# F F Teles J M

260 F.F. Teles, J.M. Lemos / Biomedical Signal Processing and Control 48 (2019) 255–264

where U(n) is the output of the bumpless transfer and anti-windup

b

system, and U(n−1) is the previous drug effect U.

The distribution of the models in S is crucial for the system per-

formance. In this work, an algorithm where the controller bank

is smaller than the model bank, i.e. where NcN, being Nc the

Fig. 8. Architecture of the bumpless transfer and anti-windup scheme.

number of local controllers, is used. Then, clustering algorithms are

implemented for creating clusters c

of models. The centroids Mc of

j

j

When changes in the plant are detected, the current controller
those clusters are the models from which the local controllers Cc

are designed.

j

is switched to a new one with a higher satisfactory performance.

However, this ideal control action can be deteriorated by the
2.4.1. Model sensitivity analysis

switching since a degradation of the transient performance can be

expected [41]. By swapping from one controller to another, abrupt

The sensitivity analysis is used to choose from the set of 11

changes in the manipulated variable U may occur, resulting in
parameters of the patient model the 2 most sensitive parameters,

doses that largely exceed the maximum values allowed. Bumpless
since clustering is performed based on them.
∈ IR11 is

transfer between different controllers can prevent this problem, for

instance by inserting an integrator common to all the controllers
the parameter vector, sensitivity Si
can be represented as

d

With this insertion, the local controllers have to be redesigned

by considering an augmented state space composed by x and the

state of the integrator Ub , resulting in

From the solution of (20), it ABT263 is concluded that the parameters I

A
B

x

x

An important factor regarding the classification and agglom-

eration of models is the choice of a good evaluation metric. The

Ub

Vinnicombe metric [45] was chosen to be used, since an evaluation

of the closed loop response is performed for comparing two distinct

models. More precisely, if the Vinnicombe metric shows that two

models are distinct, then a controller that gives satisfactory results

for one model will behave poorly or even destabilize the other.

Considering two models Mi and Mj , that are described by a

transfer function with the same number of inputs and outputs, the

Vinnicombe metric ıv(Mi , Mj ) is then given by

j

j

i

i

When there is no saturation, es is zero having no effect in the operation. If it is not null, the feedback path tries to reset the inte-grator which prevents it from winding up. A value of Tw = 10 is used, since it represents the best tested scenario.

Despite the inclusion of the bumpless transfer, there may still exist situations in which the system is not able to prevent the occur-rence of peaks, such as a switch between two models that are very distinctive. The bumpless transfer system is capable of smoothing these peaks but not of completely removing them. In this way, a cumulative moving average filter was included, which calculates for each time instant tn , the mean of all the iterations between the first instant t0 and the previous one tn−1 . Thus, the drug effect for the time instant n, U(n) , is given by

are very distinctive. Thus, values closer to 0 indicate that a con-troller designed for Mi can also stabilize Mj , having also similar closed loops gains.

Since the plant has two inputs, two Vinnicombe metrics are available. However, almost no differences were found between both metrics, being the first one used.

After creating a model data set, made of N models whose param-eters I and ˇ were generated by a log-normal distribution, two clustering methods were applied: k-means [46] and complete link-age [47].

For a test with N = 100 models and Nc = 6 clusters, the number of models Mi in each cluster ci , the mean of distances dc between Mi and its clusters, and the worst case of the same distance dc were computed for both algorithms. The biggest cluster in the k-means had 24 models, whilst the larger one in complete linkage had 69. On the other hand, the smallest one for the first method had 7 clusters, whilst for the second it had only 1. The mean of dc was for the com-plete linkage 147.4% lower than for the k-means, being the worst dc also 91.8% lower for the first method. Aside from that, by increasing