On the Control of the Geophysical System: Problem Formulation

Transcription

1 On he Conrol of he Geophysical Sysem: Problem Formulaion SERGEI SOLDATENKO 1 and RAFAEL YUSUPOV 1 Cener for Ausralian Weaher and Climae Research Earh Sysem Modeling Research Program 7 Collins Sree, Melbourne, VIC 38 AUSTRALIA s.soldaenko@bom.gov.au Sain-Peersburg Insiue of Informaics and Auomaion of RAS Laboraory of Applied Informaics 14-h Line, 39, S. Peersburg, RUSSIA yusupov@iias.spb.su Absrac: - Geophysical cyberneics is a new research area ha sudies he geophysical processes and phenomena using cyberneic mehods and approaches. A mahemaical formulaion of he problem of opimal conrol of he geophysical sysem is presened from he sandpoin of geophysical cyberneics. Furher, he essenial feaures of he geophysical sysem as a conrol objec are considered. Key-Words: - geophysical cyberneics, geophysical sysem, opimal conrol, dynamical sysem, mahemaical modeling. 1 Inroducion For many cenuries mankind, mosly uninenionally, has affeced he environmen, changing is properies. From he middle of he las cenury, modificaion of he environmen became more focused wihin he framework of modificaion of weaher and modificaion of oher hydromeeorological and geophysical processes. A he end of he h cenury, geoengineering, a new echnical and research area, was proposed as a ool for analyzing and managing he environmen, in paricular, for miigaing he impac of anhropogenic growh of GHG emissions. However, from he sandpoin of conrol heory, modificaion of geophysical processes and geoengineering aciviies represen open-loop conrol sraegies in which inpu and oupu variables are no sufficienly subsaniaed. In addiion, he boundaries of geoengineering and modificaion of geophysical processes are no clearly delineaed, heir goals and mehods of achieving he objecives are formulaed and formalized in general erms, and he expeced resuls are quie amorphous. In he lae 197s, a uniform mehodology for planning and implemenaion of mehods o conrol geophysical processes was formulaed on he basis of he ideas of cyberneics [1]. In his book he concep of geophysical cyberneics was inroduced as he new research area of a self-regulaing cyberneic sysem in which he geophysical sysem is he conrol objec, and he role of he conroller is given o human sociey as a whole. The geophysical sysem was defined as a se of objecs of inanimae naure [1]. Geocyberneics, as a novel inerdisciplinary echnical and research area, is developing on he basis of ideas and mehods of opimal conrol heory, dynamical sysems heory, echnical cyberneics, geophysics, and oher academic disciplines. Conrolling he sae of he geophysical sysem, of course, is a very problemaic issue, because he geophysical sysem is a unique physical objec wih a number of specific feaures: - The geophysical sysem is a complex ineracive sysem wih various posiive and negaive feedback mechanisms. The main componens of he geophysical sysem (he amosphere, hydrosphere, lihosphere and cryosphere) have specific physical and dynamical properies; - The geophysical sysem is also an open sysem bu is impac on he exernal environmen is negligible; - Time scales of he physical processes occurring in he geophysical sysem vary over a wide range - from seconds (urbulen flucuaions) o ens and hundreds of years; ISBN:

2 - Since he geophysical sysem is a global sysem is spaial specrum of moions covers molecular o planeary scales; - Geophysical processes oscillae due o boh inernal facors (naural oscillaions) and exernal forcing (forced oscillaions). Naural oscillaions are due o inernal insabiliy of he geophysical sysem wih respec o sochasic infiniesimal disurbances. Anhropogenic impacs on geophysical sysem, boh inenional and uninenional, belong o he caegory of exernal forcing; - Geophysical dynamical processes are nonlinear and chaoic []. Undoubedly, geophysical sysem has a number of oher specific feaures ha make i a unique physical objec, which is virually impossible o sudy using laboraory simulaions (wih rare excepions). Therefore, he main mehod of sudying he geophysical sysem is mahemaical modeling. Wih respec o he conrol of he geophysical sysem, he applicaion of cyberneic approaches and echniques developed for he sudy and opimal conrol of echnical sysems is very difficul. This is due o he following facors: - Geophysical processes are no fully idenified as conrol objecs, he associaed mahemaical models are no perfec and do no always have he sufficien adequacy and accuracy. - The geophysical sysem refers o a class of disribued parameer sysems described by parial differenial equaions, which makes he mahemaical models of geophysical processes quie complex. Synhesis of conrol sysems of such objecs requires he developmen of conrol heory, which was developed mainly for objecs wih lumped parameers (e.g. [3-7]). Mahemaical formulaion of he geophysical sysem conrol problem includes: (a) a mahemaical model of he geophysical sysem ha describes is behavior under he influence of conrol acions and exernal disurbances; (b) he formulaion of he conrol objecives, (c) a conrol model ha imposes consrains on he conrols and he sae of he geophysical sysem. In [8], he concep of conrolling he sae of he geophysical sysem has been considered in a probabilisic manner. However, in pracice, a probabilisic approach is primarily used for he developmen of conrol sraegies on he assumpion ha he geophysical sysem is in seady sae. For his reason, deerminisic mahemaical models are mainly used for modeling and conrolling he sae of he geophysical sysem (e.g. [9-13]). This paper aims o presen a deerminisic saemen of he problem of opimal conrol of he geophysical sysem. The essenial feaures of geophysical sysem as a conrol objec are also considered. Mahemaical Model Suppose ha in a bounded space-ime domain, T he sae of he geophysical sysem can be characerized by he vecor of sae variables r,, where is he infinie real space of sufficienly smooh sae funcions saisfying some problem-specific boundary condiions a he boundary of he domain, 3, T is he ime and r is a vecor of spaial variables. The domain could represen he earh s sphere, hemisphere or anoher limied area on he earh s surface. The evoluion of he geophysical sysem is mahemaically described by mulidimensional parial differenial equaions ha ake ino consideraion he specific properies of geophysical sysem as well as is processes and cycles. Formally, he evoluion of geophysical sysem in he domain can be described by he following nonlinear parial differenial equaion r, r r ( r, ), ( r, ),,, (1) where is a nonlinear operaor, is he model parameer vecor, and is a given vecor-valued funcion defining he iniial sae esimae. The space-ime specrum of he processes occurring in he geophysical sysem is exremely wide. Hence, sae-of-he-ar mahemaical models are unable o realisically simulae all of hese processes. Le be a characerisic ime-scale of a cerain geophysical process. In his case, an explici descripion of he processes wih ime scales smaller han is alered on a parameric represenaion (i.e. small scale processes are parameerized). Some of hese parameers can be considered as conrol variables wihin he cyberneic framework. In ha case, by varying he conrol parameers, we can formally conrol he dynamics of geophysical sysem. r, has an infinie dimension The sae vecor because he geophysical sysem is a coninuous medium. However, in he conrol heory, sae vecor usually is of finie dimension. In order o obain a sysem wih a finie number of degrees of freedom, and, in addiion, o use he mehods of ISBN:

3 conrol heory developed for lumped sysems, he equaion (1) is now projeced ino he subspace spanned by he orhogonal basis so ha r, series i can be represened as a normally convergen r, xi ir. () i Subsiuing () ino (1) and applying hen he Galerkin procedure, one can obain, insead of (1), he lumped sysem ha is described by he following se of ordinary differenial equaions dx d f x wih he iniial condiions x, u, x,, (3) n where xx is he sae (or phase) vecor he componens of which belong o he class of, 1 coninuously differeniable funcions funcions m uu is a conrol vecor he componens of which belong o he class of piecewise coninuous ˆ, and f n is a nonlinear vecorvalued funcion defined in he domain XU ha is coninuous wih respec o boh u and x, coninuously differeniable wih respec o х, as well as piecewise coninuous wih respec o, such ha f : X U X, and x is a given vecor-valued funcion. In he righ-hand side of (3), only conrol parameers are presened, while he unconrolled parameers are omied. We assume ha conrol parameers depend on he sae of he sysem, i.e. u g, x, which implies ha equaions (3) describe a closed-loop conrol sysem, represening he geophysical sysem. Equaions (3) canno be solved analyically for a general case, herefore some numerical mehods (specral, finie-difference ec.) should be applied o obain he soluion. 3 Iniial Condiions and Observabiliy Iniial condiions x are required o numerically solve equaions (3). The observaional daa are he primary source for formulaion of iniial condiions. Since he observaional daa are always uncerain, specific daa assimilaion procedures should be used o calculae he iniial condiions and o observe he sysem behavior. Suppose ha a = he background sae x b, i.e. he firs guess, and a cerain physical quaniies y o measured by insrumens are known. Then x x b b, y o x o, where is he projecion operaor, which is nonlinear in general, ha maps he space of model sae ino he space of observaions, ε b as well as ε o are he errors of he firs guess and observaions respecively. Using four-dimensional variaional daa assimilaion procedure, which is, in fac, he opimal conrol problem, he iniial sae esimae x is calculaed by solving he following opimizaion problem x a xx arg min J x, (4) 1 1 J x x x x y, (5) b o ( ) B R where B and R are he error covariance marices of he firs guess and observaions, respecively. Therefore, he daa assimilaion problem is simply a minimizaion problem wih consrains on x given by he model equaions (3). 4 Formulaion of he Conrol Problem Le us discuss some essenial feaures of he conrol of he geophysical sysem. This problem remains poorly sudied boh scienifically and echnically, due o is relaive novely and enormous complexiy. Indeed: - The spaial disribuion of geophysical processes requires spaially disribued conrol acions. Mehods for implemenaion of such conrols are very poorly developed; - Geophysical processes have enormous energy poenial. Implemenaion of comparable energy conrol acions is a very difficul issue. Therefore, he idenificaion of sensiive poins, which can be manipulaed o produce he desired resul, is a criical problem; - The scale and he huge energy poenial of geophysical processes impose very sric requiremens for accuracy and reliabiliy of conrol sysems; even minor errors in he conrol can be disasrous; - Processes occurring in he geophysical sysem are inerrelaed, so he changes in he dynamics of some processes can lead o unconrollable consequences. This should be aken ino accoun in ISBN:

4 he developmen of conrol sysems for geophysical processes. The inheren feaures of geophysical processes provide possible ways o conrol hem (see [8] for more deails). I is imporan o noe, ha naure iself provides he abiliy o design physical foundaions for conrol of geophysical processes based on he exising naural physical mechanisms ha conrol he behavior of geophysical sysem. Le s assume ha sysem (3) is conrollable and conrol parameers belong o he se of admissible conrols u U. I is imporan ha he se of admissible conrols is defined on he basis of physical and echnical feasibiliy, aking ino accoun he above-menioned feaures of conrol of he geophysical sysem. Furher, suppose ha conrols belong o he class of piecewise coninuous funcions wih values in U or Lebesgue measurable funcions wih values in U, hen, in accordance wih he classical Caraheodory s heorem [14], one can prove ha he Cauchy problem (3) has a unique soluion defined on an inerval in. In general, we canno deermine a priori wheher he geophysical sysem is conrollable or no. Conclusion abou he conrollabiliy of he sysem can only be made by solving a specific problem. The problem of conrol of he geophysical sysem is o synhesize he conrol law ha ensures he achievemen of he desired objecive. Since he objecive is expressed in erms of exremal problem, we are specifically ineresed in synhesizing an opimal conrol. In he mos general form, he problem of conrol of he geophysical sysem is formulaed as follows: Find he se of admissible conrols u :, T U (6) and he rajecory of he sysem (3), which is generaed by u* x :, T X (7) such ha he given objecive funcional is exremum (minimum or maximum) x, u, x, T, xt T reaches. (8) x, u, d exr, An inegrand of he objecive funcional : XU. The problem (6)-(8) includes a se X, on which he funcional is defined, and consrains on he model sae given by he subse of a se X. Noe ha he dynamic consrains are as follows x f x, u. The objecive funcional (8) is a classical cos funcional for opimizing dynamical sysems and corresponds o he Bolza problem. Problems of Mayer and Lagrange are special cases of he Bolza problem. The formulaion of an objecive funcion depends on he problem under consideraion and here are no universal approaches how i can be specified. Opimal conrol problem (6)-(8) is nonlinear and herefore can be solved only numerically. The Hamilonian funcion H : n associaed wih he opimal conrol problem (6)-(8) is x, u,, x, u, f x, u H where n is he cosae vecor., (9) There are several mehods o solve he opimal conrol problem (6)-(9): classical mehods of he variaional calculus, dynamical programming, he Ponryagin s maximum principle and oher mehods. 5 Sabilizaion Conrol Problem Sabilizaion of he geophysical sysem around he nominal rajecory represens a specific class of opimal conrol problems relevan o he conrol of geophysical processes. Sabilizaion of he climae sysem rajecory o weaken he global warming is one of he examples of his class of problems. For his class of opimal conrol problems, he differenial equaions (3) describing he evoluion of he geophysical sysem is linearized wih respec o he naural (nominal) rajecory x e () caused by exernal naural unperurbed forcing u e (): d x f f x u (1) d x wih he iniial condiions u xx* xx* uu* uu* x, (11) where δx is he perurbaion of naural rajecory of he geophysical sysem due o anhropogenic disurbances, δu is a conrol vecor o ensure he sabilizaion of he naural rajecory, f xand f u are he Jacobian marices. I is assumed ha ISBN:

5 u u e e u, u u, (1) x x e e x, x x. (13) The opimal conrol problem is formulaed as follows: Find he conrol vecor u (14) generaing he correcion of he naural rajecory such ha he cos funcional x, x e x X (15) is minimized u arg min x, u, (16) u 1 ( x, u) x ( T) G xt T, (16) 1 ( ) ( ) ( ) ( ) x W x u Qu d where W() and G are weighing posiive semidefinie n n marices, normalizing he energy of geophysical sysem per uni mass, Q() is a weighing posiive define m m marix, normalizing he energy of conrol acions per uni mass. The sabilizaion problem is solved, given he fac ha he sysem ravels along is naural rajecory ha is subjec o exernal naural forcing. The conrol goal is o keep δx() close o zero using conrol acions δu(). The informaion on he geophysical sysem sae x() is obained by measuremen devices and insrumens followed by he processing of hese informaion wihin he daa assimilaion sysem (4) - (5). 6 Concluding Remarks In his paper, he formulaion of he problem of opimal conrol of he geophysical sysem has been considered in deerminisic manner. The geophysical sysem, represening a se of objecs of inanimae naure, is a very complex and unique conrol objec. Challenges facing he humaniy oday, such as he global warming, necessiaed he developmen of mehods for conrolling he sae and dynamics of he geophysical sysem and is componens and processes. This paper provides a basis for furher research in he field of geophysical cyberneics ha is a new research area of a self-regulaing cyberneic sysem in which he geophysical sysem represens he conrol objec and he human sociey plays he role of conroller. The auhors of his paper realize ha he problem of conrol of he geophysical sysem and geophysical processes is an exremely complex issue, which has many physical, echnical, ehical and legal consrains and aspecs. However, humaniy is on he way o manipulae he climae and oher geophysical phenomena and processes. In his conex, he developmen of a heoreical framework for he conrol of geophysical sysem on he basis of mahemaical mehods seems as a problem of curren ineres. References: [1] R.M. Yusupov, Theoreical bases of conrol of geophysical processes, MHE, Moscow, [] E.N. Lorenz, Deerminisic non-periodic flow, Journal of he Amospheric Sciences, Vol., 1963, pp [3] R.E. Bellman, Dynamic programming, Princeon Universiy Press, Princeon, N.J [4] W.H. Fleming, and R. Rishel, Deerminisic and sochasic opimal conrol, Springer, Berlin, [5] L.S. Ponryagin, The mahemaical heory of opimal processes, CRC, Boca Raon, [6] J.L. Lions, Generalized soluions of Hamilon- Jacobi equaions, Piman, London, 198. [7] D.E. Kirk, Opimal conrol heory: An inroducion, Dover, New York, 4. [8] R.M. Yusupov (Ed.), An inroducion o geophysical cyberneics and environmenal monioring, S. Peersburg Sae Universiy Press, S. Peersburg, [9] S.A. Soldaenko, Some applicaions of he heory of opimal conrol of disribued parameer sysems o weaher numerical modeling. In: Research Aciviies in Amospheric and Oceanic Modelling, Geneva, Swizerland: WMO, Vol. 8, 1999, pp [1] M. Jacobson, Fundamenals of amospheric modeling, Cambridge Universiy Press, Cambridge, [11] S. Griffies, Fundamenals of ocean climae models, Princeon Universiy Press, Princeon N.J., 4. [1] W. Robinson, Modeling dynamic climae sysems, Springer-Verlag, New York, 1. [13] K.E. Trenberh, Climae sysem modeling, Cambridge Universiy Press, Cambridge, 1. [14] E.A. Coddingon and N. Levinson, Theory of ordinary differenial equaions, McGrow-Hill, New York, ISBN: