Control of the Cambridge (Hovorka) model

The next discussed complex T1DM model, namely, the Hovorka model is determined by (B.3a)-(B.3j). It originally occurred in [70]. In this Theses, I used this model in its form that was presented in [71]. The model has ten state variables that describe not just the glucose-insulin dynamics as the Minimal model does, but also capture the externally injected insulin’s absorption and distribution, the insulin effects, the insulin independent BG changes, and the internal glucose production too. Furthermore, it also contains an embedded glucose absorption model. The model equations and the descriptions of the parameters can be found in the AppendixB.

The effect chain and the approximate model

The relative order of possible kinematic type control is five, since the ”pure” control signal reflects in the 5th derivative of the controlled state Q₁(t). That means that high order derivative will appear from (B.3e) in the approximate model. This circumstance entails that high abandonment is necessary against the elements of the exact model.

Thus, the relative order of the control is 5th, the 5th derivative of the state Q₁(t) to be controlled should be determined. Two kind of approximation can be used: parameter approximation instead of exact parameters and neglecting of model elements. I applied both of these here.

First, I investigated the 5th derivative of theQ₁(t) state, which can be derived from (B.3e):

Q⁽⁵⁾₁ (t) = D⁽⁴⁾₂ (t)

V_Gτ_D −F_01,c⁽⁴⁾(t)−F_R⁽⁴⁾(t)−x⁽⁴⁾₁ (t)Q1(t)−4x⁽³⁾₁ (t) ˙Q1(t)−

−5¨x₁(t) ¨Q₁(t)−4 ˙x₁(t)Q⁽³⁾₁ (t)x₁(t)Q⁽⁴⁾₁ (t) +k₁₂Q⁽⁴⁾₂ (t)−EGP₀x⁽⁴⁾₃ (t)

. (2.28)

The (2.28) can be rearranged and completed with a new term, the Af f ine term, which represents each neglected elements. Thus, the control signal directly appears in

x⁽⁴⁾₁ (t) andx⁽⁴⁾₃ (t) states derivatives, the approximate model is:

Q⁽⁵⁾₁ (t) =−x⁽⁴⁾₁ (t)Q₁(t)−EGP₀x⁽⁴⁾₃ (t) +Af f ine term . (2.29) The 4th derivative of x⁽⁴⁾₁ (t) andx⁽⁴⁾₃ (t) can be derived from the model equation (B.3e).

However, first the route of the control signal has to be explored. Through the following equations, this can be done.

S˙₁(t) =u(t)−S₁(t)

Equations (2.33) and (2.34) can be substituted to (2.29). Further, the neglected subparts can be incorporated to theAf f ine term, as follows:

Q⁽⁵⁾₁ (t) =−k_b1 u(t)

V_Iτ_S²Q1(t)−EGP0kb3

u(t)

V_Iτ_S²+Af f ine term . (2.35) From (2.35) via rearranging the equation the connection between the control signal and the 5th derivative of Q₁(t):

u(t) = Q⁽⁵⁾₁ (t)−Af f ine term

−k_b1Q₁(t)−EGP₀k_b3 V_Iτ_S² = −Q⁽⁵⁾₁ (t) +Af f ine term

k_b1Q₁(t) +EGP₀k_b3 V_Iτ_S² . (2.36) The (2.36) is eligible to use as approximate model. I applied further approximation, since the exact model parameters are not available in every cases. The finalized and used

approximate model, what I applied in this study was the following:

u(t)≈ −Q⁽⁵⁾₁ (t) +Af f ine term_const

k_b1appQ1(t) +EGP0appk_b3app VIappτ_S²_app , (2.37) where I used 10% random deviation in the approximated parameters, moreover, I replaced the Af f ine termwith a constant,Af f ine termconst.

Control block

I assumed that the glucose distribution volume is known at this point as well which means I could write the kinematic type PID control law toQ1(t), which was the following:

Q^(5),Des₁ (t) =Q^(5),N₁ +

5

X

s=0

6 s

!

Λ^6−s d dt

!s t

Z

t0

Q^N₁ (ξ)−Q₁(ξ) dξ (2.38)

Simulation results – Hovorka model

The Hovorka model is much more complex than the Minimal model which requires not just different controller design approach, but also the evaluation of the results is different.

The Hovorka-model has several nonlinearities, attenuations and cross actions between the states.

It should be noted that I applied exactly the same feed intake during the present investigations that I used in the case of the Minimal model in2.3.1in Table2.1. Thus the glucose rate of appearance in blood were the same in both cases beside this randomized glucose load.

I have used PID-based control law.

The simulations were made in Scilab^TM and the figure plots were created with MATLAB^TM.

The initial states of the 48 hours long simulation werexini= [D1,ini, D2,ini, S1,ini, S2,ini, I_ini, x_1,ini, x_2,ini, x_3,ini, Q_1,ini, Q_2,ini]^> = [0,0,687.5,687.5,10.783,5.521e− 2,8.842e− 3,0.5607e−1,86.3,63.66]^>.

The Hovorka model behaves differently than the Minimal model as it can be seen on Fig. 2.5. Without external glucose intake, the BG level is increasing due to the glucose secretion of the liver, which is an embedded part of the model. The applied controller starts the insulin injection when the BG level is increasing, however, it turns off, when the BG level is decreasing. This switching attitude can be derived from the applied control strategy and this is a consequence that the controller cannot affect with ”negative”

control input. It can be seen that despite the continuously absorbing external glucose, the controller can manage the glycemia, namely, it is able to avoid the hypoglycemia beside minimizing hyperglycemia. The latter cannot be totally avoided because of the high and random glucose intakes.

Time [min]

0 500 1000 1500 2000 2500

BG level [mmol/L]

5 10 15

Blood glucose level

Time [min]

0 500 1000 1500 2000 2500

Insulin [mU]

0 50 100

150 Injected insulin

Time [min]

0 500 1000 1500 2000 2500

Absorbed glucose [mmol/L]

0 5

10 Blood glucose level

Figure 2.5.: Results of the 48 hours long simulation of the Hovorka model [Q^N₁ = 90 mmol/L (G^N = Q^N₁ /VG ≈ 8.036 mmol/L), Λ = 1e−4, Ac = 1/10|K_c|, Kc= 5e−1 andBc=−1]

On the CVGA plot (Fig. 2.6.), each of the white dots represent a 24 hours long simulation period of the Hovorka model.

The vertical movement alongside with the maximum BG axis comes from that the minimum BG levels of this 24 hours periods were higher than 110 mg/dL (6.05 mmol/L).

However, three points are in the B regions, which assumes a higher BG level variability – moreover, one point belongs to day 6 is totally overlapped with the point belongs to day 1. Nevertheless, the results showed that this controller configuration successfully deals with the randomized, unfavorable feed intakes and can keep the BG level among an appropriate range.

Minimum BG

>110 90 70 <50

Maximum BG

<110 180 300

>400

A U pper B U pper C

L ower B B U pper D

L ower C L ower D

E

Figure 2.6.: Results of the one week long simulations

Summary

In this Section, I have successfully demonstrated the usability of RFPT-based controller design method in case of highly complex T1DM model step-by-step. The controller was capable to avoid hypoglycemic periods, however, soft hyperglycemia occurred due to the internal attenuations and highlighted liver effects (high internal glucose load) of the model. As the text reflects, it is possible to follow the demonstrated straightforward path to use in order to realize RFPT-based controllers in complex cases as well.

2.3.3. Control of the UVA-Padova (Magni) model

In document ´Obuda University (Pldal 49-54)