LINEAR CONTROL SYSTEMS Ali Karimpour Associate Professor Ferdowsi University of Mashhad

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1 LINEAR CONTROL SYSTEMS Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad

2 Lecture Stability analyi Topic to be covered include: Stability of linear control ytem. Bounded input bounded output tability (BIBO). Zero input tability. Stability of linear control ytem through Routh Hurwitz criterion.

3 Stability analyi Topic to be covered include: Stability of linear control ytem. Bounded input bounded output tability (BIBO). Zero input tability. Stability of linear control ytem through Routh Hurwitz criterion.

4 تجزیه تحلیل پایداری Stability analyi Sytem output = zero-tate repone + zero-input repone y( t) y ( t) y ( t) z zi Input output tability BIBO tability (bounded-input bounded-output) Definition? Suitable model? Internal tability Aymptotic tability Lyapunov tability Definition? Suitable model? Stability check? Stability check?

5 Stability analyi Input output tability(bibo) g( تجزیه تحلیل پایداری Internal tability x Ax Theorem:A SISO ytem with proper rational tranfer function g( i BIBO table if and only if every pole of g( ha negative real part. قضیه: یک سیستم SISO با تابع انتقال مناسب گویای g( را پایدار BIBO گویند اگر و فقط اگر هر قطب g( دارای قسمت حقیقی منفی باشد. Theorem: x Ax i aymptotically table if and only if all eigenvalue of A have negative real part. x پایدار مجانبی قضیه: معادله Ax است اگر و فقط اگر تمام مقادیر ویژه A دارای قسمت حقیقی منفی باشد. Relation between BIBO tability and aymptotic tability?

6 Example : Dicu the tability of the ytem. R( C( x BIBO tability: G( C( R( Internal tability: There i no RHP root, o ytem i BIBO table. For internal tability we need R( tate-pace model o we have: x( x ( x x x x ( R( x x r r? x ( R( ) x x x r R ( + + x ( R( x x r ( x x x x C( C(

7 Example : Dicu the tability of the ytem. R( C( BIBO tability: There i no RHP root, o ytem i BIBO table. Internal tability: For internal tability we need tate-pace model o we have: R ( + + x x r x x c x x x x x x x r r I- A, x x ( )( ) The ytem i not internally table (neither aymptotic nor Lyapunov table). 7 C( Very important note: If RHP pole and zero between different part of ytem omitted then the ytem i internally untable although it may be BIBO table. 7

8 8 Example : Dicu the BIBO tability of a cloed loop ytem with following tranfer function. Proof your anwer. (Olympiad 9) ) ( T Example : a) Find the eigenvalue of following ytem. b) Check the BIBO tability. c) Check the Aymptotic tability. d) Check the Lyapunov tability. x y u x x Example : a) Find the eigenvalue of following ytem. b) Check the BIBO tability. c) Check the Aymptotic tability. d) Check the Lyapunov tability. x y u x x

9 تجزیه تحلیل پایداری Stability analyi Input output tability(bibo) g( Internal tability x Ax G( n( d( Let d( Let I A Find pole p,p,..., p n Find eigenvalue,,..., n For both kind of tability we need to compute the zero of ome polynomial Stability of linear control ytem through Routh Hurwitz criterion. 9

10 Stability analyi Topic to be covered include: Stability of linear control ytem. Bounded input bounded output tability (BIBO). Zero input tability. Stability of linear control ytem through Routh Hurwitz criterion.

11 Stability and Polynomial Analyi پایداری و تجزیه تحلیل چند جمله ای ها Conider a polynomial of the following form: The problem to be tudied deal with the quetion of whether that polynomial ha any root in RHP or on the jw axi. مساله این است که آیا چند جمله ای فوق ریشه ای در RHP دارد و یا خیر. و یا روی محور jw

12 Some Polynomial Propertie of Special Interet Property : The coefficient a n- atifie چند خاصیت جالب چند جمله ای ها Property : The coefficient a atifie Property : If all root of p( have negative real part, it i neceary that a i >, i {,,, n-}. Property : If any of the polynomial coefficient i nonpoitive (negative or zero), then, one or more of the root have nonnegative real plant.

13 Routh Hurwitz Algorithm The Routh Hurwitz algorithm i baed on the following numerical table. n n n n.. a n,,, a a n a n a n n,,, Routh table. n, Number of ign change =number of root in RHP

14 ) ( p Example : Check the number of zero in the RHP لاثم :.دینک نییعت ار ریز متسیس RHP رفص دادعت Two root in RHP 7 7

15 Example : Check the tability of following ytem. مثال : بخشی از جدول راث یک سیستم مرتبه 7 به شکل زیر است. بجز K مثبت باشند در مورد پایداری اظهار نظر کنید. اگر تمام پارامترها 7 A B C D E F G H I J K O واضح است که سیستم ناپایدار است. Univerity entrance exam 9

16 Routh Hurwitz pecial cae حاالت خاص روت هرویتز Routh Hurwitz pecial cae - The firt element of a row i zero. - All element of a row are zero.

17 ) ( p 7 Example 7: Check the number of zero in the RHP 7 لاثم : RHP رفص دادعت.دینک نییعت ار ریز متسیس Two root in RHP for any

18 7 8 8 ) ( p Example 8: Check the number of zero in the RHP 8 لاثم : RHP رفص دادعت.دینک نییعت ار ریز متسیس ) ( q 8 No RHP root + two root on imaginary axi 8 ) ( q Auxiliary Polynomial

19 Routh Hurwitz pecial cae Remark: If all element of the row n- are zero, there are n root with ome ymmetry to the center of the plane. It mean they can be real root with different ign or complex conjugate root. If there are no ign change in firt column(after n row), all root of the auxiliary polynomial lie on the jω axi. If the ytem ha repetitive eigenvalue on the imaginary axi then full row zero happen in repetitive time. 9

20 Example 9: Check the number of zero in the RHP تعداد صفر RHP سیستم زیر را تعیین کنید. مثال 9: p(. Auxiliary Polynomial q( q( Two RHP root + no root on imaginary axi

21 Example : Check the number of zero in the RHP تعداد صفر RHP سیستم زیر را تعیین کنید. مثال : p( 9 8 Auxiliary Polynomial q( 8 q( Two RHP root + two root on imaginary axi

22 Routh Hurwitz pecial cae Univerity entrance exam 9 مثال : در هنگام تشکیل جدول راث - هرویتز برای معادله مشخصه یک سیستم مرتبه n سطر -k به صورت کامل صفر شده است. اگر تعداد تغییر عالمت های ستون اول قبل از سطر k را با y و پس از آن را با x نشان دهیم شرط وجود قطب روی محور موهومی کدام است y < k ) x < k ( y < n-k ) x < n-k )

23 Example : Check the tability of following ytem for different value of k مثال : پایداری سیستم زیر را بر حسب مقادیر k بررسی کنید. + - k ( ) ( ) k M ( ( ) k k( ) k( ) To check the tability we mut check the RHP root of k ( k) k k k( k) k k k k k We need k>.8 for tability

24 Example : The block Diagram of a control ytem i depicted in the following figure. Find the region in K-α plane concluding the ytem table. مثال :بلوک دیاگرام یک سیستم کنترل در شکل زیرنشان داده شده است. ناحیه ای در صفحه K-α به دست آورید که سیستم پایدار باشد. R( + - K( ) Y(

25 Example : The block Diagram of a control ytem i depicted in the following figure. Find the region in K-α plane concluding the ytem table. R( + K( ) C( -

26 مثال : k را بگونه ای تعیین کنید که تنها دو قطب سیستم حلقه بسته زیر در محدوده -<Re(< قرار گیرد. + - ) k ) k 7 ) k ) It i not poible. k Univerity entrance exam 9

27 Exercie x y u x x ] [ 9 ) ( ) ) ( ) p b p a 7 - Check the internal tability of following ytem. x y u x x ] [ - a) Check the internal tability of following ytem. b) Check the BIBO tability of following ytem. - Are the real part of all root of following ytem le than -. ) ( p - Check the number of zero in the RHP in following ytem

28 Exercie - Check the internal tability of following ytem veru k. + - k ( )( ) - a)check the BIBO tability of following ytem. b) Check the internal tability of following ytem. + - x ( ) x 7- The eigenvalue of a ytem are -,,- and the pole of it tranfer function are - and -.(Midterm pring 8) a) Check the BIBO tability of following ytem. b) Check the internal tability of following ytem. 8

29 Untable Exercie 8- The open-loop tranfer function of a control ytem with negative unit feedback i: K( ) G( H( ( T( Find the region in K-T plane concluding the ytem table. Anwer: K Untable Stable.. Untable T 9- Find the number of root that the real part are in the region [-,] by ue of Routh Hurwitz criteria. 9

30 Supplementary Exercie - In the following ytem k can be poitive or negative. + R( G ( ) - c S( k G c ( i a proper controller. Show that the ytem can not be tabilized by any R( and S(. (Nice Problem) - The cloed loop tranfer function of a ytem i : k( ) M ( 7 ( k ) k For the tability of the ytem which one i true? ( k> ) ) k 8. ) < k < 8. ) k < 8. ) < k 8. Anwer: ()

31 Supplementary Exercie ) ) ) - کدام یک از سیستمهای با معادله مشخصه زیر می تواند به ازای بهره های بزرگ k پایدار باشد ( کنکور ارشد 9( (8 (78 k) (78 k) k) (78 k) 7 8 k (7 k) 8 k (7 k) 8 k هر سه گزینه دارای پنج قطب است. پس حتما به ازای بهره های مثبت بزرگ ناپایدار خواهد بود. ( Anwer: () Univerity entrance exam 9

LINEAR CONTROL SYSTEMS

LINEAR CONTROL SYSTEMS Ali Karimpour Aociate Profeor Ferdowi Univerity of Mahhad Root Locu Criteria Topic to be covered include: v v v v Root locu criterion. u u u Root loci (RL). Complement root loci