Iterative Learning Control Based on Niche Shuffled Frog Leaping Algorithm Research. Xiaohong Hao, Dongjiang Wang

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1 nd Inernaional Conference on Auomaion, Mechanical Conrol and Compuaional Engineering (AMCCE 17) Ieraive Learning Conrol Based on Niche Shuffled Frog Leaping Algorihm Research Xiaohong Hao, Dongjiang Wang School of Compuer and Communicaion, Lanzhou Universiy of Technology, Lanzhou 735, China Keywords: Nonlinear sysem; Norm opimizaion; Parameer opimizaion; Niche shuffled frog leaping Algorihm. Absrac: Aiming a slow convergence and low precision opimizaion in ieraive learning conrol, a niche shuffled frog leaping algorihm was proposed in his paper, which combined memes algorihm and paricle swarm algorihm, using he niche shuffled frog leaping algorihm based on he resricive compeiion, avoiding he paedogenesis effecively improving he convergence speed and opimizaion accuracy. In order o achieve less error and monoone convergence in he ieraive domain, ge beer ransien racking performance and esablish a fas PID parameer opimizaion ieraive learning conrol algorihm based on he discree norm performance index, he PID conroller was inroduced ino he ieraive learning conrol parameer opimizaion algorihm o expand he algorihm's dimension, increase he degree of freedom in he opimal parameers, and ulimaely promoe he learning efficiency. 1. Inroducion Ieraive learning conrol serves as a conrol echnique for improving ransien response and racking of machines, devices or sysems characerized by repeiive moion. Through modifying conrol signals shor of expecaion, i can opimize learning conrol algorihm and hus improve racking performance of he sysem. Acual and expeced oupu consisenly apply ieraive learning conrol algorihm ino racking rajecory of expeced oupu. However, exising ieraive learning conrol sraegy suffers from defecs such as low learning efficiency, slow convergence rae as well as repeiive ieraive learning demanded for saisfacory racking effecs, which may hen lead o subsanial wases in he sysem and hereby affec producion efficiency. In order o furher improve accuracy of learning conrol and boos learning efficiency, i is suggesed o employ a combinaion beween opimal algorihm and ieraive learning conrol. Amann, Owens e al [1] from he Universiy of Sheffield have suggesed an opimizaion-based ieraive learning algorihm which has brough abou new vialiy o researches on ieraive learning conrol algorihm. The evoluion process of such an algorihm has already been sysemaically elaboraed in he reference par: an opimal ieraive learning conrol algorihm advanced by Amann e al in erms of he ime-invarian linear discree sysem, which can realize monoonic and geomerical convergence of racking error []; Owens, Fang and Häönen e al have pu forward a parameer opimal ieraive learning conrol (POILC) algorihm [3]; aferwards, Häönen and Owens raised he high-order parameer opimal ieraive learning conrol algorihm [4,5]. Alhough hese algorihms are srucurally simple and can realize monoonic and geomerical convergence of error o zero, he rae and efficiency of error convergence are sill quie no saisfying. In engineering applicaion, muliple ieraions are necessary for achieving saisfacory racking accuracy by using aforesaid mehods. Hazikos and Owens hen presened an opimal ieraive learning conrol (GA-ILC) [6]based on geneic algorihm (GA) and obained good conrol resuls. Laer, Hazikos e al exended his mehod[7] and achieved fine conrol effec in conrolling nonlinear sysem. Neverheless, his mehod Copyrigh 17, he Auhors. Published by Alanis Press. This is an open access aricle under he CC BY-NC license (hp://creaivecommons.org/licenses/by-nc/4./). 166

2 failed o encode useful prior informaion ino he algorihm, which led o an exremely exensive searching space of he algorihm; moreover, weak searching capabiliy of GA algorihm has also caused a low convergence rae of he algorihm. Shuffled frog leaping algorihm (SFLA) is a cooperaive-search swarm inelligen opimal algorihm proposed by Eusuff e al [8].Individuals in he algorihm are characerized by memeic evoluion and can realize global informaion exchanges by means of memeic algorihm. Suggesed by Moscao in 1989, memeic algorihm (MA) is a swarm inelligen algorihm uilizing heurisic search in handling opimizaion problems; analogous o chromosome, memeic [9] refers o informaion deposied in human or animal brain for ransmission as well as guidance of heir own behaviors. (Paricle swarm opimizaion PSO) algorihm acs as an evoluionary compuaion mehod adoped by Kennedy e al in 1995, which aims o cope wih opimizaion problems by simulaing foraging behavior of birds [1]. For he sake of furher improving convergence effecs of ieraive learning algorihm, while also in erms of specific demands in ieraive learning conrol, his paper improves shuffled frog leaping algorihm (SFLA) which has been successfully applied in solving varied opimizaion problems, and respecively pus forward a consrained linear sysem and nonlinear sysem ha are capable of processing he inpu.. ILC problem descripion Linear discree sysem model 1) Axk ( ) + Buk ( ) xk ( + (1) y ( ) Cx ( ) k k xk,uk, yk are he saus variable oupu variable and inpu variable of he sysem running for he k-h ime respecively. he desired rajecory on a given inerval [,T], and he oupu error of he k-h ime is, hen he learning law of he ieraive learning conrol can be expressed as he following recurrence for: () uk +1 f (uk, uk 1,, uk r, ek +1, ek, ek s ) lim ek, lim uk u* k k (3) The ieraive learning conrol is convergen. The ask of ieraive learning conrol is o seek conrol inpus so ha he rajecory of he conrolled objec achieves complee zero error racking along he desired rajecory over a finie ime inerval [,T], hen he deviaion beween he acual oupu and he desired oupu is zero, and requires he enire process o be compleed quickly. Full racking here refers o he sysem oupu from sar o finish, wheher ransien or seady-sae learning conrol, and o mainain he same arge rack. 3. Opimal Ieraive Learning Conrol Based on Niche Shuffled Frog Leaping Algorihm Research 3.1 NSFLA-ILC convergence analysis and opimal soluion The ulimae goal of ieraive learning algorihm is o solve he following opimizaion problem: e yd Pu. In order o faciliae he analysis, i is assumed min J k (uk ), is consrain condiion is u U ha all he desired signals are in he range of he conrolled objec. Then here mus be an opimal inpu, making he final racking error. (4) e yd Pu The informaion ransfer mode of Niche Shuffled Frog Leaping Algorihm (NSFLA) is conduced hrough classificaion of populaion and ineracion beween local search and global mixure process. Such kind of alernaing mehod could effecively inegrae he local informaion 167

3 wih global informaion, making he algorihm ge rid of being roubled by local opimal soluion. This mehod could make he frog leaping algorihm have more efficien calculaion performance and global search capaciy. The opimal ieraive learning conrol algorihm based on NSFLA adops he norm opimal index in NOILC:As he finess funcion of Niche Shuffled Frog Leaping Algorihm, hrough search, he opimal soluion uk*+1 used for nex ieraive operaion is acquired. Where, yk +1 ( ) [ Puk +1 ]( ). Provided he inpu (i is assumed ha here is one opimal soluion for quesion (4) a leas), hen he nex ieraive compuaion could make he formula (4) rue: When he conrolled objec P in yk +1 Puk +1 is linear ime invarian sysem (LTI), he opimal inpu could be acquired from formula (3.1): (5) uk + uk + P ek +1 1 Where P is he adjoin operaor of P. I is he non-causal realizaion of algorihm. There is following conclusion for he discree linear ime-invarian sysem: 1 (6) ek +1 ek 1+ σ Where, σ > is he minimum eigen value of P. Inequaliy (7) shows: in his case, he algorihm is uniformly convergen. For he linear sysem, he above algorihm could be realized. While he nonlinear sysem does no exis possibly, which could no be realized wih he adjoin operaor P. As for he conrol problem of nonlinear sysem, his Paper deals wih i hrough solving he opimizaion problem by using NSFLA in each ieraive process.. There is a descripion ha as long as SFLA could solve one opimal soluion of quesion (3.1),a leas, formula (5) is rue. As for non-linear sysem, NSFLA-ILC could guaranee ha is racking error is monoonic convergen in ieraion domain. When he objec is discree sysem, we consider he following opimizaion problem: (7) min J (uk +1 ) ek +1 + uk +1 uk uk +1 U Is consrain condiion is yk +1 Geuk +1, Ge is he conrolled objec model defined as follows: PB PAB PB Ge PA B PAB PB B PA N 1 B PA N B PB uk : [uk (), uk (1),, uk ( N 1)]T (8) (9) yk : [ yk (), yk (1),, yk ( N 1)]T (1) In algorihm (3.3, he norm of inpu deviaion is uk +1 uk γ ek, namely, he inpu deviaion is proporional o γ. Where Where, γ k +1 uk +1 ( ) uk ( ) + γ k +1ek +1 ( + 1) is chosen as he soluion of following opimizaion problem J (γ ) γmin k +1 R k +1 k +1 γ k +1 ) ek +1 + ωγ k+1 J k +1 ( (11) (1) Where, ω >. Such kind of algorihm has same convergence characerisics wih he norm opimal ieraive learning conrol algorihm inroduced in above secion (he consrain condiion is yk +1 Geuk +1 ). 168

4 3. Ieraive learning conrol algorihm based on Niche Shuffled Frog Leaping Algorihm 3..1 Niche Shuffled Frog Leaping Algorihm While solving he opimizaion problem, he Niche Shuffled Frog Leaping Algorihm has excellen global searching abiliy and fas convergence speed. As for he ypical shuffled frog-leaping algorihm, he local search will easily make i rap in he disadvanage of local opimum. The inroduced niche echnology limiing compeiion sraegy could improve he global opimizing abiliy of algorihm and accelerae he convergence speed of algorihm. Niche echnology can mainain he diversiy of he soluion, wih a high global opimizaion and convergence rae. I divides every generaion of individual populaion ino several caegories and each caegory has a group composed of excellen represenaives. They represen he individual wih he highes finess. They hybridize and muae in he same populaion and among differen populaions, and hen generae new generaion of populaion individual. Each individual survives in he specific environmen. Compeiion exiss in he same kind of individual and among differen individuals. There are informaion exchanges beween differen kinds of individuals. The radiional shuffled frog-leaping algorihm classifies he populaions. Before he local search, i inroduces RCS niche echnology and classifies he iniial populaion ino he muually exclusive sub-populaions. I forms independen searching space dynamically, resraining he convergence due o communiy cooperaion. (1) Iniializaion The sum of iniialized populaion soluion (frog) is F, he number of sub-populaion is m, he number of each sub-populaion soluion is n, dimension is S, he number of hybrid ieraion is G, he updaing number of each sub-populaion local is N, he maximum moving sep lengh of soluion is Dmax, he radius of niche is R and finess funcion. Wihin he feasible soluion domain, he iniial populaion composed of F soluion X is consiued randomly, and hen he finess ( x) of each soluion is calculaed. Where, he a (1 a F) h soluion is represened as X a (x1a, x a, x as ). () Generaion of frogs The iniial frog populaion U {u1 ( ), u ( ),, uf ( )} is generaed randomly, he ieraion value, given each frog ui ( ), i 1,,, F, and hen he finess funcion fi ( ) f (ui ( )) (formula (11)) is calculaed; he finess of populaion is sored in degressive mode, which shall be sored in he form of X i ( ) {ui ( ), fi ( )} and he opimal frog in he frog populaion is recorded as u g ( ). (3) Niche srucure adoping RCS sraegy Sep1: he i (1 i m 1) sub-populaion is compared wih i (1 i m 1) sub-populaion. The opimal soluion xib in i sub-populaion and he opimal soluion x jb in j sub-populaion are calculaed, and hen he Euclidean disance beween he opimal soluion xib in i sub-populaion and he opimal soluion x jb in j sub-populaion is calculaed. dij is compared wih niche radius R. Sep: if dij < R, he finess ( xib ) and finess ( x jb ) of xib and x jb are compared, oherwise skip o sep 6; if finess ( xib ) < finess ( x jb ), se he finess ( x jb ) as he infiniy and iniialize x jb randomly, oherwise se finess ( xib ) as he infiniy and iniialize xib randomly. Sep3: if all niche populaion has opimal soluion, coninue sep 4; oherwise skip o sep 1, repea pairwise comparison beween he opimal soluion of subpopulaion. (4) Local search Sep4: in each sub-populaion, updae he soluion xw wih wors finess in he sub-populaion; he moving sep lengh of soluion is: π (13) sin ( xb xw ) D N 169

5 The locaion of wors soluion afer updaing is: (14) xw xw + D, Dmax D Dmax, 1,, n Where, Dmax is he maximum sep lengh, is he algebra of curren local search. Sep5: in each sub-populaion, compare he finess '( xw ) of wors soluion afer updaing wih he original finess ( xw ) of wors soluion. if finess '( xw ) < finess ( xw ), replace he original soluion wih he updaed wors soluion; oherwise, and calculae he moving sep lengh of wors soluion xw. π sin ( xb xw ), 1,, n D (15) N And updae he locaion of wors soluion. In each sub-populaion, compare he finess '(x w ) of updaed wors soluion wih he finess (x w ) of original wors soluion, if finess ''(x w ) < finess (x w ), replace he original wors soluion wih he updaed wors soluion. (5) Eliminaion mechanism of sub-populaion Afer he compleion of local search evoluion, calculae he change value of finess of opimal soluion in each sub-populaion, if i is less han a smaller value wihin he prese coninuous algebras, such sub-populaion is eliminaed and i shall be iniialized again randomly, oherwise, coninue sep 4. Sep6: mix he soluions of all sub-populaions and consiue a complee populaion conaining F soluions. Sep7: Judge wheher i mees he prese mixing ieraion number or end condiion, if no, skip o sep 8 and conduc nex round of local search, oherwise, end i. Niche Shuffled Frog Leaping Algorihm ransfers informaion according o he classificaion of populaion, alernaing he local evoluion and re-mixing process and effecively inegraing he global informaion ineracion and local evoluion search. I has highly efficien compuaion performance and excellen global searching abiliy. According o he prior informaion of conrolled objec, he advisable op and lower limi value u j max and u j min wih differen momen inpu could be se, which can grealy reduce he searching space of opimizaion algorihm and hen reduce he compuaion cos. 4. Simulaion For linear sysem wih inpu consrain and expecaion of square wave sysem, he ieraion learning of Niche Shuffled Frog Leaping Algorihm shall be applied o verify he expecaion of such improved algorihm in he non-linear sysem, which could reach a good effec. The following non-linear conrol sysems are adoped + 1) AD x j (i ) + BD u j (i ) x j (i y j (i ) CD x j (i ) i,1,,1 j,1,.95 Where, AD ; BD.197 ; CD [ 1] Bring AD BD CD ino formula (3.1), obain p [ p1 (16) (17) p p1 ] ; he funcion pc relaed o k p, T k I, k D could be obained from p, and hen J pc 1 could be obained. The parameer seing of Niche Shuffled Frog Leaping Algorihm is shown as Table 4.1. Through 17

6 Niche Shuffled Frog Leaping Algorihm, he funcion value of finess could be obained hrough Niche Shuffled Frog Leaping Algorihm. The parameers of he shuffled frog leaping algorihm are showed in he able 3.1.The finess funcion values can be obained by he algorihm. (18) J pc , k P.9, k I -.65, k D 13.6 The value of k p, k I, kd is subsiued ino PID ieraive learning conrol in (19) i +1 u j +1 (i) u j (i + 1) +.9 e j (i ).65 e j (m)+13.6[e j (i + 1) e j (i )] (19) m 1 1. y d(i) y k(i) 1.8 y(i) i Fig. 1 Tracking performance of sine funcion log ek() k Figure NSFLA-ILC Nonlinear Sysem error convergence curve log ek() k Figure 3 CSA-ILC Nonlinear Sysem error convergence curve The parameer seing of Niche Shuffled Frog Leaping Algorihm is shown as Figure 1. Through Niche Shuffled Frog Leaping Algorihm, he funcion value of finess could be obained hrough Niche Shuffled Frog Leaping Algorihm. 5. Conclusions In erms of he improved Niche Shuffled Frog Leaping Algorihm, namely he applicaion of Niche Shuffled Frog Leaping Algorihm ino he ieraion learning conrol, his Paper pus forward he opimal ieraion-learning conrol algorihm based on Niche Shuffled Frog Leaping Algorihm. Compared wih radiional opimizaion mehod, he advanage of NSFLA-ILC lies in ha i could direcly solve non-linear and consrained inpu problem. NSFLA-ILC combines he advanages of 171

7 meme algorihm based on geneic algorihm and he paricle swarm opimizaion based on swarm foraging behavior, hus resuling in few algorihm parameers, fas compuaion speed and srong local searching abiliy, which could deal wih he consrained inpu problem properly. The real number encoding of algorihm and he inflicion filers of inpu acquired by improving NSFLA primely remove he high-frequency par of inpu due o he algorihm and make NSFLA-ILC have beer convergence effec. References [1] Jari Häönen. Issues of algebra and opimaliy in ieraive learning conrol[phd. hesis]. Deparmen of Process and Environmenal Engineering, Universiy of Oulu.4. [] V Hazikos, J Häönen, D H Owens. Geneic algorihms in norm-opimal linear and non-linear ieraive learning conrol. Inernaional[J]. Journal of Conrol, 4, 77(): [3] D H Owens, J Häönen. Ieraive Learning conrol-an opimizaion paradigm[j]. Annual Reviews in Conrol, 5(9), [4] J Häönen, D H Owens. Basis Funcion and Parameer Opimizaion in High-order Ieraive Learning Conrol[J]. Auomaic, 5(4), [5]Tayebi A. Analysis of wo paricular ieraive learning conrol schemes in frequency and ime domains[j]. Auomaica, 7, 43 (9), [6] V Hazikos, J Häönen, D H Owens. Geneic algorihms in norm-opimal linear and non-linear ieraive learning conrol. Inernaional[J]. Journal of Conrol, 4, 77(): [7] Jari Häönen. Issues of algebra and opimaliy in ieraive learning conrol[phd. hesis]. Deparmen of Process and Environmenal Engineering, Universiy of Oulu. 4. [8] Eusuff M M, Lansey K E. Opimizaion of waer disribuion nework design using he shuffled frog leaping algorihm[j]. Journal of Waer Resources Planning and Managemen, 3, 19(3): 1-5. [9]Moscao P. On evoluion, search, opimizaion, geneic algorihms and marial ars: Towards memeic algorihms[j]. Calech concurren compuaion program, C3P Repor, 1989, 86: [1]Kennedy J. Paricle swarm opimizaion[m]encyclopedia of Machine Learning. Springer US, 1: