Control n Dabetc Patents Assst. Prof. Dr. Control and Systems Engneerng Department, Unversty of Technology, Baghdad-Iraq hazem..al@uotechnology.edu.q Receved: /6/3 Accepted: //3 Abstract In ths paper, the desgn of robust blood glucose controller n dabetes usng H-nfnty technque s presented. The Partcle Swarm Optmzaton (PSO) method s used to tune the specfc structure controller parameters subject to H-nfnty constrants. The Bergman model s used to represent the artfcal pancreas. Ths model s one of the more wdely used models of the effect of nsuln nfuson and glucose nputs on the blood glucose concentraton. The results show the effectveness of the desgned controller n controllng the behavor of glucose devaton to a sudden rse n the blood glucose. The proposed controller can effectvely attenuate the blood glucose devaton to.5. Ths value of attenuaton makes the proposed controller superor to the other controllers n prevous works. Matlab 7. s used to demonstrate the smulaton results. Keywords Type I dabetes, PSO, Robust control, H-nfnty control.
. Introducton Many nnate feedback control loops are found n the human body. One of these loops s the loop that regulates the blood glucose by the producton of nsuln by the pancreas. People wth dabetes have a reduced capablty of producng nsuln. Type I dabetes melltus patents cannot produce any nsuln and must admnster nsuln shots several tmes a day to help regulate ther blood glucose level. A typcal patent s then servng as a control system []. On the other hand, any patent that suffers from dabetes and not properly receves the nsuln cure can lead to complcatons such as nerve damage, bran damage, amputaton and eventually death. In the human body, the normal blood glucose concentraton level vares n a narrow range (7-) mg/dl. The dabetes s dagnosed f for some reason the human body s unable to control the normal glucose-nsuln nteracton []. The dabetes can be classfed nto four types: Type I whch s known as nsuln dependent dabetes melltus, Type II whch s known as nsuln ndependent dabetes melltus, gestatonal dabetes and the genetc deflectons [3]. The beta cells are responsble for producng the nsuln, whch s essental for uptake of glucose n muscles and storage n the lver. In dabetes, beta cells are destroyed and the human body wll be unable to control the blood glucose level. For ths reason, the blood glucose must be regulated by njectng the nsuln [4]. Therefore, several approaches have been proposed for the desgn of closed loop glucose regulaton system, whch requres three components, whch are: glucose sensor, nsuln pump and a control algorthm for determnng the necessary nsuln dosage based on the glucose measurements []. Fg. shows the block dagram of closed loop glucose regulaton system. desred blood glucose Controller Insuln Pump Blood Glucose Sensor Patent Fgure : Closed loop nsuln regulaton system block dagram []. In ths paper, a Partcle Swarm Optmzaton (PSO) method s used to tune a specfc structure controller subject to robust H-nfnty constrants. The Bergman model s used n ths work to represent the pancreas bologcal system. Ths model s the wdely used mathematcal model due to ts smplcty.. Partcle Swarm Optmzaton (PSO) method blood glucose In ths secton an overvew of the Partcle swarm optmzaton (PSO) method s presented. It s one of the powerful optmzaton methods wth hgh effcency n comparson to other methods. It s nspred from studes of socal behavour among ants and brds. The PSO mechansm s ntalsed wth a populaton of random solutons and searches for optma by updatng generatons. The potental solutons of PSO are called partcles, whch fly through the problem space by followng the current optmum partcles. Each partcle represents a canddate soluton to the problem at hand. The partcles always change ther postons by flyng around n a multdmensonal search space untl computatonal lmtatons are exceeded. The partcles keep track of ther coordnates n the problem space, whch are assocated wth the best soluton (best ftness) they have acheved so far. The best partcle n the populaton s typcally
denoted by the global best, whle the local best denotes the best poston that has been vsted by the current partcle. The global best ndvdual connects all members of the populaton to one another. That s, each partcle s nfluenced by every best performance of any member n the entre populaton. The (local best) ndvdual s conceptually seen as the ablty for partcles to remember past personal success. The partcle swarm optmzaton concept nvolves, at each tme step, changng the velocty of each partcle towards ts global best and local best locatons [5], [6], [7]. The velocty of each partcle s adjusted accordng to ts own flyng experence and the flyng experence of other partcles. The -th partcle s represented as x ( x,, x,,..., x, d) n the d-dmensonal space. The best prevous poston of the partcle, s recorded and represented as: b b b x x, x,..., x ). b (,,, d The ndex of best partcle among all of the partcles n the group s gbest. The velocty for partcle s represented as: v ( v,, v,,..., v, d ). The modfed velocty and poston of each partcle can be calculated usng the current velocty and the dstance from x b, d to gbest as shown n the followng equatons [8]-[]: k k b k g k v h v c rand ( x x ) c rand ( x x ) () k k k x x v () k k where v s the partcle velocty, x s the current partcle poston, h s the nerta weght factor, b x s the best prevous g poston of the -th partcle, x s the best partcle among all the partcles n the populaton, rand s a random functon between and, c and c are acceleraton constants, k s the ponter of teratons, =,,3,,N, where N s the number of partcles. Nowadays, the PSO algorthm has been wdely used to solve non-lnear and multobjectve problems such as optmsaton of weghts of neural network, electrcal utlty, computer games and moble robots. It s requred only a few lnes of computer code to realze PSO algorthm. Also, t s characterzed as a smple, easy to mplement, and computatonally effcent algorthm []. 3. Dabetc Mathematcal Model An adequate model s necessary to desgn an approprate control. Several models that descrbe the nsuln-glucose regulatory system and close to physologcal process have been appeared for Type I dabetes patents. For example, the most complex dabetc model proved to be the 9 th order Sorensen model []. On the other hand, one of the more wdely used models of the effect of nsuln nfuson and glucose nputs on the blood glucose concentraton s known as the Bergman s three state mnmal patent model. Ths model s descrbed by the followng dfferental equatons [], [3]: dg Gmeal pg X ( G Gb ) dt V dx p X p3i dt di U n( I Ib) dt V (3) Where G and I represent the devaton n blood glucose and nsuln concentratons respectvely. X s proportonal to the nsuln concentraton n a remote compartment. G meal and U are the meal dsturbance nput of glucose and the manpulated nsuln nfuson rate respectvely. The parameters p, p, p3, n and V are the blood volume. G s the basal base lne or steady state b 3
value of blood glucose and nsuln concentraton. I b s the Table : Dabetc model parameters []. A lnear state space model can be developed for use n control system desgn as []: x p x x 3 y Gb p x u d x x 3 x p3 x n x3 V V u d (4) where x G, x X, x3 I, u U Ub, d Gmeal and. Combnng the Laplace transformatons of (4) allows the operatng pont dependant transfer functon model of the open loop system to be wrtten as: y( s) G ( s) u( s) G ( s) d( s) (5) p d Where G p (s) s the transfer functon of the system from the devaton n blood glucose to the manpulated nsuln nfuson G d (s) s the transfer functon from the devaton n blood glucose to the meal dsturbance nput of glucose. It s mportant to refer that the states, nput and output varables are defned n devaton form. The set of parameters that are used for the modeled dabetc n (4) are gven n Table. Snce the concentratons are n mmol/lter, and the glucose dsturbance has unts of grams, the converson factor of 5.5556 mmol/g must be appled to the G meal. Also, t s more common to work wth glucose concentraton unts of mg/declter rather than mmol/lter. Snce the molecular weght of glucose s 8 g/mol, we must multply the glucose state (mmol/lter) by 8 to obtan the measured glucose output n (mg/declter). 4. Controller Desgn y Several G types of control algorthms for blood glucose regulaton have been developed n lterature. Some of these algorthms are PID and Fuzzy logc controllers [], Back steppng based PID controller [3], Sgnal correcton technque [3] and model predctve control [4]. However, n ths work the desgn of specfc structure controller s proposed. The parameters of the proposed controller are tuned usng PSO method by mnmzng the cost functon subject to H constrants to desgn a robust controller. H s one of the best known technques for robust control and t s an effectve method for attenuatng dsturbances and nose that appear n the system [4]. In ths work, the robust controller s desgned so that H -norm from the nput dsturbance d ( G meal ) to the outputs ( e and e ) s mnmzed. e and e are the weghted error sgnal and weghted control sgnal respectvely. Fg. shows the block dagram of the controlled system wth weghts and the proposed controller G c (s). The augmented plant P s derved to be: e WW 3Gd e e W3Gd Parameter W G p d W u G p Value G b ( mmol / lter) 4.5 I b ( mu / lter) 4.5 V ( lter) (mn p ). (mn p ). p3( mu / lter).3 n(mn ) 5/54 (6) 4
desred blood glucose + where u s the control sgnal, W, W and W 3 are performance weghtng functon, control weghtng functon and dsturbance flter respectvely. The lower lnear fractonal transformaton of the augmented plant and controller can be descrbed by: e Fl e WW G S WW 3GdGcS 3 d ( P, G ) d (7) c where S ( G p ( s) Gc ( s)) s the senstvty functon. - e The objectve of H technque s to fnd the controller G c (s) that nternally stablzes the system and effectvely attenuates the dsturbances. Mnmzng the followng cost functon satsfes ths objectve: WW3Gd S T WW 3GdGc S (8) where T s the transfer functon of the system from nput d to the outputs e and. e W ( s) PSO Tunng Algorth m (s) G c Fgure : Block dagram of PSO based robust controller desgn. d G p (s) W 3( s ) W ( s) (s) G d + + e e blood glucose The PSO method s used to tune the parameters of the controller G c (s) and the parameters of the selected performance weghtng functon W by mnmzng the cost functon n (8). On the other hand, the general formulaton for the controller structure s descrbed by: m m ams am s... a Gc ( s) m m bms bms... b (9) where m represents the selected controller order. In ths work, the selected order s sx order whch s a very large search doman for the requred robust controller. The obtaned optmal controller usng PSO from the search space by mnmzng the cost functon n (8) s: 4 3.867s.673s 84.39s.66s.4537 ( s) 4 3.8s 4.6s.s 3.6s 8.99 G c () The selected structure for the performance weghtng functon s a nd order structure and the obtaned performance weghtng functon usng PSO algorthm s:.385s.367s 77 ( s).77s 5.55s W () The control weghtng functon and dsturbance flter are selected by tral and error to be:. W ( s ) and W s) (). 3(.4s The parameters that have been used for carryng out the desgn of robust controller usng PSO are: populaton sze=, nerta weght factor h=, c c, the maxmum teraton= and the number of functon evaluatons s. The flowchart of PSO based robust controller desgn s shown n Fg. 3. 5. Smulaton Results and Dscusson The PSO based robust controller was desgned to regulate the blood glucose level n Type I dabetes. Fg. 4 shows the 5
meal dsturbance functon, whch represents the exogenous glucose nfuson. Ths functon has zero ntal condtons, easly mplemented and physologcally accurate functon [5]. Fg. 5 shows the resultng frequency characterstcs of the senstvty functon S compared wth the nverse of the performance weghtng functon. It s shown that the senstvty functon has magntude less than the magntude of the nverse of weghtng functon for all frequences. Ths means that the robust performance has been acheved usng the desgned controller. The frequency characterstcs of the resultng controller can be shown n Fg. 6. It s clear that the resultng controller has low magntude, whch satsfes the requrements of low control effort. Fg. 7 llustrates the nsuln nfuson rate. The blood glucose devaton response s shown n Fg. 8. It s shown that the proposed controller effectvely mproved the blood glucose response where a maxmum peak of.5 has been obtaned. Fnally, to show the effectveness of the proposed controller, the obtaned results have been compared to those acheved by prevous works that used dfferent control approaches as shown n Table. of computer code, easy to mplement and computatonally effcent method. On the other hand, t can be concluded that the proposed controller effectvely regulate the blood glucose wth a maxmum peak of.5 whch s a more desrable attenuaton for glucose devaton. Fnally, the superorty of the proposed controller to the controllers that have been proposed n the prevous works was made very clear n ths work. Star Defne the system model Intalze randomly the swarm Intalze teraton, k Intalze swarm sze, Update swarm poston Set the best local > swarm sze Yes Set the best global No 6. Conclusons In ths paper, the desgn of robust controller for regulatng the blood glucose devaton for Type I dabetes patents. The Bergman s model of the effect of nsuln nfuson and glucose nputs on the blood glucose concentraton was used as an artfcal pancreas. The Partcle Swarm Optmzaton (PSO) method was used for obtanng the optmal parameters of the proposed controller by mnmzng a cost functon subject to H constrants. It was shown that the PSO method s a powerful optmzaton method for solvng multobjectve problems wth only a few lnes Set the parameters of controller Evaluate the objectve functon crtera satsfed? No Yes Update swarm velocty k > teraton The controller optmal parameters End Yes Fgure 3: Flowchart of PSO based robust controller desgn. No 6
.4 Exogeneous glucose nfuson (mg/dl.mn.)..8.6.4. 3 4 5 6 7 8 9 Tme (mn.) Fgure 4: Dsturbance meal (exogenous glucose nfuson) functon. 6 5 4 3 Magntude (db) - - -3-4 - 4 Frequency (rad/s) Fgure 5: The resultng senstvty functon (sold) compared to the nverse performance weghtng functon (dotted).
- - -3 Magntude (db) -4-5 -6-7 -8-9 - -3 - - 3 4 Frequency (rad/s) Fgure 6: Frequency characterstcs of the resultng robust controller. 3 x -3 Manpulated nsuln nfuson rate (mu/mn.).5.5.5 -.5 3 4 5 6 7 8 9 Tme (mn.) Fgure 7: The nsuln nfuson rate..6.4. Glucose devaton (mg/dl)..8.6.4. -. 3 4 5 6 7 8 9 Tme (mn.) Fgure 8: Controlled blood glucose devaton.
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