NN-PID Based Control of MIMO Systems

Transcription

1 MIT Iteratioal oural of Electrical ad Istrumetatio Egieerig, Vol. 5, No., auary 05, pp ISSN No MIT Publicatios NN-PID Based Cotrol of MIMO Systems Taru Varsey Electrical Egieerig Departmet t_varsey@yaoo.com ABSTRACT To trai te Neural Networs (NN) stadard Bac Propagatio (BP) algoritm ad its variatios are widely used were iitial weigts are geerated as radom i ature. Te covergece of tese algoritms are very sesitive to te iitial weigts. I tis paper Particle Swarm Optimizatio (PSO) algoritm as bee used to iitialize te weigts by optimizatio. Various available algoritms ad te proposed oe ave bee tested ad compared for te implemetatio of NN-PID cotrol of two discrete-time o-liear coupled MIMO systems. Simulatio results sow tat te cotrolled performace of BP algoritm for No-liear, Keywords: Particle Swarm Optimizatio, MIMO, Bac Propagatio, Iitial weigts, PID cotroller. INRODUCTION Due to simple fuctioig of PID cotrollers it is widely used i cotrol systems. But tuig te parameters of cotroller is still a formidable callege. Also, oce te parameters are well tued satisfactory performace caot be guarateed due to process cages ad agig effect, so a regular tuig is eeded wic taes cosiderable time ad may ot be practically effective. Most of te cotrol systems are oliear coupled MIMO i ature. Due to o-liearities ad couplig effects amog te - vetioal PID cotrollers. Uiversal fuctio approximatio capabilites, learig ad geeralizatio properties maes NN a powerful cotrollig tool for suc systems []. Oe may use NN i two ways. NN ca be used eiter to adust te parameters of a covetioal PID cotroller [], [ ] or act as a direct PID cotroller tat implies emulatig te ideas of a expert cotrol tis Huaili, Xiucai ad Su used Stadard BP algoritms to update te weigts of NN based PID cotroller. But te stadard BP algoritm always suffers from slow covergece. Tere procedure ca be accelerated by icludig a mometum term i te weigt updatio law. I additio suc algoritms are muc sesitive to teir iitial weigts. Te stadard BP ad its wrog coice of iitial weigts could lead to false covergece of te algoritm. Particle Swarm Optimizatio tecique (PSO) is based o observatio of te beaviour of aimals suc as bird paper focuses te issue of weigt iitializatio of Stadard BP by te PSO algoritm ad explore its possibility to overcome te issue of covergece ad its advatage i Neural Networ based PID Cotrol of MIMO systems. Te rest of te paper is orgaised as follows: Te ext sectio describes te structure of NN based PID cotroller ad iput-output relatiosip of Proportioal Neuro, Itegral Neuro ad Derivative Neuros tese ave bee illustrated. I sectio te Stadard BP, BP wit mometum costat learig algoritm ad PSO based weigts simulatio results ad coclusios are draw i ed. NN BASED MIMO PID CONTROLLER Tis sectio describes te desig metology for a NN based. It cosists of sub-et, were represets te order of te system. Eac sub-et as tree layer etwor, wic are te iput layer, te idde layer ad te output layer. Te iput layer as two euros ad idde layer as tree euros suc as Proportioal Neuro, Itegral Neuro ad Derivative Neuro Proportioal Neuro, Itegral Neuro ad Derivative Neuro are as follows: A. Proportioal Neuro Cosider a t Proportioal Neuro, te its iput at ay istat is give by u ()= N w.x () P i i i=... ()

2 MIT Iteratioal oural of Electrical ad Istrumetatio Egieerig, Vol. 5, No., auary 05, pp ISSN No MIT Publicatios were i x i () are te outputs of N coected iput odes i iput layer ad w i are te weigts. Te te iput output relatio of Proportioal Neuro is as x () P u () were x P () is te euro output. P B. Itegral Neuro Te iput-output relatio of Itegral euro is give by x I( ) x I() u I() () were x I () is te euro output. C. Derivative Neuro Te iput-output relatio of Derivative euro is give by x D() u D() u D( ) were x D () is te euro output.... ()... () were y ( ) y ( ) r ( ) y ( ) x ( ) w u u...(8) s s ( ) ( ) xs of output variables, euros i every sub etwor, s of te sub etwor, is te traiig step istat ad x s is te output of idde layer euro. Te weigts from iput layer to output layer is updated as were w w w us ( ) us ( ) ( ) ws xsi ( ) w s ( ) s ( ) s s... () s ( ) x ( ) si... () were i etwor ad x si is te output of iput layer s euros. Fig. : NN based MIMO PID cotrol sceme Te output of sub-et U is give by U = w. X were X x P x D x I. x P x D x I ]. ALGORITHMS FOR COMPARATIVE STUDY Te cadidate algoritms cose for compariso are described below: A. BP algoritm i NN based MIMO PID cotrol Te obective of te NN based MIMO cotroller is to mimiize E r ( ) y ( ) =... (6) were r () is te iput ad y () is te output of te give system to be cotrol as metioed i Fig.. Te weigts of tis cotroller is altered by stadard BP algoritm. After te traiig te weigts from idde layer to output layer are w w s s w... (7) s B. BPAlgoritm wit Mometum i NN Based MIMO PID Cotrol Altoug te stadard BP traiig algoritm is widely used but it is wort to metio ere tat it suffers from slow covergece. Tere are several possible ways to mae it faster. Traiig procedure ca be accelerated by icludig a mometum term i w w w w s s s s ws w w w w w... () costat is a fractio of te previous weigt cage to te curret weigt cage. Te additio of suc a term ca elp i smoote te pat of decet by prevetig extreme cage i te gradiet. C. PSO Based Algoritm to Iitialze te Weigts of NN for PID Cotrol To iitialze all te weigts of NN, PSO algoritm as bee appiled. We use te idividual ad populatio to replace te particle ad group respectively. Te searcig procedure of te PSO is as follows:. Iitialize radomly te idividuals of te populatio icludig searcig poits, velocities, persoal best (pbest) ad global best (gbest).

3 MIT Iteratioal oural of Electrical ad Istrumetatio Egieerig, Vol. 5, No., auary 05, pp ISSN No MIT Publicatios. For eac iitial idividual of te populatio evaluate te itegral absolute error (IAE).. Compare evaluated value of eac idividual wit its pbest. Te best evaluated value amog te pbest is referred gbest. = X +c X rad() X pbest - x + i i i i It is desired to desig a PID cotroller so tat te time respose of a cose discrete-time, o-liear, iteractively coupled MIMO system, to a step iput as a fast covergece to step cage i ay of te iputs. Te followig PSO parameters are adopted i searcig te iitial weigts of NN +c X rad() X gbest - x i i Iter, c ad c are acceleratio costats, rad() is radom umber betwee () () is te velocity of particle at istat, x i i is te curret positio of particle at istat iertia weigt factor ad ca be selected as Iter mi X Iter mi... (6) were Iter is te imum umber of iteratio ad Iter is te curret umber of iteratio. If if, te + + i i, te + mi + mi i i 6. Update te member positio of eac idividual as: i i i x x... (7) mi mi suc tat x xi x, were x ad x refers te lower ad upper limits respectively of member of te idividual. 7. If te umber of iteratio breeces te imum te, go to step 8 else go to step. 8. Te idividual tat geerates te latest gbest is a optimal iitial weigts of NN. SIMULATION, RESULTS AND DISCUSSION To test te relative performace of proposed ad te available approaces of stadard BP algoritms two umerical examples of discrete-time o-liear coupled MIMO systems ave bee cose for simulatio study ad discussed i last sectio. Simulatios ave bee coducted i MATLAB Versio 7. ad executed o te Itel Core,.86GHz wit GB RAM. Example. A x discrete time o-liear coupled MIMO system [] is modeled by y y u u y( ) y y u u y ( )... (8) mi. Acceleratio Costat c ad c Iter A simple egative feed bac cotrol topology as bee implemeted i wic a NN based MIMO cotroller as bee desiged for set poit tracig. To trai te NN, four teciques ave bee implemeted. First tecique uses te stadard BP algorim wit radom iitializatio of etwor weigts, I secod tecique stadard BP algorim wit radom itializatio as bee used ad outputs of Proportioal, Itegral ad Derivative - tively, i tird tecique a stadard BP alog wit mometum costat wit radom itializatio as bee tested. Fially, a stadard BP algoritm wit PSO based iitializatio as bee implemeted ad tested. Te system step resposes for iput- ad iput- as bee sow i Figs. ad ad cotrolled iputs are sow i bee observed by weigts iitializatio of Stadard BP algoritm wit PSO. Rise time ad settlimg time are also oted to be muc improved. Fig. : System step respose for step cage i iput- (example )

4 MIT Iteratioal oural of Electrical ad Istrumetatio Egieerig, Vol. 5, No., auary 05, pp ISSN No MIT Publicatios Fig. : System step respose for step cage i iput- (example ) Fig. 6: Covergece of Square error (example ) Example. A discrete time o-liear coupled MIMO system wose iput output relatios represeted by: u (-) y u +u (-) u u ( ) y u +u ( ) u u (-) y u +u (-) u... ()... () Fig. 4: Cotrolled iput-(example-) Te cotrol obective remais same as before, to obtai te fast respodig tracig cotrol of system for step cages i te iputs. All parameters of PSO algoritm ave bee adopted as i example. Te time-respose of system to step cages i all iputs are feasible cotrolled iputs te proposed sceme is capable to trac te set poits faster i case of NN-based PID cotroller is traied wit BP alog wit PSO based weigts iitializatio, a Fig. 5: Cotrolled iput (example ) ta oter teciques, wit o steady-error as expected. Rise time ad settlig time are greatly reduced ad tus offer a maor advatage, as i earlier example.

5 MIT Iteratioal oural of Electrical ad Istrumetatio Egieerig, Vol. 5, No., auary 05, pp ISSN No MIT Publicatios Fig. 7: System step respose for iput-(example ) Fig. 0: Cotrolled iput (example ) Fig. 8: System step respose for iput- (example ) Fig. : Cotrolled iput (example ) Fig. 9: Step respose of system for iput- (example ) Fig. : Cotrolled iput (example )

6 MIT Iteratioal oural of Electrical ad Istrumetatio Egieerig, Vol. 5, No., auary 05, pp ISSN No MIT Publicatios REFERENCES Usig Simplest Neural Networs A Comparative Study ad Developmet of GUI System, IEEE Iteratioal Coferece o Power, Cotrol ad Embedded System, MNNIT, Allaabad, Idia, Approac for PID Cotrol of Multivariable Systems usig Diagoal Recurret Neural Networ, IEEE Iteratioal Coferece o Cotrol System, Computig ad Egieerig, Peag, Malaysia, MIMO Systems, Iteratioal oural of Automatio ad Cotrol, systems, Iteratioal Symposuim o Itellgiet Cotrol, Fig. : Covergece of Square error (example ) I tis example too, it ca be oted tat, rise-time, settlig cotroller outputs stayig witi saturatio limits witout widup. 5. CONCLUSIONS Te structure of NN based cotroller as bee revisited ad iput- output relatio of Proportioal Neuro, Itegral Neuro ad Derivative Neuro as bee explored. To trai te NN a stadard BP algoritm ad BP algoritm wit mometum costat as bee cose but te iitial weigts are ot cose radomly. To improve te covergece of BP traiig algoritm a PSO based sceme for weigts iitializatio algoritm as bee proposed. Tus a NN based MIMO-PID cotrol sceme as bee implemeted. Proposed algoritm as bee tested ad compared for te set poit cotrol of two discrete time o-liear coupled MIMO systems wit oter approaces beig used i tis class. It is cocluded tat proposed sceme is ot oly capable to cotrol te o-liear coupled MIMO systems satisfyig miimum mea square criteria wit smaller rise time ad settlig time, but also coverges faster ta a Stadard BP ad BP wit mometum costat algoritms. Tus a cosistetly satisfactory strategy for cotrol of discrete-time oliear-mimo systems is obtaied. tat guaratees covergece i feedforward etwors, IEEE Trasactio o Neural Networ IEEE Iteratioal Coferece o Idustrial ad Iformatics system, optimizatio ad its applicatio to olie-tuig PID parameters i pp based o particle swarm optimizer algoritm, oural of Systems Egieerig ad Electroics, Vol.6 Issue:, pp BPNN ad PSO algoritms, Applied Soft Computig, Volume, bac-propagatio algoritm for feed forward eural etwor multi-layer eural etwors Neural, Iteratioal coferece o Neural Iformatio Processig, Pert, Wester Australia,Vol., pp tuig PID cotroller via Quasi-Diagoal recurret wavelet Neural etwor, Iteratioal coferece o Itelliget Huma-Macie Systems ad Cyberetics