The Application of Stochastic Optimization Algorithms to the Design of a Fractional-order PID Controller

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1 008 IEEE Regon 10 Colloquum and the Thrd ICIIS, Kharagpur, INDIA December Paper Identfcaton Number: 396 The Applcaton of Stochastc Optmzaton Algorthms to the Desgn of a Fractonal-order PID Controller Mthun Chakraborty, Deepyaman Mat, and Amt Konar Department of Electroncs and Telecommuncaton Engneerng Jadavpur Unversty Kolkata, Inda mthun.chakra108@gmal.com, deepyamanmat@gmal.com, konaramt@yahoo.co.n Abstract The Proportonal-Integral-Dervatve Controller s wdely used n ndustres for process control applcatons. Fractonal-order PID controllers are known to outperform ther nteger-order counterparts. In ths paper, we propose a new technque of fractonal-order PID controller synthess based on peak overshoot and rse-tme specfcatons. Our approach s to construct an objectve functon, the optmzaton of whch yelds a possble soluton to the desgn problem. Ths objectve functon s optmzed usng two popular bo-nspred stochastc search algorthms, namely Partcle Swarm Optmzaton and Dfferental Evoluton. Wth the help of a sutable example, the superorty of the desgned fractonal-order PID controller to an nteger-order PID controller s affrmed and a comparatve study of the effcacy of the two above algorthms n solvng the optmzaton problem s also presented. Keywords-Dfferental evoluton; domnant poles; nteger-order and fractonal-order PID controllers; partcle Swarm Optmzaton I. INTRODUCTION The mert of usng a Proportonal-Integral-Dervatve (PID) controller les n ts smplcty of desgn and good performance, ncludng low percentage overshoot and small settlng tme (whch s essental for slow ndustral processes). PID controllers belong to the class of domnatng ndustral controllers and, therefore, contnuous efforts are beng made to mprove ther qualty and robustness. An elegant way of enhancng the performance of PID controllers s to use fractonal-order controllers where the I- and D-actons have, n general, non-nteger orders. In order to grasp the sgnfcance of fractonal-order PID controllers, an understandng of the theory of fractonal calculus s necessary. Fractonal calculus s that branch of mathematcal analyss [13], whch generalzes the order of the dervatve or ntegral of a functon to a real number (not necessarly an nteger). If D denotes frst-order dfferentaton, then, we know D denotes two teratons of dfferentaton. Lkewse, D 1/ may be nterpreted as some operator whch, when appled twce to a functon successvely, wll have the same effect as a sngle dfferentaton [13]. Smlar explanatons hold for fractonal ntegraton too. Just as frstorder dfferentaton (or ntegraton) of a functon n tmedoman maps to multplcaton by s 1 (or s -1 ) of the Laplace Transform of the functon n s-doman, s α ndcates tmedoman dervaton to the order α f α > 0 or tme-doman ntegraton to the order α f α < 0. The name gven to ths generalzed dfferental/ntegral operaton s dfferntegraton. Of the several defntons of fractonal dfferntegrals, the Grünwald-Letnkov and Remann-Louvlle defntons [14] are the most used. These defntons are requred for the realzaton of dscrete control algorthms. In a fractonal PID controller, besdes the proportonal, ntegral and dervatve constants, denoted by K p, T and T d respectvely, we have two more adjustable parameters: the powers of s n ntegral and dervatve actons, -λ and respectvely. As such, ths type of controller has a wder scope of desgn, whle retanng the advantages of classcal PID controllers. Fndng the approprate settngs of the values of the fve parameters {K p,t,t d,λ,} to acheve optmal system performance thus calls for optmzaton on the fve-dmensonal space. Classcal optmzaton technques are not applcable here because of the roughness of the multdmensonal objectve functon surface. We, therefore, use dervatve-free optmzaton technques: the frst one Partcle Swarm Optmzaton (PSO) draws nspraton from the ntellgent, collectve behavor of a swarm of socal nsects (partcularly bees) foragng for food together and the other Dfferental Evoluton (DE) s an evolutonary algorthm that s guded by the prncples of Darwnan Evoluton and Natural Genetcs [1]. Traces of work on fractonal-order PID controllers are avalable n the current lterature [1]-[9] on control engneerng. A frequency doman approach based on the expected crossover frequency and phase margn s mentoned n []. A method based on pole dstrbuton of the characterstc equaton n the complex plane was proposed n [5]. A state-space desgn method based on feedback poles placement can be vewed n [6]. The fractonal-order controller can also be desgned by cascadng a proper fractonal unt to an nteger-order controller /08/$ IEEE 1

2 008 IEEE Regon 10 Colloquum and the Thrd ICIIS, Kharagpur, INDIA December Paper Identfcaton Number: 396 Our desgn focuses on postonng closed loop domnant poles, and fndng the optmal set of values of the desgn parameters that satsfy the constrants thus obtaned on the characterstc equaton. The work s thus orgnal and may open up new avenues for the next generaton fractonal-order controller desgn. Moreover, as t s already proven that the performance of fractonal-order PID controllers surpasses that of the classcal ones wth ntegro-dfferental operatons of nteger orders [3], our proposed desgn s lkely to fnd extensve applcatons n real ndustral processes. II. THE INTEGER AND FRACTIONAL ORDER PID CONTROLLERS The nteger-order PID controller has the followng transfer 1 functon: Kp + Ts + Tds. Here, the orders of ntegraton and dervaton are both unty. The real objects or processes that we wsh to control are generally fractonal n order (for example, the voltage-current relaton of a sem-nfnte lossy RC lne). As, for many of them the fractonalty s very low, ntegerorder approxmatons are appled. In general, however, the nteger-order approxmaton of the fractonal systems can cause sgnfcant dfferences between the mathematcal model and the real system. The man reason for usng nteger-order models was the absence of soluton methods for fractonal-order dfferental equatons. Fgure 1. Block dagram of a unty-feedback closed loop control system A fractonal PID controller has the transfer functon: K p + T s -λ + T d s, where λ and are postve real numbers. Takng λ =1, =1, we Fgure. Expandng from pont to plane wll have an nteger-order PID controller. Thus we see that, whle the nteger-order PID controller has three parameters, ts fractonal-order counterpart has as many as fve. The fractonal-order PID controller expands the ntegerorder PID controller from pont to plane, as shown n Fg.., thereby addng flexblty to controller desgn and allowng us to control our real world processes more accurately. III. REVIEW ON PSO AND DE ALGORITHMS A. The Optmzaton Problem The optmzaton problem conssts n determnng the global optmum (n our case, mnmum) of a contnuous realvalued functon of n ndependent varables x 1, x, x 3,, x n, mathematcally represented as f ( X ), where X = ( x 1,x,x 3,...,x n ) s called the parameter vector. Then the task of any optmzaton algorthm reduces to searchng the n- dmensonal hyperspace to locate a partcular pont wth poston-vector X such that 0 f ( X 0) s the global optmum of f ( X ). B. Partcle Swarm Optmzaton PSO [10], [11], [1] developed by Eberhart & Kennedy, s n prncple a mult-agent parallel search technque. We begn wth a populaton or swarm consstng of a convenent number, say m, of partcles conceptual enttes that fly through the mult-dmensonal search space as the algorthm progresses through dscrete (unt) tme-steps t = 0, 1,,, the populatonsze m remanng constant. In the standard PSO algorthm, each partcle P has two state varables: ts current poston X () t =[X,1 (t), X, (t),, X,n (t)] and ts current velocty V ()= t [V,1 (t), V, (t),, V,n (t)], =1,,,m. The poston vector of each partcle wth respect to the orgn of the search space represents a canddate soluton of the search problem. Each partcle also has a small memory comprsng ts personal best poston experenced so far, denoted by p () t and the global best poston found so far, denoted by g() t. Here, one poston s consdered better than another f the former gves a lower value of the objectve functon, also called the ftness functon n ths context, than the latter. For each partcle, each component X, j (0) of the ntal poston vector s selected at random from a predetermned search range [X L j, X U j ], whle each velocty component s ntalzed by choosng at random from the nterval [ V jmax, V jmax ], where V jmax s the maxmum possble velocty of any partcle n the jth dmenson, j = 1,,, n, = 1,,, m; the ntal settngs for p () t and g (t) are taken as p (0) X (0), g (0) X k(0) such that f ( X = = k(0) ) f ( X (0) ). After the partcles are ntalzed, the teratve optmzaton process begns, where the postons and veloctes of all the partcles are updated by the followng recursve equatons (1) and (). The equatons are presented for the jth dmenson of the poston and velocty of the th partcle. V,j( t + 1) = ω V,j() t + C 1ϕ1. ( p,j() t X,j()) t + C ϕ. ( g j() t X,j()) t (1) X,j( t+ 1) = X,j ( t) + V,j( t+ 1) () where the algorthmc parameters are defned as : /08/$ IEEE

3 008 IEEE Regon 10 Colloquum and the Thrd ICIIS, Kharagpur, INDIA December Paper Identfcaton Number: 396 ω : nertal weght factor, C 1,C : two constant multplers called self confdence and swarm confdence respectvely, φ 1, φ : two unformly dstrbuted random numbers. U We take V jmax = X j X L j j, ω = 0.79, C 1 = C = 1.494, 0<φ 1, φ 1. C. Dfferental Evoluton DE [1], [15], [17] belongs to the class of evolutonary algorthms where each tme-varyng parameter vector (canddate soluton) n the populaton s called a chromosome and each tme-step represents a generaton. The frst step of the algorthm, as usual, s: Intalzaton: Ths step s dentcal to the random ntalzaton of poston vectors n PSO. Each teraton conssts of the followng three steps: Mutaton: For each chromosome X () t belongng to the current generaton, three other chromosomes X p () t, X q () t, and X r () t are randomly selected from the same generaton (, p, q and r are dstnct); the scaled dfference of X q () t and X r () t s added to X p () t to generate a donor vector V ( t+ 1) : V ( t + 1) = X ( t) + F X ( t) X ( t) ( ) p q r where F s a constant scalar belongng to (0,1). We take F = 0.8. Recombnaton: In ths step, a tral offsprng vector T ( t+ 1) s created for each current-generaton parent vector X () t by frst choosng a constant CR (0<CR<1) called the crossover constant and then settng the jth component T,j( t+ 1) of T ( t+ 1) accordng to the followng crteron: T,j ( t+ 1) = V,j( t) f rand j (0,1) < CR, X,j ( t) otherwse, where rand j (0, 1) s a random number selected from the nterval (0, 1), j = 1,,..., n. We take CR = Selecton: Ths step s guded by the prncple of survval of the fttest and may be mathematcally expressed as follows: T ( t 1) f f ( T ( t 1)) f ( X ( t )), ( 1) + X t + < + = X ( t) otherwse, = 1,,...,m. Thus, the next-generaton populaton s generated, keepng the populaton-sze m always unchanged. IV. FORMULATION OF THE OBJECTIVE FUNCTION AND ITS OPTIMIZATION Our approach s based on the root locus method (domnant poles method) of desgnng ntegral PID controllers [16]. As n the tradtonal root locus method, the peak overshoot M p and rse tme t rse (or, n other words, requrements of stablty and dampng levels) are specfed. From these specfcatons, we fnd out the dampng rato ζ and the undamped natural frequency ω n, makng use of the followng formulae [16]: ζ ω n = = π ln ( M p ) { ln ( M p )} + π 1 tan t rse 1 ζ 1 ζ ζ Usng these computed values of ζ and ω n, we then determne the desred postons of the domnant poles þ 1, of the closed loop system [16]: þ ζω ω ζ 1, = n ± j n 1 (5) = a ± j b, where a = ζω n, b = ω 1 ζ n, for a, b > 0. Let G p (s) be the transfer functon of the process we want to control, G c (s) the transfer functon of the controller to be desgned and H(s) the transfer functon of the feedback-path, as shown n Fg. 1. Then, the closed loop transfer functon of the controlled system s: Gs () Ts () =, 1 + GsHs ( ) ( ) where G(s) = G c (s).g p (s) s the forward path transfer functon. As we use fractonal-, and G c (s) s of the form G c (s) = K p + T s -λ + T d s (6) Therefore, n general, the characterstc equaton of the closed-loop system s: 1 + G(s)H(s) = 0 (7) Assumng unty feedback, we have H(s)=1. In ths case, the characterstc equaton becomes 1 + G(s) = 0 or, 1 + G c (s).g p (s) = 0 or, 1 + G c (s).p(s)/q(s) = 0 or, Q(s) + G c (s).p(s) = 0 (8) where P(s) and Q(s) are respectvely the numerator and denomnator polynomals of G(s), P(s) and Q(s) have no common factor. (3) (4) /08/$ IEEE 3

4 008 IEEE Regon 10 Colloquum and the Thrd ICIIS, Kharagpur, INDIA December Paper Identfcaton Number: 396 As þ 1, must be poles of the closed loop system, each of them must be a root of the characterstc equaton (7) and hence must satsfy equaton (8). Thus, puttng s = þ 1 = a ± j b n equaton (8), we obtan Q(þ 1 ) + G c (þ 1 ).P(þ 1 ) = 0 or, Q( a+jb)+[k p +T ( a+jb) -λ +T d ( a+jb) ].P( a+jb) = 0 (9) Equaton (9) s a complex equaton n fve unknowns, namely K p, T, T d, λ, and our problem of desgnng a controller, whch makes the closed loop domnant poles of the system concde wth þ 1,, now reduces to determnng the set of values of {K p, T, T d, λ, } for whch (9) holds good. But as the number of unknowns exceeds the number of equatons, there exsts an nfnte number of soluton sets and the equaton cannot be unambguously solved by tradtonal methods. Ths necesstates the applcaton of stochastc global search technques, whch, n turn, requres the formulaton of a sutable objectve functon or cost functon. Let R = real part of the L.H.S. of equaton (9), I = magnary part of the L.H.S. of equaton (9), P = tan -1 (I / R). We defne f (Kp, T, Td, λ), = R + I + P as our objectve functon. Clearly, f 0, n general, and f = 0 f and only f R = 0 and I = 0 and P = 0,.e. f and only f equaton (9) s satsfed. So, we now employ frst PSO and then DE to scour the fve-dmensonal search space K p 0, T 0, T d 0, 0 λ, 0 and home n on the optmal soluton set * * * * * {K, T, T, λ, } for whch f = f mn = 0. p d The lmts on the components of the poston-vectors of the partcles/chromosomes (.e. the controller parameters) are set by us as follows: as a practcal consderaton, we assume 0 K p, T, T d 1000, 0 λ,. In order to desgn an nteger order PID controller for controllng the same process, we smply set λ = = 1, so that the soluton space becomes three-dmensonal. All other condtons reman the same and the optmzaton process s * * * executed as before to obtan the optmal set {Kp, T, T d}. It s germane to menton here that the formulae (3), (4) and (5) hold strctly only for a second order system wth ts complex conjugate pole-par at þ 1,. However, for a hgherorder system wth a domnant pole-par (real part of ths polepar s much smaller than those of other poles), these formulae are wdely used n control engneerng applcatons wth a far degree of accuracy [16]. Nevertheless, after desgnng our controller wth the help of these formulae, we perform a smulaton to obtan the unt step response [16] of the closedloop control system, as a check. V. ILLUSTRATION A. Problem Statement The process (control objectve) has the transfer functon Gp () s =. 0.8s + 0.5s + 1 We want to desgn a controller such that the closed loop system has a peak overshoot M p 10% and rse-tme t rse 0.3 seconds. B. Soluton Usng the formulae (3) and (4), for the lmtng case, we obtan ζ = and ω n = s -1. Thus the domnant poles for the closed loop controlled system should le at þ 1 = ( j7.345) and þ = ( j7.345). As usual, ee assume unty feedback. The controller transfer functon s gven by (6). Puttng s = þ 1 = ( j7.345) n the characterstc equaton, we obtan: λ Kp + T ( j7.345) + T d ( j7.345) 1+ = ( j7.345) ( j7.345) + 1 T [Kp λ cos(. 03λ) + T d ( ) cos(. 03)] T + j[ λ sn(. 03λ) + T d ( ) sn( 03)]. = (10) After separatng the real and magnary parts, we have: R = T (Kp ) + λ cos(.03λ) + T d (9.107) cos(.03) (11) I = T λ sn (.03λ) + T d (9.107) sn (.03) (1) P = tan -1 (I / R). (13) Thus, (11), (1) and (13) gve us our objectve functon f (Kp, T, Td, λ), = R + I + P whch s mnmzed by PSO and by DE separately. If we set λ = 1 and = 1 before runnng the optmzaton algorthm, we obtan the three optmzed parameters for the nteger order PID controller. All results are presented n the next secton. Although we have constructed the objectve functon f by makng use of the by makng use of domnant pole þ 1 = a+jb n the second quadrant, we would arrve at the same f f we had started wth the thrd-quadrant domnant pole þ = a jb. Ths s because, n the latter case, the magnary part of the reduced characterstc equaton (9) would just be the negatve of what /08/$ IEEE 4

5 008 IEEE Regon 10 Colloquum and the Thrd ICIIS, Kharagpur, INDIA December Paper Identfcaton Number: 396 we have obtaned n (10) so that f, whch nvolves absolute values only, would reman unaltered. Ths s true not only for the partcular problem n queston but n general as well. TABLE I. RESULTS OF OPTIMIZATION FOR FRACTIONAL ORDER PID CONTROLLER DESIGN Algo. used Optmzed parameters for fractonal-order PID controller K p T T d λ PSO DE TABLE II. RESULTS OF OPTIMIZATION FOR INTEGER ORDER PID CONTROLLER DESIGN Algo. used Optmzed parameters for nteger-order PID controller (λ==1) K p T T d PSO DE VI. RESULTS Although we allowed a maxmum of 5000 teratons of PSO, we found that, n about teratons, the ftness value (.e. value of f) of the best partcle dropped below the tolerance value of (practcally equal to the perfect value of zero) and almost all other partcles had ftness values very close to the best. Smlar observatons were made for optmzaton by DE. Table I gves us the values of the controller parameters obtaned usng PSO and DE for fractonal order whle Table II gves us the correspondng data for the nteger-order case. From these tables, the expressons for the controller transfer functon are obtaned as shown n Table III: Fgure 3. Unt Step Responses for desgn usng PSO TABLE III. CONTROLLER TRANSFER FUNCTIONS Algo. Order of PID controller G c(s) used Fractonal Integral s s 1 + PSO s s s s 1 + DE 46.7 s s Wth each of these expressons for G c (s), we compute the overall system transfer functon T(s) and obtan the tmeresponse of the correspondng system to a unt step nput R(s)=1/s by fndng the nverse Laplace Transform of T(s)/s. Fg. 3. shows the plots of the unt step responses of the uncontrolled open-loop system gven n the example consdered, the closed-loop system controlled by the ntegral PID controller desgned usng PSO and the same system controlled by the fractonal PID controller also desgned usng PSO. Fg. 4. shows the correspondng plots for the desgn usng DE. The values of M p and t rse calculated graphcally for each of the controlled systems desgned are presented n Table IV. Fgure 4. Unt Step Responses for desgn usng DE /08/$ IEEE 5

6 008 IEEE Regon 10 Colloquum and the Thrd ICIIS, Kharagpur, INDIA December Paper Identfcaton Number: 396 TABLE IV. PEAK OVERSHOOT AND RISE-TIME DATA FOR THE DIFFERENT SYSTEMS DESIGNED Order of M p (%) t rse (seconds) controller PSO DE PSO DE ntegral fractonal VII. CONCLUSION AND FUTURE RESEARCH DIRECTIONS From Table III, many facts are apparent. Frstly, the rsetme requrement s satsfactorly met by all controllers whereas the nteger-order controllers perform poorly n terms of peak overshoot. By comparng the rows of the table, we can easly observe the sgnfcant reducton n both M p and t rse (whch translates to mproved system performance or better complance wth a gven set of user specfcatons) that may be acheved by replacng the tradtonal nteger-order PID controller wth ts fractonal counterpart. Agan, a careful comparson of the columns of the table reveals that the overall results (partcularly wth respect to M p ) gven by DE are notably better than those gven by PSO. Apart from the example shown, we also consdered several other smlar problems, solved them by our proposed method and drew smlar nferences from the results thereof. These have not been ncluded n ths paper owng to paucty of space. Hence, we may conclude that our desgn method s vable as t does produce fractonal PID controllers superor to the ones wth ntegral orders; moreover, DE appears to be a better opton than PSO as an algorthm for mnmzng the relevant objectve functon. An mportant pont s to be noted n ths context. For the fnal system to exhbt the desred performance, t s necessary that the evaluated pole-values þ 1, truly correspond to the domnant poles of the closed-loop controlled system. However, the optmzaton of our cost functon ensures only that þ 1, are poles of the system but does not guarantee that these are the domnant poles. In other words, equaton (9) embodes a necessary, but not suffcent, condton for the domnance of þ 1,. Obvously, there may be a number of possble combnatons of values of the parameters [K p, T, T d, ] for whch the resultng characterstc equaton s satsfed by þ 1, but, n each such case, the characterstc equaton wll have other roots (closed-loop system poles), too, and these poles wll also play an mportant role n determnng the overall system response. So, usng each of PSO and DE, we solved the optmzaton problem several tmes and naturally dd not obtan the same result every tme. The tme response for each soluton set was studed and the best one was reported. Ths s also a possble explanaton of the observaton that the performance of the fnal desgned system wth fractonal controller s actually better than desred. In constructng our objectve functon, we took the coeffcent of each of the terms R, I and P to be unty, whch means that we attached equal weghts to each of these terms. Intutvely, t s understood that the frst two terms are more mportant than the thrd. We are currently nvestgatng what mprovement n our method could be acheved by varyng the coeffcents of these terms or by constructng a superor cost functon (such as one that also checks whether the evaluated poles are truly domnant). We also ntend to apply other wellknown algorthms such as Genetc Algorthm and ts many varants to the same optmzaton problem and compare ther performance wth that of the two already studed. REFERENCES [1] I. Podlubny, I. Petras, B. M. Vnagre, P. O Leary, and L. Dorcak, Analogue realzatons of fractonal-order controllers, Nonlnear Dynamcs, vol. 9, pp , 00. [] B. M. Vnagre, I. Podlubny, L. Dorcak, and V. Felu, On fractonal PID controllers: A frequency doman approach, Proc. Of IFAC Workshop on Dgtal Control Past, Present and Future of PID Control, pp , 000. [3] S. Mlos and C. Martn, The fractonal-order PID controller outperforms the classcal one, 7 th Internatonal Scentfc-Techncal Conference PROCESS CONTROL 006, June 13-16, 006, Kouty nad Desnou, Czech Republc. 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Eberhart, Partcle swarm optmzaton, Proc.of the IEEE Internatonal Conference on Neural Networks, pp , [11] J. Kennedy and R. C. Eberhart, Swarm Intellgence, ISBN , Academc Press (001). [1] A. Konar and S. Das, Recent advances n evolutonary search and optmzaton algorthms, NGMS 006, January 11-13, 006, BESU, Shbpur, Howrah, Inda. [13] [14] [15] R. Storn and K. Prce, Dfferental evoluton A Smple and Effcent Heurstc for Global contnuous spaces, Journal of Global Optmzaton, 11(4), 1997, [16] I.J. Nagrath and M. Gopal, Control Systems Engneerng, Ffth Edton (007), ISBN: , New Age Internatonal Publshers, pp , , [17] S. Das, A. Konar, and U. K. Chakraborty, Two Improved Dfferental Evoluton Schemes for Faster Global Search, ACM-SIGEVO Proceedngs of Genetc and Evolutonary Computaton Conference (GECCO-005), Washngton DC, June, /08/$ IEEE 6