**3. Novel perspective in the Control of Nonlinear Systems via a Linear Param-**

**3.2. Different interpretations of quality based on LPV configurations**

**3.2.3. Usability of the development approach**

I have selected a simple DM patient model developed by Wong et al. in [85,86] for the Intensive Care Unit (ICU) treatment. The selection was plausible, since it has several benefits to use as a test model. It is a 3rd order model and contains saturation type nonlinearities. The model is described by (B.2). The model equations can be handled as in (3.1)-(3.2), which means only the state matrix depends on the parameter vector (A(p(t))).

The main goal of the model is to describe the glucose-insulin dynamics of an inpatient who suffers from T1DM and is nurtured on the ICU [85, 86]. It is expected that this simple model – after preliminary identification – can provide the current and the future blood glucose level of the patient with a precision that is good enough for the realization of the tight glycemic control (TGC). Detailed description and used parameters of the model can be found in Subsection B.2.1.

**qLPV model of the original Wong model**

I have selected the following scheduling parameters as the elements of the parameter vector based on [87]:

These selection reduces the model complexity, however, the **p(t) contains grouped**
variables. *p*1(t) and*p*2(t) have Michaelis-Menten attitude and they keep their original
physiological meaning, namely, the nonlinear kinetic effect of the appearing insulin on
the blood glucose level. Each nonlinearity causing element was selected as scheduling
variable.

Based on (B.2) and (D.16), the affine qLPV model of the matrix terms of the system matrix of the original Wong model can be described as follows:

**A(p(t)) =A**_{0}+**A**_{1}*p*1(t) +**A**_{2}*p*2(t) +**A**_{3}*p*3(t) =

After defining the border of the parameter box, i.e. the vertices of the convex polytopic space, the polytopic model form of the qLPV model can be easily obtained based on the affine qLPV form of (3.9). I have selected tight ranges in every dimension in order to

catch the dynamics as precisely as possible:

The main goal was to test the usability of the developed ”metric” without physiological constraints. In this demonstration, I did not seek the physiological validity, just to introduce the developed approach. Therefore, I have used randomized input signals and not physiologically valid ones. However, the physiologically valid input signals are much favorable (less time period, less amplitude) then the used ones. In the given circumstances, the output of the nonlinear original system and the LPV version of it can be found on Fig. 3.2. below, where it can be seen that the nonlinear original model provides the same output as the LPV version.

Time [min]

Figure 3.2.: The outputs of the models

Figure 3.3. shows the varying of the elements of the parameter vector **p(t) and the**
developed 2-norm based difference. On every diagram, the dashed line represents the
fixed value, which belongs to the*p** _{ref}* = [0.2, 0.01025, 0.98]

*. The input is selected to be a symmetrical repeating impulse (P(t) = 40 at every 140 min for 7 min long and*

^{T}*u*

*ex*(t) = 100 at every 130 min for 6.5 min long) and it can be seen that after the first period’s decay, the parameter vector have taken the same values in each cycle, which means the same LTI systems occurs over time in each cycle.

Time [min]

0 50 100 150 200 250 300 350 400 450 500

p1(t) 0 0.2 0.4

p1,fixed p

1(t)

Time [min]

0 50 100 150 200 250 300 350 400 450 500

p 2(t) 0.0095

0.01 0.0105

p2,fixed p

2(t)

Time [min]

0 50 100 150 200 250 300 350 400 450 500

p 3(t) 0.9 0.95 1

p3,fixed p

3(t)

Time [min]

0 50 100 150 200 250 300 350 400 450 500

||e||(t)

0 0.1 0.2

Figure 3.3.: Varying of the scheduling variables and the norm-based error signal

Fig. 3.4. shows the same signals as Fig. 3.3. on one diagram in order to compare the
orders of magnitudes. It can be seen that those signals that reflect mostly in the ke(t)k_{2}
have the highest amplitude. Since in this case the*p*1(t) has the highest amplitude, the
ke(t)k_{2} correlated mostly with *p*_{1}(t). If each scheduling variables have to be considered

with the same weight, normalization procedures can be done [80]. However, I did not apply such methods as the goal was only demonstration.

Time [min]

0 50 100 150 200 250 300 350 400 450 500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p1,fixed p

2,fixed p

3,fixed p

1(t) p

2(t) p

3(t) ||e(t)||

Figure 3.4.: Comparison of the magnitudes of the scheduling variables and the norm-based error signal

In order to make this demonstration complete, Fig. 3.5. shows the scheduling parameters and the 3D parameter space too. It can be seen that the parameter vector starts from a given point and ends at another. During the simulation time the parameter vector runs through a trajectory in the PS. Naturally, that means that the system matrices belonging to the different parameter vectors are varying in time, as well.

p1(t)

0.0094 0.0096 0.0098 0.01 0.0102 0.0104 0.0106
**Parameter space**

Border of the parameter space

Figure 3.5.: Evolution of the scheduling variables in the parameter space during operation

If the parameter vectors fully determine the underlying LTI systems during operation,
the parameter vector based metric can be used to compare the ”difference” between
these systems. Figure3.6. shows this issue in case of the selected model and parameter
vector, namely, instead of the Frobenius-norm based ”difference” the developed metric can
approximately provide similar results. Naturally, the signals are not the same, since the
numerical computations are different. The upper diagram presents that the signals are
covering each other. The lower diagram shows the difference between these – furthermore,
the generated maximum root mean square error (RMSE) of the was 5.8352 10^{−4} based
on the difference between the given signals.

Time [min]

0 50 100 150 200 250 300 350 400 450 500

0 0.05 0.1 0.15 0.2

||S(p

fixed)-S(p

actual)||

F ||p

fixed-p

actual||

2

Time [min]

0 50 100 150 200 250 300 350 400 450 500

e F-e 2

×10^{-4}

-6 -4 -2 0 2 4 6

Figure 3.6.: Different norm-based differences

**Summary**

In this section, I have introduced a norm based ”difference” interpretation that can be applied to LPV systems, based on the properties of the LPV parameter space. I have defined how to use these interpretations as error and quality criteria during modeling and control and demonstrated my theoretical findings by a particular example in diabetes modeling and control of ICU patients.