Generating Lower Order System using Modified Truncation and PSO

Transcription

1 Special Iue of International Journal of Computer Application ( ) International Conference on Computing, Communication and Senor Networ (CCSN) Generating Lower Order Sytem uing Modified Truncation and PSO S. R. Deai #, Rajendra Praad * # Electrical Engineering Department, Indian Intitute of Technology Rooree, Rooree, India * Electrical Engineering Department, Indian Intitute of Technology Rooree, Rooree, India ABSTRACT A novel method of generating lower order ytem i being introduced which grap the advantage of Particle Swarm Optimization (PSO) technique and Modified Truncation (MT) Method. The propoed combination of PSO and MT i applied to original fourth and ixth order Linear Time Invariant (LTI) ytem. The denominator polynomial i reduced uing MT and numerator term uing PSO. The tep repone of the reultant lower order ytem and the original ytem are compared and their performance are jutified. Keyword Lower order ytem, Particle Swarm Optimization, Modified Truncation, Linear Time Invariant Sytem.. INTRODUCTION The rapidly increaing ytem complexity along with miniaturization in ize, ha reulted in great demand for fater imulation proce during the deign validation tage []. Inpite of having high peed proceor, lowering the ytem order i one way, which i generally practiced in ytem and control engineering field and i under active reearch. Thi further end up a a neceary procedure for imulating large complex ytem. Currently a variety of order reduction algorithm are being ued [-9] but none can be judged a the univerally bet a it depend upon how well it atifie the application pecification. Moreover, the bet reduction method hould alo preerve the vital dynamic characteritic of the ytem under conideration; implify the bet available model to uit the purpoe with le error a far a poible. Here, nature inpired approach which ha proved to be fruitful i roped in to meet the requirement. Particle Swarm Optimization (PSO) in combination with Modified Truncation (MT) being a conventional method i propoed for lowering the ytem order effectively []. In the recent pat, evolutionary technique have been ued in almot all field uccefully and have become popular. Thee technique have proved to be effective in developing lower order approximation for ytem having large dimenion and controller deign of the ame [-4]. The advantage of thee optimization method i that they help in the elimination /optimizing ome of the tate variable from the original or a tranformed ytem repreentation, a ta which cannot be accomplihed eaily. Finally, thi reult in reduction of torage and computation time without affecting the vital propertie of the original ytem. Thi enure that the reultant lower order ytem i viable for ue. In pite of the current popular optimization method, there i a great vehemence for the advancement of the o called global optimization method [5]. Active reearch i going on to develop a univeral optimization method that can be applied to all multifaceted problem with equal efficiency. PSO, a ubet of evolutionary computation technique i being ued widely for quite ome time. Thi technique ha upper hand over GA in term of fat convergence; implicity, require no rigid firt gue algorithm, eae of implementation and exploration of majority of problem pace are ome of the additional feature [3]. Alo, it i uncomplicated to code and undertand it mot baic form. Hence, it i found to be ueful in olving mixed integer optimization problem that are of typical complex engineering ytem [3].In thi paper, the benefit of PSO and Modified Truncation method i ued to generate the reduced order ytem [6]. The purpoe of uing PSO i to upport in earching the bet value among the available one to uit the requirement. The propoed method turned out to be comparable with other conventional technique.. PARTICLE SWARM OPTIMIZATION (PSO) PSO, a ubet of evolutionary computation ha been popular in academia and indutry, mainly becaue of it intuitivene, handle both dicrete and continuou variable. PSO wor well with any dimenion problem and finding the optimum for ingle objective and multi-objective function (nonlinear and linear) even though the problem of being tuc in local minima exit. PSO i imilar to Genetic Algorithm due to the tochatic population baed nature, but i eaier to implement with the ame. Further, thi tochatic population baed method come with a imple memory component. In concluion, PSO ha imilar or better reult than GA [7-8]. In 995, Kennedy and Eberhart introduced PSO algorithm to the world in term of ocial and cognitive behavior [9]. Till now, many reearcher are benefited by utilizing the ame to olve variou problem belonging to varied dicipline; fairly imple computation and haring of information within the algorithm a it derive it internal communication from the ocial behavior of individual are ome of the major attraction. Thee individual referred to particle are flown through the multidimenional earch pace with each particle repreenting a poible olution to the multi-dimenional optimization problem []. Each olution fitne i baed on a performance function related to the optimization problem being olved. The PSO proce i iced off by randomly initializing the particle poition and velocity within the entire earch range. Each candidate olution (particle poition) i expreed a a poition within the dimenional pace of the problem. The particle will move into the olution pace under the influence of the information obtained from iteration-to-iteration a well a particle-to-particle. Between iteration-to-iteration, the bet olution viited o far by a particle, i tored in it memory a pbet []. Liewie, the particle-to-particle information enure that, the bet olution viited by any particle i tored in it memory and experience an attraction toward thi olution, called gbet. The pbet and gbet are updated for each particle, after every iteration, till a better or more dominating olution (in term of fitne) i found. Thee information are called a ocial and cognitive component and thi whole proce 7

2 Special Iue of International Journal of Computer Application ( ) International Conference on Computing, Communication and Senor Networ (CCSN) continue till the deired reult i found within the computational limit. The heart of the PSO algorithm i the velocity equation and expree individual particle velocity a a balance between attraction to it own peronal bet poition and the current global bet poition among all particle. The velocity of each particle i updated uing the velocity update equation given by v v c r ( p x ) c r ( p x ) id id id id gd id () and the poition i updated uing x x v id id id Where, i =,,, S repreent the particle index, S i the ize of the warm, V i = (v i, v i,,v in ) T and P i = (p i, p i,,p in ) T i the aociated velocity and previouly bet viited poition of i th the particle, g i the index of the bet particle in the warm, c and c are contant, called cognitive and ocial caling parameter repectively (uually, c = c ; r, r are random number drawn from a uniform ditribution). Equation () and () define the claical verion of PSO algorithm. Later, the concept of an inertia weight wa developed to enhance control exploration and exploitation; introduced in the literature (998) and the reulting velocity update equation [,3] i given by v w* v c r ( p x ) c r ( p x ) id id id id gd id (3) The initial value of w i et to.9 and reduced linearly to.4, allowing initial exploration followed by acceleration toward an improved global optimum. Thi i the difference between local and global earching and i one of the reaon, the algorithm i o reitant to getting tuc in local minima [3].The flowchart howing the proce of PSO i a hown in Fig.. 3. STATEMENT OF PROBLEM Conider a nth order linear time invariant ingle input ingle output (LTI-SISO) ytem decribed by the tranfer function G () n j j ( n) j a b j j j Where a j and b j are calar contant. The objective i to find the th (<n) order reduced model R(), compriing of calar contant c i and d i repreented in the form of i c i i R( ) ; d ( ) i i d (5) i i () (4) No Start Specify the parameter for PSO Initialize warm Calculate velocitie and new potion Evaluate warm and update each particle Evaluate fitne function good? 4. GENERATION OF REDUCED ORDER MODELS The following ection decribe the procedure to find out the denominator and numerator polynomial of the reduced order model. A. Denominator Polynomial The denominator polynomial D() of the original ytem G() can be rewritten a D( ) b b b b... b (6) 3 n 3 4 n The direct truncation of D(), (n-) time give the r th order model that tend to approximate the pole and zero with a mall modulu rather than thoe with large modulu. Thi reult in good approximation for ytem dominated by the pole lying near the imaginary axi. However the ame method may perform badly for the dominant pole with large magnitude. To overcome thi drawbac we ue reciprocal tranformation [6,4]. Thi technique help in revering the order of the denominator coefficient and thereby the mall magnitude pole of D() of will become large magnitude pole of and vice vera. () n D D b n n n b b3... bn (7) Thi tranformation enable that ome dominant pole having mall magnitude and ome with large magnitude root may be retained in the reduced denominator. Thi enure good time repone matching in both tranient and teady tate region. The propoed modification conit of truncation of D(). Truncate time to obtain D (). Similarly truncate time to obtain. The reduced denominator D () i then obtained a D ( ) D ( ). D ( ) (8) d d d d... d * * * * 3 * 3 4 which i normalized to give Ye Stop Fig. Optimization proce 8

3 Amplitude Special Iue of International Journal of Computer Application ( ) International Conference on Computing, Communication and Senor Networ (CCSN) D( ) d d d d... d (9) Where, d + =, D () i reciprocal of and = +. To illutrate the propoed method econd order model are obtained for ome problem taen from literature. The general form of econd order model i taen a R () c c d d Where d,d are obtained from (8). B. Numerator Polynomial () Once the denominator polynomial i found, then the numerator coefficient c, c are found by uing the PSO algorithm by minimizing the fitne function f given by M i f y( it) yr( it) () G() RCauer() R() Propoed..4.6 Time (ec)..4 Fig Comparion of tep repone The tep repone of the original ytem G(), propoed ytem R() and R cauer () are hown in figure and are comparable. It i oberved that the propoed method perform much better than R(). y(it) and y r (it) are the unit tep repone of the higher order and the reduced order model at time t=t. Uually time T i taen a ec and t =. ec. The parameter etting ued are the initial population ize ( feaible olution), the number of iteration i 5. Further the teady tate error between the repone of the original and the reduced ytem can be nullified a in any general cae. 5. NUMERICAL EXAMPLES Ex : Conider a 4 th order ytem contrived by Shamah [5] G () Conider the denominator term 4 3 ( ) ( ) By modified truncation the following reduced denominator are found for variou value of and ;, ;, ;, Conidering the denominator obtained for = and =, and uing BBBC algorithm, the numerator polynomial of the reduced ytem will be The reduced econd order ytem i then given by R () The reduced econd order ytem obtained by [8] RCAUER () Ex : Conider another 6 th order ytem having tranfer function [6] G () Conider the denominator term ( ) ( ) By modified truncation the following reduced denominator are found for variou value of and ; 3, ;, ;, ;, Conidering the denominator obtained for =, = and uing the BBBC optimization algorithm, the numerator polynomial of the reduced ytem will be Therefore the third order reduced model obtained will be R () According to the method [], the reduced third order model i R DP () Figure 3 how the tep repone of the original ytem G(), propoed reduced ytem R() and the reduced ytem uing dominant pole[] R DP (). It i een that the repone of the G() and R() are matching both in teady and tranient tate, wherea the repone of R DP (S) perform wealy. 9

4 Amplitude Amplitude Special Iue of International Journal of Computer Application ( ) International Conference on Computing, Communication and Senor Networ (CCSN).4 Step Repone.4 Step Repone G() R() RDP().6.4. G() R() propoed RDP() Fig 3 Comparion of tep repone Ex : Conider a fourth-order ytem [] decribed by the tranfer function a G () Conider the denominator term 4 3 ( ) ( ) By modified truncation the following reduced denominator are found for variou value of and ;, 35 ;, ;, Conidering the denominator obtained for = and =, and uing BBBC algorithm, the numerator polynomial of the reduced ytem will be The reduced econd order ytem i then given by R () The reduced econd order ytem obtained by [] RDP () Time (ec) Figure 4 how the tep repone of the original ytem G(), the propoed reduced ytem R() and the reduced ytem uing dominant pole[] R DP (). It i een that the repone are matching both in teady and tranient tate Fig 4 Comparion of tep repone 6. CONCLUSIONS A new method of reducing the order of the original ytem i dicued. The ta i accomplihed uing the mixed method (PSO and modified truncation method). The denominator of the reduced method i obtained by the modified truncation method and the numerator of the reduced ytem i generated uing PSO. The application of the propoed method i jutified by comparing the tep repone in the above example. It i oberved that, the tep repone of the original and the propoed reduced ytem are cloely matching and the reult are better/comparable to that of the other method. Further, the propoed method hold good for multiple input and multiple output and dicrete ytem. Variou combination of conventional technique and PSO can alo be tried for lowering the order of the ytem. 7. REFERENCES Time (ec) [] Dia Abu-Al-Nadi, Othman MK Almadi, Zaer S Abo- Hammour, Reduced order modeling of linear mimo ytem uing particle warm optimization, The Seventh International Conference on Autonomic and Autonomou Sytem ICAS,. ISBN: , pp. 6-66, IARIA,. [] Obinata, G., and Anderon, B.D.O., Model Reduction for Control Sytem Deign, London, Springer-Verlag,. [3] M. S. Mahmoud and M. G. Singh, Large Scale Sytem Modelling, Pergamon Pre, International Serie on Sytem and Control t ed., Vol. 3, 98. [4] M. Jamhidi, Large Scale Sytem Modelling and Control Serie, New Yor, Amterdam, Oxford, North Holland, Vol. 9, 983. [5] R. Praad and J. Pal, Ue of continued fraction expanion for table reduction of linear multivariable ytem, Journal of Intitution of Engineer, India, IE(I) Journal EL, Vol. 7, pp , June 99. [6] R. Praad, A. K. Mittal and S. P. Sharma, A mixed method for the reduction of multi-variable ytem, Journal of Intitution of Engineer, India, IE(I) Journal EL, Vol. 85, pp. 77-8, March 5. [7] Antonio Lepchy and Umberto Viaro, A Note on the Model Reduction Problem, IEEE Tranaction On Automatic Conixol, Vol. Ac-8, No. 4, April 983. [8] R. Geneio and M. Milanee, A note on the derivation and ue of reduced order model, IEEE Tran. Automat. Control, Vol. AC-, No., pp. 8-, February 976.

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