Controller Design of Discrete Systems by Order Reduction Technique Employing Differential Evolution Optimization Algorithm

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1 Contolle Design of Discete Systems by Ode Reduction Technique Employing Diffeential Evolution Optimization Algoithm J. S. Yadav, N. P. Patida, and J. Singhai Abstact One of the main objectives of ode eduction is to design a contolle of lowe ode which can effectively contol the oiginal high ode system so that the oveall system is of lowe ode and easy to undestand. In this pape, a simple method is pesented fo contolle design of a highe ode discete system. Fist the oiginal highe ode discete system in educed to a lowe ode model. Then a Popotional Integal Deivative (PID) contolle is designed fo lowe ode model. An eo minimization technique is employed fo both ode eduction and contolle design. Fo the eo minimization pupose, Diffeential Evolution (DE) optimization algoithm has been employed. DE method is based on the minimization of the Integal Squaed Eo (ISE) between the desied esponse and actual esponse petaining to a unit step input. Finally the designed PID contolle is connected to the oiginal highe ode discete system to get the desied specification. The validity of the poposed method is illustated though a numeical example. Keywods Discete System, Model Ode Reduction, PID Contolle, Integal Squaed Eo, Diffeential Evolution. I. INTRODUCTION HE mathematical pocedue of system modeling often T leads to detailed desciption of a pocess in the fom of high ode diffeential equations. These equations in the fequency domain lead to a high ode tansfe function. Theefoe, it is desiable to educe highe ode tansfe functions to lowe ode systems fo analysis and design puposes. Reduction of high ode systems to lowe ode models has also been an impotant subject aea in contol engineeing fo many yeas [,]. One of the main objectives of ode eduction is to design a contolle of lowe ode which can effectively contol the oiginal high ode system. The conventional methods of eduction, developed so fa, ae mostly available in continuous domain. Howeve, the J. S. Yadav is woing as an Associate Pofesso in Electonics and Communication Engg. Depatment, MANIT Bhopal, India ( jsy@ediffmail.com). N. P. Patida is woing as an Associate pofesso in Electical Engineeing Depatment, MANIT, Bhopal, India. ( nppatida@yahoo.com) J. Singhai is woing as Associate Pofesso in Electonics and Communication Engineeing Depatment, MANIT Bhopal, India. ( j_singhai@manit.ac.in) high ode systems can be educed in continuous as well as in discete domain [3-5]. Thee ae two appoaches fo the eduction of discete system, namely the indiect method and diect method. The indiect method uses some tansfomation and then eduction is caied out in the tansfomed domain. Fist the z- domain tansfe functions ae conveted into s-domain by the bilinea tansfomation and then afte educing them in s-domain, suitably, they ae conveted bac into z-domain. In the diect method the highe ode z- domain tansfe functions ae educed to a lowe ode tansfe function in the same domain without any tansfomation [6]. Thee ae two common appoaches fo contolle design. Fist appoach is to obtain the contolle on the basis of educed ode model called pocess eduction [7]. In the second appoach, the contolle is designed fo the oiginal highe ode system and then the closed loop esponse of highe ode contolle with oiginal system is educed petaining to unity feedbac called contolle eduction [8]. Both the appoaches have thei own advantages and disadvantages. The pocess eduction appoach is computationally simple as it deals with lowe ode models and contolle but at the same time eos ae intoduced in the design pocess as the eduction is caied out at the ealy stages of design. In the contolle eduction appoach eo popagation is minimized as the design pocess is caied out at the final stages of eduction but the appoaches deals with highe ode models and thus intoduces computational complexity. In ecent yeas, one of the most pomising eseach fields has been Evolutionay Techniques, an aea utilizing analogies with natue o social systems. Evolutionay techniques ae finding populaity within eseach community as design tools and poblem solves because of thei vesatility and ability to optimize in complex multimodal seach spaces applied to non-diffeentiable objective functions. Diffeential evolution (DE) is a banch of evolutionay algoithms developed by Raine Ston and Kenneth Pice in 995 fo optimization poblems [9]. It is a population-based diect seach algoithm fo global optimization capable of handling non-diffeentiable, nonlinea and multi-modal objective functions, with few, easily chosen, contol paametes. It has demonstated its usefulness and obustness in a vaiety of applications such as, Neual 64

2 netwo leaning, Filte design and the optimization of aeodynamics shapes. DE diffes fom othe evolutionay algoithms (EA) in the mutation and ecombination phases. DE uses weighted diffeences between solution vectos to change the population wheeas in othe stochastic techniques such as genetic algoithm (A) and expet systems (ES), petubation occus in accodance with a andom quantity. DE employs a geedy selection pocess with inheent elitist featues. Also it has a minimum numbe of EA contol paametes, which can be tuned effectively []. In view of the above, this pape poposes to use DE optimization technique fo both model eduction and contolle design. In this pape, contolle design of a highe ode discete system is pesented employing pocess eduction appoach. The oiginal highe ode discete system in educed to a lowe ode model employing DE technique. DE technique is based on the minimization of the Integal Squaed Eo (ISE) between the tansient esponses of oiginal highe ode model and the educed ode model petaining to a unit step input. Then a Popotional Integal Deivative (PID) contolle is designed fo lowe ode model. The paametes of the PID contolle ae tuned by using the same eo minimization technique employing DE. The pefomance of the designed PID contolle is veified by connecting the designed PID contolle with the oiginal highe ode discete system to get the desied specification. Despite significant stides in the development of advanced contol schemes ove the past two decades, the conventional lead-lag (LL) stuctue contolle as well as the classical popotional-integal-deivative (PID) contolle and its vaiants, emain the contolles of choice in many industial applications. These contolle stuctues emain an enginee s pefeed choice because of thei stuctual simplicity, eliability, and the favoable atio between pefomance and cost. Beyond these benefits, these contolles also offe simplified dynamic modeling, lowe use-sill equiements, and minimal development effot, which ae issues of substantial impotance to engineeing pactice. II. PROBLEM STATEMENT A. Model ode eduction Please iven a high ode discete time stable system of ode n that is descibed by the z -tansfe function: O N ( z) a a z... a z ( z) () n z n n = = n D( z) b b z... bn z bn The objective is to find a educed has a tansfe function ( < n ): R ( z ) N ( z ) c c z... th ode model that c = = D ( z ) d d z... d z z d z () The polynomial D(z) is stable, that is all its zeos eside inside the unit cicle z =. Whee, a i ( i n ), b i ( i n ), c i ( i ) and d i ( i ) ae scala constants. The numeato ode is given as being one less than that of the denominato, as fo the oiginal system. The R (z) appoximates ( z) in some sense and etains the impotant chaacteistics of ( z) and the tansient esponses of R(z) should be as close as possible to that of ( z) fo simila inputs. B. Contolle design All The poposed method of design of a contolle by pocess eduction technique involves the following steps: Step- Reduce the given highe ode discete system to a lowe ode model by eo minimization technique. The objective function J is defined as an integal squaed eo of diffeence between the esponses given by the expession: J = t [ y ( t) y ( t)] dt (3) Whee y ( t) and y (t) ae the unit step esponses of oiginal and educed ode systems. Step- Design a PID contolle fo the educed ode system. The paametes of the PID contolle ae optimized using the same eo same eo minimization technique employing DE. Step-3 Test the designed PID contolle fo the educed ode model fo which the PID contolle has been designed. Step-4 Test the designed PID contolle fo the oiginal highe ode model. III. PROPORTIONAL INTERAL DERIVATIVE (PID) CONTROLLER PID contolle is basic type of feedbac contolle. The basic stuctue of conventional feedbac contol systems is shown in Fig., using a bloc diagam epesentation. In this figue, the pocess is the object to be contolled. In this figue, the object to be contolled is the pocess. To mae the pocess vaiable y follow the set-point value is the main objective of contol. To achieve this pupose, the manipulated vaiable u is changed at the authoity of the contolle. The distubance d is any facto, othe than the manipulated vaiable, that influences the pocess vaiable. Fig. assumes that only one distubance is added to the manipulated vaiable. In some applications, howeve, a majo distubance entes the pocess in a diffeent way, o plual 64

3 distubances need to be consideed. The eo e is defined by e = y. e (s) u _ d C P(s) Contolle Pocess Fig.. Bloc diagam of basic feedbac contolle PID contol is the method of feedbac contol that uses the PID contolle as the main tool. PID contolle is most widely used in industial contol applications because of its stuctual simplicity, eliability, and the favoable atio between pefomance and cost. Beyond these benefits, these contolles also offe simplified dynamic modeling, lowe use-sill equiements, and minimal development effot, which ae issues of substantial impotance to engineeing pactice. A PID contolle calculates an eo value as the diffeence between a measued pocess vaiable and a desied set point. The contolle attempts to minimize the eo by adjusting the pocess contol inputs. In the absence of nowledge of the undelying pocess, PID contolles ae the best contolles. Howeve, fo best pefomance, the PID paametes used in the calculation must be accoding to the natue of the system while the design is geneic, the paametes depend on the specific system. The stuctue of a PID contolle is shown in Fig.. K P e(t) P Setpoint Eo t Output KI e( t) dt Pocess _ I o D de( t) K D dt Fig.. Stuctue of PID contolle The PID contolle involves thee sepaate paametes, and is accodingly sometimes called thee-tem contol: the Popotionality, the integal and deivative values, denoted by P, I, and D. The popotional value detemines the eaction to the cuent eo, the integal value detemines the eaction based on the sum of ecent eos, and the deivative value detemines the eaction based on the ate at which the eo has been changing. The weighted sum of these thee actions is used to adjust the pocess via a contol element. Heuistically, these values can be intepeted in tems of time: P depends on the pesent eo, I on the accumulation of past eos, and D is a pediction of futue eos, based on cuent ate of change. By tuning the thee constants in the PID contolle algoithm, the contolle can povide contol action designed fo specific pocess equiements. The esponse of the contolle can be descibed in tems of the esponsiveness of y the contolle to an eo, the degee to which the contolle oveshoot signal oveshoots the set point and the degee of system oscillation. Some applications may equie using only one o two modes to povide the appopiate system contol. This is achieved by setting the gain of undesied contol outputs to zeo. A PID contolle will be called a PI, PD, P o I contolle in the absence of the espective contol actions. PI contolles ae faily common, since deivative action is sensitive to measuement noise, wheeas the absence of an integal value may pevent the system fom eaching its taget value due to the contol action. In application, enginees have independence of implementing the thee functional elements (P, I, and D) of the PID contolle in whatsoeve gouping they conside most suitable fo thei poblems. The combination of element(s) used is called the action mode of the PID contolle. Tuning a contol loop is the adjustment of its contol paametes (gain/popotional band, integal gain/eset, deivative gain/ate) to the optimum values fo the desied contol esponse. Stability (bounded oscillation) is a basic equiement, but beyond that, diffeent systems have diffeent behavio, diffeent applications have diffeent equiements, and some desideata conflict. Futhe, some pocesses have a degee of non-lineaity and so paametes that wo well at full-load conditions don't wo when the pocess is stating up fom no-load; this can be coected by gain scheduling (using diffeent paametes in diffeent opeating egions). PID contolles often povide acceptable contol even in the absence of tuning, but pefomance can geneally be impoved by caeful tuning, and pefomance may be unacceptable with poo tuning. The analysis fo designing a digital implementation of a PID contolle equies the standad fom of the PID contolle to be discetised. Appoximations fo fist-ode deivatives ae made by bacwad finite diffeences. The integal tem is discetised, with a sampling Δt time, as follows: t ( ) dτ e( t )t Δ e τ = (4) i = The deivative tem is appoximated as, de d ( t ) () t i e = ( t ) e( t ) Δt (5) IV. DIFFERENTIAL EVOLUTION Diffeential Evolution (DE) algoithm is a stochastic, population-based optimization algoithm ecently intoduced [9]. DE wos with two populations; old geneation and new geneation of the same population. The size of the population is adjusted by the paamete N P. The population consists of eal valued vectos with dimension D that equals the numbe of design paametes/contol vaiables. The population is 64

4 andomly initialized within the initial paamete bounds. The optimization pocess is conducted by means of thee main opeations: mutation, cossove and selection. In each geneation, individuals of the cuent population become taget vectos. Fo each taget vecto, the mutation opeation poduces a mutant vecto, by adding the weighted diffeence between two andomly chosen vectos to a thid vecto. The cossove opeation geneates a new vecto, called tial vecto, by mixing the paametes of the mutant vecto with those of the taget vecto. If the tial vecto obtains a bette fitness value than the taget vecto, then the tial vecto eplaces the taget vecto in the next geneation. The evolutionay opeatos ae descibed below [9, ]., Diffeence Vecto F ( ) V =., 3, F. (, 3, i,, ) C. Cossove Thee paents ae selected fo cossove and the child is a petubation of one of them. The tial vecto U is i, developed fom the elements of the taget vecto ( i, ) and the elements of the dono vecto ( i, ).Elements of the dono vecto ente the tial vecto with pobability CR as: V j, i, if and j, i CR o j = I and U j, i, = j, i, if and j, i > CR o j I and (7) With and j, ~ U (,), I i and is a andom intege fom (,,.D) whee D is the solution s dimension i.e numbe of contol vaiables. I and ensues that V i, i,. D. Selectionn The taget vecto i, is compaed with the tial vecto V i, and the one with the bette fitness value is admitted to the next geneation. The selection opeation in DE can be epesented by the following equation:,, 3, U i, if f ( U i, ) < f ( i, = i, othewise. whee i [, N P ]. i, ) (8) 3, Fig. 3 Vecto addition and subtaction in diffeential evolution A. Initialization Fo each paamete j with lowe bound bound U j L j and uppe, initial paamete values ae usually andomly selected unifomly in the inteval [ L j, U j ]. B. Mutation Fo a given paamete vecto i,, thee vectos (,, 3, ) ae andomly selected such that the indices i,, and 3 ae distinct. A dono vecto V i, is ceated by adding the weighted diffeence between the two vectos to the thid vecto as: V =, F.(, 3, ) (6) i, Whee F is a constant fom (, ). V. NUMERICAL EAMPLE Conside the tansfe function of the plant fom efeences [, ] as: N ( z) ( z) = = D( z) (.65 z.63 z ( z z.5 z.88 z z.5 z.3z.43 ) 6.78 z.55 z.57 z z.985 z.65 z.5 ) (9) Fo which a contolle is to be designed to get the desied output. A. Application of DE fo Model Ode Reduction To educe the highe ode model in to a lowe ode model DE is employed. The objective function J defined as an integal squaed eo of diffeence between the esponses given by the equation (3) is minimized by DE. Implementation of DE equies the detemination of six fundamental issues: DE step size function, cossove pobability, the numbe of population, initialization, temination and evaluation function. eneally DE step size (F) vaies in the inteval (, ). A good initial guess to F is in the inteval (.5, ). Cossove pobability (CR) constants ae geneally chosen fom the inteval (.5, ). If the paamete is co-elated, then high value of CR wo bette, the evese is

5 tue fo no coelation []. In the pesent study, a population size of N P =, geneation numbe =, step size F=.8 and cossove pobability of CR =.8 have been used. Optimization is teminated by the pe-specified numbe of geneations fo DE. One moe impotant facto that affects the optimal solution moe o less is the ange fo unnowns. Fo the vey fist execution of the pogam, a wide solution space can be given and afte getting the solution one can shoten the solution space neae to the values obtained in the pevious iteation. The flow chat of the DE algoithm employed in the pesent study is given in Fig. 4. One moe impotant point that affects the optimal solution moe o less is the ange fo unnowns. Fo the vey fist execution of the pogam, moe wide solution space can be given and afte getting the solution one can shoten the solution space neae to the values obtained in the pevious iteation. Optimization was pefomed with the total numbe of geneations set to. Simulations wee conducted on a Pentium 4, 3 Hz, 54 MB RAM compute, in the MATLAB 7.. envionment. A typical convegence Stat Specify the DE paametes Initialize the population en.= Evalute the population Ceate offspings and evalute thei fitness.48z.58 R ( z) = (5).3465z.53953z.5634 The unit step esponses of oiginal and educed systems ae shown in Fig. 6. It can be seen that the steady state esponses of poposed educed ode models is exactly matching with that of the oiginal model. Also, the tansient esponse of poposed educed model by DE is vey close to that of oiginal model. It can be seen fom Fig. 6 that both the oiginal model and the educed model settle at a value of.7 fo a input of. (unit step input). Now, to get the desied out put i.e.., a PID contolle is designed. ISE Eo eneation Fig. 5. Convegence of fitness function en. = en Yes No Is fitness of offsping bette than fitness of paents? Yes Replace the paents by offspings in the new population Size of new population < Old population? No en. > Max. en? Yes Stop No Discad the offsping in new population Fig. 4 Flow chat of poposed DE optimization appoach of objective function with the numbe of geneation is shown in Fig.5. The optimization pocesses is un times and best among the uns ae taen as the final esult. The educed nd ode model employing DE technique is obtained as given in equation (5): B. Application of DE fo PID Contolle Design In this study, the PID contolle has been designed employing pocess eduction appoach. The oiginal highe ode discete system given by equation (4) is educed to a lowe ode model employing DE technique given by equation (5). Then the PID contolle is designed fo lowe ode model. The paametes of the PID contolle ae tuned by using the same eo minimization technique employing DE as explained in section 5.. The optimized PID contolle paametes ae: K = 6.75, K = , K =. 589 P I The unit step esponse of the educed system with DE optimized PID contolle and oiginal system with DE optimized PID contolle ae shown in Fig. 7 and 8. It is clea fom Fig. 8 that the design of PID contolle using the poposed DE optimization technique helps to obtain the designe s specifications in tansient as well as in steady state esponses fo the oiginal system. D 644

6 .6.4. Amplitude Oiginal 8th ode discete model Reduced nd ode discete model by DE Time in sec Fig. 6. Step Responses of oiginal system and educed model.5 Amplitude.5 Reduced nd ode model without PID contolle Reduced nd ode model with PID contolle Time in sec Fig. 7. Step esponse of educed model with PID Contolle.5 Amplitude.5 Oiginal 8th ode model without PID contolle Oiginal 8th ode model with PID contolle Time in sec Fig. 8. Step esponse of oiginal model with PID Contolle 645

7 VI. CONCLUSION The poposed model eduction method uses the moden heuistic optimization technique in its pocedue to deive the stable educed ode model fo the discete system. The algoithm has also been extended to the design of contolle fo the oiginal discete system. The algoithm is simple to implement and compute oiented. The matching of the step esponse is assued easonably well in this poposed method. Algoithm peseves moe stability and avoids any eo between the initial o final values of the esponses of oiginal and educed model. This appoach minimizes the complexity involved in diect design of PID Contolle. The values fo PID Contolle ae optimized using the educed model and to meet the equied pefomance specifications. The tuned values of the PID contolle paametes ae tested with the oiginal system and its closed loop esponse fo a unit step input is found to be satisfactoy with the esponse of educed ode model. REFERENCES [] J. S. Yadav, N. P. Patida, J. Singhai, S. Panda, and C. Adil A Combined Conventional and Diffeential Evolution Method fo Model Ode Reduction, Intenational Jounal of Computational Intelligence, Vol. 5, No., pp. -8, 9. [] S. Panda, J. S. Yadav, N. P. Patida and C. Adil, Evolutionay Techniques fo Model Ode Reduction of Lage Scale Linea Systems, Intenational Jounal of Applied Science, Engineeing and Technology, Vol. 5, No., pp. -8, 9. [3] Y. Shamash, Continued faction methods fo the eduction of discete time dynamic systems, Int. Jounal of Contol, Vol., pages 67-68, 974. [4] C.P. Theapos, A diect method fo model eduction of discete system, Jounal of Fanlin Institute, Vol. 38, pp. 43-5, 984. [5] J.P. Tiwai, and S.K. Bhagat, Simplification of discete time systems by impoved Routh stability citeion via p-domain, Jounal of IE (India), Vol. 85, pp. 89-9, 4. [6] J.S. Yadav, N.P. Patida and J. Singhai, S. Panda, Diffeential Evolution Algoithm fo Model Reduction of SISO Discete Systems, Poceedings of Wold Congess on Natue & Biologically Inspied Computing (NaBIC 9) 9, pp [7] D.A. Wilson and R.N. Misha, Design of low ode estimatos using educed models, Int. J. Contol, Vol. 9, pp , 979. [8] J.A. Davis and R.E. Selton, Anothe balanced contolle eduction algoithm Systems and Contolle Lettes, Vol. 4, pp , 884. [9] R Ston. And K. Pice, Diffeential Evolution A simple and efficient adaptive scheme fo lobal Optimization ove continuous spaces, Jounal of lobal Optimization, Vol., pp , 995. [] Sidhatha Panda, Diffeential Evolutionay Algoithm fo TCSC-based Contolle Design, Simulation Modelling Pactice and Theoy, Vol. 7, pp , 9. [] S. Muhjee and R.N. Misha, Optimal ode eduction of discete systems, Jounal of Institution of Enginees, Vol. 68, pp. 4-49, 988. [] K. Ramesh, A. Nimaluma and. uusamy, Design of discete contolle via a novel model ode eduction technique, Intenational Jounal of Electical Powe and Engineeing, Vol. 3, pp , 9. J S Yadav is woing as Associate Pofesso in Electonics and Communication Engineeing Depatment, MANIT, Bhopal, India. He eceived the B.Tech.degee fom EC Jabalpu in Electonics and Communication Engineeing in 995 and M.Tech. degee fom MANIT, Bhopal in Digital Communication in. Pesently he is pesuing Ph.D. His aea of inteest includes Optimization techniques, Model Ode Reduction, Digital Communication, Signal and System and Contol System. Naayana Pasad Patida is woing as Associate Pofesso in Electical Engineeing Depatment, MANIT, Bhopal, India. He eceived his PhD. degee fom Indian Institute of Technology, Rooee, in 8, M.Tech. degee fom Visvesvaaya National Institute of Technology Nagpu in Integated Powe System in 995 and B.Tech. degee fom SSITS Indoe in Electical Engineeing in 993. His aea of eseach includes voltage stability, secuity analysis, powe system stability and intelligent techniques. Optimization techniques, Model Ode Reduction, and Contol System. Jyoti Singhai is woing as Associate Pofesso in Electonics and Communication Engineeing Depatment, MANIT Bhopal, India. She eceived the PhD. degee fom MANIT Bhopal in 5, M.Tech. degee fom MANIT, Bhopal in Digital Communication in 997 and B.Tech. degee fom MANIT Bhopal in Electonics and Communication Engineeing in 99. He aea of eseach includes Optimization techniques, Model Ode Reduction, Digital Communication, Signal and System and Contol System. 646