Fuzzy Threshold Values in the Robust Design of Controllers for Probabilistic Uncertain Systems

Transcription

1 Fuzzy Threshold Values n the Robust Desgn o Controllers or Probablstc Uncertan Systems N Narman-zadeh and A Jamal Abstract n ths paper, optmal mult-obectve Pareto desgn o robust state eedbac controllers or an nverted pendulum wth parameters probablstc uncertanty s presented The obectve unctons that have been consdered are, namely, the probabltes o alure o summaton o rsng tme and overshoot o cart (SRO-C) and the probabltes o alure o summaton o rsng tme and overshoot o pendulum (SRO-P) t s also shown that mult-obectve relablty Pareto optmzaton o the robust state eedbac controllers usng a new mult-obectve unorm-dversty genetc algorthm (UGA) wth uzzy threshold values ncludes those may be obtaned by varous crsp threshold values o probablty o alure and, thus, remove the dculty o selectng sutable crsp values denng the robustness o the stochastc perormance o the controller Besdes the mult-obectve Pareto optmzaton o such robust eedbac controllers usng UGA unvels some very mportant and normatve trade-os among those obectve unctons ndex Terms Pareto, Robust control, Relablty, Uncertanty NTRODUCTON The synthess o control polces can be regarded as optmzaton problems o certan perormance measures o a controlled system A very eectve means o solvng such optmum controller desgn problems s genetc algorthms (GAs) and other evolutonary algorthms (EAs) Some early applcaton o GAs n optmum desgn o controllers are reported n [] [] n addton to the most applcatons o EAs n the desgn o controllers or certan systems, there are also much research eorts n robust desgn o controllers or uncertan systems n whch both structured or unstructured uncertantes may exst [3] ost o the robust desgn methods such as μ analyss, H or H desgn are based on derent norm-bounded uncertanty [4] As each norm has ts partcular eatures addressng derent types o perormance obectves, t may not be possble to acheve all the robustness ssues and loop perormance goals smultaneously Recently, GAs have also been deployed or mult-obectve robust control desgn consderng the robust stablty and the mxed H and H norms, smultaneously [] ndeed the desgnng robust control method or uncertan systems s a computatonally complex problem Recently, there have been many eorts or desgnng robust control methods n these robust desgn methods, probablstc uncertanty propagates through the uncertan parameter o plants These methods reduce the conservatsm nvolved wth N Narman-zadeh and A Jamal are wth the Faculty o echancal Engneerng, slamc Azad Unversty, Taestan branch, Taestan, ran, (e-mal: nnzadeh@gulanacr) such robust desgn methods or to account more or the most lely plants wth respect to uncertantes, the probablstc uncertanty, as a weghtng actor, must be consdered accordngly n act, probablstc uncertanty speces set o plants as the actual dynamc system to each o whch a probablty densty uncton (PDF) s assgned [6] Thereore, such addtonal normaton regardng the lelhood o each plant allows a relablty-based desgn n whch probablty s ncorporated n the robust desgn The noton o stochastc robustness and probablty o nstablty have been rst mentoned by Stengel [7] and Stengel and Ryan [8] The analyss o onte Carlo Smulaton (CS) has also been ntroduced by Stengel to evaluate stochastc stablty and perormance o probablstc uncertan systems n ths paper, uzzy threshold values have been used or optmal relablty-based mult-obectve Pareto desgn o state eedbac controller or an nverted pendulum wth probablstc uncertan parameters t s shown that a small change n the crsp threshold values causes to have very derent Pareto ronts and, thereore t s very dcult to select a sutable crsp threshold values whch s needed to compute the probablty o alure n RBDO desgn oreover, t s also shown that the Pareto optmal desgn ponts that are obtaned usng uzzy threshold values nclude the best Pareto desgn ponts whch are obtaned when a wde range o crsp threshold values are nstead used A mult-obectve unorm-dversty genetc algorthm (UGA) that has been proposed by authors [9], [] s used or mult-obectve optmzaton The obectve unctons that have been consdered are, namely, the probabltes o alure o summaton o rsng tme and overshoot o cart (S RO -C) and the probabltes o alure o summaton o rsng tme and overshoot o pendulum (S RO -P) Thereore, optmum robust state eedbac controllers are ound n a Pareto sense usng the probablstc measures o those obectve unctons ncludng ther probablty o alure by hybrd use o CS and UGA STOCHASTC ROBUST ANALYSS n real control engneerng practce, there exst a varety o typcal sources o uncertanty whch have to be compensated through robust control desgn approach [9] Those uncertantes nclude plant parameter varatons due to envronmental condton, ncomplete nowledge o the parameters, age, unmodeled hgh requency dynamcs, and etc Two categorcal types o uncertanty, namely, structured uncertanty and unstructured uncertanty are generally used n classcaton The structured uncertanty concerns about the model uncertanty due to unnown values o parameters

2 n a nown structure n conventonal optmum control system desgn, uncertantes are not addressed and the optmzaton process s accomplshed determnstcally n act, t has been shown that optmzaton wthout consderng uncertanty generally leads to non-optmal and potentally hgh rs soluton [] Thereore, t s very desrable to nd robust desgn whose perormance varaton n the presence o uncertantes s not hgh Generally, there exst two approaches addressng the stochastc robustness ssue, namely, robust desgn optmzaton (RDO) and relablty-based desgn optmzaton (RBDO), [] Wth the ad o ever ncreasng computatonal power, there have been a great amount o research actvtes n the eld o robust analyss and desgn devoted to the use o onte Carlo smulaton [3] n act, onte Carlo smulaton (CS) has also been used to very the results o other methods n RDO or RBDO problems when sucent number o samplng s adopted [4] onte Carlo smulaton (CS) s a drect and smple numercal method but can be computatonally expensve n ths method, random samples are generated assumng pre-dened probablstc dstrbutons or uncertan parameters The system s then smulated wth each o these randomly generated samples and the percentage o cases produced n alure regon dened by a lmt state uncton approxmately relects the probablty o alure Let be a random varable, then the prevalng model or uncertantes n stochastc randomness s the probablty densty uncton (PDF), (x) or equvalently by the cumulatve dstrbuton uncton (CDF), F (x), where the subscrpt reers to the random varable Ths can be gven by x ( x) = ( x) = ( x) F Pr dx () where Pr () s the probablty that an event ( x ) wll occur Some statstcal moments such as the rst and the second moment, generally nown as mean value (also reerred to as expected value) denoted by E ( ) and varance denoted by σ ( ), respectvely, are o the most mportant ones They can also be computed by E and ( ) xdf x = x ( x) = dx () ( ) ( ) ( ( )) ( ) ( ( ) ) Var = x E df x = x E ( x ) σ = dx (3) n the robust desgn approach, t s desred to mnmze the varablty o a stochastc response as a uncton o w (w represents tme or requency) due to the uncertan probablstc parameters about a determnstc behavor n h p, w be the random response o an ths approach, let ( ) uncertan plant due to uncertan parameters p, and let ĥ ( w) be the response o the certan determnstc plant [] The varablty s then obtaned by denng the error between each stochastc response o uncertan probablstc plant and the response o the certan determnstc plant Thereore, the sum o squared error (SSE) can be used or obtanng varablty about determnstc behavor as ollows N Nw ( h ( p, w ) hˆ ( w ) ν = (4) = = where h wth wth,, =,, KN s a random response, and w = KN s tme or requency sample w n the relablty-based desgn approach, t s requred to dene relablty-based metrcs va some nequalty constrants Thereore, n the presence o uncertan parameters o plant (p) whose PDF or CDF can be gven by p or F p, respectvely, the relablty requrements can be gven as P ( g ( ) ) = ( =,, K ) = Pr ε, p () n (), P denotes the probablty o alure (e, g ) o the th relablty measure and s the number o nequalty constrants (e lmt state unctons) and ε s the hghest value o desred admssble probablty o alure t s clear that the desred value o each P s zero Thereore, tang nto consderaton the stochastc dstrbuton o uncertan parameters (p ) as p, equaton () can now be evaluated or each probablty uncton as ( g ) = P = Pr p dp (6) g Ths ntegral s, n act, very complcated partcularly or systems wth complex g [6] and onte Carlo smulaton s alternatvely used to approxmate (6) n ths case, a bnary ndcator uncton g ( p ) s dened such that t has the value o n the case o alure ( g ) and the value o zero otherwse, = g > g (7) g Consequently, the ntegral o (6) can be rewrtten as ( G C( ) ) p P = g, dp (8) C s the controller to be desgned n the case o control system desgn problems Based on onte Carlo smulaton, the probablty usng samplng technque can be estmated usng where G s the uncertan plant model and ( )

3 P = N N = ( G( ), C( d) ) g p (9) n other words, the probablty o alure s equal to the number o samples n the alure regon dvded by the total number o samples Evdently, such estmaton o P approaches to the actual value n the lmt as N [3] n ths paper, Hammersley Sequence Samplng (HSS) has been used to generate samples or probablty estmaton o alures The bnary crsp ndcator uncton g o () that has been adopted n many research wors to compute the occurrence o alure s, however, very prone senstve to the denton o the lmt state unctons g Consequently, the optmum desgn o controllers could sgncantly lead to derent optmum results based on the boundary denton o those lmts state unctons or the obectve unctons n act, some robust non-domnated optmum desgns may be elmnated rom the Pareto ront or such speced crsp ndcator uncton g Thereore, a Gaussan uzzy membershp ndcator uncton [9] s rather used n ths wor to crcumvent to dculty o selectng the sutable crsp ndcator uncton n ths case, equaton (7) s replaced by g g μ = + er σ () where the Gaussan error uncton s dened by y t er ( y) = e dt π () n () μ and σ stand or mean and varance o the lmt state uncton g o each obectve uncton respectvely Thus, the use o the Gaussan degree o membershp g gven by () ensures gradual varaton o the probablty o alure based on selected admssble mean and varance o each lmt state uncton n relablty-based desgn ULT-OBJECTVE PARETO OPTZATON ult-obectve optmzaton whch s also called multcrtera optmzaton or vector optmzaton has been dened as ndng a vector o decson varables satsyng constrants to gve optmal values to all obectve unctons [7] n general, t can be mathematcally dened as: = x, x,, to optmze x n * * * * nd the vector [ ] T [ ( ), ( ),, ] T ( F ) = ) () ( subect to m nequalty constrants l ( ), = to m (3) and p equalty constrants h ( ) =, = to p (4) * R n Where, s the vector o decson or desgn varables, and F ( ) R s the vector o obectve unctons Wthout loss o generalty, t s assumed that all obectve unctons are to be mnmzed Such multobectve mnmzaton based on the Pareto approach can be conducted usng some dentons: Denton o Pareto Domnance: A vector U = [ u, u,, u ] R domnates to vector V = [ v, v,, v ] R (denoted by U p V ) and only {,,, }, u v {,,, } : u < v t means that there s at least one u whch s smaller than v whlst the rest u s are ether smaller or equal to correspondng v s Denton o Pareto ront: For a gven OP, the Pareto ront ƤŦ s a set o vectors o obectve unctons whch are obtaned usng the vectors o decson varables n the Pareto set, Ƥ that s, ƤŦ = { F( ) = ( ( ), ( ),, ( )): { Ƥ Thereore, the Pareto ront ƤŦ s a set o the vectors o Ƥ obectve unctons mapped rom n ths wor, a new mult-obectve unorm-dversty genetc algorthm method called UGA that has been proposed by authors [9], [] s used or mult-obectve relablty-based optmzaton V THE SNGLE NVERTED PENDULU AND STATE FEEDBACK CONTROLLER DESGN ETHOD One o the mportant benchmar o control problem s nverted pendulum that has been shown n Fg () The lnear steady state equatons o nverted pendulum are gven as x x = x 3 x 4 ( l s ) r l s r ( l s ) g l s C ( l s ) l s g C x x + u x 3 x4 l s () where = +, = ( ls ), and x, x, x 3 and x 4 stand or the poston o cart, the velocty o cart, the angle o pendulum wth the vertcal axs and the angular velocty o pendulum, respectvely And,,l s, r, and C are the mass o cart, the mass o pendulum, the length o arm, rcton coecent o cart, moment o nerta o pendulum and rotatng rcton coecent o pendulum, respectvely n the case o stochastc robust desgn, parameters o the plant gven by () vary accordng to a pror nown probablstc

4 Fg (): uzzy threshold values or calculatng the probablty o alure dstrbuton unctons around a nomnal set o parameters The nomnal values o plant are gven n Table () n ths study, lnear state-eedbac controllers are used or control o the nverted pendulum The equaton o the state eedbac controller s u = K ( + K x + K x + K x (6) x ) where the desgn vector = ( K, K, K K ) has to be 3, optmally determned based on probablstc uncertan values o those parameters n the Pareto mult-obectve optmum approach Table (): the nomnal values o sngle nverted pendulum 36g 4g l s 4m r C 4 Fg (): Bloc dagram o SP and state eedbac controller The perormance o a controlled closed-loop system s usually evaluated by varety o goals n ths paper, a 4 b-obectve optmum robust lnear state eedbac controller s consdered whlst each o those obectve uncton s the probablty o alure o S RO -C and S RO -P based on the uzzy membershp denton o alure as descrbed beore Such pre-dened uzzy membershp unctons or the probablty alure denton o overshoot and settlng tme o both cart and pendulum based on () and () are depcted n Fg (): The probablty o alure o each uncton can now be computed usng equaton () usng CS approach wth the Hammersly random dstrbuton samplng Table (): Optmum values o obectve unctons and the correspondng gans o state eedbac o Pareto ront ponts Obectve unctons Gans o the controller Prob o alure Prob o alure o K K K 3 K 4 o S SO-C S SO-P A B C D E F G H J K L N O P Q

5 Fg (3): Probablstc tme response behavors o optmum desgn pont (7 samples) V LLUSTRATVE EAPLE ult-obectve unorm-dversty genetc algorthm (UGA) [9] [] s used or Pareto mult-obectve optmzaton o lnear robust state eedbac controllers o the probablstc uncertantes sngle nverted pendulum whose parameters are vared wth Gaussan dstrbutons wthn the lmts o ± % o the nomnal values o plant parameters A populaton sze o 7 has been chosen wth crossover probablty P c and mutaton probablty P m as 8 and 9, respectvely Two obectves, namely, probablty o alure o S RO -C and the probablty o alure o S RO -P are consdered smultaneously n a Pareto optmzaton process to obtan some mportant trade-os among conlctng obectves The optmzaton process o the robust lnear stateeedbac controller gven by (6) s accomplshed by onte Carlo evaluatons usng Gaussan dstrbuton or each canddate control law durng the evolutonary process Consequently, total number o 7 non-domnated optmum desgn ponts have been obtaned and shown n Table () The optmum desgn pont Q and A smply demonstrate the best values or S RO -C and S RO -P, respectvely, whlst the optmum desgn pont s the trade-o desgn and may be compromsngly chosen rom that table o Pareto ront ponts Evdently, the desgn pont exhbts a sgncant mprovement n the probablty o alure o ether S RO -C or S RO -P Comparng to the desgn ponts Q and A, respectvely However, n order to show the robustness behavor o the robust trade-o optmum pont, a CS wth 7 evaluatons has been accomplshed to measure the probablty o alures o both S RO -C and S RO -P based o the uzzy membershp varatons gven n Fg () t s now very evdent rom the values o the probabltes gven n Table () that the robust trade-o optmum pont exhbts very good perormance Fg (3) depcts the tme step response o trade-o desgn pont n a CS wth 7 evaluatons Such robust behavor o desgn pont s very clear rom Fg (3) or both cart and pendulum n order to compare the results o probablstc desgn usng uzzy membershp denton wth those usng crsp

6 denton o thresholds, three derent Pareto ronts are obtaned correspondng to the crsp threshold values are gven n Table () t should be noted that the order o sets (), () and (3) s rom large values o the crsp threshold to small ones n act, large values o crsp threshold ncrease the possblty o ndng a desgn pont satsyng all those thresholds or the partcular onte Carlo Smulaton (CS) Ths, n turn, leads to lower values o probablty o alure or CS as shown n set (3) t s now possble to translate bac the optmum desgn pont obtaned usng uzzy membershp values o probabltes o alure nto each Pareto ront o crsp sets (), () and (3), whch are correspondngly shown n Table () t s now evdent rom ths table that all non-domnated optmum desgn ponts obtaned rom those three derent sets o crsp threshold values are almost nclusve n the optmum desgn ponts obtaned by the approach o uzzy threshold values Table (): The values o obectve uncton o the best optmal ponts obtaned by three derent sets o crsp threshold values Set No Crsp threshold values Settlng tme o cart Overshoot o cart Settlng tme o pendulum Overshoot o pendulum Best Pareto pont Prob o alure o SSO-C Prob o alure o SSO-P Cross-valdaton o desgn pont Prob o alure o SSO-C Prob o alure o SSO-P V CONCLUSON [4] LG Crespo, Optmal perormance, robustness and relablty base desgns o systems wth structured uncertanty, Proc o Amercan Control Con, Denver, Colorado, USA, Vol, 3, pp [] A Herreros, E Baeyens, and JR Persan, RCD: a genetc algorthm or mult obectve robust control desgn, Engneerng Applcaton o Artcal ntellgence, Vol,, pp 8-3 [6] LG Crespo and SP Kenny, Robust Control Degn or systems wth probablstc Uncertanty, NASA report, TP--33, arch, [7] RF Stengel, Stochastc Optmal Control: theory and applcaton, New Yor, Wley, 986 [8] RF Stengel and LE Ryan, Stochastc robustness o lnear control systems, Proc Con norm Sc Sys, 989, pp 6-6 [9] A Jamal, A Hallo, and N Narman-zadeh, Relablty-based Robust Pareto Desgn o Lnear State Feedbac Controllers usng a ult-obectve Unorm-dversty Genetc Algorthm (UGA), Expert Systems wth Applcatons, Vol 37,,, pp4-43 [] N Narman-Zadeh, Salehpour, A Jamal, and E Haghgoo, Pareto optmzaton o a ve-degree o reedom vehcle vbraton model usng a mult-obectve unorm-dversty genetc algorthm (UGA) (Artcle n Press), Engneerng Applcatons o Artcal ntellgence, do:6/engappa988 [] D Lm, YS Ong, and BS Lee, nverse mult-obectve robust evolutonary desgn optmzaton n the presence o uncertanty, GECCO, Washngton, USA,, pp -6 [] Papadraas, ND Lagaros, and V Plevrs, Structural optmzaton consderng the probablstc system response, Theoretcal Appled echancs, Vol 3, No 3-4, 4 pp [3] L Ray, RF Stengel, A onte Carlo Approach to the Analyss o Control System Robustness, Automatca, Vol 9, No, 993, pp 9-36 [4] H Kalos, PA Whtloc, onte Carlo ethod, Wley, New Yor, 986 [] LG Crespo, Probablstc Formulatons to Robust Optmal Control, 4th AAA/ASE/ASCE/AHS/ASC Structures, Structural Dynamcs & ateral Conerence, Palm Sprngs, Calorna, USA, 4 [6] Q Wang and RF Stengel, Robust control o nonlnear systems wth parametrc uncertanty, Automatca, Vol 38,, pp 9 99 [7] N Narman-Zadeh, K Atashar, A Jamal, A Plech, Yao, nverse modelng o mult-obectve thermodynamcally optmzed turboet engne usng GDH-type neural networs and evolutonary algorthms, Engneerng Optmzaton, Vol 37, 6,, pp A mult-obectve unorm-dversty genetc algorthm (UGA) has been proposed and successully used to optmally desgn lnear state eedbac controllers rom a relablty-based pont o vew n a probablstc approach wth uzzy threshold values The obectve unctons whch oten conlct wth each other were approprately dened usng some probablstc metrcs n tme doman The mult-obectve optmzaton o robust lnear state eedbac controllers led to the dscoverng some mportant trade-os among those obectve unctons The ramewor o such hybrd applcaton o mult-obectve GAs and onte Carlo Smulaton o ths wor or the Pareto optmzaton o both robust and relablty-based approach usng some non-commensurable stochastc obectve unctons s very promsng and can be generally used n the optmum desgn o real-world complex control systems wth probablstc uncertantes REFERENCES [] B Porter and AH Jones, Genetc tunng o dgtal PD controllers, Electronc Letters, 8(9), 99, pp [] DE Goldberg, Genetc Algorthms n Search, Optmzaton, and achne Learnng Addson-Wesley, 989 [3] WA Wolovch, Automatc control systems Harcourt Brace College Pub, Orlando, USA: Saunders College Publshng, 994