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11 No 7 2005 Design of Reconfiguring Control Systems via State Feedback Eigenstructure Assignment GuoSheng W ng Qi ang Lv B ng Li ang and GuangR n Duan 2 Department of Control Engineering Academy of Armored Force Engineering Beijing 100072 P R Ch ID: 24019

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International Journal of Informa tion Technology , Vol. 11, No. 7, 2005 Design of Reconfiguring Control Systems via State Feedback Eigenstructure Assignment Guo-Sheng W ng , Qi ang Lv , B ng Li ang , and Guang-R n Duan 2 Department of Control Engineering, Academy of Armored Force Engineering, Beijing, 100072, P. R. China gswang@126.com; Center for Control Theory and Guidance Technology , Harbin Institute of Technology , Harbin, 150001, P. R. China Liangbing@hit.edu.cn; Grduan@ieee.org Abstract In this paper the design of reconfiguri ng a clas of linear control s em via ate feedback eigens tructure as gnm ent is inves tigated. The des gn aim is to res nthes ze a s ate feedback control law s ch that the eigenvalues of the recon- figured closed-loop control sy stem can comp letely recover those of the original close-loop sy stem, and make the corres ponding eigenvectors of the former as clos e to thos e of the latter as pos ble. General com lete para metric expressions for the ate feedback gains are es tablis hed in term of a s t of param tric vec- tors and the closed-loop poles. The set of parametric vectors and the set of closed-loop poles represent the degrees of freedom existing in the reconfiguring design, and can be further properly chos en to m eet s des red s ecification requirem nt, such as robustness. An illu strative exam ple and the sim lation re- sults show that the proposed parametr ic method is effective and simple. Key ord: Linear control s em eigens tructure as gnm ent, ate feedback, reconfiguration. 1 Introduction Reconfigured Control System s (RCS) po sses the ability of accom odating system failures autom tically with som prior assum tions. In recent years, RCS has drawn ch at nt on of m ny researchers, and ny new m hods and schem s have been proposed (see, e. g. [1]-[7] and their refere nces). In addition to linear quadratic regu- r m hod [1] pseudo i nverse m hod [2] nverse com ponent -m ode sy nt hesi s hod [3] Ly apunov m hod [4] and LM I m hod [5] ei genst uct re assi gnm ent hod ([6] and [7] becom s m re and m re at act e. B sed on t e fact t at t e perform ances of a control system are inly det rm ed by t ei r ei genval es and t e correspondi ng ei genvect ors, t hus ei genst uc re assi gnm ent hod i conveni ent t redesi gn a new gai m i order ecover t e ei genval es of t e norm cont rol sy st em and m ke t ei r correspondi ng ei genvect ors of t e reconfi gured cl osed-l oop 61

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Guo-Sheng Wang, Qiang Lv, Bing Liang, and Guang-Ren Duan Design of Reconfiguring Control Sy stems via State Feedback Eigenstructure Assignment sy st em s as cl ose t hose of t e norm cl osed-l oop sy st em as possi bl e. Param hods for ei genst uct re assi gnm ent have been i nsi el st udi ed [8] -[19] for convent onal l ear sy st em s, descri pt or l ear sy st em s and second-order dy nam c sy st em s. The param hods gi ve t e param expressi ons of al t e cont rol ws and al t e cl osed-l oop ei genvect or m ces. These free param vect ors cl uded t ese expressi ons and t e cl osed -l oop ei genval es, offer al t e degrees of sig freed an can b fu rth r u tilized to satisfy certain sp ecificatio in so cont rol sy st em desi gns. In th is p r, we will co id er th e d sig o reco ig in lin ear system s v a state feedback ei genst uct re assi gnm ent B sed on t e resul for st at e feedback ei genst uc- re assi gnm ent proposed by Duan i [8] a param form of al t e resy nt hesi zed gai ces deri ved and a correspondi ng al gori for t reconfi gurat on i proposed. Thi param m hod offers a e degrees of desi gn freedom , whi h can u tilized to satisfyin ad itio l p rfo rm an ces in co ro l system d sig . 2 Problem Formulation Consider a linear control system in the form of Bu Ax , (1) where and are t e st at e and nput vect ors, respect el and are known m ces wi appropri di nsi ons and rank ; th e m trix p ir is co ro llab e, th at is, sI rank , . (2) Because of the outstanding va riations, the system (1) of ten becom s into the fol- lowing form , (3) where and are e st at e and i nput vect ors, respect el and are known m ces wi appropri di nsi ons and rank ; th e m trix pai is co ro llab e, th at is, sI rank , . (4) For conveni ence, we cal sy st em (1) e norm l ear sy st em and sy st em (3) t e fau lt lin ear system . Ap yin th e fo llowing state feedback control law Kx , , (5) t e sy st em (1), y el ds i cl osed-l oop sy st em as BK . (6) ecal e fact at non-defect e m ces possess ei genval es whi h are l ss i sen itiv e with resp ect to p ram ter p rtu tio , in th is r, we y co id er th e ei genval es of t e cl osed-l oop sy st em (6) are di st ct and sel f-conjugat . Denot ng e ei genval es of sy st em (6) by , where , 62

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International Journal of Informa tion Technology , Vol. 11, No. 7, 2005 , are di st ct and sel f-conjugat and ei r correspondi ng ei genvect ors by , , produces , . (7) Appl ng t e fol wi ng st at e feedback cont rol r , , (8) to th e fau lt system (3 ), o ain fc fc . (9) Due to the fact that the in rnal behavi ors of a cont rol sy st em are det rm ed by ei genval es oget er wi t ei r correspondi ng ei genvect ors, and t e perform ances of i cl osed-l oop sy st em can be i proved by odi fy ng e ei genval es and e correspondi ng ei genvect ors wi som feedback cont rol l ws, t en t e probl em of reconfi guri ng t e l ear sy st em (1) via state feedback to be so lv ed in th is p r can b stated as follows. Problem RESA: Gi ven t e cont rol bl e norm sy st em (1), i cont rol bl e faul sy st em (3), and a set of sel f-conjugat di st ct com x num bers , , then redesign a new state feedb ack controller (8) such that , (10) fc and fi , , (11) are m nim zed, where , , fi , are the eigenvectors of the closed- oop m ces and associated with , fc . Remark 1. From t e descri pt on of Probl em R SA , it is clear to see that when the relatio (1 is satisfied , th ere h , fi fi fc . (12) 3 Closed-Loop Eigenstructure Assignment Set , diag , Eq tio (7 ) is clearly red ced in to th e fo llo win fo rm : BKV AV . (13) Furt her, denot e . (14) en equat on (13) i changed i i equi val nt form BW AV . (15) Because the m trix pair is controllable, applying a series of elem ent trix tran sfo tio to m trix , we can obt ai n a pai of uni odul ar m ces and sat fy ng t e fol wi ng equat on: sI 63

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Guo-Sheng Wang, Qiang Lv, Bing Liang, and Guang-Ren Duan Design of Reconfiguring Control Sy stems via State Feedback Eigenstructure Assignment sI , . (16) Partitio in to th e fo llo win fo rm , , (17) 22 21 12 11 11 en we can obt ai n e fol wi ng l whi h gi ves t e param expressi ons of eigenstructure assignm ent via state fee ack in tim e-in ri ant linear system s. Lemma 1 [8] Gi ven m ces and B with rank( , and a group of sel f-conjugat di st ct com x num bers , , if th e trix ir is co ro llab e, th en th e p ram tric ex essio o all th e state feed ack in trix (13) can be gi ven by , (18) WV where , 11 , , (19) and , 21 , . (20) and , are a group of free param vect ors and sat fy e fol wi ng const ai nt s: Constraint 1: , ; Constraint 2: . det 4 Solution to Problem RESA Du e to th e co ro llab ility o th e m trix p ir , we can now that the eigenval- s o th e m trix can be assi gned arbi ari vi a state feedback. Thus the eigen- val es , fc , o th e m trix can be assi gned t hose via state feed ack . Th en th e relatio (1 in Pro em RESA is satisfied an th e m in task left for t e sol on t Probl em R SA i t desi gn a st at e feedback such t at (11) hol ds. fc earl , denot e , (21) fn th en eq tio in (1 can b written in to th e fo llo win co act fo rm . (22) Furt her, denot e , (23) en equat on (22) i changed i t e fol wi ng form . (24) Du e to th e co ro llab ility o th e m trix p ir , applying a series of elem ent trix tran sfo tio to m trix , we can obt ai n a pai of uni odul ar sI 64

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International Journal of Informa tion Technology , Vol. 11, No. 7, 2005 trices and satisfyin th e fo llo win eq on: , sI . (25) Partitio in to th e fo llo win fo rm , . (26) 22 21 12 11 11 By utilizing the sam m thod in Lem 1, we can obtain the following theorem whi h gi ves sol ons t equat ons (23) and (24). Theorem 1. Gi ven m ces and with rank( , and a group of sel conjugat di st ct com x num bers , , if th e m trix p ir is co ro llab e, th en th e p ram tric ex essio o all th e state feed ack in trix (23) can be gi ven by , (27) where , , fn fi 11 , (28) and , , fn fi 21 , (29) and , are a group of free param vect ors and sat fy e fol wi ng const ai nt s: Constraint 3: , ; Constraint 4: . det Su titu tin (1 an (2 in to (1 , o ain 11 11 , . (30) usi ng ort hogonal project on, we can obt ai n 11 11 11 11 , , (31) whi h m ni ze t e i ndexes i (11). Furt her, l 11 11 11 11 , , (32) en (31) i changed i , . (33) Su titu tin (3 in to (2 an (2 , yield , fi 11 , (34) and , fi 21 . (35) 65

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Guo-Sheng Wang, Qiang Lv, Bing Liang, and Guang-Ren Duan Design of Reconfiguring Control Sy stems via State Feedback Eigenstructure Assignment Thus, t e redesi gned st at e feedback gai can be gi ven by , (36) where , , (37) fn fi 11 and , . (38) fn fi 21 In order guarant ee e real ness of t e gai m in (3 , th e fo llo win const ai nt m st hol d: Constraint C1: , , ; reover, onst ai nt s 2 and 4 st hol d, and are cl earl equi val nt wi t e fol wi ng t o const ai nt s, respect el Constraint C2: det[ 11 11 11 ; Constraint C3: . det[ 11 11 11 From t e above reduct ons, we can gi ve t e fol wi ng t eorem whi h gi ves e so lu tio to Pro em RESA. Theorem 2. Gi ven t e cont rol bl e norm sy st em (1) and t e cont rol bl e faul sy s- (3), and a group of di st ct and sel f-conjugat scal ars , . Then all e desi red sol ons Probl em R SA can be gi ven by (36) wi t e param vectors , sat fy ng C onst ai nt s C -C 3. Accordi ng Theorem 2 and t e above deduct ons, t e fol wi ng al gori for Probl em R SA can be proposed as fol ws. Algorithm RESA: 1. ul at e a pai of uni odul ar m ces and sat fy ng (16), and part on as in (1 ; 2. ul at e a pai of uni odul ar m ces and sat fy ng (25), and rtitio as in (2 ; 3. Fi nd a group of param rs , sat fy ng C onst ai nt s C -C 3, and calculate the m trices and according to (38) and (37), respectively; 4. Calculate the state f eed ack g in m trix according to (36). 5 An Illustrative Example onsi er a norm l ear sy st em and i correspondi ng faul l ear sy st em wi t e following param ters: 66

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International Journal of Informa tion Technology , Vol. 11, No. 7, 2005 , , . Easily, we can fin th at th e trix ir and are bot h cont rol bl e. In exam pl e, we choose t e ei genval es of t e norm cl osed-l oop sy st em as , . Alg ith RESA is u tilized to so lv e th is reco ig atio problem . The results of each step are given as follows. Ob tain th e fo llo win m trices satisfyin (1 as , . Ob tain th e fo llo win m trices satisfyin (2 as , . 3) -4) Denoting , . Then from (37) and (38), we can get t e param expressi ons of and . Thus we can get t e param expressi on of the redesigned state feedback gain m trix from (36). Speci al , choosi ng a group of t e param vect ors as , , then we can calculate that , , 13 13 15 19 17 19 17 37 33 35 33 35 and . 23 10 25 77 30 91 oreover, from (18) we obtain . 77 30 13 21 67

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Guo-Sheng Wang, Qiang Lv, Bing Liang, and Guang-Ren Duan Design of Reconfiguring Control Sy stems via State Feedback Eigenstructure Assignment For convenience, we call the closed-l oop sy stem s of the norm al sy stem under as sy stem 1, the closed-loop sy stem of the fault sy stem under as system 2 and the closed-loop sy stem of the fault sy stem under as sy stem 3, respectively The errors between the outputs of sy stem 1 and sy stem 2 are given in Fig.1 and Fig.2, respec- tively and the sim lation results s how that the redesigned feedback is effective. oreover, the eigenvalues of sy stem 3 are 67.4133, -0.2066+0.9042i, and -0.2066- 0.9042i, in which 67.4133 is an unstable eige nvalue, while the eigenvalues of sy stem 1 are the sam with those of system 2. 10 12 14 16 18 20 -6 -4 -2 x 1 -1 t/s en t he f ou ut Fig. 1. Errors between the first output s of sy ste 1 a nd sy ste 2 10 12 14 16 18 20 -2 -1 -1 -0 0. 1. x 1 -1 t/s bet en t e sec ond ou ut Fig. 2. Errors between the second output s of sy stem 1 and sy stem 2 6 Conclusions In this paper reconfiguring linear control sy st em s via state feedback eigenstructure assignm ent is investigated. By utilizing the freedom degrees offered by a param tric 68

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International Journal of Informa tion Technology , Vol. 11, No. 7, 2005 result of eigenstructure assignm ent in linear control sy stem s, a param tric expression for all the state feedback gain m trices, which can recover the eigenvalues of the norm al closed-loop sy stem and m ke the ei genvectors of the fault closed-loop sy stem as close to those of the norm al closed-loop sy stem as possible, is established and an algorithm for this design is proposed. The pa ram tric m thod offers all the design degrees of freedom , which can be further utili zed to satisfy certain specifications in control system designs, such as robustness etc. An illustrative exam ple and the sim lation figures show the effect of the proposed algorithm References [1] Looz D. P. We iss, J. L. Ba rre tt, N. M. : An automatic redesign approach for restructurable control sy stem s. IEEE Control Sy stem Magazine. Vol. 5 (1985), pp. 1621-1627. [2] Gao, Z., Artsaklis, P.J.: Stability of the ps eudo-inverse m thod for reconfigurable control sy stems. Int. J. of Control. Vol. 53 (1991), pp. 520-528. [3] Takewaki, I.: Inverse component-mode sy nthe sis method for redesign of large structural sy stems. Comput. Methods. Appl. Mech . Engrg. Vol. 166 (1998), pp. 201-209. [4] Parks, P.C.: Ly apunov redesign of m odel re ference adaptive control sy stem s. IEEE Trans- actions On Automatic Control. AC-11 (1966), pp. 362-367. [5] Chang, W ., P rk, J B., Lee, H ., J oo, Y : LMI approach to digital redesign of linear tim e- invariant sy stems. IEE, Proc-Control Theory Appl. Vol. 149 (2002), pp. 297-302. [6] Jiang, J.: Design of reconfi gurable control sy stems. Int. J. of Control. Vol. 59 (1994), pp. 395-401. [7] Ren, Z., Tang, X.J., Chen, J.: Reconfigurab le control sy stem design by output feedback eigenstructure assignment. Journal of Contro l Theory and Applications. Vol. 19 (2002), pp. 356-362. [8] Duan, G.R.: Solutions to trix equation AV+BW=VF and their application to eigenstruc- ture assignm ent in linear sy stem s. IEEE Tran s. on Autom tic Control. Vol. 38 (1993), pp. 276-280. [9] Duan, G.R.: Eigens tructure as gnm ent by decentralized output feedback--A com lete param tric approach. IEEE Trans. on Autom tic Control. Vol. 39 (1994), pp. 1009-1014. [10] Duan, G.R.: Eigenstructure assignment in descriptor linear sy stems via output feedback. Int. J. Control. Vol. 72 (1999), pp. 345-364. [11] Duan, G.R., Patton, R.J.: Eigenstructure a ssignme t in de sc riptor sy ste s via sta fe back---A new complete parametric approach. In t. J. Sy stems Science. Vol. 29 (1998), pp. 167-178. [12] Duan, G.R., Patton, R.J.: Eigenstructure assignment in descriptor sy stems via proportional plus derivative state feedback. Int. J. Control. Vol. 68 (1997), pp. 1147-1162. [13] Duan, G.R.: Eigenstructure assignment in descriptor linear sy stems by output feedback, IEE Proceeding Part D: Control Theory and Applications, Vol. 142 (6) (1995), pp. 611-616. [14] Duan, G.R., Liu, G.P.: Com lete param tric approach for eigenstructure assignment in a class of second-order linear sy stems. Automatica. Vol. 38 (2002), pp. 725-729. [15] Duan, G.R., Wang, G.S., Liu, G.P.: Eigens tructure assignment in a class of second-order linear sy stem s: A com lete param tric appr oach, Proceedings of the 8th Annual Chinese Automation and Computer Society Confer ence. Manchester, UK. (2002), pp. 89-96. [16] W ng, G.S ., Duan, G.R.: S ate feedback ei genstructure assignment wi mi ni mum c ont rol effort. The 5th World Congress on Intelligent Control and Autom tion. Hangzhou, China. 1 (2004), pp. 35-38. 69

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Guo-Sheng Wang, Qiang Lv, Bing Liang, and Guang-Ren Duan Design of Reconfiguring Control Sy stems via State Feedback Eigenstructure Assignment [17] Wang, G.S., Duan, G.R.: Robust pole assignm ent via P-D feedback in a class of second- order dy namic sy stems. International Confer ence of Automation, Robots and Computer Vi- sion. Kunming, China (2004), pp. 1152-1156. [18] Duan, G.R., W ng, G.S : P D feedback ei genstructure assignment wi mi ni mum c ont rol effort in second-order dy nam c sy stem s. I EEE International Sy posium of Com puter Aid Control Sy stem Design. Taibei, China (2004), pp. 344-349. [19] Wang, G.S., Liang, B., Duan, G.R.: R econfiguring second-order dy namic sy stems via ate feedback eigens tructure as gnm ent. Inte rnational Journal of Control, Automation, and Sy stems. Vol. 3 (2005), pp. 109-116. Guo-Sheng Wang received both the B. S. and M. S. degrees in 1999 and 2001, respectively and the Ph. D. degree in C on- trol Theory and C ontrol Engineering from Harbin Institute of Technology in 2004. He is curren tly a lecture of Departm nt of C ontrol Engineering at cadem of Arm ored Force Engi- neering. His research intere sts include robust control, eigen- structure assignm ent, and second-order linear sy stem s. Qiang Lv received the B. S. degree, the M. S and Ph. D. de- grees in Control System s Th eory from Harbin Institute of Technology . He is currently a professor of Departm nt of ontrol Engineering at Academ of Arm ored Force Engineer- ing. His m in research inte rests include robust control, ar- ored force control and neural network control. Bing Lia received both the B. S. and M. S. degrees in them atics from Hebei University in 1999 and 2002, respec- tively She is currently working toward PhD degree in ontrol System s Theory of Harbin Ins titute of Technology. Her re- search interests include descript or sy stem s, fault-tolerant con- trol and robust control. Guang-Ren Duan received the B. S. degree in Applied them atics, and both the M S and Ph. D. degrees in ontrol System s Theory from Harbin Ins titute of Technology. He is currently the director of the C nter for ontrol Sy stem s The- ory and Guidance Technology at Harbin Institute of Technol- ogy . His m in research intere sts include robust control, eigen- structure assignm ent, descript or system s, m ssile autopilot control and m gnetic bearing control. 70