Localization of Eigenvalues in Small Specified Regions of Complex Plane by State Feedback Matrix

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1 Jounal of Sciences, Islamic Republic of Ian (): - () Univesity of Tehan, ISSN - Localization of Eigenvalues in Small Specified Regions of Complex Plane by State Feedback Matix H Ahsani Tehani * epatment of Applied Mathematics, Faculty of Mathematical Sciences, Shahood Univesity, Shahood, Islamic Republic of Ian Received: Apil / Revised: Febuay / Accepted: May Abstact This pape is concened with the poblem of designing discete-time contol systems with closed-loop eigenvalues in a pescibed egion of stability Fist, we obtain a state feedback matix which assigns all the eigenvalues to zeo, and then by elementay similaity opeations we find a state feedback which assigns the eigenvalues inside a cicle with cente and adius This new algoithm can also be used fo the placement of closed-loop eigenvalues in a specified disc in z-plane fo discete-time linea systems Some illustative examples ae pesented to show the advantages of this new technique Keywods: iscete-time systems; State feedback matix; Localization of eigenvalues; isc; Closed-loop system Intoduction In many applications, mee stability of the contolled obect is not enough, and it is equied that the poles of the closed-loop system should lie in a cetain esticted egion of stability Seveal design methods have been epoted which utilize the LQ technique to achieve the desied pole allocation The closed-loop poles can be placed exactly as specified Kawasaki and Shimemua () have deived a method of allocating all the closed-loop poles in a pefeable egion athe than exact location Howeve, the continuous-time esults cannot be diectly extended to the discete-time case Fuinaka and Katayama () descibe a method fo designing discete-time optimal contol systems with closed-loop poles in a pescibed egion, Yuan and Achenie and Jiang () addessed the poblem of linea quadatic egulato (LQR) synthesis with egional closed-loop pole constaints Benne and Castillo and Quintana-Oti () pesented the method fo patial stabilization of lage-scale discete-time linea contol systems Gammont and Lagillie () employed an appoach to localize matix eigenvalues in the sense that they build a sufficiently small neighbohood fo each eigenvalue (o fo a cluste) Recently, Ayatollahi () obtained a method fo Maximal and minimal eigenvalue assignment fo discete-time peiodic systems by state feedback Zhou, Cai and uan () obtained a method fo Stabilisation of time-vaying linea systems via Lyapunov diffeential equations Fanke () pesented the method fo Eigenvalue assignment by static output feedback on a new solvability condition and the computation of low gain feedback matices A well-known desied egion fo discete systems is * Coesponding autho: Tel: +; Fax: +; hasani@shahoodutaci

2 Vol No Sping H Ahsani Tehani J Sci I R Ian a disc ( c, ) centeed at (c,) with adius, in which c, as shown in Fig In this pape, the aim is to pesent a method fo localization of eigenvalues in small specified egions of complex plane by state feedback contol fo discete-time linea contol systems Mateials and Methods Poblem Statement The poblem of localization of eigenvalues in a small specified egion has been the subect of many investigatos in the last decades [,] Conside a contollable linea time-invaiant system defined by the state equation t Axt But x () x o its discete-time vesion t Axt But () n m whee xt, ut and the matices A nn and B ae eal constant matices of dimensions and n m espectively, with ank( B) m The aim of eigenvalue assignment in a specified egion is to design a state feedback contolle, K, poducing a closed-loop system with a satisfactoy esponse by shifting contollable poles fom undesiable to desiable locations Kabassi and Bell [,], have intoduced an algoithm fo obtaining an explicit paametic contolle matix K by pefoming similaity opeations on the contollable pai B, A In fact, K is chosen such that the closed-loop system eigenvalues A BK () lie in the self conugate eigenvalue spectum Recently, Kabassi and,, n, Tehani [] extended the pevious esults as to obtain an explicit fomula involving nonlinea paametes in the contol law The stabilization poblem consists in mn finding a feedback matix K such that the input u K x, k,,,, yields a stable closed k loop system x k ( A BK) xk xk, k,,, k () In case the spectum (o set of eigenvalues) of the closed-loop matix, denoted by ( ), is contained in the open unit disk we say that is (Schu) stable o convegent (in othe wods, fo all ( ) i ) The stabilization poblem aises in contol poblems such as, the computation of an initial appoximate solution in Newton s method fo solving discete-time algebaic Riccati equations, simple synthesis methods to design contolles, and many moe [,, ] The stabilization poblem can in pincipal be solved as a pole assignment poblem Pole assignment methods compute a feedback matix such that the closed-loop matix of system () has a pespecified spectum In this pape, we pesent an efficient appoach fo localization of eigenvalues in small specified egions fo linea discete-time systems Ou assignment pocedue is composed of two stages We fist obtain a pimay state feedback matix F which assigns all the eigenvalues p of closed-loop system to zeo, then poduce a state feedback matix K which assigns all the closed-loop system eigenvalues in a small specified disc o discs Synthesis Conside the state tansfomation x t T x t () whee T can be obtained by elementay similaity opeations as descibed in [] In this way, A T AT and B T B ae in a compact canonical fom known as vecto companion fom: G A I n m nm m i B B nmm () Hee G is an m n matix and B is an m m uppe tiangula matix Note that if the Konecke invaiants of the pai B, A ae egula, then A and B ae always in the above fom [] In the case of iegula Konecke invaiants, some ows of in I n m A ae displaced [] It may also be concluded that if the vecto companion fom of A obtained fom similaity opeations has the above stuctue, then the Konecke invaiants associated with the pai B, A ae egula [] The state feedback matix which assigns all the

3 Localization of Eigenvalues in Small Specified Regions of eigenvalues to zeo, fo the tansfomed pai B, A, is then chosen as u B G x Fx () Which esults in the pimay state feedback matix fo the pai B, A defined as FT Fp () The tansfomed closed-loop matix A BF with zeo eigenvalues assumes a compact Jodan fom Poof: The pimay compact Jodan fom in the case of egula Konecke invaiants is in the fom m n I nm nm m The sum of with has the fom: H () () m n I nm nm m () mn I nm nm m k () Theoem : Let be a block diagonal matix in the fom fom whee each k (), (,,, k) is eithe of the () ( to designate the complex conugate eigenvalues ) i o in case of eal eigenvalues d ] () [ If such block diagonal matix with self conugate eigenvalue spectum is added to the tansfomed closedloop matix,, then the eigenvalues of the esulting matix is the eigenvalues in the spectum I whee I l l k () I s, s,,, is the unit matix of size n m is even In case n m is odd only one in case Is takes the fom of a unit matix of size one By expanding det( H I ) along the fist ow it is obvious that the eigenvalues of H ae the same as the eigenvalues of Fo the case of iegula Konecke invaiants [] only some of the unit columns of ae displaced, In m since the unit elements ae always below the main diagonal, the poof applies in the same manne Then H Results can be obtained fom H by pefoming elementay similaity opeations

4 Vol No Sping H Ahsani Tehani J Sci I R Ian Column ( ) Column ( i) () followed by Row ( i) Row ( ) () fo = n, n-,, m, i=-m Hence, the matix H thus obtained will be in pimay vecto companion fom such that: H H I n m nm m H is an n () whee m matix Because of similaity opeation, the eigenvalues of the matix H ae the same as the eigenvalues of H and that of Now the feedback matix of the pai ( A, B) is defined by: K F B H B ( G H () Theoem : The state feedback matix K assigns the eigenvalues of closed-loop matix A BK inside a cicle with cente c and adius if in case cicle intesects axis of abscissas we suppose, be in the fom: sqt ( Im( c) ) * andom(,) Re( c) () ( sqt( l ) Im( c) ) * andom(,) ) () and in case cicle does not intesect axis of abscissas we suppose * andom(,) Re( c) () sqt( l ) andom(,) Im( c) () whee we take l Re(c) if * Re( c) and othewise we take l Re(c) and fo assigning eal valued eigenvalues in the cicle c and adius we choose d sqt( Im( c) ) * andom(,) Re( c) () Poof: The eigenvalues of matix defined above fall inside a cicle with cente c and adius Let () A BK I nm () G BB I nm H G nm, m G B B H B n Clealy, since m, m nm, m B ( G I nm H H ) nm, m H is simila to the matix n H and the eigenvalues of matix H ae the same as that of matix and elementay similaity opeations do not change the eigenvalues, then the eigenvalues of closedloop matix A BK fall inside a cicle with cente c and adius Remak: Since K assigns the eigenvalues of the closed-loop matix A BK inside a cicle with cente c and adius, it is obvious that the state feedback contolle matix, K KT B ( G H ) T also assigns the eigenvalues of the closed-loop matix A BK inside a cicle with cente c and adius too Note that fo assigning the eigenvalues of the closed-,,, we loop matix in spectum suppose,, n (), An algoithm fo assignment of eigenvalues in a disc ( c, ) In this section we fist give an algoithm fo finding a state feedback matix which assigns zeo eigenvalues to the closed-loop system Then we detemine a gain matix which assigns the closed-loop eigenvalues in a cicle with cente c and adius ( A, B Input: The contollable pai ), the pimay state feedback F, B p and T which ae calculated by the algoithm poposed by Kabassi and Bell [,], the cente c and adius of the taget cicle Step Constuct the block diagonal matix in

5 Localization of Eigenvalues in Small Specified Regions of the fom (), in which fo assigning complex valued eigenvalues in the cicle with cente c and adius if cicle intesects axis of abscissas we suppose sqt ( ( sqt( Im( c) ) * andom(,) Re( c) l othewise we choose * andom(,) Re( c) sqt( ( ) Im( c) ) * andom(,) Re( c) ) ) andom(,) Im( c) whee we take l Re(c) if * Re( c) and othewise we take l Re(c) and fo assigning eal valued eigenvalues in the cicle c and adius we choose d sqt( Step Set Im( c) ) * andom(,) Re( c) H Step Tansfom H to pimay vecto companion fom H as in () using elementay similaity opeations as specified in coollay of theoem step Now compute K Fp B H T the equied state feedback matix Illustative Examples Conside a discete-time system given by x t Ax t Bu t Whee A and B ae andomly geneated with n and m A B The open loop eigenvalues ae { i, i, i,,,, } which ae widely spead in the complex plane In ode to locate them in small discs inside the unit cicle, we employ the above algoithm step by step Fist, the pimay state feedback matix which locates all the eigenvalues of the closed-loop system to the oigin of the complex plane is found to be: F p

6 Vol No Sping H Ahsani Tehani J Sci I R Ian a) K b) K c) K d) K Now we conside the following diffeent cases: a) It is desied to locate the closed-loop eigenvalues inside the unit cicle centeed at oigin By using the algoithm, the state feedback matix obtained is (above): It can be veified that the closed-loop eigenvalues ae { i, i, i,,,, }, clealy all ae inside the unit cicle b) In this case, we find the state feedback matix which assigns the closed-loop eigenvalues in the disc (, ) By using the algoithm, the state feedback matix obtained is (above): The closed-loop eigenvalues ae { i, i, i, i,,}, all of which ae inside the disc (, ) c) In this case, we find the state feedback matix which assigns the closed-loop eigenvalues in the disc (, ) By using the algoithm, the state feedback matix obtained is: The closed-loop eigenvalues ae in the set { i, i,, i,,,, } all of which ae inside the disc (, ) Clealy, localization of eigenvalues in small specified egions of complex plane by state feedback contol is achieved d) In this case, we find the state feedback matix which assigns the closed-loop eigenvalues in the disc ( i,) By using the algoithm, the state feedback matix obtained is (above):

7 Localization of Eigenvalues in Small Specified Regions of e) K The closed-loop eigenvalues ae in the set { i, i, i, i,, }, all of which ae inside the disc ( i,) e) In this case, we find the state feedback matix which assigns the closed-loop eigenvalues in the discs: ( i,), ( i,), ( i,), (,) By using the algoithm, the state feedback matix obtained is (above): The closed-loop eigenvalues ae { i, i } which ae inside the discs ( i,), ( i,), and also in the sets i,, and, which ae inside the discs ( i,) and (, ) espectively iscussion A simple algoithm was given fo localization of eigenvalues in small specified egions of complex plane by state feedback contol This method was achieved by implementing popeties of vecto companion foms The meit of this appoach is that it can be achieved by elementay similaity opeations which ae significantly simple to ealize computationally than the existing methods In the existing liteatue only location of eigenvalues in discs centeed on the eal axis is pesented wheeas location of eigenvalues in any abitay specified disc o discs inside the unit cicle can be achieved by the pesented algoithm easily The numeical examples which wee tested showed that the algoithm woks pefectly and the numbe of aithmetic opeations of the poposed method is less than the method of assignment with application of the Geschgoin Theoem [], although the system matices and the location of discs wee chosen andomly It is claimed that the tansfomations Figue Specified discs ( c, )

8 Vol No Sping H Ahsani Tehani J Sci I R Ian obtained by similaity opeations educe accuacy of the computations [], howeve, othe methods such as LQR methods [,,] and the method pesented in [,] ae moe complicated Refeences Ayatollahi M Maximal and minimal eigenvalue assignment fo discete-time peiodic systems by state feedback Optim Lett, : - () Benne P, Castillo M, and Quintana-oti ES Patial stabilization of lage-scale discete-time linea contol systems in: Technical Repot, Univesity of Bemen, Gemany () agan V, and Halanay A Stabilization of linea systems in: Basel, Switzeland () Fanke M Eigenvalue assignment by static output feedback on a new solvability condition and the computation of low gain feedback matices Int J Contol, : - () Fuinaka T, and Katayama T iscete-time optimal egulato with closed-loop poles in a pescibed egion Int J Contol, : - () Gammont L, and Lagillie A Kylov method evisited with an application to the localization of eigenvalues Nume Func Anal Opt, : - () Kabassi SM, and Bell J Paametic time-optimal contol of linea discete-time systems by state feedback- Pat : Regula Konecke invaiants Int J Contol, : - () Kabassi SM, and Bell J Paametic time-optimal contol of linea discete-time systems by state feedback- Pat : Iegula Konecke invaiants Int J Contol, : - () Kabassi SM, and Tehani HA Paameteizations of the state feedback contolles fo linea multivaiable systems Comput Math Appl, : - () Kawasaki N, and shimemua E etemining quadatic weighting matices to locate poles in a specified egion Automatica, : - () Mehmann V The autonomous linea quadatic contol poblem theoy and numeical solution, In: Lectue notes in contol and infomation sciences, Spinge-velag, Heidelbeg () Sima V Algoithms fo linea quadatic optimization Pue and applied mathematics, New Yok () Tehani HA Assignment of Eigenvalues in a isc (c,) of Complex Plane with Application of the Geschgoin Theoem Wold Appl Sci J, (): () Yuan L, Achenie LEK, and Jiang W Linea quadatic optimal output feedback contol fo systems with poles in a specified egion Int J Contol, : - () Zhou B, Cai G, and uan G Stabilisation of time-vaying linea systems via Lyapunov diffeential equations Int J Contol, : - ()