Regularization and Stabilization of the Rectangle Descriptor Decentralized Control Systems by Dynamic Compensator

Transcription

1 Modern Appled ene Vol. 5, o. 2; Aprl 2 Regularzaon and ablzaon of he Reangle Desrpor Deenralzed Conrol ysems by Dynam Compensaor Xume Tan Deparmen of Eleromehanal Engneerng, Heze Unversy Heze 274, Chna E-mal: showmelwood@63.om Reeved: Deember 6, 2 Aeped: January 7, 2 do:.5539/mas.v5n2p244 Absra The regularzaon, he mpulse-free, and he sablzaon of he reangle desrpor deenralzed onrol sysem by dynam ompensaon are suded n hs arle. The neessary and suffen ondon of he regularzaon and he free mpulse of he losed-sysem afer ompensaon, and he neessary and suffen ondon ha he reangle desrpor deenralzed onrol sysem ould make real sable by he dynam feedbak are provded and he resuls keep onssen wh he square sysem n form. Keywords: Reangle desrpor deenralzed onrol sysems, Dynam ompensaor, Regularzaon, Impulse-free, ablzaon. Inroduon In reen years, he researh on he desrpor deenralzed onrol sysem has been developed largely, and espeally for he deenralzed (pulse) fxed modes and he mpulse onrollably/ observably, a seres of resuls have ourred (. H. Wang, 973, P & Q. L. Zhang, 989, P & T.. Chang, 986, P & T.. Chang, 2, P & Xe, 986, P & Xe, 995, P & L. Da, 989). Formerly, he oeffen marx of he sysem was square,.e. he lne number equaled he olumn number, bu no maer n heory or n prae, he oeffen marx may be reangular, for example, he reangular marx penl ( sm ) s used o denoe he sruure haraer of he sysem, and desrbe some dfferenal equaon n he prae, whh s very mporan o sudy ommon sysem. Q. L. Zhang defned he fne fxed modes and he mpulse fxed modes of he ommon desrpor sysem (nludng enralzed sysem and deenralzed sysem) (Q. L. Zhang, 989, P ). Ishhara J Y redefned and suded he pulse onrollably and mpulse observably of he non-square (reangular) desrpor sysem (Ishhara J Y, 2, P ), and Hou M defned he onrollably of anoher knd of mpulse modulus o he non-square desrpor sysem, and proved hs defnon was equvalen o elmnang he mpulse modulus by feedbak (Hou M, 24, P ). The arge of hs arle s o desgn dynam ompensaors o make he whole losed-loop sysems o be regular, losed, gradually sable, and mpulse-free. 2. Bas knowledge For he desrpor deenralzed onrol sysem wh loal onrol saons desrbed by Ex () Ax() Bu() () y() Cx (), {,2,, } Where x R n q s he sae, p u R and y R are he npu and he oupu, respevely, of he h loal onrol saon, he mares mn EA, R m q, B R p n, C R are real and onsan, {, 2,, }. Assume he mares B, C are full rank, and rank( E) r, r mn{ m, n} obvously. If m n and de( se A), s,we all () s regular, oherwse, f m n or de( se A), we all () s non-regular. If m n, we also all () s a square sysem. If no, we all a 244 I E-I

2 Modern Appled ene Vol. 5, o. 2; Aprl 2 reangular sysem or a non- square sysem. We somemes denoe sysem () by ( EABC,,, ) for shor. For he square sysem, followng lemmas are fundamenal. Lemma (D. Wang, 989, P.27-3): For he square sysem (), he onrol rules exs, u ky v, {,, }, and he suffen and neessary ondon o make he losed-loop sysems Ex [ A BKC ] x Bv, be regular and pulse-free s ha for any non-nerse dvsons P {,, k } and P { k,, } of he se {, 2,, }, E rank E A BP n rank[ E] C P Lemma 2 (Hou M, 24, P ): The suffen and neessary ondon o make he square desrpor sysem Ex Ax o be regular and mpulse-free s E rank n rank E E A (3) m n Lemma 3 (X. K. Xe, 988, P ): upposed ha A R m q, B R p n, and C R are fxed q p marxes, and K R s varable marx, so A gr. A BKCmn{ rank A, B, rank } K C, where, gr. denoes he gener rank of he marx,.e. denoes he maxmum rank of he marx n he square brakes when he marx K hanges. 3. Desrpon of problem The deenralzed sablzaon s o fnd loal oupu onrol rules wh dynam ompensaor E z () () () z Ry (5) u() Qz () Ky(), {,2,, }, o make he losed-loop sysems omposed by he sysem () and he sysem (5) be regular, sable, and n pulse-free. Where, z () R s he sae of he h onroller, and he szes of he marxes E,, R, Q and K respevely are m n, m n, m p, q n, and q p. The formula (5) an be smplfed as Ez () z () Ry () u () Qz () Ky () Where, E blok dag E, E,, E 2 blok dag, 2,, Q blok dag Q, Q2,, Q R blok dag R, R2,, R K blok dag K, K2,, K z [ z T, z T 2,, z T ] T The whole losed-loop sysems omposed by he feedbak rule (6) and he sysem () an be denoed as (2) (4) (6) Publshed by Canadan Cener of ene and Eduaon 245

3 Modern Appled ene Vol. 5, o. 2; Aprl 2 E x () A BKC BQx() E z() RC z(), where, C C C2 C T T T T oe: m, B B B B,,, 2 m, n n and r 4. Man resul 4. Regularzaon and mpulse-free r. The problems whh should be frs solved s he regularzaon and mpulse-free of he sysem. Aordng o he defnon n Xe Xuka s arle (Xe, 986, P.85-88), he man dea s o ompensae he sysem () o be a normal square sysem by he dynam ompensaor, and hen solve he problems of regularzaon and mpulse-free of he square losed-loop sysems. The sysem (7) should fulfll he ondon of regularzaon,.e. hs sysem should be square sysem, so he dmenson of he desgned ompensaor should fulfll nn m m (8) Theorem : For he sysem (), he dynam ompensaor (5) exss, and he suffen and neessary ondon o make he losed-loop sysems (7) be regular and pulse-free s ha, for any paron of he se {, 2,, } no dsjon subses P {,, k } and P { k,, }, he followng nequaly s rue E rank E A B n n r m n C P Defne: m m, n n. P P P P P P P Proof: By usng he Lemma 2, he suffen and neessary ondon ha he sysem (7) o be regular and pulse-free s he followng equaly holds: (9) (7) E E rank n n r r E A BKC BQ E RC upposed ha E blok dag E, E,, E 2 blok dag, 2,, Q blok dag Q, Q2,, Q R blok dag R, R2,, R K blok dag K, K2,, K C C C C T T T T 2, The lef of he formula () an be wren as B B, B,, B 2 () 246 I E-I

4 Modern Appled ene Vol. 5, o. 2; Aprl 2 E E E E E E A BKC BQ E A BKC BQ E RC E RC E K Q C B R I n I m Aordng o he Lemma 3, he suffen and neessary ondon ha he formula () exss s E E rank E m E A BKC BQ B E E E RC gr. mn E E A BKC BQ K E E RC E A BKC BQ rank n E RC E C nn r r (2) The suffen and neessary ondon ha he formula (2) exss s ha E E rank E m n n r r (3) E A BKC BQ B E RC, and () Publshed by Canadan Cener of ene and Eduaon 247

5 Modern Appled ene Vol. 5, o. 2; Aprl 2 E E E A BKC BQ rank n n n r r (4) E RC E C, all exs. By repeaedly usng he deomposon of he formula () and he Lemma 3 o above wo formulas, he suffen and neessary ondon ha he formula () exs s E rank E A BP n n r mp n, P P (5) C P, where, mp m, n P n, P P, P P,.e. o any non-nerse dvsons P and P P P, he above formula exss. End. oe : In he ondon (9), when m n, here s n mp n, whh s jus he ondon (2),.e. he P ondon ha he square desrpor deenralzed onrol sysem s regular and mpulse-free. Tha means he Theorem keeps onssen wh he suffen and neessary ondon ha he square sysem s regular and mpulse-free n form,.e. s he suffen and neessary ondon ha he ommon desrpor sysem s regular and mpulse-free. 4.2 ablzaon Lemma 5 (G. C. Verghese, 98, P.8-83): For he square sysem (), he suffen and neessary ondon ha s C e s unsable fne deenralzed fxed modes or pulse deenralzed fxed modes s ha, for any non-nerse dvsons P {,, k } and P { k,, } of he se {, 2,, }, se A BP rank n C P, exss, and here, C C { } e, and C denoes he rgh-half omplex plane. Lemma 6 (Z. W. Gao, 997, P ): When and only when he sysem () has no unsable deenralzed fne fxed modes and deenralzed pulse fxed modes, he square sysem () ould be real sable by he normal ( E I ) deenralzed dynam ompensaor,.e. he orrespondng losed-loop sysems s nerorly real sable. Lemma 7 (Yang, 24): The suffen and neessary ondon ha he square sysem () s losed and regular n he sa oupu feedbak s ha, s C (omplex plane) exss, and for he eran one non-nerse dvsons {,, } P {,, } of he se {, 2,, }, P k and k se A BP rank n C P Theorem 2: The suffen and neessary ondon ha he desrpor deenralzed sysem () s real sablzed by he desrpor dynam ompensaor (5) s ha, for any, here s s C e (6) (8) 248 I E-I

6 Modern Appled ene Vol. 5, o. 2; Aprl 2 se A B se A Bj B rank m C and rank C j n (9) C, where, j {, 2,, }, and are any non-nerse dvsons of he se {, 2,, } { j}. j Prove: To sudy he sablzaon of he reangular sysem (), he ompensaor should be used o ompensae he sysem o be square sysem frs, and here, he mehod o buld he ompensaor an be desrbed as follows. One sngular dynam ompensaor s added o he j h subsysem frs, and for he onvenene, supposed ha he followng sngular ompensaor s added o he frs subsysem. E z () z() Ry() u() Qz() Ky() To ompensae he whole sysem no he square sysem, he orrespondng losed-loop sysems s E () () x ABKC BQx B u E z () RC z() 2 x () y C z() {2,3,, } upposed E ABKC BQ B E A B C C ', ', ', ' E RC, and supposed ha he dmenson number of he sub-ompensaor fulflls nn m m (22) The sably of he losed-loop sysems (2) s suded as follows. From he Theorem, one dynam ompensaor (2) exss o make he sysem (2) be losed and regular,.e. all parameer marxes n he sysem (2) s relavely fxed. By usng he square sysem as referenes, he fne fxed modes of he sysem (2) ould be defned as follows. Defnon : If max rank[ se ' A' B ' KC '] n n exss, and s s fne, or K R de [ se ' A ' B ' KC '] K R, so s s one fxed modes or fne fxed modes of he sysem (2). Defnon 2: upposed ha ( se ' A') has nfne zero pons, and f max rank[ se ' A' B ' KC '] n n, and s s nfne, K R, he sysem (2) has he pulse fxed modes or he nfne fxed modes. From he Lemma 5, for any se A BP rank n C P s C e, when and only when, exss, he square sysem () has no unsable fne deenralzed fxed modes and he mpulse deenralzed () (2) (2) (23) Publshed by Canadan Cener of ene and Eduaon 249

7 Modern Appled ene Vol. 5, o. 2; Aprl 2 fxed modes. Aordng o he Lemma 7, f he square sysem has no unsable fne deenralzed fxed modes and he mpulse deenralzed fxed modes, he sysem eranly s losed-loop and regular n he feedbak of sa oupu. Aordng o above dsussons, ha he sysem (2) s losed-loop and regular ould exs obvously, and based on ha, he real sably s suded as follows (D. Wang, 989, P.27-3), and ( ) normal dynam ompensaors are added o he sysem (2). z () z() Ry() u() Qz () Ky(), {2,3,, }, here, n n R, R n p R, Q q n R q p, K R o he obaned losed-loop sysems A BKC BQ BQ E x ( ) x ( ) E z ( ) RC z( ) I z () RC z(), s sable. Where, blok dag,, 2 Q blok dag Q,, Q 2 R blok dag R,, R 2 z [ z,, z ] T T T 2 C C C T T T 2, B B B 2 (24), {2,3,, }. Beause he sysem (2) s losed and regular, so he losed-loop sysems (25) mus be regular, so he exsene and unqueness of he soluon ould be guaraneed. From he Lemma 5 and he Lemma 6, he suffen and neessary ondon ha he square sysem (2) an be real sablzed by he normal ompensaor (24) s ha he sysem (2) has no unsable deenralzed fne fxed modes and he deenralzed mpulse fxed modes. The suffen and neessary ondon ha he sysem (2) has no unsable deenralzed fne fxed modes and he deenralzed mpulse fxed modes s dedued as follows. Aordng o he defnon of he fxed modes, for any s, C e (25) (26) 2 gr. [ se ' A' B ' K C '] n n K, exss, and he sysem has no unsable deenralzed fne fxed modes and he deenralzed mpulse fxed modes. By repeaedly applyng he Lemma 3, se ' A' B rank n n C Here, and are eran non-nerse dvsons { 2,, k } and { k,, }. Tha s jus equvalen wh he Lemma 5. The formula (27) an be denoed as follows. (27) 25 I E-I

8 Modern Appled ene Vol. 5, o. 2; Aprl 2 seabk se ' A' B C BQ B rank rank RC se C C se A B B I K Q C rank se m R I n C From he Lemma 3 and he formula (28), se A B B rank m, C se ' A' B rank mn se A B n n C rank C n C o he suffen and neessary ondon ha he formula (29) exss s ha se A B B rank C, and se A B rank C n C m, all exs. o he suffen and neessary ondon ha he sysem (2) has no unsable deenralzed fne fxed modes and he deenralzed mpulse fxed modes an be obaned,.e. he suffen and neessary ondon ha he sysem () an be real sablzed by he ompensaor (2) and he ompensaor (24). And he ompensaor (2) and he ompensaor (24) an be wren as followng form. E z () z () R y ( ) z() I z() R y () u() Q z() K y( ) u () Q z() K y ( ), where, he defnons of z,, R, Q are same n he former of he arle, and n addon K blok dag K,, K 2 y [ y,, y ] T T T 2 u [ u,, u ] T T T 2 In hs way, he obaned dynam ompensaor (32) an real sablze he reangular sngular sysem (), and obvously, he ompensaor (32) keeps onssen wh he ompensaor (5) n form. By he same way, he suffen and neessary ondon ha one sngular dynam ompensaor s added o he (28) (29) (3) (3) (32) Publshed by Canadan Cener of ene and Eduaon 25

9 Modern Appled ene Vol. 5, o. 2; Aprl 2 j h subsysem of he sysem ould be dedued. End. oe 2: In he Theorem 2, when m n, he formula (9) s equvalen o he Lemma 5 n meanng. 5. Example For one double-hannel sysem, 2 Ex () Ax() Bu() y() Cx (), {,2}, where, E, A, B 6. Conlusons The dynam ompensaor s adoped o hange he reangular desrpor deenralzed sysem no normal square sysem, and he orrespondng losed-loop sysems s suded n hs arle, and he suffen and neessary ondon ha he losed-loop sysems s regular and mpulse-free s obaned, whh keeps onssen wh he orrespondng suffen and neessary ondon of he square sysem, as seen n he formula (9). For he sably, he sngular dynam ompensaor s added o one subsysem, and hen he normal dynam ompensaor s added o orrespondng losed-loop sysems, n order o sablze he reangular sngular sysem, and he suffen and neessary ondon of he orrespondng square sysem keeps onssen n form, as seen n he formula (9). In hs way, he whole sysem s no only gradually sable, bu also has no desruve pulse behavors, whh s very mporan o perfe he desrpor deenralzed sysem heory and apply he desrpor deenralzed sysem n he prae. Referenes D. Wang and C. B. oh. (989). On regularzng sngular sysems by deenralzed oupu feedbak. IEEE Trans. Auoma. Conr. Vol. 44. P G. C. Verghese, B.C. Levy, and T. Kalah. (98). A desrpor sae-spae for sngular sysems. IEEE Trans. Auoma. Conr.. o. 26 (4). P Hou M. (24). Conrollably and elmnaon of mpulsve modes n desrpor sysems. IEEE Trans. Auomaa, Conr.. o. 49(). P Ishhara J Y, Terra M H. (2). Impulse onrollably and observably of reangular desrpor sysems. IEEE Trans. Auoma. Conr. o. 46(6). P L. Da. (989). ngular Conrol ysems-leure oes n Conrol and Informaon ene. Berln, Germany: prnger-verlag. Q. L. Zhang. (989). Algebraal Charaerzaons of Fxed Modes n Lnear Deenralzed Desrpor ysems. Conferene on Deson and Conrol. P H. Wang and E. J. Davson. (973). On he sablzaon of deenralzed onrol sysems. IEEE Trans. Auoma. Conr. Vol. AC-8. P O T.. Chang and E. J. Davson. (2). Deenralzed onrol of desrpor sysems. IEEE Trans. Auoma. Conr. Vol. AC-46. P O. 2. T.. Chang and E. J. Davson. (986). Deenralzed onrol of desrpor ype sysems. Pro. of he 25h IEEE Conferene on Deson and Conrol. P Xe, Xuka e al. (995). Unform Judgemen of he Fxed Mode of he Desrpor Deenralzed Conrol ysem. Aa Auoma na. o. 2(2). P Xe, Xuka & Jn, Hayng. (986). Fxed Mode of he Deenralzed Conrol ysem. Aa Auoma na. o. 2(2). P I E-I

10 Modern Appled ene Vol. 5, o. 2; Aprl 2 X. K. Xe. (988). On fxed modes n sngular sysems. Pro. of Ameran Conrol Conferene. P Yang, Dongme & Zhang, Qnglng e al. (24). Desrpor ysem. Bejng: ene Press. Zhang, Guoshan. (26). Regularzaon and Pole-plaemen of Desrpor ysems by Dynam Compensaon. Conrol and Deson. o.. Z. W. Gao and X. L. Wang. (997). Inernal properness and sably n sngular deenralzed onrol sysems. Pro. of Ameran Conrol Conferene. P Publshed by Canadan Cener of ene and Eduaon 253