Internatonal Book Seres "Informaton Scence and Comutng" 89 Industral Control and Montorng APPLICATION OF GENETIC ALGORITHMS TO VECTOR OPTIMIZATION OF THE AUTOMATIC CONTROL SYSTEMS Valery Severn Abstract: Methods for calculaton of qualty ndexes for automatc control systems are resented. For the otmzaton of qualty ndexes defned only n a stablty doman a vector objectve functon of vared arameters of the system s roosed. The stewse rncle of successve satsfacton of constrants for the assage nto the defnton doman of qualty ndexes s consdered as well as a ratonal mechansm of ts realzaton n the form of the rorty otmzaton of the vector objectve functon. For the otmzaton of the vector objectve functons genetc algorthms as vector otmzaton methods are resented. Ther alcaton allows one to steer the otmzaton rocess from any ntal ont of the sace of vared arameters nto the stablty doman of the system and to fnd the otmum of the qualty ndexes n ths doman. The effcency of the roosed alcaton of vector genetc algorthms for the qualty ndexes otmzaton s confrmed by comutatonal exerments. Keywords: genetc algorthms vector otmzaton methods automatc control systems qualty ndexes. ACM Classfcaton Keywords: G..6 Otmzaton - Nonlnear rogrammng Introducton For the synthess of automatc control systems (ACS) qualty of ther functonng can be resented by dfferent crtera [Besekersky Poov 004]. There are many dffcultes even at the synthess of the lnear control systems and the results of synthess substantally deend on the aled crtera [Poljak Scherbako 005]. Yet the task of synthess of the nonlnear systems s even more dffcult. Qualty crtera for both lnear and nonlnear ACS can be reflected by the drect qualty ndexes (DQI) and mroved ntegral quadratc estmatons (IQE) [Severn Nkulna 004] [Severn 004] [Severn 005]. The otmzaton tasks of these crtera have an dentcal feature ther objectve functons doman s lmted by stablty condtons [Severn 008]. Therefore the standard otmzaton methods can not be effectvely used for the otmzaton of ACS qualty ndexes. For the synthess of the lnear control systems by means of DQI and mroved IQE otmzaton the vector objectve functons are offered and for ther otmzaton the drect methods of unconstraned mnmzaton are modfed [Severn 005]. The methods otmzaton laboratory OPTLAB s develoed n MATLAB system [Severn 009]. However the offered vector otmzaton methods do not allow to fnd global extremes. The decsons search of otmzaton tasks n very large and dffcult domans s executed by genetc algorthms whch are evolutonary comutatons varety and behave to heurstc search methods [Voronovsky Makhotlo Petrashev Sergeev 997] [Setlak 004]. Genetc algorthms are used for otmzaton tasks decson based on the rncles and mechansms remndng bologcal evoluton [Panchenko 007] [Wese 008].
90 No:3 Intellgent Informaton and Engneerng Systems The urose of ths aer conssts of conceton develoment of synthess for lnear and nonlnear ACS on the bass of DQI mroved IQE and modfcatons of genetc algorthms for vector otmzaton. The methods of DQI and mroved IQE calculaton are consdered for lnear and nonlnear systems. Statements of systems otmzaton tasks are resented wth usng vector objectve functons. Modfcaton of genetc algorthms s offered for vector otmzaton of ACS qualty ndexes. Qualty Indexes Calculaton of Lnear Automatc Control Systems Let lnear model of ACS deendng on a vector of vared arameters x R n state sace looks lke: t A( + B( u( y ( C( () where t ) s a state vector wth an ntal condton X 0 0 ; u ( s entrance nfluence y ( s control outut co-ordnate; A ( B ( C ( are matrces of the control system arameters. For the stable watchng system at standard nut ste sgnal u ( ( the matrx of outut C ( s set so that the condton of outut coordnate scalng was executed: y (. At the fxed arameters vector value x wll buld transent rocesses n model () on the quantum of tme [ 0 T f ]. For ths urose at L ntegraton stes of constant length h T f L wth numbers k L wll enter denotatons: We wll desgnate matrx exonent and t s ntegral: t k kh X k ( tk ) yk ( C( X k (. () h A( τ A( h ϕ ( e Φ( e dτ g( Φ ( B(. (3) 0 Then at an nut sgnal u ( ( transent rocess n ACS wth model () t s ossble to buld on the recurrent formulas of matrx method [Severn 008]: X ( ϕ( X k ( g( k L. (4) k + For a devaton z ( y( y( there are ts values and ther ncrements: u zk ( yk ( y( k 0 L (5) zk ( zk ( urk ( zk ( zk ( k L. (6) lk ( If the followng condton meanng that both successve ncrements are of the same sgn s met then wth usng of quadratc nterolaton the extreme value s calculated e ( : d uk ulk ( urk ( > 0 (7) ( [ ulk ( urk ( ] suk ( ulk ( + urk ( ruk ( duk ( suk ( (8) e ( zk ( duk ( ruk ( (9) where ne ( n e ( s extreme s number on segment [ 0 T f ]. By extreme s values of transent rocess the drect qualty ndexes are calculated: overshoot σ ( vbratons scoe ζ ( vbratons damng ndex λ (. Let ( v ) + max{ v0} s a cuttng functon of otonal varable v. For watchng system wth y ( drect ndexes are determned on formulas:
Internatonal Book Seres "Informaton Scence and Comutng" 9 0 σ( [max e ( ] n ( 0 e + ne ( > 0 (0) 0 ζ( max e ( e ne ( 0 ( n ( ) > e x 0 λ( max{ e ( e ne ( 0 ( } n ( ) > e x. For the stablzaton system wth y ( 0 a rocess n whch has even one extreme e ( () σ( max e ( () and at a calculaton ζ ( and λ ( n formulas () not taken nto account e (. For the calculaton of ACS control tme the entry tmes of devaton z ( t ) n the set segment [ δ z δ z ] of the steady-state value z ( 0 are determned by verfcaton of entrance condton: z k ( δ z zk ( < δ z. (3) At mlementaton of ths condton takng nto account denotatons (5) (8) auxlary values are calculated: u ( uk + δ zsgn zk ( zk ( v0 ( ruk ( h (4) s ( h r ( u ( s ( uk v0 ( + s ( v ( 0 v ( v0 ( s ( v ( > 0. (5) The moment of tme roer entrance of devaton functon z ( t ) n area of steady-state value s determned: Control tme t c ( and ts relatve value τ ( are calculated on formulas: t ( tk ( + v (. (6) t ( max t ( c τ ( x ) t ( T. (7) c f On formulas (3) (7) for calculaton of DQI σ ( ζ ( λ ( t c ( τ ( the algorthms are obtaned. For the synthess of watchng ACS n lace of few drect qualty ndexes t s ossble to use ther summarzng sngle ndex mroved IQE. On the model of knd () one can buld a transfer functon (TF) n 0 n m W ( β( α( α( α ( s β( β ( s. (8) For the watchng systems a method s offered for formng of mroved IQE I ( of error e ( : [ e( ] 0 l k 0 ( l k) I ( dt e( w z ( (9) k t m 0 where l s an order of estmaton l < n m ; w k are weghtng coeffcents: wk μ l k γ k s l k 0 l k l k k 0 l ; μ t e ts γ( ) γ ( s ) w k s. (0) k s w l k 0 Here t e and t s are control tmes of etalon and standard rocesses γ ( and w ( are standard and weghtng olynomals. On TF (8) Lalace reresentaton of error s formed E( δ( α( where δ ( x [ α( β( w( ] [Severn 005]. s. On the bass of ths reresentaton IQE calculaton algorthm s develoed
9 No:3 Intellgent Informaton and Engneerng Systems Otmzaton Tasks of Lnear Automatc Control Systems Takng nto account the hgh scoe values σ m ζ m λ m for DQI σ ( ζ ( λ ( and requrements of maxmal ACS resonse seed the system otmzaton task can be formulated as task of the constraned otmzaton whch requres mnmzaton of control tme at mlementaton of lmts on the other ndexes: mn τ( x σ ( σ ζ ( ζ m λ ( λ m. () m Usng the mroved ntegral estmaton the task of control system otmzaton conssts n mnmzaton IQE: mn I( x. () However statements of otmzaton control system tasks () and () take nto account nether rorty of drect ndexes nor lmtaton of ther defntonal doman and defntonal doman of ntegral estmaton. The analyss of automatc control system requrements allows to set the followng reference order of drect qualty ndexes: σ ( ζ ( λ ( τ (. The feature of these ndexes as rvate qualty crtera of the automatc control systems s the lmtaton of ther defntonal doman by stablty condtons. On Routh crteron for stablty of lnear ACS wth transfer functon (8) there are necessary and suffcent condtons: α ( > 0 0 n ; ρk ( > 0 k n (3) where ρ k ( are elements of the frst column of Routh table. The analyss of Routh crteron and research of roertes of functons ρ k ( justfy the stewse scheme of assage to the stablty doman: f some from elements ρ k ( s not ostve t s suggested to ncrease frst of them to the ostve value by the change of arameters values vector x and then to ncrease subsequent elements. To smlfy the scheme of assng to the stablty doman and to meet the condtons of drect qualty ndexes () the arameter sace R s dvded nto three doman sequences. The nequaltes (3) and () are satsfed on the followng domans of lmtatons: Ω { x α ( > 0 0 } Ωk { x ρk ( > 0 } k n (4) n Ω n { x σ( σm } Ω n + { x ζ( ζ m } Ω n + { x λ( λ m }. (5) On these m n + domans the derved ntersecton domans D k and domans of lmtatons levels H k are formed: D Ω ; Dk D k Ωk k m ; (6) \ + H 0 R \ D ; H k Dk Dk k m ; H m Dm. (7) The domans of levels dvde arameters sace nto the sequence of dsjont domans. The degree of volaton of the frst grou of nequaltes (3) s resented by enalty functon ( ) n x [ α ( 0 x P )]. (8) Stewse rncle of transferrng to the stablty doman and satsfacton of all lmtatons of drect qualty ndexes s based on the followng: from any ont x of arameters sace R t s necessary to ass consstently to the level doman wth greater ndex by mnmzng n the current level doman usng ts corresondng enalty functon. Takng nto account the amount of levels domans there wll be no more such stes of transton than the number of lmtatons m. For realzaton of stewse rncle of satsfacton of lmtatons n the task of ACS synthess wth otmzaton of drect qualty ndexes on the bass of levels domans a vector objectve functon s ntroduced +
Internatonal Book Seres "Informaton Scence and Comutng" 93 (0; P( ) x H 0; ( k; ρk+ ( ) x H k k n ; ( n ; σ( σ m ) x H n ; F ( ( n; ζ( ζ m ) x H ; (9) n ( n + ; λ( λ m ) x H n+ ; ( n + ; τ( ) x H n+. Denote the frst coordnate of ths functon as the functon of level F ( and the second coordnate as the functon of enalty F (. The vector objectve functon (9) can be calculated algorthmcally. Algorthm for calculaton of the vector objectve functon for drect qualty ndexes otmzaton. Inut arameters: x s a vector of varable arameters T f s the uer lmt of ntegraton nterval L s a number of stes of ntegraton σ m ζ m and λ m maxmum accetable values of DQI. Outut arameter: F s a value of vector objectve functon.. On model () calculate TF (8) wth the characterstc olynomal α ( α( of degree n.. If the necessary stablty condtons are volated calculate enalty functon (8) let F ( 0; P ) and go to. 3. Let k. 4. On Routh chart calculate ρ k + ρk+ (. 5. If ρ k + 0 let F ( k ; ρk+ ) and go to. 6. If k < n let k k + and go to 4. 7. On formulas () (7) by numercal ntegraton wth quadratc nterolaton calculate values of DQI ζ ζ( λ λ( tc tc ( τ τ(. 8. If σ > σm let F ( n ; σ σm ) and go to. 9. If ζ > ζm let F ( n; ζ ζ m ) and go to. 0. If λ > λm let F ( n + ; λ λ m ) and go to.. Let F ( n + ; τ).. Ext the algorthm. Lke functon (9) a vector objectve functon s bult for mnmzaton of the mroved IQE (9): (0; P( ) x H 0; F ( ( k ; ρk+ ( ) x H k k n ; (30) ( n ; I( ) x H n. The goal of control systems otmzaton usng vector objectve functons (9) and (30) can be resented as mnmzaton of the functon of enalty F ( wth the rorty condton of maxmzaton of functon of level F ( whch n turn can be resented as a sngle task of vector otmzaton: mn F( x. (3) Unlke the tasks of scalar otmzaton () and () the task of vector otmzaton (3) takes nto account the stablty condtons and order of reference of lmtatons. The rocess of otmal synthess of ACS s grounded by mnmzaton F ( wth rorty maxmzaton F ( as otmzaton of vector functons (9) and (30) on the bass of comarson of ther two arbtrary values U ( U ; U ) and V ( V ; V ) by the bnary oeratons: U < V 0 U V 0 U U U U > V < V > V < V U U U U V V V V U U U U < V U V > U U < V U V U > V U > V 0 U > V U V U U (3) U < V U V U V U V 0 U > V U V U < U. (33) These oeratons allowng to determne whch of the two values of vector objectve functon s «better» «worse» «not worse» or «not better» can be used n the numercal methods of unconstraned otmzaton.
94 No:3 Intellgent Informaton and Engneerng Systems Calculaton of Qualty Indexes of Nonlnear Control Systems For nonlnear models the state vector and control coordnate wll deend nonlnearly on the value of nut nfluence u u(. Unlke the lnear model of ACS n state sace () the nonlnear model can be resented as: u t f [ u u t] y ( u C( u. (34) For the stable watchng system at an nut ste sgnal u( us( wth magntude u s [ u mn ; umax ] the outut matrx C ( u ) scales an outut coordnate y ( u. At a fxed value of arameters vector x let s buld transent rocesses n model (34) on the quantum of tme [ 0 T f ] and calculate the Jacoban matrx of vector functon of equaton (34) by dfferentatng t on state vector coordnates: A ( f [ u u t] X ( u x 0 u 0 t 0. (35) Let s ntroduce notaton smlar to () and (3) but takng nto account system nonlnearty: h A( τ Φ( e dτ. (36) 0 Transent rocess n the control system on a model (34) t s ossble to buld on recurrent formulas for k L : X k ( X k ( + Φ( f [ X k ( u tk ]. (37) As a result of alcaton of formulas smlar to formulas (4) (7) but wth functons deendng both on x and u we can calculate the drect ndexes of qualty σ ( ζ ( λ ( t c ( τ (. Unlke lnear ACS for the nonlnear systems an ntegral estmaton (9) can be calculated only by numercal ntegraton of the nonlnear system of dfferental equatons (34) together wth dfferental equaton of estmaton: l k 0 ( l k) I ( u t w z ( u. (38) For the extended system Jacoban matrx (35) and ntegral of matrx exonent (36) are calculated. The mroved IQE I ( u T f ) wll be a result of ntegraton of such system of dfferental equatons usng formula (37) on the san of tme [ 0 T f ] requred for the convergence of mroer ntegral. Otmzaton Tasks of Nonlnear Control Systems In the frst aroachng stablty of the nonlnear control system can be defned on a lnearzed model. For ths urose we dfferentate the vector functon of equaton (34) on nut acton: k t B ( f [ u u t] u x 0 u 0 t 0. (39) Takng nto account matrx (35) let s resent the lnearzed model of the nonlnear system (34): u t A( u + B( u y ( u C( u. (40) On ths model let s buld a transfer functon n 0 n m W ( u β( u α( u α( u α ( s β( u β ( s. (4) On a characterstc olynomal α ( u let s defne the enalty functon P ( of knd (8) and elements of the frst column of Routh table ρ k (. Vector objectve functons for qualty crtera otmzaton (9) and (30) also wll deend both on the vector of the vared arameters x and on nut acton u. Let s desgnate these m 0
Internatonal Book Seres "Informaton Scence and Comutng" 95 functons through F ( u ) and ntroduce n u nut ste sgnals u ( us( nu wth magntudes u s [ u mn ; umax ]. Changng the value of nut acton u s at the fxed value of vector x we wll get dfferent ( ) values of vector functon F ( F[ u ( ] and usng comarson oeratons (3) fnd the worst value s G( max F ( ) (. (4) By analogy wth task (3) for lnear ACS the task of otmzaton of the nonlnear systems can be resented as: mn G( x. (43) The soluton of ths task gves the otmal vector of the vared arameters resultng to the best qualty of transent rocesses for the secfed set of nut actons. Modfcaton of Genetc Algorthms for Vector Otmzaton of Control Systems For otmzaton of vector objectve functons we offer modfcatons of genetc algorthms. Intal oulaton from M ndvduals s generated by ntroducng a set of random vectors x j M wth real coordnates n the sace of arameters R of the control system or vectors of bnary values. In the second case t s necessary to ( j) reresent every bnary vector n sace R and convert them to the vectors x j M. Usually for ths urose a bnary-to-decmal code or Gray code s used. The values of vector objectve functons (9) (30) or (4) ( j) j F F( x ) j M are calculated for all ndvduals usng systems models equatons () (34) (35) (39) (4) qualty ndexes calculaton formulas () (0) (36) (38) and defnng exressons of vector functons (3) (8). To rank ndvduals by the degree of ftness t s suggested to use the vector objectve functon sortng algorthms on the bass of oeratons of comarson of ts values (3) (33). The ftness level of ndvduals s subsequently scaled by the nverse square root j of ther rank j n the sorted sequence. The scaled ftness level s used n the casual mechansm of selecton. Alcaton of genetc oerators to the aternal ndvduals and generaton of descendants s made as well as n scalar genetc algorthms wth the use of dfferent tyes of crossover mutaton nverson. For all got descendants the values of vector objectve functon are calculated and ther ranks are obtaned smlarly to the stage of formng the ntal oulaton. The new oulaton s formed based on the results of sortng. Concluson The researches results allow to formulate next conclusons.. The calculaton methods of drect qualty ndexes and mroved ntegral quadratc estmatons have been studed for the lnear automatc control systems. These qualty ndexes are defned only n stablty doman of the systems.. The otmzaton tasks of qualty ndexes of the lnear automatc control systems are resented as the tasks of otmzaton of vector objectve functons takng nto account the condtons of stablty of the systems requrements to the qualty ndexes and rorty of system requrements. For modfcaton of otmzaton methods the set of comarson oeratons for vector objectve functons s ntroduced. 3. The methods of qualty ndexes calculaton have been also consdered for the nonlnear automatc control systems. These qualty ndexes are the functons of not only varyng arameters but also nut acton of control system. ( j)
96 No:3 Intellgent Informaton and Engneerng Systems 4. Through deendence of qualty ndexes on nut acton of nonlnear control systems for one value of vector of varyng arameters the several vector functons values are calculated at dfferent values of nut acton. By the choce from these vector values the worst value of the vector objectve functon of nonlnear system s determned. 5. Vector modfcatons of genetc algorthms for otmzaton of vector objectve functons allowng to solve the tasks of synthess for the lnear and nonlnear control systems have been develoed. The effcency of the roosed alcaton of genetc algorthms for the vector otmzaton of qualty ndexes of control systems has been confrmed by comutatonal exerments on the test and aled tasks. Bblograhy [Besekersky Poov 004] V. A. Besekersky E. P. Poov Teorja system avtomatcheskogo uravlenja St. Petersburg Professja 004 (n Rus.) 75. [Poljak Scherbakov 005] B. T. Poljak P. S. Scherbakov. Trudne zadach lnejnoj theor uravlenja. Nekotore odhod k reshenju // Automatka telemehanka 005 5 (n Rus.). 7-46. [Severn Nkulna 004] V. P.Severn E. N. Nkulna. Algortm vchslenja rjamh okazatelej kachestva hunkzj vesa sstem avtomatcheskogo uravlenja // Radoelectronca nformatca 004 (n Rus.). 5-59. [Severn 004] V. P. Severn. Mnmzaton of Integral Square Estmates of Automatc Control Systems. Part II. Ste by Ste Aroach // Journal of Automaton and Informaton Scences 004 Vol. 36; N 9. -9. [Severn 005] V. P. Severn. Vector Otmzaton of the Integral Quadratc Estmates for Automatc Control Systems // Journal of Comuter and Systems Scences Internatonal 005 Vol. 44 N. 07 6. [Severn 008] V. P. Severn. Parametrcheskj synthes sstem automatcheskogo uravlenja methodam vectornoj otmzac // Tehncheskaja elektrodnamka 008 Vol. 4 (n Rus.). 47-5. [Severn 009] Severn V. P. Struktura laborator methodov otmzac OPTLAB v systeme MATLAB // Trud IV Vserossjskoj nauchnoj konferenc «Proektrovane ngenernh nauchnh rlogenj v srede MATLAB» Astrakhan Izdatelskj dom «Astrakhanskj unverstet» 009 (n Rus.). 35 67. [Voronovsky Makhotlo Petrashev Sergeev 997] G. K. Voronovsky K. V. Makhotlo S. N. Petrashev S. A. Sergeev. Kharkov. Genetcheske algortm skusstvenne nejronne set roblem vrtualnoj realnost / Bass 997 (n Rus.). [Setlak 004] G. Setlak. Intellgent Decson Suort System Kev LOGOS 004 (n Rus.) 5. [Panchenko 007] T. V. Panchenko. Genetcheske algortm. Uchebnoe osobe / Pod red. J. J. Tarasevcha Astrakhan Izdatelskj dom «Astrakhanskj unverstet» 007 (n Rus.) 88. [Wese 008] T. Wese. Global Otmzaton Algorthms Theory and Alcaton 008 88. Authors' Informaton Valery Severn Professor Deartment of Comuter Scence and Control Natonal Techncal Unversty Kharkov Polytechncal Insttute ul. Frunze 300 Kharkov Ukrane; e-mal: severnv@mal.ru