Identification and low-complexity regime-switching insulin control of type I diabetic patients

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1 J. Biomedical cience and Engineering,, 4, doi:.436/jbise..444 Published Online Aril (htt:// Identification and low-comlexity regime-switching insulin control of tye I diabetic atients Ali Hariri, Le Yi Wang DTE Energy, One Energy Plaza, Detroit, Michigan, UA; EE Deartment, Wayne tate University, Detroit, UA. hariria@eenergy.com, U..lywang@wayne.edu Received 3 June ; revised 9 June ; acceted June. ABTRAT This aer studies benefits of using simlified regime-switching adative control strategies in imroving erformance of insulin control for Tye I diabetic atients. Tyical dynamic models of glucose levels in diabetic atients are nonlinear. Using a linear time invariant controller based on an oerating condition is a common method to simlify control design. On the other hand, adative control can otentially imrove system erformance, but it increases control comlexity and may create further stability issues. This aer investigates atient models and resents a simlified switching control scheme using PID controllers. By comaring different switching schemes, it shows that switched PID controllers can imrove erformance, but frequent switching of controllers is unnecessary. These findings lead to a control strategy that utilizes only a small number of PID controllers in this scheduled adatation strategy. Keywords: Insulin ontrol; Diabetes; witching ontrol; Modeling; Adatation. INTRODUTION Insulin is a hormone that is necessary for converting the blood sugar, or glucose, into usable energy. The human body maintains an aroriate level of insulin. Diabetes are caused by lack of insulin in the body. There are two major tyes of diabetes, called tye I and tye II diabetes. Tye I diabetes are called Insulin Deendent Diabetes Mellitus (IDDM), or juvenile onset diabetes mellitus. Tye II diabetes are known as Non-Insulin Deendent Diabetes Mellitus (NIDDM) or Adult-Onset Diabetes (AOD) [-7]. This aer is focused on tye I diabetes. Tye I diabetes develo when the ancreas sto roducing insulin. onsequently, insulin must be rovided through injection or continuous infusion to control glucose levels. The insulin infusion rate to a diabetic atient can be administrated based on the glucose (sugar) level inside the body. Over the years many mathematical models have been develoed to describe the dynamic behavior of human glucose/insulin systems. The most commonly used model is the minimal model introduced by Bergman, et al. [6,8-3]. The minimal model consists of a set of three differential equations with unknown arameters. ince diabetic atients differ dramatically due to variations of their hysiology and athology characteristics, the arameters of the minimal model are significantly different among atients. Based on such models, a variety of control technologies have been alied to glucose/ insulin control roblems [4-7]. This aer studies benefits of using simlified adatation control strategies in imroving erformance of insulin control for Tye I diabetic atients. Tyical dynamic models of glucose levels in diabetic atients are nonlinear. Using a linear time-invariant controller based on an oerating condition is a common method to simlify control design. On the other hand, adative control can otentially imrove system erformance, but it increases control comlexity and may create further stability issues. This aer investigates atient model identification and resents a simlified switching control scheme using PID controllers. By comaring different switching schemes, it shows that switched PID controllers can imrove erformance, but frequent switching is unnecessary. These findings lead to a control strategy that utilizes only a small number of PID controllers in this scheduled adatation strategy. Many methods and techniques have been investigated, tested, and studied for controlling the glucose level in tye I diabetes atients. Lynch and Bequette [4] tested the glucose minimal model of Bergman [] to design a Model Predictive ontrol (MP) to control the glucose level in diabetes atients. Also in that study the nonlinear Published Online Aril in cires. htt://

2 98 A. Hariri et al. / J. Biomedical cience and Engineering 4 () mathematical model was linearized about the steadystate values of time variant variables. Fisher [5] used the glucose insulin minimal model to design a semi closed-loo insulin infusion algorithm based on three hourly lasma glucose samlings. The study concentrated on the glucose level, and did not take in consideration of some imortant factors such as the free lasma insulin concentration and the rate at which insulin is roduced as the level of glucose rises. Furler [6,8] modified the glucose insulin minimal model by removing the insulin secretion and adding insulin antibodies to the model. The algorithm calculates the insulin infusion rate as a function of the measured lasma glucose concentration. The linear interolation was used to find the insulin rate. The algorithm neglected some imortant variations in insulin concentration and other model variables. Ibbini, Masadeh and Amer [7] tested the glucose minimal model to design a semi closed-loo otimal control system to control the glucose level in diabetes atients. The rest of this aer is organized as follows. ection discusses basic model structures, exerimental data, and model simulations. ection 3 is focused on model arameter identification. The Levenberg-Marquar algorithm is emloyed to obtain model arameters iteratively. ontrol design and switching strategies are resented in ection 4. By comaring fixed PID controllers with switching strategies, we first show that adatation can be beneficial. Further studies show that using a small set of PID controllers in switching control is a feasible and desirable aroach, which simlifies switching scheme without erformance degeneration.. MODEL ince the level of the glucose inside the human being body changes significantly in resonse to food intake and other hysiological and environment conditions, for control design it is necessary to derive mathematics models to cature such dynamics [8-,9-]. Many mathematical models have been develoed to describe the human glucose/insulin system. uch models are highly nonlinear and usually very comlex. The most commonly used and simlified model is the Minimal Model introduced by Bergman, et al. [8-], which is still nonlinear. To further simlify the model for control design, a common ractice is to locally linearize the Minimal Model under a given oerating condition... Model tructures The minimal model of the glucose and insulin is erhas the simlest model that is hysiologically based and reresents well for the observed glucose/insulin dynamics of a diabetic atient. The insulin enters or exits the interstitial insulin comartment at a rate that is roortional to the difference I t Ib of lasma insulin I t and the basal insulin level Ib [,3]. If the level of insulin in the lasma is below the insulin basal level, insulin exits the interstitial insulin comartment; and if the level of insulin in the lasma is above the insulin basal level, insulin enters the interstitial insulin comartment. Insulin also can flee the interstitial insulin comartment through another route at a rate that is roortional to the insulin amount inside the interstitial insulin comartment. On the other hand, glucose enters or exits the lasma comartment at a rate that is roortional to the difference t b of the lasma glucose level t and the basal glucose level b. If the level of glucose in the lasma is below the glucose basal level, the glucose exits the lasma comartment; and if the level of glucose in the lasma is above the glucose basal level, glucose enters the glucose comartment. lucose also can flee the lasma comartment through another route at a rate that is roortional to the glucose amount inside the interstitial insulin comartment. urrently, the most widely used model in hysiological research on the metabolism of glucose is the Minimal Model. This model structure describes exerimental data well with the smallest set of identifiable and meaningful arameters. The Minimal Model consists of two arts: the minimal model of glucose disaearance and the minimal model of insulin kinetics. P x t t P b () dxt Px t P3 I t Ib () di t ni t + t ht (3) where t (mg/dl) is the blood glucose level in lasma; I t (µu/ml) is the insulin concentration level in lasma; xt (min ) is the variable which is roortional to insulin in the remote comartment, b (mg/dl) is the basal blood glucose level in lasma; I b (µu/ml) is the basal insulin level in lasma; t (min) is the time interval from the glucose injection. Eqs. and reresent the glucose disaearance and Eq.3 reresents the insulin kinetics. The initial conditions of the above three differential equations are as the following:, x, I I The model arameters carry some hysiological meanings that can be summarized as follows. P (min ) describes the glucose effectiveness which reresents the ability of blood glucose to enhance its own disosal at the basal insulin level. P (min ) describes the oyright cires.

3 A. Hariri et al. / J. Biomedical cience and Engineering 4 () decreasing level of insulin action with time. P 3 ( min UmL ) describes the rate in which insulin action is increased as the level on insulin deviates from the corresonding baseline. ((µu/ml) (mg/dl) min ) denotes the rate at which insulin is roduced as the level of glucose rises above a target glycemia level. n (min ) reresents fractional insulin clearance. h (mg/dl) is the ancreatic target glycemia level. (mg/dl) is the theoretical glucose concentration in lasma extraolated to the time of glucose injection t = [8-,4]. I (µu/ml) is the theoretical lasma insulin concentration at t =. µu/ml is the conventional unit to measure the insulin level and has the following conversion formula: micro-unit/ milliliter = 6 icomole/liter (µu/ml = 6 mol/l) [5,6]. P, P, P 3, n,, h, and I are the arameters to be estimated. A fourth differential equation will be added to the set of the Minimal Model equations to reresent a first-order um dynamics: du t U t u t (4) a where U t is the infusion rate, u t is the inut command, and a is the time constant of the um... Exerimental Data A new aroach was develoed by Bergman, et al. [8-] to comute the ancreatic resonsiveness and insulin sensitivity in the intact organism. This aroach uses comuter modeling to investigate the lasma glucose and insulin dynamics during a Frequently amled Intravenous lucose Tolerance (FIT). An amount of glucose was injected at t = over a eriod of time equal to 6 seconds [8-,7]. The blood samles were taken from a fasting subject at regular intervals of time, and then analyzed for glucose and insulin content. lucose was measured in trilicate by the glucose oxidize technique on an automated analyzer. The coefficient of variation of a single glucose determination was about ±.5%. Insulin was measured in dulicate by radioimmunoassay, with dextrin-charcoal searation using a human insulin standard. Table shows the FIT test data for a normal individual..3. Model imulation Imlementation of the minimal model can be achieved by using comuter simulation tools. The two differential Eqs. and of the minimal model that corresond to the glucose kinetics are modeled here by using the MAT- LAB/imulink software. In this model, the insulin I t is considered as an inut and the glucose t as an outut. The values of the inut I t at a time interval are given in Table. The simulation diagram of the minimal model for the glucose kinetic is shown in Figure. The outut of the system, glucose t, is Table. FIT test data for a normal individual. amling time (minutes) lucose level (mg/dl) Insulin level (µu/ml) shown in Figure for a normal individual with the following arameters: P = 3.8, P =.93, P 3 =.6 5, = 35, b = 9, I b =, [8-]. 3. PARAMTER ETIMATION 3.. Parameter Estimation Parameter estimation is to determine the values of model arameters that rovide the best fit to measured data, based on some error criteria such as least-squares. The method of least squares assumes that the best-fit curve of a given set of data is the curve that has the minimal sum of the deviations squared from a given set of data [8-3]. iven a set of data (x, y ), (x, y ), (x 3, y 3 ),, (x N, y N ), where the indeendent variable is x and the deendent variable is y, under a selected model function form f x the least squares (L) estimation seeks to minimize N yi f xi (5) i When the function is an m-th degree olynomial we have m m (6) f x a a x a x a x N i N i yi f xi y a a x a x a x m i i i m i The unknown coefficients a, a, a,, am, can be estimated to yield a least squares error. (7) oyright cires.

4 3 A. Hariri et al. / J. Biomedical cience and Engineering 4 () Figure. imulation diagram of the glucose kinetic model. Figure. The outut of the minimal model for the glucose kinetics Model arameters can be obtained iteratively to reduce comutational comlexity. It starts with an initial guess of the unknown arameters. Each iteration udates the current estimate based on new observations. uose there are m base functions f, f,, fm of n arameters,,, n. The functions and the arameters can be reresented as follows: T f f, f,, fm (8) T,,, n The least square method is to find the values of the unknown arameters,,, n for which the cost function is minimum, i.e. m T P f f fi (9) i The Levenberg-Marquar algorithm is an iterative technique that seeks the minimum of a multivariate function that is exressed as the sum of squares of nonlinear real-valued functions [8]. It has become a standard technique for nonlinear least-squares roblems. Levenberg-Marquar can be thought of as a combination of steeest descent and the auss-newton method. When the current solution is far from the correct one, the algorithm behaves like a steeest descent method which is guaranteed to converge. When the current solution is close to the correct solution, it becomes a auss-newton method. The Levenberg-Marquar algorithm is an iterative rocedure. Let xˆ f be the arametrized model function. The minimization starts after an initial guess for the arameter vector is rovided. The algorithm is locally convergent, namely it converges when the initial guess is close to the true values. In each iteration ste, the arameter vector is udated by a new estimate, where is a small correction term that can be determined by a Taylor eries exansion which leads to the following aroximation: f f J () oyright cires.

5 A. Hariri et al. / J. Biomedical cience and Engineering 4 () where, J is the Jacobian of f at f J () Levenberg-Marquar iterative initiates at the starting oint, and roduces a series of vectors,, 3 etc, that converge towards a local minimizer of f. At each ste, it is required to find the which minimizes the value of f x f x J That gives the following: ˆ x f x xj ej () where is the solution to a linear least squares roblem. The minimum is achieved when the term J e is orthogonal to column sace J. Based on that, the following can be concluded T J J e (3) Eq.3 can be rearranged as the following T T J J J (4) e e The Levenberg-Marquar algorithm solves a slight variation of Eq.4, which is known as the augmented normal equation T N J e (5) where the diagonal elements of N are comuted T as N ii J J for, while the rests of the ii T matrix N are identical to those of the matrix J J. is called the daming arameter. If the udated arameter vector,, where is comuted from Eq.5, yields a reduction in the residual value or error e, then the udate is valid and the rocess reeats with a decreased daming arameter. Otherwise, the daming arameter is increased and the augmented normal Eq.5 is solved again. Then the rocess iterates until a value of that reduces error is found. The MATLAB oftware has the Otimization Toolbox which has a command called Lsqnonlin for this algorithm. This algorithm is alied to our roblem here. The FIT data samle in Table consists of 4 samles. The FIT samles were taken over a eriod of 8 minutes. The unknown arameters of the minimal model Eq., Eq., Eq.3 were estimated by utilizing the Levenberg-Marquadrt Algorithm. The arameters to be estimated were given an initial guess, then the algorithm was used to udate the arameters using the sequential data in Table. A MATLAB rogram was written to estimate the unknown arameters. The estimated values of those arameters are shown in Table. Table. Estimated Parametres. Parameters P P P 3 n γ h I Normal Individual # e Normal Individual # e e The values of the arameters shown in Table were imlemented in the simulation diagram of the minimal model shown on Figure. The values of the glucose levels of both individuals are shown in Table 3. The grahs of both exerimental and simulated data for normal individual # and # are shown in Figure 3 and Figure 4 resectively. These lots show that the two grahs (exerimental and simulated) are close to each other, leading to the conclusion that the estimated values of arameters are close to the actual values. In general, the relative errors indicate how good an estimate is, relative to the true values. Although absolute errors are useful, they do no necessarily give an indication of the imortance of an error. If the exerimental value is denoted by, and the estimated (or simulated) value is denoted by, then the relative error is defined as ˆ Relative Error (6) Table 3. imulated data for normal individuals # and #. amling time during the test (minutes) Normal Individual # lucose Level (mg/dl) Normal Individual # lucose Level (mg/dl) oyright cires.

6 3 lucose mg/dl) A. Hariri et al. / J. Biomedical cience and Engineering 4 () Basal Level Exerimental Data imulated Data time (min) Figure 3. Plot of glucose level (t) for normal individual #. lucose mg/dl) Basal Level Exerimental Data imulated Data time (min) Figure 4. Plot of glucose level (t) for normal individual #. And the quare Relative Error can be exressed as ˆ quare Relative Error (7) When the data is samled over a certain eriod of time, the mean squared relative error (MRE) can be used. The MRE is defined as n ˆ i i MRE, for i,,, n (8) n i i where is the exerimental value at samle i. ˆ is i i the estimated value at samle i. n is the number of samles of a data set. The quare Relative Error between the exerimental data and the simulated data of the glucose level for individual # and # are calculated and shown in Table 4 and Table 5 resectively. Normally the Mean quare Relative Error is exressed in ercentage format. Below is the ercentage error for both individuals: the ercent- Table 4. Error data for normal individuals #. Exerimental Data, (t) imulated Data, t ˆ () quare Relative Error e e e e e e Table 5. Error data for normal individuals #. Exerimental Data, (t) imulated Data, t ˆ () quare Relative Error.3463e e e 5.359e e e e e e e 5.496e e e e 5.794e age MRE for individual # =.998%; the ercentage MRE for individual # =.4596%. 3.. Model Linearization Now let us recall the four differential Eqs.-4 that define the roosed mathematical model and denote them as f U. The result is f, f x, f I, and 3 4 oyright cires.

7 A. Hariri et al. / J. Biomedical cience and Engineering 4 () d t f Px t t P b (9) dxt fx Px t P3 I t Ib () di t f3i ni t t htu t () du t f4u U t U t a () The above equations can be written and arranged as follows. The above equations can be further simlified as f P t x t t P (3) b f x Px t PI t PI (4) 3 3 b f I ni t t t ht U t (5) 3 f4u U t U t a a (6) The above system is a nonlinear system due to the resence of the nonlinear term that aears in Eq.3. The nonlinear term is x t t. Now let us make the following definitions,,, x t t x t x t x t I t x t u t 3 4 Then Eqs.3 to 6 become dx t Px t x t x t P b (7) dx t Px t Px 3 3 t PI 3 b (8) dx3 t t tx t nx t x t ht (9) 3 4 dx4 x4 t u t (3) a a The Jacobian matrices ( J x and can be derived as and J x J u ) of the model Px x P P3 t n a x x, uu J u (3) a x x, uu where the oint (x, u ) is the equilibrium oint. The equilibrium oint can be calculated by setting the state equations to zero and solving Px xxp b (3) Px Px 3 3 PI 3 b (33) tx nx x ht (34) 3 4 x 4 u a a (35) where x, x, x 3, x 4 and u are the values of the state variables and the inut at the oerating oint (i.e. the equilibrium oint). At the equilibrium oint, u =, Eq.35 becomes x4, that gives a x 4 = (36) ubstituting the value of x 4 in Eq.34 results in tx nx ht and 3 x h t x3 (37) n The value of x 3 can be substituted in Eq.33 x h t Px P3 PI 3 b n to obtain P3tx P3th PI 3 x b (38) Pn Pn P Now, by substituting the value of x in Eq.3 we have P3t P3th PI 3 b x P x P b (39) Pn Pn P The above equation is a nd order equation and can be solved by using the quadratic formula where x b b 4ac a Pt Pth PI a, bp, c P Pn Pn P b b (4) There are ossible values of x. ince x and x 3 are exressed in term of x, there will be ossible values for each. Based on that, the controllability test will be studied to check which value of x is acceted A ase tudy The simulation diagram shown in Figure was used to simulate the data of a diabetic atient. The value of the arameters for a diabetic atient is shown below. P =, P =.8/, P 3 = 4./, I = 9, = 337, =.4/, h = 93, n =.3, a =, b = 99, I b = 8, []. oyright cires.

8 34 A. Hariri et al. / J. Biomedical cience and Engineering 4 () The outut of the simulated system is shown in Figure 5. By examining Figure 5, it can be clearly seen that the glucose level does not come down to the basal level after injecting an amount of 337 mg/dl of glucose inside a diabetic atient. Figure 5 shows that the level of the glucose inside a diabetic atient decreases for the first minutes, starts increasing afterward, and reaches the value of around 3 mg/dl after 3 hours from the time the glucose was injected. The goal is to bring down the value of the glucose inside a diabetic atient to the normal level or at least into a small neighborhood of the basal level. The above goal can be achieved by designing a PID feedback controller. The controller is to regulate the infusion rate and inject the required amount of the insulin inside the diabetic atient, and in turn the insulin will work inside the atient to bring down the level of glucose to the normal level or at least to the neighborhood of the normal level. 4. ONTOL DEIN We start with a linearized state sace model for our systems. The general form of the state sace is defined in the following equation: x Ax Bu (4) y x Du The roosed mathematical model at the equilibrium oint (x, u ) can be written in the state sace form as shown below: Px x P P3 x t n x u (4) a a y x Figure 5. Outut of the simulated system for diabetic. where u is the inut and y is the outut of the system. The data of a diabetic erson shown in ection 3.3 was used and the equilibrium oint (x, u ) was calculated as time varies from t = min to t = 8 min. The two values for x that were calculated before were substituted in Eq.4 and the controllability test was erformed. It was found that only one value (the one obtained from the sign) makes the system to be controllable, hence only this value is used in the subsequent develoment. 4.. Design of PID ontrollers for Diabetic Patient When designing a controller, the designer must define the secifications that need to be achieved by the controller. Normally the maximum overshoot (M) of the system ste resonse should be small. ommonly a range between % and % is accetable. Also the settling time (ts), is an imortant factor. The objective here is to design a PID controller, so that the closed-loo system has the following secifications: small steady-state error for a ste inut; less than % overshoot; settling time less than 6 minutes. The atient dynamic system was exressed in the state sace reresentation in (4). For an overshoot less than %, a daming ratio must be greater than.59, and a settling time less than 6 minutes imlies that n must be greater than.67. The PID ontrollers can be designed based on the models at different oerating oints. The following list contains the models at t =,, 4, 6, 9,, 5 and 8 minutes, and the corresonding PID controllers. ince B and do not change with time, they are fixed in all cases as: B and.5 The control design is done by alying the root locus method and then evaluated by using the ste resonse. For examle, if the model at t = min is used, after substituting the above numerical values (4), the root locus lot can be generated by the Matlab functions MATLAB Program Plotting the oen loo system Root locus using MATLAB [num, den] = sstf(a,b,,d]; rlocus(num, den) axis([ ]) sgrid(.59,) sigrid(.67) oyright cires.

9 A. Hariri et al. / J. Biomedical cience and Engineering 4 () The oen loo oles are shown in Figure 6. These oles are located at j,.4.45j,.3,.5. The four oles are stable, but the first two oles are very close to the imaginary axis and hence reresent the slowest dynamics. The controller takes the form c K z z where K is the value of the gain where the root locus intersects with the line of the daming ratio. The z and z reresent the value of the zeros to be added and may be selected to cancel the slowest oles of the dynamic system. Hence, select z = j, z =.4.45j. The controller is * (lease see the equation below) resulting in the system matrix A and the PID controller as A, K The PID Parameters are: K K K 4.444, i.94, d 5.59 The design secifications of the system require the maximum overshoot to be less than % and the settling time to be less than 6 minutes. After inserting the PID controller in series with the atient system and connecting them in a unity feedback, we get the maximum overshoot 5.7% and the settling time 47.5 minutes (see Figure 7). t = minutes: A , K.76. Figure 6. Outut of the simulated system for Root Locus. Figure 7. Outut of the unit ste resonse, using the model at t= minute with K = 5.5. t = 4 minutes: A4, K.7. 4 The PID Parameters are: K =.6, K.3, K = 8.4 i d * c The PID Parameters are: K =.374, K.6, K = 3.7 i d K j j oyright cires.

10 36 A. Hariri et al. / J. Biomedical cience and Engineering 4 () A A t = 6 minutes: , K.7.3 The PID Parameters are: K K K 4.444, i.94, d 5.59 t = 9 minutes: 9 9 A.5.8.4, K.64.4 The PID Parameters are: K =.33, K.38, K = 36.4 i d t = minutes: A , K.58.5 The PID Parameters are: K =.34, K.87, K = 37.9 t = 5 minute: 5 5 i d , K.54.6 The PID Parameters are: K =.3, K.9, K = 37.7 i d A t = 8 minute: , K.48.7 The PID Parameters are: K =.87, K.8, K = 38.5 i d 4.. Individual PID ontrollers Non-adative PID controllers use a fixed PID controller for the entire control eriod and rely on its robustness to maintain control erformance. For each individual PID controller (with its transfer function found in the revious subsection for t =,, 4, 6, 9,, 5 and 8 min) the simulation results on t and ut are shown below in Figures 8-5. Under the individual PID controllers, the outut t, the glucose level, did not really meet the design secification, and the glucose level is not near or at least in a small neighborhood of the glucose basal level. The overshoot of the system was too high and beyond the accetable level. Also the settling time was not even close to where it should be as er the design requirement. And the steady state error was not satisfactory. A new method should be develoed and imlemented to meet all the design secifications. This method is exlained in detail in the following section witched PID ontrollers The individual PID controllers could not lower the glucose level t of the atient to the neighborhood of the glucose basal level. onsequently, we introduce a new control-switching scheme that adats controllers to meet design secifications. The switching control consists of the following items: One time clock, one switch case block, one if action case block, one merge block, and eight off-on switches. All these blocks are connected together to form the wiring diagram of the ontrol-witching cheme. The functions of the control switching scheme is detailed in Figure 6. The Time lock is to rovide the witch ase block with time as a signal inut to activate it. The witch ase block receives a single inut from the clock, which it uses to form case conditions that determine which subsystem to execute. Each outut ort case condition is attached to a witch ase Action sub- oyright cires.

11 A. Hariri et al. / J. Biomedical cience and Engineering 4 () Figure 8. t and u t using. Figure 9. t and u t using. Figure. t and u t using 4. oyright cires.

12 38 A. Hariri et al. / J. Biomedical cience and Engineering 4 () Figure. t and u t using 6. Figure. t and u t using 9. Figure 3. t and u t using. oyright cires.

13 A. Hariri et al. / J. Biomedical cience and Engineering 4 () Figure 4. t and u t using 5. Figure 5. t and u t using 8. Figure 6. ontrol-witching cheme imulation Diagram. oyright cires.

14 3 A. Hariri et al. / J. Biomedical cience and Engineering 4 () system. The cases are evaluated to-down, starting with the to case. If a case value corresonds to the actual value of the inut, its witch ase Action subsystem is executed. The witch ase model is divided into eight time interval zones as shown in Table 6. The If Action ase block consists of eight If- Action ubsystem. The If-Action ase imlements Action ubsystems used in if-statement and switch control flow statements. Action ubsystems execute their rogramming in resonse to the conditional oututs of an If-statement or witch ase block. A schematic diagram of the If Action ase block is shown in Figure 7. The Merge block combines its inuts into a single outut line whose value at any time is equal to the most recently comuted outut of its driving blocks. The number of inuts can be secified by setting the block s inuts arameter. The Off-On witches are used to turn the PID controllers OFF or ON. In general when the time clock is running, it feeds the witch ase block with an inut signal which in turn switches on the If-Action ase block as er the time interval that was secified in Table 6. Based on the status of the If-Action ase, a secific PID controller will be turned on and executed to control the outut of the system. For zone, the time interval is between - minute, during this eriod of time the witch ase is enabling inut In9 of the If-Action ase. When inut In9 is enabled, it will only execute the inut In to the outut Out. The inut In is connected to the first PID ontroller. That means only the first PID ontroller, (PID ontroller ), is working. At the end of the first minute, the witch ase will switch to zone which runs from the beginning of the minute number and will last until the end of the minute number. During this eriod of time, the witch ase is enabling inut In of the If-Action ase. When inut In is enabled, it will only execute the inut In to the outut Out. The inut In is connected to the second PID ontroller. That means only the second PID ontroller, (PID ontroller ), is working. The same rocedure will be followed until the witch ase switches between the eight time zones that were secified in Table 6. In turn the PID ontrollers will be executed based on the status of the If-Action ase. The ontrol-witching cheme Diagram shown in Figure 6 was simulated with all the PID controllers executed (connected to the circuit). The outut t of the system is shown in Figure 8. It can be clearly seen that the PID controllers are able to bring the glucose level from 337 mg/dl to the basal level (99 mg/dl) within 4 minutes. But in about 7 minutes the value of the glucose starts going below the basal level, and it Table 6. withing Time Interval. Zone number Time Interval (minutes) went further below the minimum value that the glucose can be at. In this case the erson will be classified as a atient with the hyoglycemia (low sugar), and that is not accetable. The ontrol-witching cheme Diagram was then simulated in which all the PID controllers are executed 8 excet the eighth PID ontroller. The grah of the outut, t of the system is shown in Figure 9. It can be seen the same roblem still exists. Again in this case the erson will be classified as a atient with the hyoglycemia. The same rocedure was reeated but with all the PID controllers executed excet the PID ontrollers 5 8 and with the grah of the outut shown in Figure ; and excluding 5,, and 8 with the simulation result shown Figure. Again in both cases the erson will be classified as a atient with the hyoglycemia When we exclude,,, and, the outut t of the system, shown in Figure,. reaches the glucose basal level (99 mg/d/l) within 4 minutes, and it stays in that neighborhood. The inut of the PID ontroller system is shown in Figure 3. For verification, the same control strategy is evaluated on atient #. Following the same modeling rocedure that was erformed for the diabetic atient #, the model arameters are identified as P =, P =.4/, P 3 =.56/, I = 9, = 97, = 3.7/, h = 54, n =., a =, b =, I b = 8, []. Without control, the above data was imlemented in model simulation. The outut of the simulation diagram is shown in Figure 4, which shows that without roer control the glucose level does not come down to the basal level after injecting an amount of 97 mg/dl of glucose inside a diabetic atient. The level of the glucose inside a diabetic atient decreases for the first minutes and starts increasing afterward and reaches the value of about 7 mg/dl after 3 hours from the time the glucose was injected. The same control-switching scheme that was erformed for diabetic atient # is reeated for diabetic atient #. The values of the arameters for the first four PID controllers (at t =,, 4 and 6 minutes) are oyright cires.

15 A. Hariri et al. / J. Biomedical cience and Engineering 4 () In9 In In In In 3 In3 case: { } In Out If Action ubsystem case: { } In Out If Action ubsystem case: { } In Out If Action ubsystem 4 Out Out 3 Out3 PID PID ontroller PID PID ontroller PID PID ontroller 4 Out Out 3 Out3 In 4 In4 case: { } In Out If Action ubsystem 6 4 Out4 In PID PID ontroller 6 4 Out4 3 In3 5 In5 4 In4 6 In6 case: { } In Out If Action ubsystem 9 case: { } In Out If Action ubsystem 5 Out5 6 Out6 PID PID ontroller 9 PID PID ontroller 5 Out5 6 Out6 5 In5 7 In7 6 In6 8 In8 case: { } In Out If Action ubsystem 5 case: { } In Out If Action ubsystem 8 7 Out7 8 Out8 Figure 7. If Action ase system and PID controller switching function module. PID PID ontroller 5 PID PID ontroller 8 7 Out7 8 Out8 summarized in Table 7. The control-switching scheme was simulated for diabetic atient # by using only the first four PID controllers. The outut t of the system is shown in Figure t reaches the glucose basal level 5. The outut ( mg/d/l) within 6 minutes and it stays in that neighborhood. The grah of the inut of the PID ontroller system is shown in Figure 6. Based on the simulation results, although adative control can otentially imrove control erformance, it is oyright cires.

16 3 A. Hariri et al. / J. Biomedical cience and Engineering 4 () Figure 8. Plot of (t) when all PID ontrollers are executed. Figure. Plot of (t) when all PID ontrollers excet 5 controller, and 8 are executed. Figure 9. Plot of (t) when all PID ontrollers excet controller are executed. 8 Figure. Plot of (t) when all PID ontrollers excet controller,, and are executed. Figure. Plot of (t) when all PID ontrollers excet controller and 8 5 are executed. Figure 3. Plot of u(t) when all PID ontrollers excet controller,, and are executed. oyright cires.

17 A. Hariri et al. / J. Biomedical cience and Engineering 4 () Table 7. PID ontroller for diabetic atient #. PID ontrollers Parameters PID at t = min PID at t = min PID at t = 4 min PID at t = 6 min K K i K d Figure 4. Outut of the imulation Diagram for diabetic atient #. Figure 5. Plot of (t) when PID ontrollers,, 4 and 6 are executed. Figure 6. Plot of u(t) when PID ontrollers,, 4 and 6 are executed. sometimes unnecessary, or even harmful when switching overly frequently. Our results show that when the switching scheme is limited to the first four PID controllers, the erformance is in fact enhanced. This may be related to the fact that some PID controllers are more robust with resect to the model variations. On the other hand, in comarison to individual controllers, the control-switching scheme achieves design secification while all individual controllers fail to deliver the required erformance. 5. ONLUION This study reveals that tyical PID controllers may not be sufficient to deliver satisfactory control erformance in glucose level control roblems. This is mainly due to the nonlinear nature of atient dynamic models and limited robustness of the PID controllers. An adative control that switches controllers based on oerating conditions can otential enhance control erformance. However, the switching control scheme must be carefully designed to ensure that control secifications be met. Our results show that overly frequent switching of controllers may have detrimental effects on control erformance. We show that by reducing the number of PID controllers in the switching scheme, not only control comlexity is reduced, but erformance is actually enhanced. REFERENE [] Karam, J.H., rodsky,.m. and Forsham, P.H. (963) Excessive insulin resonse to glucose in obese subjects as measured by immunochemical assay. Diabetes,, [] insberg, H., Olefsky, J.M. and Reaven,.M. (974) Further evidence that insulin resistance exists in atients with chemical diabetes. Diabetes, 3, [3] Reaven,.M. and Olefsky, J.M. (977) Relationshi between heterogeneity of insulin resonses and insulin resistance in normal subjects and atients with chemical diabetes. Diabetologia, 3, -6. doi:.7/bf97 [4] Lerner, R.L. and Porte, D. (97) Acute and steady state insulin resonses to glucose in nonobese, diabetic subjects. Journal of linical Investigation, 5, doi:.7/ji6963. [5] Reaven,.M. (98) Insulin-indeendent diabetes mellitus: Metabolic characteristics. Metabolism linical and Ex- oyright cires.

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