PROBLEMS OF DESIGNING STATE FEEDBACK CONTROLLERS FOR OBJECTS WITH TRANSFER FUNCTION ZEROS*

Transcription

1 POWER ELECRONICS AND DRIVES 2(37), N. 2, 207 DOI: /PED70203 PROBLEMS OF DESIGNING SAE FEEDBACK CONROLLERS FOR OBJECS WIH RANSFER FUNCION ZEROS* VALENIN DROZDOV, ARUR ABDULLIN, ALEKSANDR MAMAOV IMO University, Krnverkskiy pr., 49, 970 St. Petersburg, Russia, addresses: Abstrat: he analysis f the influene f transfer funtin zers n the parameters f state feedbak ntrllers has been nduted. If a transfer funtin f a ntrl bjet has zers whih are lated lsely t ples, the ntrl bjet tends t singularity, and the influene f the input ntrl signal t the states f the ntrl bjet bemes weaker. he prblem f the state feedbak ntrller synthesis bemes ill-nditined, whih leads t the appearane f extremely large state feedbak effiients. In this ase, the state feedbak effiients are sensitive t the parameters f the ntrl bjet. As a result, the parametri rbustness f the ntrl system is redued. Knwn methds f strutural analysis f ntrl bjet mdels are inluded amngst different methds f the numerial evaluatin f the ntrllability and the bservability, as well as methds f the mdel rder redutin. hese methds have sme disadvantages, suh as dependene n the state spae representatin frm f the ntrl bjet, ignring a part f the ntrl bjet mdel. In this paper, sme ways f the preliminary strutural analysis f the state spae mdels f ntrl bjets have been prpsed. he singular (Hankel) matrix is prpsed fr analyzing the prperties f ntrl bjet mdels. he singular matrix is the invariant harateristi f the ntrl bjet in varius state spae frms and it haraterizes the prperty f the ntrl bjet mpleteness. As a result f the researh, it was fund that the effiients f the state feedbak ntrller are inversely prprtinal t the determinant f the singular matrix, and the determinant f the singular matrix is equal t the resultant f the transfer funtin plynmials. hus the value f the determinant f the singular matrix depends n the latin f the zers f the transfer funtin. he methd f the strutural transfrmatin (dempsitin) f the ntrl bjet fr the defining the need f the reduing the rder f the ntrl bjet mdel is prpsed. Keywrds: zers f a transfer funtin, bservability, ntrllability, Hankel matrix, state feedbak ntrller, resultant, residues. INRODUCION he main attentin f researhers invlved in the designing f the state feedbak ntrllers has traditinally fused n the analysis f the eigenvalues plaement f the *Manusript reeived: June 22, 207; aepted: September 25, 207.

2 78 V. DROZDOV et al. ntrl bjet and the hie f the desired eigenvalues plaement fr the lsed lp system whih an prvide the required quality f the ntrl system [ 3]. Hwever, ignring the presene f transfer funtin zers in the mdel f the ntrl bjet an lead t undesirable nsequenes and an negatively affet parameters f the designed state feedbak ntrller. his fat, in partiular, drew the attentin f speialists wrking with diret drive serv systems. Using diret drive serv mtrs instead f gear reduers aused a signifiant influene f the elasti prperties f the ntrl bjets n the prperties f the drive system [3 5]. Elasti nstrutins are the systems with distributed parameters, whih mdels have the frm f partial differential equatins. In pratie, mathematial mdels f suh bjets are apprximated t the frm f rdinary differential equatins [3 6]. he feature f finite-dimensinal mdels f bjets with elasti uplings is that they have transfer funtin zers. he synthesis f state feedbak ntrllers fr suh bjets ften leads t the alulatin f exessively high and sensitive t bjet s parameters abslute values f the state feedbak effiients, whih leads t dereasing f the parametri rbustness f the ntrl system [7]. Cntrl systems with high values f the state feedbak effiients an beme unstable under the nise in measurements and bservatins f the state vetr r under the hanging f the bjet s parameters during the peratin. S, the preliminary strutural analysis f state spae mdels f ntrl bjets is needed, taking int aunt the latin f transfer funtin zers. Cnsider the state spae mdel f the diret drive serv system with tw mass lad 2, x where i is the urrent f the mtr, angular velity f the first mass, whih an be alulated frm the indiatins f the ender, trsin angle between the first mass angle and the send mass angle, 2 angular velity f the send mass. Matries f the state spae mdel have the fllwing frm: with the ender n the first mass and the state vetr i R Ce 0 0 L L Cm kd C2 k d L A J J J J, B0, С [0 0 0] kd C2 k d 0 0 J2 J2 J2 where R the resistane f the phase f the mtr, L the indutane f the phase f the mtr, C e the bak-emf nstant f the mtr, C m the trque nstant f the mtr, J inertia f the first mass, k d vissity effiient f the elasti upling, C 2 rigidity effiient f the elasti upling, J 2 inertia f the send mass.

3 Prblems f designing state feedbak ntrllers fr bjets with transfer funtin zers 79 Magnitude, db Imaginary Axis (sends - ) Frequeny, Hz Fig.. ypial harateristis f the ntrl bjet (): а) magnitude frequeny harateristi, b) ple-zer map Real Axis (sends - ) a) b) Figure shws a typial magnitude frequeny harateristi and ple-zer map fr the suh kind f ntrl bjets. It has tw mplex zers lated near the ples. Distane between these ples and zers is determined by the values f k d, C 2, J, J 2. Smetimes ntrl systems with state feedbak ntrllers designed using the full state spae mdel have insuffiient quality r beme unstable beause f high abslute values f the state feedbak effiients. Preliminary strutural analysis f suh mdels is needed t estimate the ntrllability and the bservability f the ntrl bjet, t find ut the neessity f mdel redutin, t make the deisin abut the type and struture f the ntrller. It is suppsed that the state feedbak effiients beme higher when the zer apprahes t ples. Prblems f the strutural analysis f mathematial mdels in the designing f state feedbak ntrllers are nsidered in [6 7]. In [6, 8], it is suggested t use the mdal dminane measures whih are intrdued in these wrks fr assigning the desired eigenvalues f the lsed lp system. he pssibility f using sme numerial measures f the ntrllability and the bservability fr hsing desired eigenvalues f the lsed lp system is disussed in [9 2]. Suggested numerial measures are based n the use f sme numerial harateristis f ntrllability and bservability matries, r ntrllability and bservability Gramians. In [2, 3], it is suggested t hange the state spae representatin f the mathematial mdel f the bjet, using sme prperties f ntrllability and bservability Gramians. he use f mdal dminane measures as well as varius numerial measures f the ntrllability and the bservability ften des nt prvide psitive results, beause f their dependene n the hie f the state spae representatin and ignring zers f the transfer funtin. Signifiant spread in the designing f state feedbak ntrllers was btained by the predure f the reduing the rder f the ntrl bjet mdel [3 5]. he redutin f the rder f the ntrl bjet mdel simplifies the press f designing the state feedbak ntrller. Hwever, if the synthesis f state feedbak ntrllers is based n the redued mdel, it bemes neessary t ndut additinal researhes f the ntrl

4 80 V. DROZDOV et al. system prperties with the redued-rder ntrller with pssible rretin f the state feedbak ntrller parameters. One f pssible slutins is the nstrutive develpment f the ntrl bjet, fr example, by inreasing the stiffness f the mehanial part. In this ase, it is pssible t ahieve an inrease f the distane between zers and ples f the transfer funtin. he methds f frming the desired spetrum f transfer zers are nsidered in [6], these methds als require nstrutive hanges f the ntrl bjet at the design stage. Hwever, in pratie this apprah is nt always pssible. he researh is devted t the analysis f the influene f the latin f transfer funtin zers n the values f the state feedbak effiients and als t the develpment f sme new ways f preliminary strutural analysis f ntrl bjets. he singular matrix whih is the invariant harateristi f the ntrl bjet in different state spae representatins is intrdued. he nnetin between the determinant f the singular matrix and the state feedbak is defined. he dependene f the determinant f the singular matrix n the relative latin f zers and ples f the transfer funtin is prved. he numerial example is given fr the simplest ntrl bjet. he methd f strutural transfrmatin (dempsitin) f the ntrl bjet whih an be used in the designing f the state feedbak ntrllers fr bjets with zers f the transfer funtin fr the defining the need f the reduing the rder f the ntrl bjet mdel is prpsed. 2. CONROLLABILIY AND OBSERVABILIY OF OBJECS WIH RANSFER FUNCION ZEROS Cntrllability and bservability matries f linear bjets x=ax+b u y Cx with a single input and a single utput (SISO), arding t [7] are equal t () n n P B AB A B, P C CA CA he transfrmatin f the state spae frm arding t the equatin x Mx is perfrmed in the state spae f the bjet (). Matries f the new state spae frm f the bjet have the fllwing frms: A MAM, B MB, C CM Cntrllability and bservability matries in the new state spae frm P MP, P PM, (2)

5 Prblems f designing state feedbak ntrllers fr bjets with transfer funtin zers 8 Chsing different state spae representatins, different prperties f ntrllability and bservability matries an be btained. It fllws that using expressins bjet ntrllability r bjet bservability is nt entirely rret. It will be mre rret t nsider the ntrllability r the bservability f the pair f matries (A, B) r (A, C) hus it an be nluded that numerial measures f the ntrllability and the bservability an be used fr the synthesis f state feedbak ntrllers nly in ertain ases. Instead f the ntrllability and the bservability it is advisable t nsider sme invariant prperty f the bjet that des nt depend n the hie f the state spae representatin. he harateristi f suh a prperty an be the matrix whih has the fllwing frm: P P P (3) In ardane with (2) P P P P M MP P P P Matries f this struture are alled Hankel matries. his matrix is invariant harateristi f the ntrl bjet in all pssible state spae representatins. Hankel matrix has a lear physial meaning. It desribes the Hankel experiment, whih haraterizes the nnetin between spaes f past input signals and future utput signals f the system [6, 8]: y P Pu P u Hankel peratr with the matrix (3), unlike the transfer funtin, nt nly desribes the nnetin f the spae f input signals and the spae f utput signals, but als haraterizes the degree (intensity) f the partiipatin f eah ple f the bjet in the pwer transmissin frm the input t the utput f the bjet. his degree f the partiipatin is estimated by the singular numbers f the matrix (3). he matrix (3), arding t [9], is alled the mpleteness matrix f the bjet. In the ase when the matrix P is nt singular, the bjet is alled mplete. In this ase, the bjet is mpletely ntrlled and mpletely bserved, regardless f the state spae representatin. If the rank P is smaller, then the state spae dimensin the bjet bemes singular. he degree f the denminatr f the transfer funtin bemes less than the dimensin f the state spae. Fr this reasn, the matrix P an als be alled a singular matrix. If the determinant f the singular matrix tends t zer, the nnetin between input and utput spaes f the bjet bemes weaker. 3. INFLUENCE OF HE HANKEL MARIX ON CONROL COEFFICIENS he mathematial mdel f the linear statinary bjet with a single input and a single utput (SISO) () is nsidered.

6 82 V. DROZDOV et al. he prblem f designing the state feedbak ntrller that prvides desired dynami prperties f the lsed lp ntrl system is nsidered. Fr slving this prblem, the priniple f separating predures fr alulating mdal ntrl and bserving rdinates f the state vetr f the bjet () is used. he mdal ntrller synthesis is mst nvenient t implement in the ase when the mdel f the ntrl bjet is presented in the ntrllable annial frm: x = Ax +Bu (4) y Cx he transfer funtin f the bjet () is a linear fratinal funtin f the mplex variable: n ys ( ) bn s... bs b0 us ( ) n n s an s... asa0 W() s C sia B (5) Matries haraterizing the mdel f the ntrl bjet in the ntrllable annial frm (4), in ardane with the frm f the transfer funtin (5), are expressed as fllws: b b A , B..., C (6) b a 0 a a2... a b n n he representatin f the mdel in the ntrllable annial frm ensures the existene f the ntrl algrithm that prvides the desired assignment f eigenvalues f the harateristi plynmial f the lsed lp system [6]. he ntrl law fr the bjet () an be written as fllws: u Kx (7) he effiients f the matrix K in the ntrllable annial frm are alulated as fllws: K =aсg a (8) where the matrix a g is the rw f effiients f the desired harateristi plynmial f the lsed lp system, and the matrix a is the rw f effiients f the harateristi plynmial f the matrix A. he state vetrs f mdels () and (4) are related by the transfrmatin matrix x =M x (9)

7 Prblems f designing state feedbak ntrllers fr bjets with transfer funtin zers 83 wherein:,, A = M AM B = M B C =CM (0) he transfrmatin matrix M an be represented using ntrllability matries in different state spae frms. he ntrllability matrix in the state spae frm () is: P = n... B AB A B he ntrllability matrix in the state spae frm (4) is fllwing: n P... = B A B A B Using the equatin (0) the fllwing relatin an be btained: P = MP Cntrllability matries are knwn in bth frms, s the transfrmatin matrix an be represented as fllws: M = P P () Using Eqs. (9) and (), the ntrl law (7) an be rewritten in the fllwing frm: u KPP x (2) In mst pratial ases, sme f the rdinates f the state vetr x f the bjet () are nt measured, s it bemes neessary t design the bserving devie. he representatin f the mdel in the bservable annial frm ensures the existene f the algrithm that perfrms the estimatin f the state vetr f the bjet. Using the Kalman duality priniple [7], matries f the bservable annial frm are hsen as fllws: a a A = A = a2, B =C, C =B (3) an he mathematial mdel f the state bserver in the bservable annial frm an be written as fllws: x = Ax +BuL( ycx) y Cx (4)

8 84 V. DROZDOV et al. In ardane with the struture f matries A, C, the matrix f the bserver input by the bserver errr has the fllwing frm: a g is the lumn f effiients f the desired harateristi plyn- where the matrix mial, and the matrix g a L =a a is the lumn f effiients f the harateristi plynmial f the matrix A. State vetrs f mdels () and (3) are related as fllws: wherein x =M x (5),, A =M AM B =M B C =CM (6) he transfrmatin matrix M an be represented using bservability matries f the bjet in different state spae frms. he bservability matrix in the state spae frm () is: P = C CA CA CA he bservability matrix in the frm (3) is fllwing: 2 n... 2 n... P = C C A C A C A Frm relatins (6) it an be written: P = P M Observability matries are knwn in bth frms. herefre the transfrmatin matrix t the bservable annial frm is determined by the fllwing expressin: M = P P (7) Fr the implementatin f the ntrl law (2), bserved states x =Mx are used. In ardane t this fat the state vetr an be written as fllws: x=m x Substituting this value int the expressin (2), the fllwing equatin fr the ntrl law an be btained: u KPP M x

9 Prblems f designing state feedbak ntrllers fr bjets with transfer funtin zers 85 In ardane with the expressin (7), the last expressin is nverted t the fllwing frm: u KPP P Px u KP(PP) Px aking int aunt the expressin fr the singular matrix (3), the last expressin is rewritten in the fllwing frm: u KP(P ) Px (8) Arding t the expressin (8), the singular matrix P must be reversed. hus, the state feedbak effiients are inversely prprtinal t the determinant f the singular matrix. When the determinant f this matrix tends t zer, the abslute values f the state feedbak effiients tend t infinity. he harateristi f tw plynmials nneting distanes between their rts (fr the transfer funtin between zers and ples) is the resultant f plynmials: n us ys i j res ( ), ( ) where i are ples f the transfer funtin, and j are zers f the transfer funtin. he resultant f the transfer funtin plynmials is equal t the prdut f all distanes between their zers and ples. he equality f the determinant f the Hankel matrix f the bjet () and the resultant f transfer funtin (5) plynmials an be prved using the Krneker methd f alulating the resultant [20]. hus, if the zer f the transfer funtin (5) apprahes t any ple, the determinant f the singular matrix dereases and abslute values f the state feedbak effiients inrease. Ardingly, the presene f zers lated lsely t ples f the transfer funtin affets the values f the state feedbak effiients. An attempt t rret suh ples leads t inreasing f the parametri sensitivity f the ntrl system. j 4. HE MEHOD OF CONROL OBJEC DECOMPOSIION he mst ften enuntered in pratie ase when the transfer funtin (5) has n multiple ples is nsidered, taking int aunt that the resultant f numeratr and denminatr plynmials is equal t the determinant f the singular matrix. Under this nditin, the linear fratinal transfer funtin an be represented by the sum f elementary fratins: r r2 rn W( s)... (9) s s s 2 n

10 86 V. DROZDOV et al. where i are ples f the transfer funtin (5), inluding mplex njugate ples. Arding t Eq. (9), the numeratrs f elementary fratins an be alulated as fllws: r lim( s ) W( s) (20) i si he value r i alulated frm (20) is alled the residue f the funtin W(s) at the ple i. Perfrming alulatins arding t Eq. (20), we btain: b r i n n i j j n j ji i i where i are ples f the transfer funtin, and j are zers f the transfer funtin. he abslute value f the numeratr f the residue at the ple i is prprtinal t the prdut f distanes between this ple and all zers f the transfer funtin. he abslute value f the denminatr f the residue at the ple is equal t the prdut f distanes between this ple and ther ples f the transfer funtin. If the ple is equal t sme zer f the transfer funtin, the value f the residue at this ple bemes zer. S, if the value f the ple apprahes t the value f any zer f the transfer funtin, abslute value f the rrespnding residue bemes smaller. In the strutural diagram, the sum f transfer funtins rrespnds t the parallel nnetin f rrespnding strutural elements. Fr this reasn, the residue at the ple is the transmissin effiient f the rrespnding branh f the strutural diagram. If the abslute value f the rrespnding residue bemes smaller, the part f the pwer transmitted by the rrespnding branh f the strutural diagram frm the bjet s input t its utput bemes smaller as well. S, attempts t hange the value f this ple using the state feedbak ntrllers lead t the appearane f exessively high state feedbak effiients. he strutural diagram f the bjet () with the transfer funtin (5) is represented in the frm shwn in Fig. 3. j, Fig. 2. Dempsed diagram f the ntrl bjet In this interpretatin, the strutural element with the transfer funtin (s) an be nsidered as the multipliative unertainty fr the bjet with the transfer funtin W (s).

11 Prblems f designing state feedbak ntrllers fr bjets with transfer funtin zers 87 he sum f abslute values f residues whih related t the transfer funtin (s) is defined as m, and the sum f abslute values f all residues f the series (9) is defined as m n r. he hie f (s) shuld be arried ut using the fllwing nditin: i i m m where the number haraterizes the part f the transmitting pwer, whih has a little effet n the verall behavir f the bjet when the ntrl signal is applied. he value f this number has t be seleted frm perating nditins f the bjet and requirements fr the designed ntrl system. he representatin f the ntrl bjet in the frm shwn in Fig. is typial f the thery f rbust ntrl. Methds f the rbust ntrl designing, inluding thse based n the MALAB sftware pakage, were nsidered in [2, 22]. Suggested methd f the ntrl bjet dempsitin an be used fr defining the need f the reduing the rder f the ntrl bjet mdel. Residues f the transfer funtin an be used as numerial harateristis f this predure. he methd requires experimental verifiatin n a large sale f real bjets with different parameters t find ut sme dependenes and regularities in predures f dempsitin and redutin f the ntrl bjet. It shuld be nted that the mre detailed researh devted t the state feedbak ntrller designing fr prpsed representatin f the ntrl bjet is needed. 5. NUMERICAL EXAMPLE Cnsider the mdel f the linear ntrl bjet with the fllwing transfer funtin: s с W() s ( s a)( s b) (2) he ntrl bjet is represented in three state spae annial frms: diagnal, ntrllable, and bservable, then ntrllability and bservability matries are mpsed fr eah frm f the bjet representatin. Residues at the ples f the bjet (2): a a, r r b b b a a b

12 88 V. DROZDOV et al. he diagnal annial frm: A a 0 r, a, 0 b r b C a a( a) ab ab P, P b b( b ) a b ab ab he ntrllable annial frm: A 0 0 с, с, с ab ( a b) B C P 0, ab P ab ba he bservable annial frm: A 0 ab,, 0 ( ab) B C P ab 0 2, 2 ba P ab he singular matrix f the ntrl bjet is mpsed. he determinant f the singular matrix, and als the resultant f the transfer funtin plynmials are alulated: ( ab) P 2 2 ( ab) a abab b det( P ) ( b)( a), res( W( s)) ( b)( a) It is easy t verify that the singular matrix is the same in all the annial frms whih have been nsidered. Als, it is bvius that the determinant f the singular matrix and the resultant f the plynmials are equal up t a sign. Let the desired harateristi plynmial f a lsed lp system be as fllws: 2 Dg s as a2

13 Prblems f designing state feedbak ntrllers fr bjets with transfer funtin zers 89 Suppse that the state bserver, whih restres the state variables f the bjet, is designed. he state feedbak effiients fr bserver states are alulated in ardane with Eqs. (8) and (8): a ab a b a2 KU () ( b)( a) K U (2) ( b)( a ) aa aab a2 ab ab a b a b ab It fllws frm abve equatins that the values f the state feedbak effiients are inversely prprtinal t the determinant f the singular matrix. he numerial simulatin f the system using fllwing parameters is perfrmed: a = 25, b = 5, a = 00, a 2 = 2400, = Desired eigenvalues f the lsed lp system are [ 40, 60]. Figure 3 shws the dependene f the abslute value f the determinant f the singular matrix n the latin f the bjet s zer. Figure 4 shws the dependene f abslute values f the state feedbak effiients n the latin f the bjet s zer. 80 ) 60 det(p ) Fig. 3. Abslute value f the determinant f the singular matrix C K() u 00 K(2) u C C а) б) b) Fig. 4. Abslute values f the state feedbak effiients: а) effiient f the first state KU(), b) effiient f the send state KU(2)

14 90 V. DROZDOV et al. Submitted dependenes learly demnstrate that the state feedbak effiients are inversely prprtinal t the determinant f the singular matrix. he determinant f the singular matrix bemes smaller and the state feedbak effiients beme higher when zer apprahes t ples. Clsed lp ntrl systems with suh high values f the state feedbak effiients an be unstable under the nise in measurements and bservatins f the state vetr r under the hanging f the bjet s parameters during the peratin. S these dependenes als demnstrate the negative influene f the transfer funtin zers n ntrller parameters. Figure 5 shws the dependene f abslute values f residues n the latin f the zer..5 ra r rb C Fig. 5. Abslute values f the residues f the ntrl bjet Obviusly, if the ple apprahes t zer, the value f the rrespnding residue dereases. his means that the part f the pwer transmitted t the utput f the system by the subspae rrespnding t this ple dereases as well. 6. CONCLUSIONS Zers f the transfer funtin an negatively affet parameters f the state feedbak ntrllers. Sme ways f preliminary strutural analysis f state spae mdels were nsidered. Singular (Hankel) matrix f the ntrl bjet was intrdued. Influene f the singular matrix n the press f ntrller designing was analytially researhed. here was fund ut that the state feedbak effiients are inversely prprtinal t the determinant f the singular matrix f the ntrl bjet. Als the determinant f the singular matrix is equal t the prdut f all distanes between ples and zers f the transfer funtin, whih is alled the resultant f transfer funtin plynmials. S, the attempt t replae the ple lated lsely t any zer f transfer funtin entails the exessive grwth f abslute values f the state feedbak effiients. On the ther hand, suh a ple has a little effet n the nnetin between state spaes f input and utput signals. In this ase, it is advisable t dempse the ntrl bjet by islating

15 Prblems f designing state feedbak ntrllers fr bjets with transfer funtin zers 9 ples lated near zers as the transfer funtin f the multipliative unertainty f the ntrl bjet. Suh ples an be defined by mparing abslute values f residues at all transfer funtin ples. Suggested representatin f the ntrl bjet an be used fr designing the rbust ntrller r reduing the rder f the state spae mdel. Represented simple numerial example learly demnstrates analytial dependenes whih were fund ut during the researh. Suggested ways f preliminary strutural analysis f state spae mdels requires experimental verifiatin n a large sale f real bjets with different parameters. ACKNOWLEDGMENS his wrk was finanially supprted by the Gvernment f Russian Federatin, Grant N. 074-U0. REFERENCES [] ANDRIEVSKIJ B.R., FRADKOV A.L., Seleted hapters f ntrl thery with examples in MALAB, Nauka, Saint-Petersburg 2000 (in Russian). [2] CHU E.K., Optimizatin and ple assignment in ntrl system design, Int. J. Appl. Math. Cmp. Si., 200, (5), [3] BALKOVOJ A.P., CACENKIN V.K., Preisin eletri drive with brushless mtr, MPEI Publ., Msw, Russia, 200 (in Russian). [4] ABDULLIN A.A., DROZDOV V.N., PLOICYN A.A., Mdified design methd f an ptimal ntrl system fr preisin mtr drive, WSEAS rans. Syst. Cntrl, 204, 9(), [5] ABDULLIN A.A., DROZDOV V.N., Cntrl systems fr bjets with elasti upling, J. St. Petersburg State University f ehnlgy and Design, Natural Eng. Si., 202, (2), (in Russian). [6] BALONIN N.A., New urse n the thery f mtin ntrl, Saint Petersburg Izd., 2000 (in Russian). [7] ANISIMOV A.A., KOOV D.G., ARARYKIN S.V., YUIKOV V.V., Analysis f parametri sensitivity and strutural ptimizatin f mdal ntrl systems with state ntrllers, J. Cmp. Syst. Si. Int., 20, 50(5), [8] EFIMOV S.V., GAYVORONSKIY S.A., ZAMYAIN S.V., SUHODOEV M.S., Determinatin f the desirable arrangement area f dminating ples f the lsed system in view f its zeres, Bull. msk Pl. Univ., 2008, 32(5), [9] USHAKOV A.V., BIRJUKOV D., DUDARENKO N.A., Cntrl ver dynami bjets and systems degeneratin. Gramian apprah, J. Inst. Eng., 203, 56(4), (in Russian). [0] VORONIN A.V., Qualimetry f aessibility and disturbane f linear dynami systems, Pr. msk Pl. Univ., 203, 323(5), (in Russian). [] CHUKANOV S.N., Quantitative evaluatin f ntrllability and bservability f mpliated dynami netwrks, Inf. Cntrl Syst., 205, 4(46), (in Russian). [2] ARARYKIN S.V., ANISIMOV A.A., Synthesis f rbust ntrl systems with state bservers fr eletrmehanial bjets based n Gramian methd, Pr. 9th Int. Cnf. Pwer Drives Systems, ICPDS, Perm, Russia, 206, 5. [3] RERGIS C.M., MESSINA A.R., BEANCOUR R.J., Mdel-rder redutin using trunated mdal balaned realizatin, Nrth Amerian Pwer Sympsium, NAPS, Charltte, USA, 205, 6. [4] APOLONSKII V.V., ARARYKIN S.V., Methds fr the synthesis f redued state ntrllers f linear dynami systems, J. Cmp. Syst. Si. Int., 204, 53(6),

16 92 V. DROZDOV et al. [5] ROMANOVA I.K., Mdern methds f nn-linear systems rder redutin and their appliatins fr frming mdels f mving bjets, Herald f the Bauman Msw State ehnial University, Ser. Mehanial Engineering, speial issue Speial Rbtis and Mehatrnis, 202, (in Russian). [6] ASANOV A.Z., DEMYANOV D.N., Assigning the set f zers in ntrl systems with parallel mpensatin, J. Cmp. Syst. Si. Int., 203, 52(5), [7] KALMAN R., On the general thery f ntrl systems, Clleted papers f the Internatinal Federatin f Autmati Cntrl Cngress, Msw 96, 2, (in Russian). [8] PELLER V.V., Hankel peratrs and their appliatins, Regular and Chati Dynamis, Institute f Cmputer Siene, Msw 2005 (in Russian). [9] UONJEM M., Linear multi-dimensinal ntrl systems. Gemetri apprah, Nauka, Msw 980 (in Russian). [20] KALININA E.A., UESHEV A.J., Eliminatin thery. utrial, Izd. NII Khimii SPbGU, Saint Petersburg 2002 (in Russian). [2] Methds f rbust, neur-fuzzy and adaptive ntrl. utrial, N.D. Egupv (Ed.), Bauman Msw State ehnial University Publ., Msw 200 (in Russian). [22] RUSAMOV G.A., Abslutely rbust ntrl systems, Autm. Cntrl Cmp. Si., 203, 47(5),