THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 13, Number 3/2012, pp

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1 THE PUBISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume 3 Number 3/22 pp HIGH-ACCURACY AGORITHMS TO THE SOUTION OF THE OPTIMA OUTPUT FEEDBACK PROBEM FOR THE INEAR SYSTEMS * FA AIEV NI VEIEVA YS GASIMOV NA SAFAROVA F AGAMAIEVA Baku State Unversty Insttute of Appled Mathematcs Z Khallov 23 AZ48 Baku Azerbajan E-mal: f_alev@yahoocom On the bass of evne-athans algorthm the hgh accuracy algorthm for the constructon of the output optmal lnear regulator s proposed Intal approxmaton s chosen by applyng the penalty functon method that allows developng algorthms whch produce hgh accuracy soluton usng symbolc calculatons Results are extended for the dscrete case and llustrated on the examples Key words: output feedback numercal methods algebrac yapunov euaton symbolc calculatons INTRODUCTION One of the mportant branches of the optmal control theory s a constructon of the optmal regulator over all phase coordnates For the soluton of ths problem there exst varous calculaton algorthms basng on the fndng of the non-negatvely defned soluton of the matrx algebrac Rccat euatons (MARE) [ 2] Another mportant branch of these nvestgatons s a constructon of the optmal output regulator [3-9] To solve ths problem n [8] the convex analyss method n [5] Newton method n [3 4] penalty functon method had been appled In [3 4 8] the algorthms are proposed whch don t reure choce of the ntal condtons But they don t allow obtanng a soluton wth enough accuracy However there exsts a class of problems reurng the constructon of the regulator over a part of the phase coordnates Soluton of these problems s reduced to the soluton of two essentally nonlnear euatons [5] Recently n connecton wth developng of the Symbolc Toolbox of the package Matlab allowng to carry out calculaton wth hgh accuracy the algorthms are developed for the soluton of MARE both n statonary [2] and perodc [] cases The smlar stuaton appears n the soluton of the optmal output regulator problem where Newton method s applcable In the present paper on the base of gven n [5] method a calculaton algorthm s proposed Ths algorthm allows one to solve the consdered problem wth hgh accuracy usng Symbolc Toolbox of package Matlab The ntal condton s chosen usng the penalty functon method [3] An essental dfference of accuracy of the soluton of the problem obtaned by the prevous algorthms whch use calculatons by usual procedures of Matlab s llustrated on the examples 2 STATEMENT OF THE PROBEM et the movement of object be descrbed by the followng system of the lnear dfferental euatons wth constant coeffcents x ( t) = Fx( t) + Gu( t) x() = x () ( t) y = Cx (2) * The work s supported by Azerbajan Scence Development Foundaton Grant N EIF-2-(3-82)/25/-M-29

2 28 FA Alev NI Veleva YS Gasmov NA Safarova F Agamaleva 2 It needs to mnmze the functonal wth the regulaton law J =< ( xqx + uru )dt> (3) ( t) Ky( t) u = (4) under the condton that the closed-loop system ( 4) be asymptotcally stable e the condton Re( λ ( F + GKC)) < has to be satsfed Here K s sought ( m l) dmensonal C-gven ( l n) dmensonal constant matrces x s a state vector of the system u -vector of controllng nfluences y -output vector (measurement) x s a random vector of the ntal condtons wth < x >= X =< x x > covarance matrx < > s a sgn of mathematcal expectaton FG Q= Q ; R= R > constant matrces of the correspondng dmensons a prme herenafter means operaton of transpose It s known [5] that the soluton of the problem ( 4) may be reduced to fndng K = R G SUC ( CUC ) (5) where S = S > and U = U > are the solutons of the followng nonlnear matrx algebrac euatons ( F + GKC) S + S( F + GKC) + Q + C K RKC = (6) ( F + GKC) U + U ( F + GKC) + X = (7) The sought matrx K may be determned by solvng the euatons (5 7) Note that usng the results of [] one can carry out the senstvty analyss of the euatons (6 7) It wll allow one the use the problem ( 4) for the soluton of some appled problems 3 MAIN RESUTS The soluton of the stated above problem may be found by the followng teratve scheme [5] Algorthm The matrces F G Q = Q > R = R > C are gven 2 Choose ntal approxmaton K that provdes asymptotcal stablty of the matrx ( F + GK nc) Suppose ( n ) teratons have been done 3 Solvng algebrac yapunov euatons S ( F + GK n C) + ( F + K nc) Sn + Q + C K n RK nc U F GK C) + ( F + GK C) U + X n = n ( + n n n = fnd S n > U n > Ths step may be solved by the help of the procedure alyapm The algorthm and realzaton of ths procedure s gven below (see algorthm 3) 4 Check up the crtera Sp ( S n ) Sp( S n ) where Sp ( ) s a trace of the matrx If t s satsfed the teraton s stopped otherwse: 5 Calculate K = R G S U C ( CU C ) n n n n take K n = K n and go to step 3 The most dffcult procedure here s the fndng the ntal approxmaton For the soluton of ths problem we offer below the method that uses penalty functon [3] It stmulated by the fact that the other methods [4 8] do not admt the use of Symbolc calculatons technue et the controllng nfluence be searched as

3 3 Hgh-accuracy algorthms to the soluton of the optmal output feedback problem for the lnear systems 29 u = x (8) e let s consder an optmal stablzaton problem over all phase coordnates It means that the optmal regulator s sought as a functon of all phase coordnates It s known [ 2] that the soluton of the problem () (3) (8) ndeed s = R G S (9) where S = S > s a soluton of the matrx algebrac Rccat euaton (ARE) F S + SF SGR G S + Q = () It s not dffcult to prove that the problem () (3) (8) may be reduced to the problem ( 4) wth the help of Sngular Value Decomposton (SVD) of the matrx C The procedure MATAB [ V D U ] = svd( C) produces a dagonal matrx D of the same dmenson as C that has a form D = [ σ e ] wth nonnegatve dagonal elementsσ e n decreasng order and untary matrces V and U such that [3] C = VDU V V = E U U = E () where E -unt matrx of the correspondng dmenson After some smple transformatons the problem () (3) (8) s reduced to the soluton of the followng output optmal control problem z = Fz + Gu J = ( z ( t) Qz( t) + u ( t) Ru( t))dt mn where z Ux = z F= UFU G= UG Q= UQU h e V =σ y z = z = h u = Kσ e V y z2 The specfc character of ths problem s that here only part of the coordnates s measured e z = z( z ) et us represent the matrx n the form = [ 2] where s m l dmensonal 2 m ( n l) dmensonal matrces If to the problem () (3) (8) to add the addtonal condton then 2 = (2) x u = x = [ ] = x x2 Thus n ths case K = For fulfllment the condton (2) to the functonal (3) we add the term [ 2 ] [ 2 ] wth scalar weghtα > Then replacng the functonal (3) by ()( [ 2][ 2]) ()d (3) I = x t Q+ R+α x t t one may easly prove that 2 as α and the functonal (3) reaches ts mnmum on u = x e J ( u) I( u) It s known [ 2] that the value of the functonal (3) on the trajectory x = ( F + G) x( t) s calculated as J = Sp( S( α ) X )

4 2 FA Alev NI Veleva YS Gasmov NA Safarova F Agamaleva 4 where Sp ( ) means a trace of the matrx S( α ) s a soluton of the matrx algebrac yapunov euaton (MAE) ( F + G) S( α ) + S( α )( F + G) = ( Q+ R+α [ ] [ ]) (4) 2 2 It s dffcult to express the solutons of the euaton (4) through the elements of the matrx For ths purpose the adjont gradent method s used Intal approxmaton of the matrx s calculated wth the use of the stablzng soluton of the euaton () that provdes stablty of the closed-loop system ( 4) The soluton of (4) by α we fnd n the form = [ ] K = Thus the followng algorthm s proposed to the soluton of the problem ( 4) Algorthm 2 Gven matrces F G Q C R fnd the stablzng soluton S from MARE () 2 Calculate ntal from (9) Suppose that the ( ) teratons have been done 3 Take enough large number α and solve the euaton ( 2 2 ) ( F + G ) S + S( F + G ) = Q + ( ) R +α [ ] [ ] ( U F + G ) + ( F + G ) U = X 4 Calculate Sp( S( α )) ( F + G) 5 Construct the vector dsp( S) = d = 23 pˆ pˆ p p ' = Sp 2 SU + U ( Q + R +α[ 2] [ 2]) kj kj kj = = 6 Construct the matrx ˆ 2 = m + 7 The next s calculated by the relaton + pˆ + + = pˆ + = γ ˆ 2 22 m2 + β ( where γ β are calculated by the "Golden secton" method 8 Takng small real number ε check up the condton n 2n mn = 2 pˆ pˆ + pˆ + ) m n = 2 p+ ˆ 2 pˆ < ε where s a matrx norm If the condton s satsfed the procedure stops otherwse choose another ˆp take 2 pˆ + 2 as and go to step 3 4 HIGH ACCURACY AGORITHM FOR THE SOUTION OF THE AGEBRAIC YAPUNOV EQUATIONS The euatons (6) (7) are the algebrac yapunov euatons on the set K [4] As one can see both proposed above algorthms reure solvng of these euatons For ther soluton there exst varous computng algorthms as well as method of nfnte seres [2] Schur s method matrx sgn-functon

5 5 Hgh-accuracy algorthms to the soluton of the optmal output feedback problem for the lnear systems 2 method From these the matrx sgn functon method s the most comprehensble for use of Symbolc Toolbox procedures of package Matlab Therefore we shall use further ths method for the soluton of the yapunov euaton Here we descrbe ths method et s the matrx algebrac yapunov euaton be gven AX + XA = C (5) and t s reured to fnd a symmetrc matrx X on the set of Hurwtz matrces A (any suare matrx) and C (any negatvely defned symmetrc matrx) It s known that f A s Hurwtz matrx then the soluton of the euaton (5) exsts Here we gve the algorthm from [2] for the soluton the euaton (5) wth the help of matrx sgn functon Matrx sgn functon sgn A of the matrx A s determned as follows [4] sgna= lm A A ( A A + = + ) A = A 2 Here s the algorthm from [2] to the soluton of the euaton (5) Algorthm 3 Take A = A'; C = C Suppose n teratons have been done 2 Calculate A+ =α A +β A where α = + det A β = α n -dmenson of the matrx A ' + ( ) C =α C +β A C A ; n 3 If A+ + E <ε then X = C+ and the procedure s stopped Otherwse take A = A + C = C+ 2 and go to step 2 Here ε s a gven constant Ths algorthm s easly realzed n package Matlab by use of procedures Symbolc Toolbox Note that there exst varous algorthms to calculate the nverse matrx [5] For the purpose and to calculate the determnant of the matrx we use the procedures nvm and detm of the Matlab package Symbolc Toolbox The procedure alyapm s developed on the base of offered algorthm that s also realzable by Symbolc Toolbox Matlab Thus the followng algorthm for the soluton of the problem ( 4) may be offered Algorthm 4 The matrces F G Q C R are gven 2 Use algorthms 2 and 3 for choce the ntal approach K 3 By chosen K usng the algorthms and 3 calculate the sought soluton by N All calculatons are done by the Symbolc Toolbox Matlab The example below llustrates ths result K where K N K Example The ntal data of ths example are taken from [8 6] In ths case the matrces F G N D ndeed are = F = ; G 4 47 ; N = ; D = Appearng n (3) matrces Q and R are taken as Then the observable vector s ' Q = N N R = D D ' N

6 22 FA Alev NI Veleva YS Gasmov NA Safarova F Agamaleva 6 y = x = Cx Solvng ths problem by Matlab usng the procedure Symbolc Toolbox by the help of Algorthm 2 the coeffcent of the optmal regulator and mnmal value of the correspondng functonal are obtaned as K = = J Usng the Algorthm the followng values are obtaned K A = = J A Gradent of the functonal for the gven case s calculated as J = 354e-29 KA The euatons (6) (7) whch contan K are solved wth accuracy of order 533e-29 n [6] In [8] the coeffcent of the optmal regulator s calculated as K G = Correspondng mnmum value of the functonal s J = and the gradent s G J K = 372 Such essental devatons of the mnmum values of the functonal and gradents demonstrates effcency [6] of the offered here algorthms Example 2 et s take a F = ; ; = ; = ; = [ ] Q = R G C a a Results of the soluton of ths problem for varous values of a solved by usual Matlab and Symbolc Toolbox are gven n Table Table Comparson table a λ - egenvalue of the closed-loop system K coeffcent of the optmal regulator Usual calculatons Symbolc calculatons Usual calculatons Symbolc calculatons 48±92 48± e+4; 598e+4 e+4; 58e e-6; e ; e e-8; 346e e Here --- means that the condton of the asymptotcal stablty s not satsfed

7 7 Hgh-accuracy algorthms to the soluton of the optmal output feedback problem for the lnear systems 23 Soluton of the dscrete analogue of the problem ()-(4) s reduced to the soluton of the dscrete algebrac yapunov euaton From exstng methods for the soluton of the dscrete algebrac yapunov euaton the method of nfnte seres s more easly realzable n the package Matlab by use of procedures Symbolc Toolbox et s descrbe ths method [2] Consder dscrete algebrac yapunov euaton S Ψ SΨ = R (6) It s reured to fnd a symmetrc matrx S on the set of matrces Ψ (any suare matrx) and R (any symmetrc matrx) Here the followng statement s vald: f egenvalues of the matrx Ψ are nsde of unt crcle then 2 2 Y = R + ( Ψ ) RΨ + ( Ψ ) RΨ + (7) Seres (7) converges to the soluton of the euaton (6) Partal sums of ths seres are calculated by the followng teratve formulas 2k k + k k = ( Ψ Y k ) RΨ = 2k 2k Y = R Y = Y + ( Ψ ) Y Ψ All steps of ths scheme may be easly solved wth the help of symbolc arthmetc Another method s to reduce the euaton (6) wth the help of Cayley transformaton A = ( Ψ E)( E + Ψ) ; C = 2( Ψ + E) R( Ψ + E) to the contnuous algebrac yapunov euaton (4) that then may be solved by the sgn-functon method et us llustrate ths by the followng example Example 3 In [9] the matrces Ψ Γ C Q R are taken as 2 Ψ = ; Γ = 3 ; Q = ; R = ; C = In ths case solvng the dscrete problem t s obtaned F = The mnmal value of the functonal s J = Solvng ths problem n package Matlab by use of procedures Symbolc Toolbox we obtan the optmalty condton as J K = e - 24 Correspondng euatons by the obtaned K are solved wth accuracy 7423е-23 In [9] for the same problem s obtaned 9 37 K = J = Comparson of these two results demonstrates that the offered here algorthms mprove the results of the work [9] 5 CONCUSION In the work hgh accuracy algorthms are offered for the soluton of the optmal regulator output problem both n the statonary and perodc cases that allows one to obtan solutons wth necessary accuracy

8 24 FA Alev NI Veleva YS Gasmov NA Safarova F Agamaleva 8 REFERENCES ARIN VB Hgh-accuracy algorthms for soluton of dscrete perodc Rccat euatons Appl Comput Math 6 pp VARGA A On computng hgh accuracy solutons of a class of Rccat euatons Control Theory and Advanced Technology 4 Part 5 pp AIEV FA VEIEVA NI MARDANOV MD An algorthm for solvng the synthess problem for the optmal stablzaton system wth respect to output varable Engneerng Smulat 3 pp ARIN VB Stablzaton of System by Statc Output Feedback Appl Comput Math 2 pp EVINE WS ATHANS M On the determnaton of the optmal constant output feedback gans for lnear multvarable systems IEEE Trans Autom Control AC-5 pp AIEV FA ARIN VB Stablzaton problems for the output feedback systems (survey) Internatonal Appled Mechancs 47 3 pp TOSCANO R YONNET P Stablzaton of systems by statc output feedback va heurstc Kalman algorthm Appl Comput Math 5 2 pp PERES PD GEROME JG An alternate numercal soluton to the lnear uadratc problem IEEE Trans Autom Control 39 pp GEROME JC YAMAKAMI A ARMENTAMO VA Structural constrans controllers for dscrete tme lnear systems Jour of Optmzaton Theory and Applcatons 6 pp KONSTANTINOV MM PETKOV PH POPCHEV IP ANGEOVA VA Senstvty of the matrx euaton k A * + p σ A X A = σ = ± Appl Comput Math 3 pp = KWAKERNAAK H SIVAN P near optmal control systems New York Wley AIEV FA ARIN VB Optmzaton of lnear control systems n Analytcal methods and computatonal algorthms Amsterdam Gordon and Breach FORSYTHE GE MACOMN MA MOER CB Computer methods for mathematcal computatons Prentce-hall Inc Englewood Clffs NJ AIEV FA ARIN VB On the Objectvty of Scentfc Ctaton TWMS J Pure Appl Math 2 pp SADEGHI A IZANI A Md ISMAI AHMAD A A specal Newton teraton computng nverse matrx roots Appl Comput Math 3 pp AIEV FA GASIMOV YS VEIEVA NI Comments on An alternate numercal soluton to the lnear uadratc problem by Peres PD and Geromel J G Appl Comput Math 9 pp Receved December 7 2