Multivariable Control Systems

Transcription

1 Multivaiable Contol Sytem Ali Kaimpou Aociate ofeo Fedowi Univeity of Mahhad Refeence ae appeaed in the lat lide.

2 Stability of Multivaiable Feedback Contol Sytem Topic to be coveed include: Well - oedne of Feedback Loop ntenal Stability The Nyquit Stability Citeion The Genealized Nyquit Stability Citeion Nyquit aay and Gehgoin band Copime Factoization ove Stable Tanfe Function Stabilizing Contolle Stong and Simultaneou Stabilization 2

3 Well - oedne of Feedback Loop Aume that the plant and the contolle K ae fixed eal ational pope tanfe matice. The fit quetion one would ak i whethe the feedback inteconnection make ene o i phyically ealizable. Let, 2 K u 2 n d) 3 3 Hence, the feedback ytem i not phyically ealizable! d i 3

4 Well - oedne of Feedback Loop Definition 7- A feedback ytem i aid to be well-poed if all cloed-loop tanfe matice ae well-defined and pope. Now uppoe that all tanfe matice fom the ignal, n, d and d i u ae epectively well-defined and pope. Thu y and all othe ignal ae alo well-defined and the elated tanfe matice ae pope. to So the ytem i well-poed if and only if the tanfe matix fom d i and d to u exit and i pope. 4

5 Well - oedne of Feedback Loop So the ytem i well-poed if and only if the tanfe matix fom d i and d to u exit and i pope. Theoem 7- The feedback ytem in Figue i well-poed if and only if K ) ) i invetible d oof u K K K d i Thu well - poedne i equivalent to the condition that and i pope. K And thi i equivalent to the condition that the contant tem of the tanfe matix K ) ) i invetible. exit 5

6 6 Well - oedne of Feedback Loop Tanfe matix i invetible. ) ) K i equivalent to eithe one of the following two condition: invetible i ) ) K invetible i ) ) K The well- poedne condition i imple to tate in tem of tate-pace ealization. ntoduce ealization of and K: D C B A D C B A K ˆ ˆ SO well- poedne i equivalent to the condition that invetible i D D

7 Stability of Multivaiable Feedback Contol Sytem Well - oedne of Feedback Loop ntenal Stability The Nyquit Stability Citeion The Genealized Nyquit Stability Citeion Nyquit aay and Gehgoin band Copime Factoization ove Stable Tanfe Function Stabilizing Contolle Stong and Simultaneou Stabilization 7

8 ntenal Stability Aume that the ealization fo ) and K) ae tabilizable and detectable. Let x and xˆ denote the tate vecto fo and K, epectively. x Ax Bu xˆ Ax ˆ ˆ By ˆ y Cx Du u Cx ˆ ˆ Dy ˆ Definition 7-2 The ytem of Figue i aymptotically table, d, d i and i aid to be intenally table if the oigin x, xˆ),) i.e. the n tate x, xˆ) go to zeo fomall initial tate when 8

9 9 ntenal Stability x x C C D D y u ˆ ˆ ˆ x x A x x C C D D B B A A x x ˆ ~ ˆ ˆ ˆ ˆ ˆ ˆ Theoem 7-2 The ytem of above Figue with given tabilizable and detectable ealization fo and K i intenally table if and only if i a Huwitz matix All eigenvalue ae in open left half plane). A ~ A ~ e

10 ntenal Stability x x C C D D y u ˆ ˆ ˆ x x A x x C C D D B B A A x x ˆ ~ ˆ ˆ ˆ ˆ ˆ ˆ A ~ e What about tability in the ene of Lyapunov? The ytem of above Figue with given tabilizable and detectable ealization fo and K i table in the ene of Lyapunov if and only if all eigenvalue of be in A ~

11 ntenal Stability d e u K i p Theoem 7-3 The ytem in Figue i intenally table if and only if the tanfe matix ) ) ) ) K K K K K K K fom d i, -) to u p, - e) be a pope and table tanfe matix. e

12 2 ntenal Stability ) ) ) ) K K K K K K K Note that to check intenal tability, it i neceay and ufficient) to tet whethe each of the fou tanfe matice i table., Let K d e u i p 2 2 2) ) 2 Stability cannot be concluded even if thee of the fou tanfe matice ae table.

13 ntenal Stability Theoem 7-4 Suppoe K itable.then the ytem in Stable the figue i intenally table if K ) i table. and only if Remembe K K ) K ) K K ) K ) oof: The neceity i obviou. To pove the ufficiency let Q K) K K) KQ Stable K K) K QK) Stable K) Q Stable K) K) K) K QK Stable 3

14 ntenal Stability Theoem 7-5 Stable Suppoe itable.then the the figue i intenally table if K K ) i table. ytem in and only if Remembe K K ) K ) K K ) K ) oof: The neceity i obviou. To pove the ufficiency let Q K K) K K) Q Stable K K) Q Stable K) Q) Stable K) K) K K) Q Stable 4

15 ntenal Stability Theoem 7-6 Suppoe and K ae both table.then the ytem in the figue i intenally table if and only if K ) i table. Stable Stable K K ) Remembe K ) K ) K K ) oof: The neceity i obviou. To pove the ufficiency let Q K) K K) KQ Stable K K) KQ K) Q K) Q Stable Stable Stable 5

16 ntenal Stability No pole zeo cancellation Theoem 7-7 The ytemin the figue i intenally table if and only if it iwell - poed and i) the numbe of open RH pole of ) K ) n ii) K ) i table. K K ) Remembe K ) K n K K ) K ) n K i the numbe of RH poleof K n i the numbe of RH poleof oof: See: Eential of Robut contol witten by Kemin Zhou 6

17 Stability of Multivaiable Feedback Contol Sytem Well - oedne of Feedback Loop ntenal Stability The Nyquit Stability Citeion The Genealized Nyquit Stability Citeion Nyquit aay and Gehgoin band Copime Factoization ove Stable Tanfe Function Stabilizing Contolle Stong and Simultaneou Stabilization 7

18 The Nyquit Stability Citeion Now we ae going to check the tability of the above ytemfo vaiou eal value of k. Let det[ kg )]have Then o pole and c zeoin the RH plane. jut a SSO ytem,the Nyquit plot of ) det enciclethe oigin, c o time. kg )) Howeve, we would have to daw the Nyquit locu of fo each value of k in which we wee inteeted, wheea in the claical Nyquit citeion we daw a locu only once, and then infe tability popetie fo all value of k. 8

19 The Nyquit Stability Citeion det[ kg )] [ k )] Whee i i ) i an eigenvalue of G ) det[ kg )] k ) i i i - 9

20 The Nyquit Stability Citeion Theoem 7-8 Genealized Nyquit theoem) f G) with no hidden untable mode, ha o untable Smith-McMillan) pole, then the cloed-loop ytem with etun atio kg) i table if and only if the chaacteitic loci of kg), taken togethe, encicle the point -, o time anticlockwie. Z N kg ) kg ) kg ) G ) 2 Z N

21 The Nyquit Stability Citeion Example 7-: Let G ) 2.5 ) 3 ) 3 ) ).5 ) ) Suppoe that G) ha no hidden mode, check the tability of ytem fo diffeent value of k. G ).5 ) ) 2 3 2

22 The Nyquit Stability Citeion.5 ) 2 ) 3.25 Since G) ha one untable pole, we will have cloed loop tability if thee loci give one net enciclement of -/k ccw) when a negative feedback k i applied. - -/ k / k.5.5 / k -/k no two one no enciclement o thee i one RH cloedlooppole. enciclement o thee i thee RH cloedlooppole. enciclement o thee i two RH cloedlooppole. enciclement o thee i one RH cloedlooppole. 22

23 The Nyquit Stability Citeion Example 7-2: Let G ).25 ) 2) 6 2 Suppoe that G) ha no hidden mode, check the tability of ytem fo diffeent value of k. G ) ) 2) ) 2) 23

24 The Nyquit Stability Citeion ) 2) ) 2) Since G) ha no untable pole, we will have cloed loop tability if thee loci give zeo net enciclement of -/k when a negative feedback k i applied. / k.8,.4 / k and.53 / k no enciclement..8 / k.4 one enciclement o one RH pole in cloed loop ytem. / k.53 two enciclement o two RH pole in cloed loop ytem. 24

25 The nvee Nyquit Stability Citeion Theoem 7-8 Genealized Nyquit theoem) f G) with no hidden untable mode, ha o untable Smith-McMillan) pole, then the cloed-loop ytem with etun atio kg) i table if and only if the chaacteitic loci of kg), taken togethe, encicle the point -, o time anticlockwie. Z Z invg )/ k invg )/ k N kg ) kg ) N Z N kg ) G ) Theoem 7-9 Genealized nvee Nyquit theoem) f G) with no hidden untable mode, ha Z o untable Smith-McMillan) zeo, then the cloed-loop ytem with etun atio kg) i table if and only if the chaacteitic loci of invg)/k, taken togethe, encicle the point -, Z o time anticlockwie. Z Z N kg ) G ) 25

26 The Nyquit Stability Citeion Example 7-3: Let G ) 5) ) Suppoe that G) ha no hidden mode, check the tability of ytem fo diffeent value of k. G ) 5) ) 5) ) k 75 k 75) Two untable pole. 75 k k 75) Stable. -75 k k ) One RH pole. 26

27 Nyquit aay and Gehgoin band The Nyquit aay of G) i an aay of gaph not neceaily cloed cuve), the ij th gaph being the Nyquit locu of g ij ). Theoem 7- Gehgoin theoem) Let Z be a complex matix of dimenion m m.the eigenvalue of Z ae contained in two union of cicleaound the diagonal elementa follow: i m i zii zij, i, 2,..., m j ji m i zii z ji, i, 2,..., m j ji The band obtained in thi way ae called Gehgoin band, each i compoed of Gehgoin cicle. 27

28 Nyquit aay and Gehgoin band Nyquit aay, with Gehgoin band fo a ample ytem f all the Gehgoin band exclude the point -, then we can ae cloed-loop tability by counting the enciclement of - by the Gehgoin band, ince thi tell u the numbe of enciclement made by the chaacteitic loci. f the Gehgoin band of G) exclude the oigin, then we ay that G) i diagonally dominant ow dominant o column dominant). The geate the degee of dominant of G) o +G) ) that i, the naowe the Gehgoin band- the moe cloely doe G) eemble m non-inteacting SSO tanfe function. 28

29 Nyquit aay and Gehgoin band And in geneal: 29

30 Nyquit aay and Gehgoin band Diagonal dominance of a mm matix G): Row Diagonal dominance: f fo all on the Nyquit contou, m gii ) gij ) i,2,..., j ji m Column Diagonal dominance: f fo all on the Nyquit contou, m gii ) g ji ) i,2,..., j ji m 3

31 Stability of Multivaiable Feedback Contol Sytem Well - oedne of Feedback Loop ntenal Stability The Nyquit Stability Citeion The Genealized Nyquit Stability Citeion Nyquit aay and Gehgoin band Copime Factoization ove Stable Tanfe Function Stabilizing Contolle Stong and Simultaneou Stabilization 3

32 Copime Factoization ove Stable Tanfe Function Two polynomial m) and n), with eal coefficient, ae aid to be copime if thei geatet common divio i a contant numbe o they have no common zeo o thee exit polynomial x) and y) uch that x ) m ) y ) n ) Execie7- : Let n)= and m)= find x) and y) if n and m ae copime. Execie 7-2 : Let n)= and m)=+2 how that one cannot find x) and y) in x)m)+y)n)= Similaly, two tanfe function m) and n) in the et of table tanfe function ae aid to be copime ove table tanfe function if thee exit x) and y) in the et of table tanfe function uch that x ) m ) y ) n ) Bezout identitie 32

33 Copime Factoization ove Stable Tanfe Function Definition 7-3 Two matice M and N in the et of table tanfe matice ae ight copime ove the et of table tanfe matice if they have the ame numbe of column and if thee exit matice X and Y in the et of table tanfe matice.t. M N X Y X M Y N Similaly, two matice and in the et of table tanfe matice M ~ ~ N ~ ae left copime ove the et of table tanfe matice if they have the ame numbe of ow and if thee exit two matice X l and Y l in the et of table tanfe matice uch that ~ X ~ ~ l Y l l M N MX NY l 33

34 Copime Factoization ove Stable Tanfe Function Now let be a pope eal-ational matix. A ight-copime factoization cf) of i a factoization of the fom NM whee N and M ae ight-copime in the et of table tanfe matice. Similaly, a left-copime factoization lcf) of ha the fom ~ M ~ N n cf and lcf ~ M ~, N, M and N ae copime in the et of table tanfe function matice 34

35 Copime Factoization ove Stable Tanfe Function Remembe: Matix Faction Deciption MFD) Right matix faction deciption RMFD) Left matix faction deciption LMFD) Let G) i a mm matix and it the Smith McMillan i ~ G ) Let define: diag ),..., ),,..., D ) diag ),..., ),,..., N ) ~ ~ G ) N ) D ) o G ) D ) N ) n Matix Faction Deciption D) and N) ae polynomial matice 35

36 36 Copime Factoization ove Stable Tanfe Function Theoem 7- Suppoe ) i a pope eal-ational matix and D C B A ) i a tabilizable and detectable ealization. Let F and L be uch that A+BF and A+LC ae both table, and define Then cf and lcf of ae: N M ~ ~ NM Execie 7-3 : Let )=+)/+2) find two diffeent cf fo. D DF C F L B BF A X N Y M l l D C F L LD B LC A M N Y X ) ~ ~

37 Copime Factoization ove Stable Tanfe Function The ight copime factoization of a tanfe matix can be given a feedback contol intepetation. x Ax Bu y Cx Du x u y A BF ) x Bv Fx v C DF ) x Dv the tanfe matix fom v to u i M ) A BF F B u v Fx and that fom v to y i N ) A BF C DF B D y ) N ) v ) N ) M ) u ) ) u ) ) N ) M ) Execie7-4 : Deive a imila intepetation fo left copime factoization. 37

38 Stability of Multivaiable Feedback Contol Sytem Well - oedne of Feedback Loop ntenal Stability The Nyquit Stability Citeion The Genealized Nyquit Stability Citeion Nyquit aay and Gehgoin band Copime Factoization ove Stable Tanfe Function Stabilizing Contolle Stong and Simultaneou Stabilization 38

39 Stabilizing Contolle Theoem 7-2 Suppoe i table. Then the et of all tabilizing contolle in Figue can be decibed a K Q Q) fo any Q in the et of table tanfe matice and ) Q ) non ingula. oof: K Q Q) K Q) Q Q K K) Sytem i table. Now uppoe the ytem i table, o K K) i table,then define Q K K) K) K Q KQ K ) Q ) i noningula o K Q Q) 39

40 Stabilizing Contolle Example 7-4 Fo the plant ) ) 2) Suppoe that it i deied to find an intenally tabilizing contolle o that y aymptotically tack a amp input. Solution: Since the plant i table the et of all tabilizing contolle i deived fom K Q S T Q) K fo any K ) table Q Q uch that ) Q ) a b Q 3 a b ) 2) 3) 6 Q 3 i noningula, 4 o let ) 2) 3) a b) ) 2) 3)

41 Stabilizing Contolle Theoem 7-3 Let K matix and Then thee Y X l Whee be l a pope X NM Y eal- ational ~ ~ M N. exit a tabilizing contolle X ~ N Y M ~ M N Y X l l oof. See Multivaiable Feedback Deign By Maciejowki 4

42 42 Theoem 7-emembe) Suppoe ) i a pope eal-ational matix and D C B A ) i a tabilizable and detectable ealization. Let F and L be uch that A+BF and A+LC ae both table, and define D DF C F L B BF A X N Y M l l D C F L LD B LC A M N Y X ) ~ ~ Stabilizing Contolle

43 Stabilizing Contolle Theoem 7-4 oof. See Multivaiable Feedback Deign By Maciejowki 43

44 44 Example 7-5 Fo the plant 2) ) ) The poblem i to find a contolle that. The feedback ytem i intenally table. 2. The final value of y equal when i a unit tep and d=. 3. The final value of y equal zeo when d i a inuoid of ad/ and =. To deive copime factoization let F = [ -5] and L = [-7-23] T clealy A+BF and A+LC ae table. D DF C F L B BF A X N Y M l l D C F L LD B LC A M N Y X ) ~ ~ ] [ 3 2 Clealy D C B A Stabilizing Contolle

45 45 Solution: The et of all tabilizing contolle i: ) ) l l l l NQ X MQ Y K D DF C F L B BF A X N Y M l l D C F L LD B LC A M N Y X ) ~ ~ ) 38 9, ) 72 8, ) ) 2), ) X Y M N l l ) 38 9, 2) 72 8, 2) ) 2) ~, 2) ~ X Y M N Stabilizing Contolle

46 Stabilizing Contolle Solution: The et of all tabilizing contolle i: K Y l MQ l ) X l NQ l ) Clealy fo any table Q the condition atified To met condition 2 the tanfe function fom to y mut atify ~ y ) N Y ~ Q M ) ) N ) Y ) Q ) M )) Q ) 36.5 To met condition 3 the tanfe function fom d i to y mut atify ~ y ) N X ~ Q N) di ) N j) X j) Q j) N j)) Q j) 629 j Now define Q ) x x2 x3 2 ) Execie 7-5: Deive tanfe function fom to y. Execie 7-6: Deive tanfe function fom d i to y. Execie 7-7: Deive Q 46

47 Stability of Multivaiable Feedback Contol Sytem Well - oedne of Feedback Loop ntenal Stability The Nyquit Stability Citeion The Genealized Nyquit Stability Citeion Nyquit aay and Gehgoin band Copime Factoization ove Stable Tanfe Function Stabilizing Contolle Stong and Simultaneou Stabilization 47

48 Stong and Simultaneou actical contol enginee ae eluctant to ue untable contolle, epecially when the plant itelf i table. f the plant itelf i untable, the agument againt uing an untable contolle i le compelling. Howeve, knowledge of when a plant i o i not tabilizable with a table contolle i ueful fo anothe poblem namely, imultaneou tabilization, meaning tabilization of eveal plant by the ame contolle. Simultaneou tabilization of two plant can alo be viewed a an example of a poblem involving highly tuctued uncetainty. A plant i tongly tabilizable if intenal tabilization can be achieved with a contolle itelf i a table tanfe matix. 48

49 Stong and Simultaneou Theoem 7-5: i tongly tabilizable if and only if it ha an even numbe of eal pole between evey pai of eal RH zeo including zeo at infinity). oof. See Linea feedback contol By Doyle. Example 7-6: Which of the following plant i tongly tabilizable? ) 2) ) ) ) 2 3 2) ) Solution: i not tongly tabilizable ince it ha one pole between z= and z= But 2 i tongly tabilizable ince it ha two pole between z= and z= 49

50 Execie 7-8 Find two diffeent lcf fo the following tanfe function matix. 7-9 Find a lcf and a cf fo the following tanfe matix. G ) Find a lcf and a cf fo the following tanfe matix. G ) G ) 2) 3) ) 4 2 ) 7- By ue of MMO ule in peviou chapte how that the following figue intoduce the contolle in the fom: 5

51 Refeence Refeence Web Refeence Multivaiable Feedback Deign, J M Maciejowki, Weley,989. Multivaiable Feedback Contol, S.Skogetad,. otlethwaite, Wiley,25. Contol Configuation Selection in Multivaiable lant, A. Khaki-Sedigh, B. Moaveni, Spinge Velag, 29. تحليل و طراحی سيستم های چند متغيره دکتر علی خاکی صديق 5