سیستمهای کنترل خطی پاییز 1389

سیستمهای کنترل خطی

1389پاییز

بسم ا... الرحمن الرحيم

دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

مرور

2

( استخراج معادالت ديفرانسيل از 1( اس#تخراج م#دل رياض#ي سيس#تم و خالص#ه ک#ردن نتيج#ه 2 مدل فيزيكي سيستم.

.بصورت يك بلوك دياگرام نتيجه خالصه شدن يك (3

.سيگنال فلوگراف

)(

)()(

sD

sNsT

مثال :

اعمال وروديهای تست

اعمال وروديهای تست

تحلي###ل پاس###خ سيستم

پاي#داري مطل#ق و نسبي

: طراحیتنظیم

پارامترها

جبران سازها

تك -تك ورودي( 1خروجي

تابع تبديل در (2sحوزة

روشهاي (3 رسم مكان (4 فركانسي

هندسي ريشه ها

حصول اهداف کنترلی

حصول اهداف کنترلی

طراحی جبرانساز

ها

طراحی جبرانساز

ها

5

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

6

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 Routh-Hurwitz 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

7

Topics CoveredModeling of control systems using ode, block diagrams, and transfer

functionsBlock diagrams and signal flow graphsModeling and analysis of control systems using state space methodsAnalysis of dynamic response of control systems, including transient

response, steady state response, and tracking performance.Closed-loop stability analysis using the Routh-Hurwitz criteriaStability and performance analysis using the Root Locus techniquesControl system design using the Root Locus techniquesStability and performance analysis using the frequency response

techniquesControl system design using the frequency response techniques

8

References for reading

1. R.C. Dorf and R.H. Bishop, Modern Control Systems,10th Edition, Prentice Hall, 2008,

2. Golnaraghi and Kuo, Automatic Control Systems,, ninth edition, Wiley, 2009

9

Grading

• Midterm 40% • Final 40%• Quiz 10%• H.W. 10%

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

16

Introduction

• To understand and control complex systems we must obtain quantitative mathematical models of system.

17

• 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

18

19

Approach to dynamic systems

• 1. Define the system and its components.• 2. Formulate the MM and list the necessary

assumptions.• 3. Write the differential equations describing

the model.• 4. Solve the equations for the desired output

variables.• 5. Examine the solutions and the

assumptions.• 6. If necessary, reanalyze or redesign the

system.

20

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.

21

This approach applies equally well to;• Mechanical,• Electrical,• Fluid,• Thermodynamic systems.

22

Physical laws

The physical laws define relationships between fundamental quantities and are usually represented by equations.

بط###ور كلي دو دي###دگاه جهت مدلس###ازي :وجود دارد

تقسيم نمودن سيستم به اجزاء تشكيل دهنده و الف: .مدلسازي آن توسط روابط رياضي

در اين حالت : شناسايي پارامتري سيستمب :بررسي آزمايشهايي سيستم انجام مي پذيرد و با

نتايج حاصله يك مدل رياضي براي سيستم تعيين مي شود. در راس##تاي پايه گ##ذاري و تب##يين سيس##تم، م##دل

بدست آمده بايد مبين پارامترهای زير باشد:

# ارتباط ديناميكي بين پارامترهاي دستگاه

# ورودي كارانداز

# خروجي قابل اندازه گيري باشد.

جمعبندی :اولیه

نکاتی که در مدلسازی سيستمها بايد در نظر داشت

مدلس##ازي دربرگيرن##ده اطالع##ات دروني سيس##تمبينب#وده و همچ#نين ارتب#اط effect , cause متغيره#اي

.سيستم مي باشد پاي#ه و اس#اس اص#لي جهت انج#ام ك#ار اس#تفاده از

قوانين فيزيكي حاكم بر سيستم مي باشد.

انتخ###اب متغيره###اي ح###الت در روش متغيره###ايف#يزيكي براس#اس عناص#ر موج#ود نگهدارن#ده ان#رژي

سيستم بنا مي شود. متغ#ير ف#يزيكي در معادل#ة ان#رژي ب#راي ه#ر عنص#ر

نگهدارن##ده ان##رژي مي توان##د بعن##وان متغ##ير ح##الت سيس#تم انتخ#اب ش#ود. الزم ب#ه ي#ادآوری اس#ت ک#ه متغيره#اي ف#يزيكي باي#د بگون#ه ای انتخ#اب ش#وند ك#ه

.ناوابسته باشند

عناصر نگهدارندة انرژي

عناصر نگهدارندة انرژي

• 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: J a = St

= t KDq = t C _Dq dot

In GeneraL:Common Physical Laws

28

Symbols and units

Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors

Ex.1; Find ODE eqn.

RLC network dt

tdiL

)()(tRi

dttiC

)(1

)()(1

)()(

tvdttiC

tRidt

tdiL

vcdttiC

)(1

را بنویس#ید. odeمع#ادالت :2مث#ال

dt

dvCiii

iRdt

diLtV

tVdt

diLtiRte

cc

C

C

21

222

2

1111

)(

)()()(

سيستمهای مکانيکی

( انتق##الي : مجموع##ة نيروه##ا براب##ر اس##ت ب##ا 1)(N) حاصلضرب شتاب در جرم

maF

دوراني : مجموع##ة گش##تاورها براب##ر اس##ت ب##ا ( 2)حاصلضرب ممان اينرسي در شتاب زاويه اي

.I

اجزای اصلی سيستمهای مکانيکی

مث##ال 3:

)(tfKxxBxm )(1

tfm

xm

Kx

m

Bx

)(tfm

xm

Kx

m

Bx

xx1

122

21

:

مث##ال 4:

21112111222222 YKYKYBYBYBYKYN

)()()( tfyykxxBYN 21121111

36

MECHANICAL ROTATIONAL SYSTEMS

37

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.

38

• 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.

39

40

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.

41

42

• 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).

General solution procedure:

43

Example 5

)(2)(342

2

trtydt

dy

dt

yd

y(0) 1,dy

dt(0) 0, r(t) 1, t 0

)(2)(3)]0()([4)]0()([ 2 sRsYyssYsysYs

44

Example 5

)34(

2

)34(

)4()(

22

sssss

ssY

0)3)(1(34)( 2 sssssq

)()()(3/2

)3(

3/1

)1(

1

)3(

2/1

)1(

2/3)( 321 sYsYsY

ssssssY

45

Example 5

3

2

3

11

2

1

2

3)( 33

tttt eeeety

limt y(t) 2

3

46

Disadvantage:

• 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.

47

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.

48

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.

49

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.

50

• A transfer function can be derived only

for a LTI differential equation model.

51

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.

52

EX. 6: An automobile shock absorber

Spring-mass-damper Free-body diagram

53

The automobile shock absorber

)()()()(

2

2

trtkydt

tdyb

dt

tydM

Output

InputG(s)

Y (s)

R(s)

1

Ms2 bs k

54

EX.7:Transfer function of the RC network

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)

55

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

Ex.8; Find G(S)?

RLC network dt

tdiL

)()(tRi

dttiC

)(1

)()(1

)()(

tvdttiC

tRidt

tdiL

2

2

C C

2C C

C2

2C C C

C2

2

( ) 1( ) ( ) ( )

q(t) i(t)dt

( ) ( ) 1( ) ( )

v , q(t) Cv ( )

v ( ) v ( )v ( ) ( )

V ( ) V ( ) V ( ) ( )

1V ( ) 1

( )( ) 1

di tL Ri t i t dt v t

dt C

as

d q t dq tL R q t v t

dt dt Coutput t

d t d tLC RC t v t

dt dt

LCs s RCs s s V s

s LCG sRV s LCs RCs s sL

1LC

Ex. 9: Mesh analysis

Mesh 1 Mesh 2

01

)()(

01

2

)(

1

221

211

12222

2111

ICs

RLsLsI

sVLsIILsR

LsIICs

IRLsI

mesh

sVLsILsIIR

mesh

Sum of impedance around mesh 1

Sum of impedance around mesh 2

Sum of impedance common to two meshes

Sum of applied voltages around the mesh

Write equations around the meshes

)(0

)(

'

0

)(1

1

2

2

1

2

1

sLsVLs

sVLsR

I

RulesCramer

sV

I

I

CsRLsLs

LsLsR

Determinant

1212

21

2

2

1212

21

322

21

221

)(

)(

)(

1

1

RsLCRRsRRLC

sVLCsI

Cs

RsLCRRsRRLC

Cs

CsLCsRLCsLsR

LsCs

RLsLsR

65

• A transfer function of LTI system is defined as the Laplace transform of the impulse response, with initial conditions set to zero.

66

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.

67

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.

68

Summary

3. The transfer function allows to determine the response of the system to various input signals.

Y(s) = X(s) G(s)

69

70

71

مدرن روش

از معادالت ديفرانسي

ل

از معادالت ديفرانسي

ل

معادالت فضاي حالت

معادالت فضاي حالت

پيش گفتار:

يادآوراك#ثر روش#هاي ط#راحي سيس#تم هاي كن#ترل مبت#ني ب#ر ن#وعي م#دل ی :

رياضي از سيستم فيزيكي مي باشد. طراحي ه###اي كالس###يك سيس###تم هاي كن###ترل از

روش#هايي مانن#د مك#ان، پاس#خ فركانس#ي جهت تحلي#ل کنیم. و طراحي سيستم ها استفاده مي فع##اليت ،ش##ايان توج##ه اس##ت ك##ه در اين دي##دگاه است. متمركز بر استفاده از تابع تبديل

قاب##ل SISOاين روش ب##راي سيس##تمهاي ص##نعتي ( 1 نت#ايج مطل#وبي را ب#دنبال د و مي توان#میباش#دبه#ره وري

داشته باشد

تحلي#ل دقي#ق سيس#تمهاي ص#نعتي پيش#رفته م#دلهاي ( 2 كاملتري را طلب مي كند.

سيس##تم هاي ص##نعتي پيچي##ده ب##راي دقت، س##رعت ( 3عم#ل و ك#ارايي بيش#تر نيازمن#د ب#ه طراحي ه#اي م#درن

سيستم هاي كنترل مي باشند.

معايب روشهای کالسيک

معايب روشهای کالسيک

مدلس##ازي سيس##تم هاي كن##ترل ب##ا اس##تفاده ازمتغيره#اي ح#الت در راس#تاي تحق#ق اه#دافي اس#ت ك#ه

به آن اشاره كرده ايم.

متغيره#اي ح#الت در واق#ع مي توانن#د دين#اميكي ازسيس#تم را ش#امل ش#وند ك#ه در م#دل خ#روجي # ورودي ظ#اهر نمی ش#وند. از اين جهت م#دل متغيره#اي ح#الت

را مدل داخلي نيز مي گويند.

توص#يف فض#اي ح#الت، تص#وير ك#املي را از س#اختارداخلي سيس##تم ف##راهم مي كن##د. اين م##دل نش##ان مي ده#د ك#ه متغيره#اي ح#الت چگون#ه ب#ا يك#ديگر ت#داخل نم#وده، ورودي سيس#تم چگون#ه ب#ر متغيره#اي ح#الت ت##أثير مي گ##ذارد و چگون##ه ب##ا تركيبه##اي متف##اوت

مي توان يك سيستم خاص را نشان داد.

نتيجه