MODEL ORDER REDUCTION AND CONTROLLER DESIGN OF DISCRETE SYSTEM EMPLOYING REAL CODED GENETIC ALGORITHM J. S. Yadav, N. P. Patidar, J.

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1 ABSTRACT Research Artcle MODEL ORDER REDUCTION AND CONTROLLER DESIGN OF DISCRETE SYSTEM EMPLOYING REAL CODED GENETIC ALGORITHM J. S. Yadav, N. P. Patdar, J. Sngha Address for Correspondence Maulana Azad Natonal Insttute of Technology, Bhopal, Madhya Pradesh, Inda, Pn: 465 e-mals: One of the man objectves of order reducton s to desgn a controller of lower order whch can effectvely control the orgnal hgh order system so that the overall system s of lower order and easy to understand. In ths paper, a smple method s presented for controller desgn of a hgher order dscrete system. Frst the orgnal hgher order dscrete system n reduced to a lower order model. Then a Proportonal Integral Dervatve (PID) controller s desgned for lower order model. An error mnmzaton technque s employed for both order reducton and controller desgn. For the error mnmzaton purpose, Real-Coded Genetc Algorthm (RCGA) has been employed. RCGA method s based on the mnmzaton of the Integral Squared Error (ISE) between the desred response and actual response pertanng to a unt step nput. Fnally the desgned PID controller s connected to the orgnal hgher order dscrete system to get the desred specfcaton. The valdty of the proposed method s llustrated through a numercal example. KEYWORDS Dscrete System, Model Order Reducton, PID Controller, Integral Squared Error, Real Coded Genetc Algorthm.. INTRODUCTION The mathematcal procedure of system modelng often leads to detaled descrpton of a process n the form of hgh order dfferental equatons. These equatons n the frequency doman lead to a hgh order transfer functon. Therefore, t s desrable to reduce hgher order transfer functons to lower order systems for analyss and desgn purposes. Reducton of hgh order systems to lower order models has also been an mportant subject area n control engneerng for many years []. One of the man objectves of order reducton s to desgn a controller of lower order whch can effectvely control the orgnal hgh order system. The conventonal methods of reducton, developed so far, are mostly avalable n contnuous doman [-5]. However, the hgh order systems can be reduced n contnuous as well as n dscrete doman [6-8]. There are two approaches for the reducton of dscrete system, namely the ndrect method and drect method. The ndrect method uses some transformaton and then reducton s carred out n the transformed doman. Frst the z- doman transfer functons are converted nto s-doman by the blnear transformaton and then after reducng them n s-doman, sutably, they are converted back nto z-doman. In the drect method the hgher order z- doman transfer functons are reduced to a lower order transfer functon n the same doman wthout any transformaton. There are two common approaches for controller desgn. Frst approach s to obtan the controller on the bass of reduced order model called process reducton. In the second approach, the controller s desgned for the orgnal hgher order system and then the closed loop response of hgher order controller wth orgnal system s reduced pertanng to unty feedback called controller reducton. Both

2 the approaches have ther own advantages and dsadvantages. The process reducton approach s computatonally smpler as t deals wth lower order models and controller but at the same tme errors are ntroduced n the desgn process as the reducton s carred out at the early stages of desgn. In the controller reducton approach error propagaton s mnmzed as the desgn process s carred out at the fnal stages of reducton but the approaches deals wth hgher order models and thus ntroduces computatonal complexty. In recent years, one of the most promsng research felds has been Evolutonary Technques, an area utlzng analoges wth nature or socal systems. Evolutonary technques are fndng popularty wthn research communty as desgn tools and problem solvers because of ther versatlty and ablty to optmze n complex multmodal search spaces appled to non-dfferentable objectve functons.recently, Genetc Algorthm (GA) appeared as a promsng evolutonary technque for handlng the optmzaton problems. GA has been popular n academa and the ndustry manly because of ts ntutveness, ease of mplementaton, and the ablty to effectvely solve hghly nonlnear, mxed nteger optmsaton problems that are typcal of complex engneerng systems. When a GA s appled to optmzaton problems, characterstc preservng n desgnng codng/representaton s very mportant. Over the years, several researchers have concentrated on usng Real-Coded Genetc Algorthm (RCGA). It s reported that, for some problems, real-valued encodngs and assocated technques outperform conventonal bt strng approaches. In vew of the above, ths paper proposes to use RCGA optmzaton technque for both model reducton and controller desgn. In ths paper, controller desgn of a hgher order dscrete system s presented employng process reducton approach. The orgnal hgher order dscrete system n reduced to a lower order model employng RCGA technque. RCGA technque s based on the mnmzaton of the Integral Squared Error (ISE) between the transent responses of orgnal hgher order model and the reduced order model pertanng to a unt step nput. Then a Proportonal Integral Dervatve (PID) controller s desgned for lower order model. The parameters of the PID controller are tuned by usng the same error mnmzaton technque employng RCGA. The performance of the desgned PID controller s verfed by connectng the desgned PID controller wth the orgnal hgher order dscrete system to get the desred specfcaton. Despte sgnfcant strdes n the development of advanced control schemes over the past two decades, the conventonal lead-lag (LL) structure controller as well as the classcal proportonalntegral-dervatve (PID) controller and ts varants, reman the controllers of choce n many ndustral applcatons. These controller structures reman an engneer s preferred choce because of ther structural smplcty, relablty, and the favorable rato between performance and cost. Beyond these benefts, these controllers also offer smplfed dynamc modelng, lower user-skll requrements, and mnmal development effort, whch are ssues of substantal mportance to engneerng practce.

3 Problem Statement Model order reducton Gven a hgh order dscrete tme stable system of order n that s descrbed by the z -transfer functon: G O N ( z) ( z) = = D( z) b a + a z a + b z b The objectve s to fnd a reduced that has a transfer functon ( r< n ): N r ( z) R( z) = = D ( z) d r c + c z d z d n n n z n + bn z z n () th r order model r + cr z r r z + d r r z () The polynomal D(z) s stable, that s all ts zeros resde nsde the unt crcle z =. Where, a ( n ), b ( n ), c ( r ) and ( r) are scalar constants. d The numerator order s gven as beng one less than that of the denomnator, as for the orgnal system. The R (z) approxmates G ( z ) n some sense and retans the mportant characterstcs of G ( z ) and the transent responses of R(z) should be as close as possble to that of G ( z ) for smlar nputs. Controller desgn The proposed method of desgn of a controller by process reducton technque nvolves the followng steps: Step- Reduce the gven hgher order dscrete system to a lower order model by error mnmzaton technque. The objectve functon J s defned as an ntegral squared error of dfference between the responses gven by the expresson: J = t [ y ( t) y ( t)] dt (3) r Where y ( ) and y r (t) are the unt step responses t of orgnal and reduced order systems. Step- Desgn a PID controller for the reduced order system. The parameters of the PID controller are optmzed usng the same error same error mnmzaton technque employng RCGA. Step-3 Test the desgned PID controller for the reduced order model for whch the PID controller has been desgned. Step-4 Test the desgned PID controller for the orgnal hgher order model. 3. Proportonal Integral Dervatve (PID) Controller PID controller s basc type of feedback controller. The basc structure of conventonal feedback control systems s shown n Fgure, usng a block dagram representaton. In ths fgure, the process s the object to be controlled. In ths fgure, the object to be controlled s the process. To make the process varable y follow the set-pont value r s the man objectve of control. To acheve ths purpose, the manpulated varable u s changed at the authorty of the controller. The dsturbance d s any factor, other than the manpulated varable, that nfluences the process varable. Fgure assumes that only one dsturbance s added to the manpulated varable. In some applcatons, however, a major dsturbance enters the process n a dfferent way, or plural dsturbances need to be consdered. The error e s defned by e = r y.

4 r e + + (s) u + d C P(s) Controller Process y Fg.. Block dagram of basc feedback controller K P e(t) P Setpont + Error t Output KI e( t) dt Process + + I o D de( t) K D dt + Fg.. Structure of PID controller PID control s the method of feedback control that uses the PID controller as the man tool. PID controller s most wdely used n ndustral control applcatons because of ts structural smplcty, for best performance, the PID parameters used n the calculaton must be accordng to the nature of the system whle the desgn s generc, the parameters depend on the specfc system. The relablty, and the favorable rato between structure of a PID controller s shown n Fgure. performance and cost. Beyond these benefts, these The PID controller nvolves three separate controllers also offer smplfed dynamc modelng, parameters, and s accordngly sometmes called lower user-skll requrements, and mnmal three-term control: the Proportonalty, the ntegral development effort, whch are ssues of substantal and dervatve values, denoted by P, I, and D. The mportance to engneerng practce. A PID proportonal value determnes the reacton to the controller calculates an error value as the dfference between a measured process varable and a desred set pont. The controller attempts to mnmze the error by adjustng the process control nputs. In the absence of knowledge of the underlyng process, current error, the ntegral value determnes the reacton based on the sum of recent errors, and the dervatve value determnes the reacton based on the rate at whch the error has been changng. The weghted sum of these three actons s used to PID controllers are the best controllers. However, adjust the process va a control element.

5 Heurstcally, these values can be nterpreted n terms of tme: P depends on the present error, I on the accumulaton of past errors, and D s a predcton of future errors, based on current rate of change. By tunng the three constants n the PID controller algorthm, the controller can provde control acton desgned for specfc process requrements. The response of the controller can be descrbed n terms of the responsveness of the controller to an error, the degree to whch the controller overshoot sgnal overshoots the set pont and the degree of system oscllaton. Some applcatons may requre usng only one or two modes to provde the approprate system control. Ths s acheved by settng the gan of undesred control outputs to zero. A PID controller wll be called a PI, PD, P or I controller n the absence of the respectve control actons. PI controllers are farly common, snce dervatve acton s senstve to measurement nose, whereas the absence of an ntegral value may prevent the system from reachng ts target value due to the control acton. In applcaton, engneers have ndependence of mplementng the three functonal elements (P, I, and D) of the PID controller n whatsoever groupng they consder most sutable for ther problems. The combnaton of element(s) used s called the acton mode of the PID controller. Tunng a control loop s the adjustment of ts control parameters (gan/proportonal band, ntegral gan/reset, dervatve gan/rate) to the optmum values for the desred control response. Stablty (bounded oscllaton) s a basc requrement, but beyond that, dfferent systems have dfferent behavor, dfferent applcatons have dfferent requrements, and some desderata conflct. Further, some processes have a degree of non-lnearty and so parameters that work well at full-load condtons don't work when the process s startng up from noload; ths can be corrected by gan schedulng (usng dfferent parameters n dfferent operatng regons). PID controllers often provde acceptable control even n the absence of tunng, but performance can generally be mproved by careful tunng, and performance may be unacceptable wth poor tunng. The analyss for desgnng a dgtal mplementaton of a PID controller requres the standard form of the PID controller to be dscretsed. Approxmatons for frst-order dervatves are made by backward fnte dfferences. The ntegral term s dscretsed, wth a samplng t t tme, as follows: k = ( ) dτ e( t ) t e = τ 4) The dervatve term s approxmated as, de d ( tk) ( t) e = ( t ) e( t ) k t k 4. Real Coded Genetc Algorthm (RCGA) (5) Genetc algorthm (GA) has been used to solve dffcult engneerng problems that are complex and dffcult to solve by conventonal optmzaton methods. GA mantans and manpulates a populaton of solutons and mplements a survval of the fttest strategy n ther search for better solutons. The fttest ndvduals of any populaton tend to reproduce and survve to the next generaton thus mprovng successve generatons. The nferor ndvduals can also survve and reproduce. Implementaton of GA requres the determnaton of sx fundamental ssues: chromosome representaton, selecton functon, the genetc operators, ntalzaton, termnaton and evaluaton functon. Bref descrptons about these ssues are provded n the followng sectons. 4. Chromosome representaton Chromosome representaton scheme determnes how the problem s structured n the GA and also determnes the genetc operators that are used. Each ndvdual or chromosome s made up of a sequence of genes. Varous types of representatons of an

6 ndvdual or chromosome are: bnary dgts, ntegers, floatng pont, matrces, etc. Generally natural representatons are more effcent and produce better solutons. Floatng pont representaton of real numbers s more effcent n terms of CPU tme and offers hgher precson wth more consstent results. 4. Selecton functon To produce successve generatons, selecton of ndvduals plays a very sgnfcant role n a genetc algorthm. The selecton functon determnes whch of the ndvduals wll survve and move on to the next generaton. A probablstc selecton s performed based upon the ndvdual s ftness such that the superor ndvduals have more chance of beng selected. There are several schemes for the selecton process: roulette wheel selecton and ts extensons, scalng technques, tournament, normal geometrc, eltst models and rankng methods. The selecton approach assgns a probablty of selecton P j to each ndvduals based on ts ftness value. In the present study, normalzed geometrc selecton functon has been used. In normalzed geometrc rankng, the probablty of selectng an ndvdual P s defned as: r P = q'( q) (6) ' q q = (7) P ( q) where, q = probablty of selectng the best ndvdual r = rank of the ndvdual (wth best equals ) P = populaton sze 4.3 Genetc operator The basc search mechansm of the GA s provded by the genetc operators. There are two basc types of operators: crossover and mutaton. These operators are used to produce new solutons based on exstng solutons n the populaton. Crossover takes two ndvduals to be parents and produces two new ndvduals whle mutaton alters one ndvdual to produce a sngle new soluton. The followng genetc operators are usually employed: smple crossover, arthmetc crossover and heurstc crossover as crossover operator and unform mutaton, non-unform mutaton, mult-non-unform mutaton, boundary mutaton as mutaton operator. Arthmetc crossover and non-unform mutaton are employed n the present study as genetc operators. Smple crossover generates a random number r from a unform dstrbuton from to m (m s the dmenson of row vector representng the ndvduals n the populaton) and creates two new ndvduals usng equatons: X ' and Y ' from parents X and Y by X, f < r X ' = (8) Y otherwse Y, f < r Y ' = (9) X otherwse Arthmetc crossover produces two complmentary lnear combnatons of the parents, where r = U (, ): = r X + ( r Y X ' ) () = ry + ( r X Y ' ) () Non-unform mutaton randomly selects one varable j and sets t equal to an non-unform random number. X + ( b X ) f ( G) f = X + ( X + a ) f ( G) f X, otherwse ' X r <.5, r.5, ()

7 where, methods. Evaluaton functons or objectve G f ( G) ( r ( )).65 z b = (3) Gmax r, r = unform random nos. between to. G = current generaton. G max = maxmum no. of generatons. b = shape parameter. a and b = lower and upper bounds for each varable. 4.4 Intalzaton, evaluaton functon and stoppng crtera An ntal populaton s needed to start the genetc algorthm procedure. The ntal populaton can be randomly generated or can be taken from other N ( z) G ( z) = = D( z) z z +.5 z z.5 z z +.55 z z.63 z 4 functons of many forms can be used n a GA so that the functon can map the populaton nto a partally ordered set. The GA moves from generaton to generaton untl a stoppng crteron s met. The stoppng crteron could be maxmum number of generatons, populaton convergence crtera, lack of mprovement n the best soluton over a specfed number of generatons or target value for the objectve functon. 5 NUMERICAL EXAMPLE Consder the transfer functon of the plant from Mukherjee and Mshra (988) as: G (z)= z For whch a controller s to be desgned to get the desred output. 5. Applcaton of RCGA for model reducton 3.88 z z +.3z z+.5 (4) IS E E rror Generaton Fg. 3. Convergence of ftness functon

8 To reduce the hgher order model n to a lower order model RCGA s employed. The objectve functon J defned as an ntegral squared error of dfference between the responses gven by the equaton (3) s mnmzed by RCGA. For the mplementaton of RCGA, normal geometrc selecton s employed whch s a rankng selecton functon based on the normalzed geometrc dstrbuton. Arthmetc crossover takes two parents and performs an nterpolaton along the lne formed by the two parents. Non unform mutaton changes one of the parameters of the parent based on a nonunform probablty dstrbuton. Ths Gaussan dstrbuton starts wde, and narrows to a pont possble that the same parameters for GA do not gve the best soluton and so these can be changed accordng to the stuaton. One more mportant pont that affects the optmal soluton more or less s the range for unknowns. For the very frst executon of the program, more wde soluton space can be gven and after gettng the soluton one can shorten the soluton space nearer to the values obtaned n the prevous teraton. Optmzaton was performed wth the total number of generatons set to. A typcal convergence of objectve functon wth the number of generaton s shown n Fgure 3. The optmzaton processes s run tmes and best among the runs are taken as the fnal result. dstrbuton as the current generaton approaches the The reduced nd order model employng RCGA maxmum generaton. For dfferent problems, t s technque s obtaned as follows: z R ( z) = z z (5).8.6 Orgnal Conventnal RCGA.4. Ampltude Tme n sec. Fg. 4. Step Responses of orgnal system and reduced model

9 The unt step responses of orgnal and reduced systems are shown n Fgure 4. It can be seen that the steady state responses of proposed reduced order models s exactly matchng wth that of the orgnal model. Also, the transent response of proposed reduced model by RCGA s very close to that of orgnal model. It can be seen from Fgure 4 that both the orgnal model and the reduced model settle at a value of.7 for a nput of. (unt step nput). Now, to get the desred out put.e.., a PID controller s desgned. 5. Applcaton of RCGA for PID controller desgn In ths study, the PID controller has been desgned employng process reducton approach. The orgnal hgher order dscrete system gven by equaton (4) s reduced to a lower order model employng RCGA technque gven by equaton (5). Then the PID controller s desgned for lower order model. The parameters of the PID controller are tuned by usng the same error mnmzaton technque employng RCGA as explaned n secton 5.. The optmzed PID controller parameters are: K = , P K = , I K D = The unt step response of the reduced system wth RCGA optmzed PID controller and orgnal system wth RCGA optmzed PID controller are shown n Fgures 5 and 6. It s clear from Fgure 6 that the desgn of PID controller usng the proposed RCGA optmzaton technque helps to obtan the desgner s specfcatons n transent as well as n steady state responses for the orgnal system..8.6 Reduced second order model wthout PID controller Reduced second order model wth PID controller.4. Ampltude Tme n sec. Fg. 5. Step response of reduced model wth PID Controller

10 .8.6 Orgnal 8th order model wthout PID controller Orgnal 8th order model wth PID controller.4. Amltude CONCLUSION Tme n sec. Fg. 6. Step response of orgnal model wth PID Controller The proposed model reducton method uses the modern heurstc optmzaton technque n ts procedure to derve the stable reduced order model for the dscrete system. The algorthm has also been extended to the desgn of controller for the orgnal dscrete system. The algorthm s smple to mplement and computer orented. The matchng of the step response s assured reasonably well n ths proposed method. Algorthm preserves more stablty and avods any error between the ntal or fnal values of the responses of orgnal and reduced model. Ths approach mnmzes the complexty nvolved n drect desgn of PID Controller. The values for PID Controller are optmzed usng the reduced model and to meet the requred performance specfcatons. The tuned values of the PID controller parameters are tested wth the orgnal system and ts closed loop response for a unt step nput s found to be satsfactory wth the response of reduced order model. 7 REFERENCES. J. S. Yadav, N. P. Patdar, J. Sngha, S. Panda, and C. Ardl A Combned Conventonal and Dfferental Evoluton Method for Model Order Reducton, Internatonal Journal of Computatonal Intellgence, Vol. 5, No., pp. -8, 9.. S. Panda, J. S. Yadav, N. P. Patdar and C. Ardl, Evolutonary Technques for Model Order Reducton of Large Scale Lnear Systems, Internatonal Journal of Appled Scence, Engneerng and Technology, Vol. 5, No., pp. - 8, Y. Shamash, Contnued fracton methods for the reducton of dscrete tme dynamc systems, Int. Journal of Control, Vol., pages 67-68, C.P. Therapos, A drect method for model reducton of dscrete system, Journal of Frankln Insttute, Vol. 38, pp. 43-5, J.P. Twar, and S.K. Bhagat, Smplfcaton of dscrete tme systems by mproved Routh stablty crteron va p-doman, Journal of IE (Inda), Vol. 85, pp. 89-9, J.S. Yadav, N.P. Patdar and J. Sngha, S. Panda, Dfferental Evoluton Algorthm for Model Reducton of SISO Dscrete Systems, Proceedngs of World Congress on Nature & Bologcally Inspred Computng (NaBIC 9) 9, pp D.A. Wlson and R.N. Mshra, Desgn of low order estmators usng reduced models, Int. J. Control, Vol. 9, pp , J.A. Davs and R.E. Skelton, Another balanced controller reducton algorthm Systems and Controller Letters, Vol. 4, pp , 884.

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