
Summary
1. The differential equations describing the dynamic
performance of physical systems were utilized to construct a
mathematical model. The physical systems included
mechanical, electrical, fluid, and thermodynamic systems.
2. For linear systems we apply the Laplace transformation
and its related input-output relationship given by the transfer
function.
67
Summary
3. The transfer function allows to determine the response
of the system to various input signals.
Y(s) = X(s) G(s)
68
69
70
71
‫روش مدرن‬
‫پيش گفتار‪:‬‬
‫از معادالت‬
‫ديفرانسيل‬
‫معادالت فضاي‬
‫حالت‬
‫يادآوری ‪:‬‬
‫‪ ‬اكثر روشهاي طراحي سيستمهاي كنترل مبتني بر نوعي مدل رياض ي از سيستم فيزيكي ميباشد‪.‬‬
‫‪ ‬طراحيهاي كالسيك سيستمهاي كنترل از روشهايي مانند مكان‪ ،‬پاسخ فركانس ي جهت‬
‫تحليل و طراحي سيستمها استفاده ميکنیم‪.‬‬
‫‪ ‬شايان توجه است كه در اين ديدگاه‪ ،‬فعاليت متمركز بر استفاده از تابع تبديل است‪.‬‬
‫معايب روشهای کالسيک‬
‫‪ )1‬اين روش براي سيستمهاي صنعتي ‪ SISO‬قابل بهرهوري میباشد و ميتواند نتايج‬
‫مطلوبي را بدنبال داشته باشد‬
‫‪ )2‬تحليل دقيق سيستمهاي صنعتي پيشرفته مدلهاي كاملتري را طلب ميكند‪.‬‬
‫‪ )3‬سيستمهاي صنعتي پيچيده براي دقت‪ ،‬سرعت عمل و كارايي بيشتر نيازمند به‬
‫طراحيهاي مدرن سيستمهاي كنترل ميباشند‪.‬‬
‫نتيجه‬
‫‪ ‬مدلسازي سيستمهاي كنترل با استفاده از متغيرهاي حالت در راستاي تحقق اهدافي‬
‫است كه به آن اشاره كردهايم‪.‬‬
‫‪ ‬متغيرهاي حالت در واقع ميتوانند ديناميكي از سيستم را شامل شوند كه در مدل‬
‫خروجي ـ ورودي ظاهر نمی شوند‪ .‬از اين جهت مدل متغيرهاي حالت را مدل داخلي نيز‬
‫ميگويند‪.‬‬
‫‪ ‬توصيف فضاي حالت‪ ،‬تصوير كاملي را از ساختار داخلي سيستم فراهم ميكند‪ .‬اين‬
‫مدل نشان ميدهد كه متغيرهاي حالت چگونه با يكديگر تداخل نموده‪ ،‬ورودي سيستم‬
‫چگونه بر متغيرهاي حالت تأثير ميگذارد و چگونه با تركيبهاي متفاوت ميتوان يك‬
‫سيستم خاص را نشان داد‪.‬‬
```