AUTOMATIC CONTROL SYSTEMS - ???????? - FUMcpanelprofsite.um.ac.ir/~karimpor/autocontrol/lec5_auto.pdf · Lecture 5 Ali Karimpour Sep 2012 2 Lecture 5 Stability analysis Topics to

AUTOMATIC CONTROLAUTOMATIC CONTROLSYSTEMSSYSTEMS

Ali KarimpourAssociate Professor

Ferdowsi University of Mashhad

Lecture 5

Ali Karimpour Sep 2012

2

Lecture 5

Stability analysisTopics to be covered include:

Stability of linear control systems.

Bounded input bounded output stability (BIBO).

Zero input stability.

Stability of linear control systems through Routh Hurwitz criterion.

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3

The response of linear systems can always be decomposed as the zero-state response and zero-input response. We study

1. Input output stability of LTI system is called BIBO (bounded-input bounded-output) stability ( the zero-state response )

2. Internal stability of LTI system is called Asymptotic stability ( the zero-input response )

Stability analysisتجزیه تحلیل پایداري

Suitable for TF model

Suitable for SS model BuAxx

451)( 2

ss

sT

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Input output stability of LTI system LTIپایداري ورودي خروجی سیستمهاي

Definition: A system is said to be BIBO stable (bounded-inputbounded-output) if every bounded input excited a boundedoutput. This stability is defined for zero-state response and isapplicable only if the system is initially relaxed.

گویند اگر هر ورودي محدود خروجی محدود BIBOرا پایدار یک سیستم : تعریفاین پایداري براي پاسخ حالت صفر تعریف شده و سیستم در ابتدا آرام . را تولید کند

. است

451)( 2

ss

sTSuitable for TF model

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Theorem: A SISO system with proper rational transfer function T(s) is BIBO stable if and only if every pole of T(s)

has negative real part.

گویند BIBOرا پایدار T(s)با تابع انتقال مناسب گویاي SISOیک سیستم : قضیه.داراي قسمت حقیقی منفی باشد T(s)فقط اگر هر قطب اگر و

Input output stability of LTI system LTIپایداري ورودي خروجی سیستمهاي

451)( 21

ss

sT 1

1)( 22 s

sTstable is 45

1)( 21

sssT unstable is

11)( 22

s

sT

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Internal stability پایداري داخلی

Definition: The zero-input response of is stable in thesense of Lyapunov if every finite initial state x0 excites abounded response. In addition if the response approaches tozero then it is asymptotically stable.

را به مفهوم لیاپانوف پایدار گویند اگر پاسخ ورودي صفر سیستم : تعریفعالوه بر این اگر پاسخ به . بوجود آوردمحدودي را پاسخ x0هر حالت اولیه محدود

.صفر میل کند پایداري مجانبی حاصل می شود

Axx

Axx

The BIBO stability is defined for the zero-state response. Now we study the stability of the zero-input response.

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Internal stabilityپایداري داخلی

Theorem: The equation isasymptotically stable if and only if alleigenvalues of A have negative realparts.

معادله پایدار مجانبی است اگر و: قضیهداراي قسمت حقیقی Aفقط اگر تمام مقادیر ویژه

.منفی باشد

Axx

Axx

u 11

x3-201

x u

11

x56-10

xstable isu

11

x3-201

x unstable isu

11

x56-10

x

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Different type of stabilityپایداري هاي مختلف

Relation between BIBO stability and asymptotic stability?

Theorem: A SISO system with proper rational transfer function g(s) is BIBO stable if and only if every pole of g(s)

has negative real part.

45

1)( 2

sssT

Theorem: The equation is asymptotically stable ifand only if all eigenvalues of A have negative real parts.

Axx

u 11

x3-201

x

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Example 1: Discuss the stability of the system .

9

11s

)(sC

21

ss)(sR

21

)()()(

ssR

sCsT There is no RHP root , so system is BIBO stable.

BIBO stability:

Internal stability:For internal stability we need state-space model so we have: 1

1s

)(sC

21

ss)(sR 1x 2x

)(1

1)( 12 sxs

sx

212 xxx

)(21)(1 sR

sssx

rrxx 11 2

? )()(1

1)( 12 sRsxs

sx

rxxx 212

)(2

3)(1 sRs

sx

rxx 32 11 1

1s

)(sC

23s

)(sR ++1x 2x

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Example 1: Discuss the stability of the system .

10

11s

)(sC

21

ss)(sR

There is no RHP root , so system is BIBO stable.BIBO stability:

Internal stability:For internal stability we needstate-space model so we have:

rxxx 212rxx 32 11 11s

)(sC

23s

)(sR ++1x 2x

2

1

2

1

2

1

10

13

1102

xx

c

rxx

xx

22,11

)2)(1(11

02A-sI

sss

s

The system is not internally stable (neither asymptotic nor Lyapunov stable).

Very important note: If RHP poles and zeros between different part of system omitted then the system is internally unstable although it may be BIBO stable.

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Review مرور

How can we check BIBO stability?

How can we check asymptotic stability?

)()()(sdsnsT 0)(Let sd p,...,p,p poles Find n21

LHPp,...,p,p If n21 System is BIBO stable

0Let AsI ,...,, seigenvalue Find n21

LHP,...,, If n21 System is asymptotically stable

For both kind of stability we need to compute the zero of some polynomial

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Different regions in S planeSنواحی مختلف در صفحه

RHP planeLHP planeUnstableStable

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Stability and Polynomial Analysis

Consider a polynomial of the following form:

The problem to be studied deals with the question of whether that polynomial has any root in RHP or on the jw axis.

jwو یا روي محور RHPمساله این است که آیا چند جمله اي فوق ریشه اي در .دارد و یا خیر

پایداري و تجزیه تحلیل چند جمله اي ها

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Some Polynomial Properties of Special Interest

Property 1: The coefficient an-1 satisfies

Property 2: The coefficient a0 satisfies

Property 3: If all roots of p(s) have negative real parts, it is necessary that ai > 0, i {0, 1, …, n-1}.

Property 4: If any of the polynomial coefficients is nonpositive (negative or zero), then, one or more of the roots have nonnegative real plant.

چند خاصیت جالب چند جمله اي ها

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Routh Hurwitz Algorithm

The Routh Hurwitz algorithm is based on the following numerical table.

Routh’s table

آلگوریتم روت هرویتز

ns1ns

11na

2na ..............3na

4na5na ..............

2ns3ns

.

.

.0s

1,2

1,3 2,32,2 3,2

3,3

............................

1,n

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Routh’s table

ns1ns

11na

2na ..............

3na4na5na ..............

2ns3ns

.

.

.0s

1,2

1,3 2,32,2 3,2

3,3

............................

1,n

Routh Hurwitz Algorithmآلگوریتم روت هرویتز

1

3121,2

1

n

nnn

aaaa

1

5142,2

1

n

nnn

aaaa

1

7163,2

1

n

nnn

aaaa

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Result

Consider a polynomial p(s) and its associated table.Then the number of roots in RHP is equal to the numberof sign changes in the first column of the table.

تعداد ریشه هاي . و جدول متناظر آن را در نظر بگیرید p(s)چند جمله اي

.برابر با تعداد تغییر عالمت در ستون اول جدول است RHPواقع در

نتیجه

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Routh’s table

ns1ns

11na

2na ..............

3na4na5na ..............

2ns3ns

.

.

.0s

1,2

1,3 2,32,2 3,2

3,3

............................

1,n

Routh Hurwitz Algorithmآلگوریتم روت هرویتز

Number of sign changes=number of roots in RHP

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Example 2: Check stability of following system.پایداري سیستم زیر را تعیین کنید: 2مثال

105323)( 234

ssssssT

0511032

3

4

ss

Two roots in RHP

01072 s

007451s

00100s

010532 234 ssss

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Routh Hurwitz special casesحاالت خاص روت هرویتز

Routh Hurwitz special cases

1- The first element of a row is zero. (see example 2)

2- All elements of a row are zero. (see example 3)

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0322)( 234 sssssp

021321

3

4

ss

0302s

00321

s

0030s3

Two roots in RHPfor any


3221)( 234

ssss

sT

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22

047884)( 2345 ssssssp

484781

4

5

ss

0663s0442s0001s

044)( 2 ssq

0081s0040s

No RHP roots + two roots on imaginary axis

08)( ssq

Auxiliary Polynomial


478841)( 2345

sssss

sT

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Example 4: Check the stability of following system for different values of k

. بررسی کنید kپایداري سیستم زیر را بر حسب مقادیر : 4مثال

42)13(

3 ssssk

+-

42)13(1

42)13(

)(3

3

sssskssssk

sM)13(42

)13(3

skssssks

04)2(3 23 skkssTo check the stability we must check the RHP roots of

4321

2

3

ksks

03

4)2(31

kkks

040s

03

46303

2

kkk

kWe need k>0.528

for stability

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Exercises

xy

uxx

]122[43

6

661221463

1- Check the internal stability of following system.

xy

uxx

]021[010

141230312

2- a) Check the internal stability of following system. b) Check the BIBO stability of following system.

3- Are the real parts of all roots of following system less than -1.

42654)( 2345 ssssssp

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Exercises (Cont.)

)5)(10(10

sssk

+-

4- Check the internal stability offollowing system versus k.

5- a)Check the BIBO stability of following system.b) Check the internal stability of following system.

)10(1

sss+

- 11s

1x2x

6- The eigenvalues of a system are -3,4,-5 and the poles of its transfer function are -3 and -5.(Midterm spring 2008)a) Check the BIBO stability of following system. b) Check the internal stability of following system.

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Exercises (Cont.)

7- Find the number of roots in the region [-2,2]. 015115 23 sss

8- The closed loop transfer function of a system is :

kskssssksG

2)25(157)2()( 234

For the stability of the system which one is true? ( k>0 )

1) 0 ≤ k ≤ 28.12 2) 0 < k < 28.12

3) 0 ≤ k < 28.12 4) 0 < k ≤ 28.12

Documents

AUTOMATIC CONTROL SYSTEMS - ???????? - FUMcpanelprofsite.um.ac.ir/~karimpor/autocontrol/lec5_auto.pdf · Lecture 5 Ali Karimpour Sep 2012 2 Lecture 5 Stability analysis Topics to