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Interval type-2 fuzzy neural network controller for a multivariable anesthesia system based on a hardware-in-the-loop simulation Ahmad M. El-Nagar ∗ , Mohammad El-Bardini Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menofia University, Menouf 32852, Egypt
a r t i c l e
i n f o
Article history: Received 15 January 2013 Received in revised form 11 March 2014 Accepted 11 March 2014 Keywords: Hardware-in-the-loop Back-propagation algorithm Interval type-2 fuzzy neural network Anesthesia Analgesia Muscle relaxation
a b s t r a c t Objective: This manuscript describes the use of a hardware-in-the-loop simulation to simulate the control of a multivariable anesthesia system based on an interval type-2 fuzzy neural network (IT2FNN) controller. Methods and materials: The IT2FNN controller consists of an interval type-2 fuzzy linguistic process as the antecedent part and an interval neural network as the consequent part. It has been proposed that the IT2FNN controller can be used for the control of a multivariable anesthesia system to minimize the effects of surgical stimulation and to overcome the uncertainty problem introduced by the large inter-individual variability of the patient parameters. The parameters of the IT2FNN controller were trained online using a back-propagation algorithm. Results: Three experimental cases are presented. All of the experimental results show good performance for the proposed controller over a wide range of patient parameters. Additionally, the results show better performance than the type-1 fuzzy neural network (T1FNN) controller under the effect of surgical stimulation. The response of the proposed controller has a smaller settling time and a smaller overshoot compared with the T1FNN controller and the adaptive interval type-2 fuzzy logic controller (AIT2FLC). The values of the performance indices for the proposed controller are lower than those obtained for the T1FNN controller and the AIT2FLC. Conclusion: The IT2FNN controller is superior to the T1FNN controller for the handling of uncertain information due to the structure of type-2 fuzzy logic systems (FLSs), which are able to model and minimize the numerical and linguistic uncertainties associated with the inputs and outputs of the FLSs. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The type-1 fuzzy neural network (T1FNN) controller combines the capability of fuzzy reasoning to handle uncertain information and the capability of artificial neural networks to learn from processes [1]. These controllers have been successfully applied in many fields [2–4]. The T1FNN controller was introduced to handle the uncertainties found in real systems, but it has been demonstrated to be limited in the handling of the uncertainties of fuzzy membership sets and rule-based type-1 fuzzy logic systems (T1FLSs). Therefore, a type-2 fuzzy set (T2FS) was used. A T2FS is characterized by a fuzzy membership function (MF) (i.e., the membership grade for each element of this set is a fuzzy set in [0,1]), unlike a type-1 fuzzy set, the membership grade of which is a crisp number in [0,1] [5]. Therefore, a T2FS provides
∗ Corresponding author. Tel.: +20 1006469369. E-mail addresses: ahmed elnagar@menofia.edu.eg (A.M. El-Nagar), [email protected] (M. El-Bardini).
additional degrees of freedom that make it possible to model and handle the uncertainties directly [6]. A type-2 fuzzy logic system (T2FLS) is also characterized by IF–THEN rules, but its antecedent or consequent sets are type-2 sets. A T2FLS can be used when the circumstances are too uncertain to exactly determine the membership grades, and these have been used in many applications, particularly in the control system [7–9]. The interval type-2 fuzzy logic system (IT2FLS) is a special case of the T2FLS [10] in which the IT2FLS is simpler to work with than a general T2FLS and distributes the uncertainty evenly among all admissible primary memberships [11]. The IT2FLS has been applied to various fields with great success [12–17]. The purpose of this study was to develop an interval type-2 fuzzy neural network (IT2FNN) controller that consists of an interval type-2 fuzzy linguistic process as the antecedent and an interval neural network as the consequent. The parameters of the IT2FNN controller were trained using the back-propagation (BP) method to minimize the difference between the desired and actual outputs. The multivariable anesthetic model, which represents modern general anesthesia, consists of muscle relaxation (MR)
http://dx.doi.org/10.1016/j.artmed.2014.03.002 0933-3657/© 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: El-Nagar AM, El-Bardini M. Interval type-2 fuzzy neural network controller for a multivariable anesthesia system based on a hardware-in-the-loop simulation. Artif Intell Med (2014), http://dx.doi.org/10.1016/j.artmed.2014.03.002
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(i.e., paralysis), unconsciousness (i.e., hypnosis), and analgesia (i.e., pain relief) [18]. Two drugs, namely isoflurane and atracurium, are commonly used for general anesthesia. These drugs elicit the anesthesia and MR signs, which are represented by the mean arterial blood pressure (MABP) and the evoked electromyogram (EMG), respectively [19]. There are two main problems associated with multivariable anesthesia systems [20]. First, the nonlinear structure of the pharmacodynamics representing the relaxant drug behavior may make the MR level saturate with any large control dose. Second, there is great uncertainty inherited from the large interindividual variability of the patient parameters, and a large delay is associated with this process. Hence, these problems make the multivariable anesthesia system a very challenging one. The T1FNN controller has been previously used to control the anesthesia system [21–24]. The parameters of the T1FNN controller were trained using the BP algorithm. Tosun and Güntürkün [24] tested a T1FNN controller using 10 datasets obtained from four different patients. In our previous work [25], the interval type-2 fuzzy logic controller (IT2FLC) and the adaptive interval type-2 fuzzy logic controller (AIT2FLC) were proposed for controlling the multivariable anesthesia system. Our results showed that the AIT2FLC rather than the IT2FLC is able to respond to the uncertainty introduced by the large inter- and intra-individual variability of patient parameters. In this paper, the IT2FNN controller was proposed for controlling the multivariable anesthesia system. The test was performed using a hardware-in-the-loop (HIL) simulation. The results of the proposed IT2FNN controller were compared with those obtained with a T1FNN controller and an AIT2FLC. The robustness of the IT2FNN controller was expected to provide some performance improvements compared with the performance achieved with a T1FNN controller and an AIT2FLC due to the reduced effects of the inter-individual variability of patient parameters and surgical stimulations. This paper is organized as follows. In Section 2, the IT2FNN controller is presented. The description of the mathematical model of the multivariable anesthesia system is presented in Section 3. The HIL simulation of the multivariable anesthesia system is described in Section 4. Section 5 details the experimental results, and Section 6 presents the conclusions.
Fig. 1 shows a 2-D interval type-2 Gaussian MF with a fixed mean, m, and an uncertain standard deviation in [ 1 , 2 ]. This MF can be expressed as Eq. (1) [26]: 1 A˜ (x) = exp − 2
x − m 2
,
∈ [, ¯ ]
(1)
The T2FS is found in a region called the footprint of uncertainty and is bounded by an upper membership and a lower membership, ¯ A˜ (x) and ˜ (x ), respectively. which are denoted A The network structure of the IT2FNN controller is shown in Fig. 2. This controller consists of an interval type-2 fuzzy linguistic process as the antecedent and a three-layer interval neural network as the consequent. In the following derivation, the superscripts of all of the symbols shown in Eqs. (2)–(7) represent the number of the layer of the IT2FNN controller. The IF–THEN rule for the IT2FNN controller can be expressed as f
f
membership grade of the secondary MF set to unity, which can be called the weighting interval set and is derived from IT2FSs in the consequent partition. This centroid set refers to the collection of centroids from all of the embedded T1FLSs. The IT2FNN controller is introduced as follows in each layer [27]: (1) Layer 1 – Input layer: For every node i in this layer, the node input and the node output are represented as neti1 (N) = xi1 ,
u1i = fi1 (neti1 (N)) = neti1 (N),
i = 1, 2
(3)
where N denotes the number of iterations.Layer 2 Membership layer: In this layer, each node performs an interval type-2 fuzzy MF, as shown in Fig. 1. For the jth node, ˜ j (x2 ) = f 2 (net 2 (N)) = exp(net 2 (N)) u2j (N) = M j j j i i
=
u¯ 2j (N) as
ij = ¯ ij
u2j (N)
ij = ij
as
j = 1, . . ., s
(4)
(2)
1 are the inputs of the where f = 1, 2, . . ., n is the rule number, x11 . . .xm f f ˜ . . .M ˜ m are the interval type-2 fuzzy sets IT2FNN controller, and M 1
where netj2 (N) = −(1/2)(xi2 − mij ) /(ij )2 , mij and ij are the mean and the standard deviation, respectively, of the Gaussian MF in the jth term of the ith input linguistic variable xi2 to the node of layer 2, and s is the number of the linguistic values with respect to each input node. As shown in Fig. 1, a type-2 MF can be repre¯ A˜ (x) and a lower MF sented as an interval bound by an upper MF ˜ (x). Therefore, the output of layer 2, u2j (N), is also represented as
A
u2j (N), u¯ 2j (N) .
Layer 3 Rule layer: Each node k in this layer is denoted by which multiplies the input signals and outputs the result. For the kth rule node, netk3 (N) =
3 x3 (N) wjk j
j
⎧ n ⎪ 3 2 3 ⎪ ¯ u (N) u¯ j ) = (wjk ⎪ k ⎨ u3k (N) = fk3 (netk3 (N)) = netk3 (N)
j=2 n
⎪ ⎪ 3 2 3 ⎪ ⎩ uk (N) = (wjk uj )
k = 1, . . ., n
(5)
j=1
f
1 ˜ , and x1 is M ˜ , and. . .and xm ˜m is M Rf : IF x11 is M 2 1 2 4 4 THEN u1 is wRf , wLf
4 , w 4 is a centroid set with the (IT2FSs) of the antecedent part. wRF Lf
2
2. IT2FNN controller
Fig. 1. Interval type-2 fuzzy set with uncertain mean.
3 are the where xj3 represents the jth input to the node of layer 3, wjk weights between the membership layer and the rule layer and are set to unity to simplify the implementation for real-time control, and n is the number of rules. Similar to layer 2, the output of layer 3 is represented as [u3k (N), u¯ 3k (N)].
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Fig. 2. Structure of the IT2FNN.
Layer 4 Type-reduction layer: This layer is used to implement the type-reduction. The center-of-set type-reduction based on Karnik–Mendel (KM) algorithms [28,29] was adopted in this study. The process of this layer is described as follows: netl4 (N)
n w4 u3 (N) k=1 k k = n 3 u (N)
k=1 k
n 4 3 ⎧ w u (N) ⎪ k=1 Rk Rk 4 ⎪ ⎨ uRl = n u3 (N) = WRT UR n k=1 4 Rk 3 u4l (N) = fl4 (netl4 (N)) = netl4 (N) = , w u (N) ⎪ k=1 Lk Lk T ⎪ 4 W U = ⎩ uLl = L L n 3
l=1
(6)
u (N)
k=1 Lk
wk4
4 w4 ] ∈ [wRk where Lk 4 quent set, WR = [wR1
u3R1 (N)
n
UR =
u3 (N) k=1 Rk
UL =
u3L1 (N)
n
is the centroid of the type-2 interval conseT
u3R2 (N)
T
u3Rn (N)
n
u3 (N) k=1 Rk
u3L2 (N)
u3 (N) k=1 Lk
T
4 4 ] W = [w 4 4 4 ] , wR2 . . . wRn wL2 . . . wLn L L1
. . . n
u3 (N) k=1 Rk
u3Ln (N)
n
u3 (N) k=1 Lk
,
and
T
2
=
. . . n
u3 (N) k=1 Lk
1 1 (WRT UR + WLT UL ) = W T U(X 1 , m, ) 2 2
J=
(7)
1 (y − y)2 2 d
,
(8)
where yd is the desired output, namely, the teacher’s signal, and y is the actual output. 4 4 4 wRk (N + 1) = wRk (N) + wRk (N)
4 wRk (N) =
set before the computation of u4l (N). Moreover, the computations of u4Rl and u4Ll are performed using the type-reduction algorithms described by Karnik and Mendel [28] and Mendel [29], respectively. Layer 5 Output layer: This layer performs the linear combination of u4Rl and u4Ll using the following equation: u4Rl + u4Ll
T
U = [UR UL ]T , X 1 = x11 x21 , m = [m11 . . .m1s m21 . . .m2s ], and = [11 . . .1s 21 . . .2s ]. The fuzzy neural network (FNN) requires the adjustment of the connection weights of all of the layers such that the output of the controlled object can approach the desired output. A previous study [2] used the BP algorithm to adjust the parameters of a T1FNN controller. This idea was extended to update the parameters of the IT2FNN controller. The target function is defined as follows:
4 wRk = −1
4 , w4 The weighting interval set wRK (k = 1, . . ., n) should be LK
u5o =
where u5o is the output of the IT2FNN controller, W = WRT WLT
∂J 4 ∂wRk
= −1
(9)
∂J ∂y ∂u5o . . ∂y ∂u5o ∂w4
1 1 (yd − y) · ıu · 2
(10)
Rk
u3 (N)
n Rk
u3 (N) k=1 Rk
,
k = 1, 2, . . ., n, (11)
where 1 is the learning rate for tuning the weighting interval factor 4 is the increment of weight w 4 , and ıu is defined in of FNN, wRk Rk Eq. (12) as ıu =
∂y ∼ y(N) − y(N − 1) = 5 uo (N) − u5o (N − 1)
∂u5o
(12)
4 can be changed Based on the same principle, the weight wLk as follows: 4 4 4 wLk (N + 1) = wLk (N) + wLk (N)
(13)
Please cite this article in press as: El-Nagar AM, El-Bardini M. Interval type-2 fuzzy neural network controller for a multivariable anesthesia system based on a hardware-in-the-loop simulation. Artif Intell Med (2014), http://dx.doi.org/10.1016/j.artmed.2014.03.002
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4 wLk (N) =
1 1 (yd − y) · ıu · 2
u3 (N)
n Lk
u3 (N) k=1 Lk
k = 1, 2, . . ., n.
,
(14)
For the same principle, the mean and standard deviation of the IT2FNN controller can be changed as follows: 1 2 (yd − y) · 2
m1j (N + 1) = m1j (N) +
·
js s
4 wLk (N) · u¯ 21j
ıu
n
u3 (N) k=1 Lk
· u¯ 22v
x11 − m1j (N)
·
2 (N) ¯ 1j
k=1+is v=1
1 2 (yd − y) · 2
,
The transfer function that describes the effect component with the atracurium concentration is given by the following equation:
k
G11 (S) =
u3 (N) k=1 Lk
j+s(s−1) s
·
(15)
ıu
n
4 wLk (N) · u¯ 21v · u¯ 22j ·
x21 − m2j (N) 2 (N) ¯ 2j
v=1
This section describes the multivariable anesthesia model, which combines the depth of anesthesia (DOA) and the MR. Two drugs were used as inputs to the model: isoflurane (for DOA) and atracurium (for producing MR). The individual pathways are described as follows [20]: 3.1. Atracurium mathematical model
j = 1, 2, . . ., s and i = 0, 1, 2, . . ., s − 1.
m2j (N + 1) = m2j (N) +
3. The multivariable anesthesia model
j = 1, 2, ..., s and k = j, j + s, j + 2s, . . ., j + s(s − 1) (16)
js s
·
n
ıu
u3 (N) k=1 Lk
4 wLk (N) · u¯ 21j · u¯ 22v ·
(x11 − ¯ 1j (N))
2
3 (N) ¯ 1j
k=1+is v=1
j = 1, 2, . . ., s and i = 0, 1, 2, . . ., s − 1 1 ıu 3 (yd − y). n 2 u3 (N) k=1 Rk
1j (N + 1) = 1j (N) +
.
(17)
js s
4 wRk (N).u21j .u22v .
(x11 − 1j (N))
2
1j 3 (N)
k=1+is v=1
j = 1, 2, ..., s and i = 0, 1, 2, ..., s − 1
¯ 2j (N + 1) = ¯ 2j (N) +
1 3 (yd − y) · 2
n
·
k
4 wLk (N) · u¯ 21v
(x21 − ¯ 2j (N))
·
2
(19) 1 3 (yd − y) · 2
n
·
u3 (N) k=1 Rk
k
v=1
4 wRk (N) · u21v · u22j ·
MABP K2 e−2 s = U2 (1 + T5 S)
(23)
where MABP is the change in the mean arterial blood pressure, U2 is the isoflurane drug input, and the values of parameters 2 , T5 and K2 are calculated from the pharmacokinetics parameters which are explained in detail by Mahfouf and Linkens [20]. 3.3. Interactive component model
ıu
j+s(s−1) s
where Eeff , Emax , and XE (50) are the drug effects produced (paralysis), the maximum drug effect (i.e., 100% paralysis), and the drug concentration that produces 50% of the maximal effect, respectively. The values of XE (50) and ˛ depend on the patient and vary from one patient to another. The overall nonlinear atracurium mathematical model was then obtained by combining Eq. (21) with the Hill equation, which is shown in Eq. (22). Fig. 3 shows a series of Hill equations for different values of ˛ and XE (50). For the nominal Hill equation values, the linearized gain for the operating points range from 0.85 to 0.95 for paralysis can lead to difficulties due to the curved shapes around this region. Additionally, the patientto-patient parameter variability can affect the nonlinearity shape (uncertainty) by making it steeper or flatter. All of these considerations make the MR process a very challenging one.
G22 (S) =
j = 1, 2, ..., s and k = j, j + s, j + 2s, . . ., j + s(s − 1)
2j (N + 1) = 2j (N) +
(22)
The transfer function that describes the variation of MABP with small changes in inhaled isoflurane concentration is given by:
3 (N) ¯ 2j
v=1
1 + ((XE (50)˛ )/XE˛ )
3.2. Isoflurane mathematical model
u3 (N) k=1 Lk
· u¯ 22j
Emax
(18)
ıu
j+s(s−1) s
(21)
where XE is the drug concentration, U1 is the atracurium drug input, and the values of parameters K1 , 1 , T1 , T2 , T3 , and T4 were calculated from the pharmacokinetics parameters, which are explained in detail by Mahfouf and Linkens [20]. Many pharmacokinetic–pharmacodynamics models have extensively used the Hill equation to describe the nonlinear drug dose–response relationships [30]. The Hill equation may be used to relate the effect to a specific blood drug concentration, Eeff =
1 ¯ 1j (N + 1) = ¯ 1j (N) + 3 (yd − y) · 2
XE K1 e−1 s (1 + T4 S) = U1 (1 + T1 S)(1 + T2 S)(1 + T3 S)
(x21 − 2j (N))
3.3.1. Atracurium to MABP interaction The atracurium to MABP interaction has been investigated in human beings, and there seem to be small, clinically insignificant changes in blood pressure.
2
2j 3 (N)
j = 1, 2, . . ., s and k = j, j + s, j + 2s, . . ., j + s(s − 1) (20) where 2 and 3 are the learning rate parameters of the mean and standard deviation of the IT2FNN controller, and s is the number of linguistic values with respect to each input node.
3.3.2. Isoflurane to MR interaction The overall model that describes the effect of isoflurane on MR is given by: G12 (S) =
K3 e−3 s (1 + T6 S)(1 + T7 S)
(24)
Please cite this article in press as: El-Nagar AM, El-Bardini M. Interval type-2 fuzzy neural network controller for a multivariable anesthesia system based on a hardware-in-the-loop simulation. Artif Intell Med (2014), http://dx.doi.org/10.1016/j.artmed.2014.03.002
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Fig. 3. Graph showing the shape of the various nonlinearity curves used in the system and model [25].
where the values of parameters K3 , 3 , T6 and T7 are calculated from the pharmacokinetics parameters which are explained in detail by Mahfouf and Linkens [20]. 4. HIL simulation of the multivariable anesthesia system HIL has been developed in industrial control for testing of systems comprised of some physical and some simulated components [31–33]. Simulation is used to represent processes that are physically unavailable or whose use would be too costly, dangerous or time-consuming. Proven benefits of HIL include reproducibility of experiments and the ability to perform tests which would otherwise be impossible, impractical or unsafe. Using HIL simulation of biological systems could provide: (1) the possibility to test for a wide range of simulated patient geometries and pathologies, (2) repeated testing on a consistent model, and (3) facilitated
MABP Signal
numerical quantification of performance by recording simulated physiological parameters [34]. Fig. 4 shows the block diagram of the IT2FNN for controlling the multivariable anesthesia system. The IT2FNN controller block has four inputs: the error signals (e1 and e2 ), the change in error signals (e1 and e2 ) and two control signal outputs (u1 and u2 ). An EMG sensor is used to produce the EMG signal, which represents the MR (paralysis level). Then, it is compared with set-point Ref1 to obtain e1 . The blood pressure sensor is used to produce the MABP signal that is compared with set-point Ref2 to obtain e2 . The output signal of the BP algorithm is used for tuning the weighting interval factor and the mean and standard deviation of the IT2FNN controller. The IT2FNN block shown in Fig. 4 has been explained in detail in Section 2, and it is implemented using a personal computer (PC). Fig. 5 shows the block diagram of the HIL simulation for the multivariable anesthesia system. The PC is used as a controller, and the anesthesia
Blood Pressure Sensor
EMG Signal
EMG Sensor
Computer Ref1
e1
+
Ref2
U1
e1
d/dt -
e2
+
FNN Controller
U2
Multivariable Anesthesia Model
e2
d/dt
BP Algorithm
+
-
+
-
Fig. 4. Block diagram of the IT2FNN controller for the multivariable anesthesia system.
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Sending data
PC based Controller
Receiving data
EMG Serial Communication
PIC18F452 (Anesthesia Model)
D/A MABP Converter
Osc.
LCD
Fig. 5. Block diagram of the HIL simulation of the multivariable anesthesia system.
model is implemented using a PIC microcontroller (PIC18F452). The communication between the PC and the PIC microcontroller is performed by serial communication via RS232. An analog to digital (A/D) converter is used to convert the digital outputs of the anesthesia system to analog values to be displayed on an oscilloscope. The values of two outputs of the anesthesia system are displayed on liquid crystal display (LCD) to monitor the numerical values in each sample. Implementation of the multivariable anesthesia system is shown in Fig. 6. 4.1. Hardware component The hardware used in the proposed architecture includes the following items: • PC Pentium 4. CPU 2.4 GHz/128 Mb cache, 256 MB random access memory. • A PIC development kit including the PIC18f452, LCD display, serial communication, and A/D converter using DAC0808. • An oscilloscope to show the response of the system in real time.
EMG) and −30 mmHg, respectively. Control signal 1 was limited between 0 and 1 for the atracurium drug input, and control signal 2 was limited between 0% and 5% for the isoflurane input [20]. For the anesthesia system, the control objective is to attain and maintain an adequate anesthetic level without an overshoot. This is done by maintaining a steady level of the MR and the MABP with a minimum deviation from the set-point [20]. The values of the learning rate parameters of the weighting interval factor, mean, and standard deviation of the IT2FNN controller are defined between [0,1]. The value of the learning rate is affected by the speed of the drug injection. Thus, when the learning rate is increased, the injection of the drug increased. The anesthetist chooses the learning rate based on the patient’s condition. The interval type-2 membership functions are initialized as follows: Output 1: MR m11 = −0.15, m12 = 0, m13 = 0.14, m14 = 0.4, m15 = 0.6, m21 = −0.06, m22 = −0.04, m23 = −0.02, m24 = 0, m25 = 0.045 ¯ 11 = 0.06, ¯ 12 = 0.07, ¯ 13 = 0.075, ¯ 14 = 0.08, ¯ 15 = 0.1, ¯ 21 = 0.009, ¯ 22 = 0.009, ¯ 23 = 0.009, ¯ 24 = 0.009, ¯ 25 = 0.017,
4.2. Software implementation The PC uses a Microsoft Windows XP operating system, and the MATLAB package includes some of the toolboxes that are used to perform the type-reduction algorithm. 5. Experimental results To determine the improvement of the IT2FNN, the previously proposed T1FNN proposed was also implemented for comparison purposes. In these experimental tasks, the initial conditions were 0% paralysis and 1 min sampling period. The set-point command signals for the MR and the MABP were 0.8 (80% paralysis – 20%
11 = 0.03, 12 = 0.04, 13 = 0.045, 14 = 0.05, 15 = 0.07, 21 = 0.006, 22 = 0.006, 23 = 0.006, 24 = 0.006, 25 = 0.012
Output 2: MABP m11 = −30, m12 = −20, m13 = −10, m14 = −1, m15 = 4, m21 = −3, m22 = 0, m23 = 4, m24 = 7, m25 = 12, ¯ 11 = 4.5, ¯ 12 = 4.5, ¯ 13 = 4.5, ¯ 14 = 3.1, ¯ 15 = 1.8, ¯ 21 = 1.9, ¯ 22 = 1.9, ¯ 23 = 1.9, ¯ 24 = 1.9, 11 = 3, 12 = 3, 13 = 3, 14 = 1.8, 15 = 1, 21 = 1.3, 22 = 1.3, 23 = 1.3, 24 = 1.3, 25 = 1.3
Fig. 6. System implementation.
All of the above initial values of the IT2FNN controller parameters are obtained from knowing the universe of discourse for the inputs of the IT2FNN controller (the error signal and the change of error signal). The universe of discourse is defined based on experience. For the multivariable anesthesia model, the universe of discourse for the error signal is [−0.3: 1] for the MR output and [−40: 10] mmHg for the MABP output [20]. The universe of discourse for the change of the error signal is [−0.1: 0.1] for the MR output and [−10: 20] mmHg for the MABP output [20]. After the initial values of the network parameters are chosen based on the limitations of the universe of discourse, the online learning algorithm shown in Eqs. (9)–(20) is updated with the network parameters. Any other values for the initial parameters within the limitations of the universe of discourse are acceptable because the online learning algorithm is updated with the final values, which guarantee the set-point tracking.
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5.1. Disturbance in blood pressure There are many disturbances for the patient during an operation. They can be divided into a short period (e.g., intubations, small incisions) and a long period (e.g., minor and major operations). There are two types of incision: surface incision and deep incision. A surface incision means that a patient is lightly stimulated, but a deep incision means a patient is severely stimulated. A minor operation is defined as an operation that contains surface incisions only, but an operation that contains both surface and deep incisions is called a major operation [35]. In this study, the disturbances considered were due to surface incisions and deep incisions that tend to make the MABP rise to undesired values. This affects the controller performance when the multivariable parameters are set to its nominal values, which are K1 = 1.0, 1 = 1 min, T1 = 4.81 min, T2 = 34.42 min, T3 = 3.08 min, T4 = 10.64 min, 2 = 25 s, T5 = 2 min, K2 = −15 mmHg/%, K3 = 0.27, 3 = 1 min, T6 = 1.25 min, T7 = 2.83 min, XE (50) = 0.404 g ml−1 , and ˛ = 2.98. Fig. 7 shows the effects of a light surgical stimulation occurring at t = 30 min and a severe surgical stimulation occurring at t = 60 min. For the MR output, (1) the performance under the effect of a light surgical stimulation is good for both the T1FNN and the IT2FNN controllers because the peak value after the effect of stimulation is very small, and (2) the performance under the effect of the severe surgical stimulation for the IT2FNN controller has a smaller peak value and a smaller settling time versus the T1FNN controller. For the MABP output, (1) the performance has a small settling time for both the T1FNN controller and the IT2FNN controller under the effect of the light surgical stimulation, and (2) the performance of the T1FNN controller under the effect of the severe surgical stimulation has a large peak value that increases in the values greater than 0 mmHg, which means that the blood pressure was increased, but the controller decreased the blood pressure. Additionally, the performance of the T1FNN controller has a large settling time after the effect of stimulation. (3) The performance of the IT2FNN controller under the effect of the severe surgical stimulation has a small peak value and settling time. Thus, the IT2FNN controller is superior to the T1FNN controller for minimizing the effect of surgical stimulation. 5.2. The inter-individual variability of the patient parameters In this test, the parameters change from patient to patient for the T1FNN controller and the IT2FNN controller. The model parameters are chosen in a random manner using the Monte-Carlo method [20]. The model parameters can be chosen according to the following formula: par = parmin + Random(parmax − parmin )
(25)
where 0 < Random < 1 and is obtained from a random number generator. The parmin and parmax values for each parameter were chosen to reflect probable pharmacological ranges known to exist. In this way, many combinations could be produced. According to this method, we study two cases for parameter changes. Case 1. The controller performance is investigated when the multivariable parameters are K1 = 1.69, 1 = 1 min, T1 = 2.45 min, T2 = 16.44 min, T3 = 4.28 min, T4 = 7.30 min, 2 = 25 s, T5 = 1.54 min, K3 = 0.27, 3 = 1 min, T6 = 1.33 min, K2 = −17.70 mmHg/%, T7 = 3.01 min, XE (50) = 0.404 g ml−1 , and ˛ = 2.98. Fig. 8 shows the performance of the system for case 1. For the MR output, the performance is without any deviation from the set-point and without an overshoot for both the T1FNN and IT2FNN controllers. For the MABP output, the performance of the IT2FNN controller has a smaller settling time and a smaller
7
Table 1 The ISE, the ITAE, and the RMSE values of the T1FNN, the AIT2-FLC and IT2FNN for the muscle relaxation output. Tasks
T1FNN
AIT2-FLC [25]
IT2FNN
ISE
Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2)
3.111 3.454 3.069 306 96 45 0.125 0.186 0.175
2.956 3.281 3.050 56 26 196 0.120 0.181 0.174
ITAE
RMSE
2.503 3.130 2.954 146 29 20 0.112 0.177 0.171
Table 2 The ISE, the ITAE, and the RMSE values of the T1FNN, the AIT2-FLC and IT2FNN for the blood pressure output.
ISE
ITAE
RMSE
Tasks
T1FNN
AIT2-FLC [25]
IT2FNN
Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2)
10153 4380 2068 31241 12574 1983 7.623 6.618 4.548
2567 2014 1995 1194 522 9560 4.137 4.688 4.463
2132 1949 1934 6494 863 450 3.243 4.415 4.397
overshoot versus the performance of the T1FNN controller. Therefore, for this case, the IT2FNN controller is better than the T1FNN controller at minimizing the effect of inter-individual variability of patient parameters on the MABP output. Case 2. The controller performance is investigated when the multivariable parameters are K1 = 2.01, 1 = 1 min, T1 = 2.54 min, T2 = 27.88 min, T3 = 4.17 min, T4 = 14.89 min, 2 = 25 s, T5 = 1.57 min, K3 = 0.24, 3 = 1 min, T6 = 1.19 min, K2 = −17.26 mmHg/%, T7 = 2.79 min, XE (50) = 0.5 g ml−1 , and ˛ = 2.98. Fig. 9 shows the performance of the system for case 2. For the MR output, the performance of the IT2FNN controller is better than the performance of the T1FNN controller because there is a peak value at time equal 5 min and 95 min for the response of the T1FNN controller. For the MABP output, (1) the performance of the T1FNN controller has a large peak value at time equal 5 min and 95 min. Additionally, the performance has a deviation from the set-point, and (2) the performance of the IT2FNN controller has no peak value or steady state error. Thus, the IT2FNN controller is better than the T1FNN controller at minimizing the effect of inter-individual variability of patient parameters on the MR and the MABP. To show the visual indications of the control performance, an objective measure of error of performance over the simulation run was made using the integral of square of errors (ISE), integral of time and absolute error (ITAE), and the root mean square error (RMSE). The ISE, ITAE and RME are defined in Eqs. (26)–(28), respectively. To show the robustness of the proposed IT2FNN controller, the experimental results are compared with the T1FNN controller and the AIT2FLC, which was proposed previously for the multivariable anesthesia system [25]. Tables 1 and 2 list the ISE, ITAE and RMSE values for the MR and the MABP, respectively, for the T1FNN controller, the AIT2FLC and the IT2FNN controller for all of the above experimental tasks.
∞ [e(t)]2 dt
ISE =
(26)
0
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Fig. 7. Response of the multivariable anesthesia system (task 1).
Fig. 8. Response of the multivariable anesthesia system (task 2 – case 1).
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Fig. 9. Response of the multivariable anesthesia system (task 2 – case 2).
∞ ITAE =
t e(t) dt
(27)
0
N 1 RMSE = (e(t))2 N
(28)
i=1
In general, the ISE, ITAE and RMSE values were greater for the MABP than for the MR simply because of the non-normalized values for the MABP. As shown in Tables 1 and 2, all of the ISE, ITAE and RMSE values for the MR output and the MABP output obtained using the IT2FNN controller are lower than those obtained using the T1FNN controller and the AIT2FLC. Thus, the proposed IT2FNN controller is superior to the T1FNN controller and the AIT2FLC for responding to the effects of the surgical stimulation and the inter-individual variability of the patient parameters. 6. Conclusions In this paper, the IT2FNN controller has been proposed for controlling the multivariable anesthesia system. The test is performed using HIL simulation. The proposed controller has been tested using two experimental tasks, namely the inter-individual variability of the patient parameters and the effect of surgical stimulation, such as the disturbance in blood pressure related to the use of the surgeon’s knife, which tends to make the blood pressure rise to undesired values. Experimental results for the proposed controller were compared with the T1FNN controller, which was implemented previously for controlling the multivariable anesthesia system. Results show that the proposed controller
is able to respond to the uncertainty introduced by a large interindividual variability of patient parameters. Additionally, under the effect of surgical stimulation, the proposed controller performs better than the T1FNN controller because the blood pressure increases to a value outside of the limit of constraint when applying the T1FNN controller. To show the robustness of the proposed IT2FNN controller, the results are compared with the AIT2FLC, which was designed previously for controlling the multivariable anesthesia system. The test is performed using three performance indices (ISE, ITAE, and RMSE). All values obtained by the proposed IT2FNN controller are lower than the values obtained by the T1FNN controller and the AIT2FLC. Thus, the proposed controller is superior to the T1FNN controller and the AIT2FLC for responding to the inter-individual variability of the patient parameters and the effect of surgical stimulation. For the calculations of the proposed IT2FNN controller, the KM algorithms are used to perform the typereduction, which increases the time of computation. Therefore, in future work, we can propose other type-reduction methods, which decrease the time of computations for the output processing of the IT2FNN controller. The major contributions of this study are the successful application of the IT2FNN controller using the HIL simulation to control the multivariable anesthesia system and to reduce the effect of the uncertainty of the patient parameters and the surgical stimulation and the success of the IT2FNN control system to reduce the ISE, ITAE and RMSE, resulting in smaller values than those obtained using the T1FNN controller and the AIT2FLC. References [1] Fung RF, Lin FJ, Wai RJ, Lu PY. Fuzzy neural network control of a motor-quickreturn servomechanism. Mechatronics 2000;10:145–67.
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Interval type-2 fuzzy neural network controller for a multivariable anesthesia system based on a hardware-in-the-loop simulation Ahmad M. El-Nagar ∗ , Mohammad El-Bardini Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menofia University, Menouf 32852, Egypt
a r t i c l e
i n f o
Article history: Received 15 January 2013 Received in revised form 11 March 2014 Accepted 11 March 2014 Keywords: Hardware-in-the-loop Back-propagation algorithm Interval type-2 fuzzy neural network Anesthesia Analgesia Muscle relaxation
a b s t r a c t Objective: This manuscript describes the use of a hardware-in-the-loop simulation to simulate the control of a multivariable anesthesia system based on an interval type-2 fuzzy neural network (IT2FNN) controller. Methods and materials: The IT2FNN controller consists of an interval type-2 fuzzy linguistic process as the antecedent part and an interval neural network as the consequent part. It has been proposed that the IT2FNN controller can be used for the control of a multivariable anesthesia system to minimize the effects of surgical stimulation and to overcome the uncertainty problem introduced by the large inter-individual variability of the patient parameters. The parameters of the IT2FNN controller were trained online using a back-propagation algorithm. Results: Three experimental cases are presented. All of the experimental results show good performance for the proposed controller over a wide range of patient parameters. Additionally, the results show better performance than the type-1 fuzzy neural network (T1FNN) controller under the effect of surgical stimulation. The response of the proposed controller has a smaller settling time and a smaller overshoot compared with the T1FNN controller and the adaptive interval type-2 fuzzy logic controller (AIT2FLC). The values of the performance indices for the proposed controller are lower than those obtained for the T1FNN controller and the AIT2FLC. Conclusion: The IT2FNN controller is superior to the T1FNN controller for the handling of uncertain information due to the structure of type-2 fuzzy logic systems (FLSs), which are able to model and minimize the numerical and linguistic uncertainties associated with the inputs and outputs of the FLSs. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The type-1 fuzzy neural network (T1FNN) controller combines the capability of fuzzy reasoning to handle uncertain information and the capability of artificial neural networks to learn from processes [1]. These controllers have been successfully applied in many fields [2–4]. The T1FNN controller was introduced to handle the uncertainties found in real systems, but it has been demonstrated to be limited in the handling of the uncertainties of fuzzy membership sets and rule-based type-1 fuzzy logic systems (T1FLSs). Therefore, a type-2 fuzzy set (T2FS) was used. A T2FS is characterized by a fuzzy membership function (MF) (i.e., the membership grade for each element of this set is a fuzzy set in [0,1]), unlike a type-1 fuzzy set, the membership grade of which is a crisp number in [0,1] [5]. Therefore, a T2FS provides
∗ Corresponding author. Tel.: +20 1006469369. E-mail addresses: ahmed elnagar@menofia.edu.eg (A.M. El-Nagar), [email protected] (M. El-Bardini).
additional degrees of freedom that make it possible to model and handle the uncertainties directly [6]. A type-2 fuzzy logic system (T2FLS) is also characterized by IF–THEN rules, but its antecedent or consequent sets are type-2 sets. A T2FLS can be used when the circumstances are too uncertain to exactly determine the membership grades, and these have been used in many applications, particularly in the control system [7–9]. The interval type-2 fuzzy logic system (IT2FLS) is a special case of the T2FLS [10] in which the IT2FLS is simpler to work with than a general T2FLS and distributes the uncertainty evenly among all admissible primary memberships [11]. The IT2FLS has been applied to various fields with great success [12–17]. The purpose of this study was to develop an interval type-2 fuzzy neural network (IT2FNN) controller that consists of an interval type-2 fuzzy linguistic process as the antecedent and an interval neural network as the consequent. The parameters of the IT2FNN controller were trained using the back-propagation (BP) method to minimize the difference between the desired and actual outputs. The multivariable anesthetic model, which represents modern general anesthesia, consists of muscle relaxation (MR)
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(i.e., paralysis), unconsciousness (i.e., hypnosis), and analgesia (i.e., pain relief) [18]. Two drugs, namely isoflurane and atracurium, are commonly used for general anesthesia. These drugs elicit the anesthesia and MR signs, which are represented by the mean arterial blood pressure (MABP) and the evoked electromyogram (EMG), respectively [19]. There are two main problems associated with multivariable anesthesia systems [20]. First, the nonlinear structure of the pharmacodynamics representing the relaxant drug behavior may make the MR level saturate with any large control dose. Second, there is great uncertainty inherited from the large interindividual variability of the patient parameters, and a large delay is associated with this process. Hence, these problems make the multivariable anesthesia system a very challenging one. The T1FNN controller has been previously used to control the anesthesia system [21–24]. The parameters of the T1FNN controller were trained using the BP algorithm. Tosun and Güntürkün [24] tested a T1FNN controller using 10 datasets obtained from four different patients. In our previous work [25], the interval type-2 fuzzy logic controller (IT2FLC) and the adaptive interval type-2 fuzzy logic controller (AIT2FLC) were proposed for controlling the multivariable anesthesia system. Our results showed that the AIT2FLC rather than the IT2FLC is able to respond to the uncertainty introduced by the large inter- and intra-individual variability of patient parameters. In this paper, the IT2FNN controller was proposed for controlling the multivariable anesthesia system. The test was performed using a hardware-in-the-loop (HIL) simulation. The results of the proposed IT2FNN controller were compared with those obtained with a T1FNN controller and an AIT2FLC. The robustness of the IT2FNN controller was expected to provide some performance improvements compared with the performance achieved with a T1FNN controller and an AIT2FLC due to the reduced effects of the inter-individual variability of patient parameters and surgical stimulations. This paper is organized as follows. In Section 2, the IT2FNN controller is presented. The description of the mathematical model of the multivariable anesthesia system is presented in Section 3. The HIL simulation of the multivariable anesthesia system is described in Section 4. Section 5 details the experimental results, and Section 6 presents the conclusions.
Fig. 1 shows a 2-D interval type-2 Gaussian MF with a fixed mean, m, and an uncertain standard deviation in [ 1 , 2 ]. This MF can be expressed as Eq. (1) [26]: 1 A˜ (x) = exp − 2
x − m 2
,
∈ [, ¯ ]
(1)
The T2FS is found in a region called the footprint of uncertainty and is bounded by an upper membership and a lower membership, ¯ A˜ (x) and ˜ (x ), respectively. which are denoted A The network structure of the IT2FNN controller is shown in Fig. 2. This controller consists of an interval type-2 fuzzy linguistic process as the antecedent and a three-layer interval neural network as the consequent. In the following derivation, the superscripts of all of the symbols shown in Eqs. (2)–(7) represent the number of the layer of the IT2FNN controller. The IF–THEN rule for the IT2FNN controller can be expressed as f
f
membership grade of the secondary MF set to unity, which can be called the weighting interval set and is derived from IT2FSs in the consequent partition. This centroid set refers to the collection of centroids from all of the embedded T1FLSs. The IT2FNN controller is introduced as follows in each layer [27]: (1) Layer 1 – Input layer: For every node i in this layer, the node input and the node output are represented as neti1 (N) = xi1 ,
u1i = fi1 (neti1 (N)) = neti1 (N),
i = 1, 2
(3)
where N denotes the number of iterations.Layer 2 Membership layer: In this layer, each node performs an interval type-2 fuzzy MF, as shown in Fig. 1. For the jth node, ˜ j (x2 ) = f 2 (net 2 (N)) = exp(net 2 (N)) u2j (N) = M j j j i i
=
u¯ 2j (N) as
ij = ¯ ij
u2j (N)
ij = ij
as
j = 1, . . ., s
(4)
(2)
1 are the inputs of the where f = 1, 2, . . ., n is the rule number, x11 . . .xm f f ˜ . . .M ˜ m are the interval type-2 fuzzy sets IT2FNN controller, and M 1
where netj2 (N) = −(1/2)(xi2 − mij ) /(ij )2 , mij and ij are the mean and the standard deviation, respectively, of the Gaussian MF in the jth term of the ith input linguistic variable xi2 to the node of layer 2, and s is the number of the linguistic values with respect to each input node. As shown in Fig. 1, a type-2 MF can be repre¯ A˜ (x) and a lower MF sented as an interval bound by an upper MF ˜ (x). Therefore, the output of layer 2, u2j (N), is also represented as
A
u2j (N), u¯ 2j (N) .
Layer 3 Rule layer: Each node k in this layer is denoted by which multiplies the input signals and outputs the result. For the kth rule node, netk3 (N) =
3 x3 (N) wjk j
j
⎧ n ⎪ 3 2 3 ⎪ ¯ u (N) u¯ j ) = (wjk ⎪ k ⎨ u3k (N) = fk3 (netk3 (N)) = netk3 (N)
j=2 n
⎪ ⎪ 3 2 3 ⎪ ⎩ uk (N) = (wjk uj )
k = 1, . . ., n
(5)
j=1
f
1 ˜ , and x1 is M ˜ , and. . .and xm ˜m is M Rf : IF x11 is M 2 1 2 4 4 THEN u1 is wRf , wLf
4 , w 4 is a centroid set with the (IT2FSs) of the antecedent part. wRF Lf
2
2. IT2FNN controller
Fig. 1. Interval type-2 fuzzy set with uncertain mean.
3 are the where xj3 represents the jth input to the node of layer 3, wjk weights between the membership layer and the rule layer and are set to unity to simplify the implementation for real-time control, and n is the number of rules. Similar to layer 2, the output of layer 3 is represented as [u3k (N), u¯ 3k (N)].
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3
Fig. 2. Structure of the IT2FNN.
Layer 4 Type-reduction layer: This layer is used to implement the type-reduction. The center-of-set type-reduction based on Karnik–Mendel (KM) algorithms [28,29] was adopted in this study. The process of this layer is described as follows: netl4 (N)
n w4 u3 (N) k=1 k k = n 3 u (N)
k=1 k
n 4 3 ⎧ w u (N) ⎪ k=1 Rk Rk 4 ⎪ ⎨ uRl = n u3 (N) = WRT UR n k=1 4 Rk 3 u4l (N) = fl4 (netl4 (N)) = netl4 (N) = , w u (N) ⎪ k=1 Lk Lk T ⎪ 4 W U = ⎩ uLl = L L n 3
l=1
(6)
u (N)
k=1 Lk
wk4
4 w4 ] ∈ [wRk where Lk 4 quent set, WR = [wR1
u3R1 (N)
n
UR =
u3 (N) k=1 Rk
UL =
u3L1 (N)
n
is the centroid of the type-2 interval conseT
u3R2 (N)
T
u3Rn (N)
n
u3 (N) k=1 Rk
u3L2 (N)
u3 (N) k=1 Lk
T
4 4 ] W = [w 4 4 4 ] , wR2 . . . wRn wL2 . . . wLn L L1
. . . n
u3 (N) k=1 Rk
u3Ln (N)
n
u3 (N) k=1 Lk
,
and
T
2
=
. . . n
u3 (N) k=1 Lk
1 1 (WRT UR + WLT UL ) = W T U(X 1 , m, ) 2 2
J=
(7)
1 (y − y)2 2 d
,
(8)
where yd is the desired output, namely, the teacher’s signal, and y is the actual output. 4 4 4 wRk (N + 1) = wRk (N) + wRk (N)
4 wRk (N) =
set before the computation of u4l (N). Moreover, the computations of u4Rl and u4Ll are performed using the type-reduction algorithms described by Karnik and Mendel [28] and Mendel [29], respectively. Layer 5 Output layer: This layer performs the linear combination of u4Rl and u4Ll using the following equation: u4Rl + u4Ll
T
U = [UR UL ]T , X 1 = x11 x21 , m = [m11 . . .m1s m21 . . .m2s ], and = [11 . . .1s 21 . . .2s ]. The fuzzy neural network (FNN) requires the adjustment of the connection weights of all of the layers such that the output of the controlled object can approach the desired output. A previous study [2] used the BP algorithm to adjust the parameters of a T1FNN controller. This idea was extended to update the parameters of the IT2FNN controller. The target function is defined as follows:
4 wRk = −1
4 , w4 The weighting interval set wRK (k = 1, . . ., n) should be LK
u5o =
where u5o is the output of the IT2FNN controller, W = WRT WLT
∂J 4 ∂wRk
= −1
(9)
∂J ∂y ∂u5o . . ∂y ∂u5o ∂w4
1 1 (yd − y) · ıu · 2
(10)
Rk
u3 (N)
n Rk
u3 (N) k=1 Rk
,
k = 1, 2, . . ., n, (11)
where 1 is the learning rate for tuning the weighting interval factor 4 is the increment of weight w 4 , and ıu is defined in of FNN, wRk Rk Eq. (12) as ıu =
∂y ∼ y(N) − y(N − 1) = 5 uo (N) − u5o (N − 1)
∂u5o
(12)
4 can be changed Based on the same principle, the weight wLk as follows: 4 4 4 wLk (N + 1) = wLk (N) + wLk (N)
(13)
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4
4 wLk (N) =
1 1 (yd − y) · ıu · 2
u3 (N)
n Lk
u3 (N) k=1 Lk
k = 1, 2, . . ., n.
,
(14)
For the same principle, the mean and standard deviation of the IT2FNN controller can be changed as follows: 1 2 (yd − y) · 2
m1j (N + 1) = m1j (N) +
·
js s
4 wLk (N) · u¯ 21j
ıu
n
u3 (N) k=1 Lk
· u¯ 22v
x11 − m1j (N)
·
2 (N) ¯ 1j
k=1+is v=1
1 2 (yd − y) · 2
,
The transfer function that describes the effect component with the atracurium concentration is given by the following equation:
k
G11 (S) =
u3 (N) k=1 Lk
j+s(s−1) s
·
(15)
ıu
n
4 wLk (N) · u¯ 21v · u¯ 22j ·
x21 − m2j (N) 2 (N) ¯ 2j
v=1
This section describes the multivariable anesthesia model, which combines the depth of anesthesia (DOA) and the MR. Two drugs were used as inputs to the model: isoflurane (for DOA) and atracurium (for producing MR). The individual pathways are described as follows [20]: 3.1. Atracurium mathematical model
j = 1, 2, . . ., s and i = 0, 1, 2, . . ., s − 1.
m2j (N + 1) = m2j (N) +
3. The multivariable anesthesia model
j = 1, 2, ..., s and k = j, j + s, j + 2s, . . ., j + s(s − 1) (16)
js s
·
n
ıu
u3 (N) k=1 Lk
4 wLk (N) · u¯ 21j · u¯ 22v ·
(x11 − ¯ 1j (N))
2
3 (N) ¯ 1j
k=1+is v=1
j = 1, 2, . . ., s and i = 0, 1, 2, . . ., s − 1 1 ıu 3 (yd − y). n 2 u3 (N) k=1 Rk
1j (N + 1) = 1j (N) +
.
(17)
js s
4 wRk (N).u21j .u22v .
(x11 − 1j (N))
2
1j 3 (N)
k=1+is v=1
j = 1, 2, ..., s and i = 0, 1, 2, ..., s − 1
¯ 2j (N + 1) = ¯ 2j (N) +
1 3 (yd − y) · 2
n
·
k
4 wLk (N) · u¯ 21v
(x21 − ¯ 2j (N))
·
2
(19) 1 3 (yd − y) · 2
n
·
u3 (N) k=1 Rk
k
v=1
4 wRk (N) · u21v · u22j ·
MABP K2 e−2 s = U2 (1 + T5 S)
(23)
where MABP is the change in the mean arterial blood pressure, U2 is the isoflurane drug input, and the values of parameters 2 , T5 and K2 are calculated from the pharmacokinetics parameters which are explained in detail by Mahfouf and Linkens [20]. 3.3. Interactive component model
ıu
j+s(s−1) s
where Eeff , Emax , and XE (50) are the drug effects produced (paralysis), the maximum drug effect (i.e., 100% paralysis), and the drug concentration that produces 50% of the maximal effect, respectively. The values of XE (50) and ˛ depend on the patient and vary from one patient to another. The overall nonlinear atracurium mathematical model was then obtained by combining Eq. (21) with the Hill equation, which is shown in Eq. (22). Fig. 3 shows a series of Hill equations for different values of ˛ and XE (50). For the nominal Hill equation values, the linearized gain for the operating points range from 0.85 to 0.95 for paralysis can lead to difficulties due to the curved shapes around this region. Additionally, the patientto-patient parameter variability can affect the nonlinearity shape (uncertainty) by making it steeper or flatter. All of these considerations make the MR process a very challenging one.
G22 (S) =
j = 1, 2, ..., s and k = j, j + s, j + 2s, . . ., j + s(s − 1)
2j (N + 1) = 2j (N) +
(22)
The transfer function that describes the variation of MABP with small changes in inhaled isoflurane concentration is given by:
3 (N) ¯ 2j
v=1
1 + ((XE (50)˛ )/XE˛ )
3.2. Isoflurane mathematical model
u3 (N) k=1 Lk
· u¯ 22j
Emax
(18)
ıu
j+s(s−1) s
(21)
where XE is the drug concentration, U1 is the atracurium drug input, and the values of parameters K1 , 1 , T1 , T2 , T3 , and T4 were calculated from the pharmacokinetics parameters, which are explained in detail by Mahfouf and Linkens [20]. Many pharmacokinetic–pharmacodynamics models have extensively used the Hill equation to describe the nonlinear drug dose–response relationships [30]. The Hill equation may be used to relate the effect to a specific blood drug concentration, Eeff =
1 ¯ 1j (N + 1) = ¯ 1j (N) + 3 (yd − y) · 2
XE K1 e−1 s (1 + T4 S) = U1 (1 + T1 S)(1 + T2 S)(1 + T3 S)
(x21 − 2j (N))
3.3.1. Atracurium to MABP interaction The atracurium to MABP interaction has been investigated in human beings, and there seem to be small, clinically insignificant changes in blood pressure.
2
2j 3 (N)
j = 1, 2, . . ., s and k = j, j + s, j + 2s, . . ., j + s(s − 1) (20) where 2 and 3 are the learning rate parameters of the mean and standard deviation of the IT2FNN controller, and s is the number of linguistic values with respect to each input node.
3.3.2. Isoflurane to MR interaction The overall model that describes the effect of isoflurane on MR is given by: G12 (S) =
K3 e−3 s (1 + T6 S)(1 + T7 S)
(24)
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Fig. 3. Graph showing the shape of the various nonlinearity curves used in the system and model [25].
where the values of parameters K3 , 3 , T6 and T7 are calculated from the pharmacokinetics parameters which are explained in detail by Mahfouf and Linkens [20]. 4. HIL simulation of the multivariable anesthesia system HIL has been developed in industrial control for testing of systems comprised of some physical and some simulated components [31–33]. Simulation is used to represent processes that are physically unavailable or whose use would be too costly, dangerous or time-consuming. Proven benefits of HIL include reproducibility of experiments and the ability to perform tests which would otherwise be impossible, impractical or unsafe. Using HIL simulation of biological systems could provide: (1) the possibility to test for a wide range of simulated patient geometries and pathologies, (2) repeated testing on a consistent model, and (3) facilitated
MABP Signal
numerical quantification of performance by recording simulated physiological parameters [34]. Fig. 4 shows the block diagram of the IT2FNN for controlling the multivariable anesthesia system. The IT2FNN controller block has four inputs: the error signals (e1 and e2 ), the change in error signals (e1 and e2 ) and two control signal outputs (u1 and u2 ). An EMG sensor is used to produce the EMG signal, which represents the MR (paralysis level). Then, it is compared with set-point Ref1 to obtain e1 . The blood pressure sensor is used to produce the MABP signal that is compared with set-point Ref2 to obtain e2 . The output signal of the BP algorithm is used for tuning the weighting interval factor and the mean and standard deviation of the IT2FNN controller. The IT2FNN block shown in Fig. 4 has been explained in detail in Section 2, and it is implemented using a personal computer (PC). Fig. 5 shows the block diagram of the HIL simulation for the multivariable anesthesia system. The PC is used as a controller, and the anesthesia
Blood Pressure Sensor
EMG Signal
EMG Sensor
Computer Ref1
e1
+
Ref2
U1
e1
d/dt -
e2
+
FNN Controller
U2
Multivariable Anesthesia Model
e2
d/dt
BP Algorithm
+
-
+
-
Fig. 4. Block diagram of the IT2FNN controller for the multivariable anesthesia system.
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Sending data
PC based Controller
Receiving data
EMG Serial Communication
PIC18F452 (Anesthesia Model)
D/A MABP Converter
Osc.
LCD
Fig. 5. Block diagram of the HIL simulation of the multivariable anesthesia system.
model is implemented using a PIC microcontroller (PIC18F452). The communication between the PC and the PIC microcontroller is performed by serial communication via RS232. An analog to digital (A/D) converter is used to convert the digital outputs of the anesthesia system to analog values to be displayed on an oscilloscope. The values of two outputs of the anesthesia system are displayed on liquid crystal display (LCD) to monitor the numerical values in each sample. Implementation of the multivariable anesthesia system is shown in Fig. 6. 4.1. Hardware component The hardware used in the proposed architecture includes the following items: • PC Pentium 4. CPU 2.4 GHz/128 Mb cache, 256 MB random access memory. • A PIC development kit including the PIC18f452, LCD display, serial communication, and A/D converter using DAC0808. • An oscilloscope to show the response of the system in real time.
EMG) and −30 mmHg, respectively. Control signal 1 was limited between 0 and 1 for the atracurium drug input, and control signal 2 was limited between 0% and 5% for the isoflurane input [20]. For the anesthesia system, the control objective is to attain and maintain an adequate anesthetic level without an overshoot. This is done by maintaining a steady level of the MR and the MABP with a minimum deviation from the set-point [20]. The values of the learning rate parameters of the weighting interval factor, mean, and standard deviation of the IT2FNN controller are defined between [0,1]. The value of the learning rate is affected by the speed of the drug injection. Thus, when the learning rate is increased, the injection of the drug increased. The anesthetist chooses the learning rate based on the patient’s condition. The interval type-2 membership functions are initialized as follows: Output 1: MR m11 = −0.15, m12 = 0, m13 = 0.14, m14 = 0.4, m15 = 0.6, m21 = −0.06, m22 = −0.04, m23 = −0.02, m24 = 0, m25 = 0.045 ¯ 11 = 0.06, ¯ 12 = 0.07, ¯ 13 = 0.075, ¯ 14 = 0.08, ¯ 15 = 0.1, ¯ 21 = 0.009, ¯ 22 = 0.009, ¯ 23 = 0.009, ¯ 24 = 0.009, ¯ 25 = 0.017,
4.2. Software implementation The PC uses a Microsoft Windows XP operating system, and the MATLAB package includes some of the toolboxes that are used to perform the type-reduction algorithm. 5. Experimental results To determine the improvement of the IT2FNN, the previously proposed T1FNN proposed was also implemented for comparison purposes. In these experimental tasks, the initial conditions were 0% paralysis and 1 min sampling period. The set-point command signals for the MR and the MABP were 0.8 (80% paralysis – 20%
11 = 0.03, 12 = 0.04, 13 = 0.045, 14 = 0.05, 15 = 0.07, 21 = 0.006, 22 = 0.006, 23 = 0.006, 24 = 0.006, 25 = 0.012
Output 2: MABP m11 = −30, m12 = −20, m13 = −10, m14 = −1, m15 = 4, m21 = −3, m22 = 0, m23 = 4, m24 = 7, m25 = 12, ¯ 11 = 4.5, ¯ 12 = 4.5, ¯ 13 = 4.5, ¯ 14 = 3.1, ¯ 15 = 1.8, ¯ 21 = 1.9, ¯ 22 = 1.9, ¯ 23 = 1.9, ¯ 24 = 1.9, 11 = 3, 12 = 3, 13 = 3, 14 = 1.8, 15 = 1, 21 = 1.3, 22 = 1.3, 23 = 1.3, 24 = 1.3, 25 = 1.3
Fig. 6. System implementation.
All of the above initial values of the IT2FNN controller parameters are obtained from knowing the universe of discourse for the inputs of the IT2FNN controller (the error signal and the change of error signal). The universe of discourse is defined based on experience. For the multivariable anesthesia model, the universe of discourse for the error signal is [−0.3: 1] for the MR output and [−40: 10] mmHg for the MABP output [20]. The universe of discourse for the change of the error signal is [−0.1: 0.1] for the MR output and [−10: 20] mmHg for the MABP output [20]. After the initial values of the network parameters are chosen based on the limitations of the universe of discourse, the online learning algorithm shown in Eqs. (9)–(20) is updated with the network parameters. Any other values for the initial parameters within the limitations of the universe of discourse are acceptable because the online learning algorithm is updated with the final values, which guarantee the set-point tracking.
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5.1. Disturbance in blood pressure There are many disturbances for the patient during an operation. They can be divided into a short period (e.g., intubations, small incisions) and a long period (e.g., minor and major operations). There are two types of incision: surface incision and deep incision. A surface incision means that a patient is lightly stimulated, but a deep incision means a patient is severely stimulated. A minor operation is defined as an operation that contains surface incisions only, but an operation that contains both surface and deep incisions is called a major operation [35]. In this study, the disturbances considered were due to surface incisions and deep incisions that tend to make the MABP rise to undesired values. This affects the controller performance when the multivariable parameters are set to its nominal values, which are K1 = 1.0, 1 = 1 min, T1 = 4.81 min, T2 = 34.42 min, T3 = 3.08 min, T4 = 10.64 min, 2 = 25 s, T5 = 2 min, K2 = −15 mmHg/%, K3 = 0.27, 3 = 1 min, T6 = 1.25 min, T7 = 2.83 min, XE (50) = 0.404 g ml−1 , and ˛ = 2.98. Fig. 7 shows the effects of a light surgical stimulation occurring at t = 30 min and a severe surgical stimulation occurring at t = 60 min. For the MR output, (1) the performance under the effect of a light surgical stimulation is good for both the T1FNN and the IT2FNN controllers because the peak value after the effect of stimulation is very small, and (2) the performance under the effect of the severe surgical stimulation for the IT2FNN controller has a smaller peak value and a smaller settling time versus the T1FNN controller. For the MABP output, (1) the performance has a small settling time for both the T1FNN controller and the IT2FNN controller under the effect of the light surgical stimulation, and (2) the performance of the T1FNN controller under the effect of the severe surgical stimulation has a large peak value that increases in the values greater than 0 mmHg, which means that the blood pressure was increased, but the controller decreased the blood pressure. Additionally, the performance of the T1FNN controller has a large settling time after the effect of stimulation. (3) The performance of the IT2FNN controller under the effect of the severe surgical stimulation has a small peak value and settling time. Thus, the IT2FNN controller is superior to the T1FNN controller for minimizing the effect of surgical stimulation. 5.2. The inter-individual variability of the patient parameters In this test, the parameters change from patient to patient for the T1FNN controller and the IT2FNN controller. The model parameters are chosen in a random manner using the Monte-Carlo method [20]. The model parameters can be chosen according to the following formula: par = parmin + Random(parmax − parmin )
(25)
where 0 < Random < 1 and is obtained from a random number generator. The parmin and parmax values for each parameter were chosen to reflect probable pharmacological ranges known to exist. In this way, many combinations could be produced. According to this method, we study two cases for parameter changes. Case 1. The controller performance is investigated when the multivariable parameters are K1 = 1.69, 1 = 1 min, T1 = 2.45 min, T2 = 16.44 min, T3 = 4.28 min, T4 = 7.30 min, 2 = 25 s, T5 = 1.54 min, K3 = 0.27, 3 = 1 min, T6 = 1.33 min, K2 = −17.70 mmHg/%, T7 = 3.01 min, XE (50) = 0.404 g ml−1 , and ˛ = 2.98. Fig. 8 shows the performance of the system for case 1. For the MR output, the performance is without any deviation from the set-point and without an overshoot for both the T1FNN and IT2FNN controllers. For the MABP output, the performance of the IT2FNN controller has a smaller settling time and a smaller
7
Table 1 The ISE, the ITAE, and the RMSE values of the T1FNN, the AIT2-FLC and IT2FNN for the muscle relaxation output. Tasks
T1FNN
AIT2-FLC [25]
IT2FNN
ISE
Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2)
3.111 3.454 3.069 306 96 45 0.125 0.186 0.175
2.956 3.281 3.050 56 26 196 0.120 0.181 0.174
ITAE
RMSE
2.503 3.130 2.954 146 29 20 0.112 0.177 0.171
Table 2 The ISE, the ITAE, and the RMSE values of the T1FNN, the AIT2-FLC and IT2FNN for the blood pressure output.
ISE
ITAE
RMSE
Tasks
T1FNN
AIT2-FLC [25]
IT2FNN
Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2) Task 1 Task 2 (Case 1) Task 2 (Case 2)
10153 4380 2068 31241 12574 1983 7.623 6.618 4.548
2567 2014 1995 1194 522 9560 4.137 4.688 4.463
2132 1949 1934 6494 863 450 3.243 4.415 4.397
overshoot versus the performance of the T1FNN controller. Therefore, for this case, the IT2FNN controller is better than the T1FNN controller at minimizing the effect of inter-individual variability of patient parameters on the MABP output. Case 2. The controller performance is investigated when the multivariable parameters are K1 = 2.01, 1 = 1 min, T1 = 2.54 min, T2 = 27.88 min, T3 = 4.17 min, T4 = 14.89 min, 2 = 25 s, T5 = 1.57 min, K3 = 0.24, 3 = 1 min, T6 = 1.19 min, K2 = −17.26 mmHg/%, T7 = 2.79 min, XE (50) = 0.5 g ml−1 , and ˛ = 2.98. Fig. 9 shows the performance of the system for case 2. For the MR output, the performance of the IT2FNN controller is better than the performance of the T1FNN controller because there is a peak value at time equal 5 min and 95 min for the response of the T1FNN controller. For the MABP output, (1) the performance of the T1FNN controller has a large peak value at time equal 5 min and 95 min. Additionally, the performance has a deviation from the set-point, and (2) the performance of the IT2FNN controller has no peak value or steady state error. Thus, the IT2FNN controller is better than the T1FNN controller at minimizing the effect of inter-individual variability of patient parameters on the MR and the MABP. To show the visual indications of the control performance, an objective measure of error of performance over the simulation run was made using the integral of square of errors (ISE), integral of time and absolute error (ITAE), and the root mean square error (RMSE). The ISE, ITAE and RME are defined in Eqs. (26)–(28), respectively. To show the robustness of the proposed IT2FNN controller, the experimental results are compared with the T1FNN controller and the AIT2FLC, which was proposed previously for the multivariable anesthesia system [25]. Tables 1 and 2 list the ISE, ITAE and RMSE values for the MR and the MABP, respectively, for the T1FNN controller, the AIT2FLC and the IT2FNN controller for all of the above experimental tasks.
∞ [e(t)]2 dt
ISE =
(26)
0
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Fig. 7. Response of the multivariable anesthesia system (task 1).
Fig. 8. Response of the multivariable anesthesia system (task 2 – case 1).
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Fig. 9. Response of the multivariable anesthesia system (task 2 – case 2).
∞ ITAE =
t e(t) dt
(27)
0
N 1 RMSE = (e(t))2 N
(28)
i=1
In general, the ISE, ITAE and RMSE values were greater for the MABP than for the MR simply because of the non-normalized values for the MABP. As shown in Tables 1 and 2, all of the ISE, ITAE and RMSE values for the MR output and the MABP output obtained using the IT2FNN controller are lower than those obtained using the T1FNN controller and the AIT2FLC. Thus, the proposed IT2FNN controller is superior to the T1FNN controller and the AIT2FLC for responding to the effects of the surgical stimulation and the inter-individual variability of the patient parameters. 6. Conclusions In this paper, the IT2FNN controller has been proposed for controlling the multivariable anesthesia system. The test is performed using HIL simulation. The proposed controller has been tested using two experimental tasks, namely the inter-individual variability of the patient parameters and the effect of surgical stimulation, such as the disturbance in blood pressure related to the use of the surgeon’s knife, which tends to make the blood pressure rise to undesired values. Experimental results for the proposed controller were compared with the T1FNN controller, which was implemented previously for controlling the multivariable anesthesia system. Results show that the proposed controller
is able to respond to the uncertainty introduced by a large interindividual variability of patient parameters. Additionally, under the effect of surgical stimulation, the proposed controller performs better than the T1FNN controller because the blood pressure increases to a value outside of the limit of constraint when applying the T1FNN controller. To show the robustness of the proposed IT2FNN controller, the results are compared with the AIT2FLC, which was designed previously for controlling the multivariable anesthesia system. The test is performed using three performance indices (ISE, ITAE, and RMSE). All values obtained by the proposed IT2FNN controller are lower than the values obtained by the T1FNN controller and the AIT2FLC. Thus, the proposed controller is superior to the T1FNN controller and the AIT2FLC for responding to the inter-individual variability of the patient parameters and the effect of surgical stimulation. For the calculations of the proposed IT2FNN controller, the KM algorithms are used to perform the typereduction, which increases the time of computation. Therefore, in future work, we can propose other type-reduction methods, which decrease the time of computations for the output processing of the IT2FNN controller. The major contributions of this study are the successful application of the IT2FNN controller using the HIL simulation to control the multivariable anesthesia system and to reduce the effect of the uncertainty of the patient parameters and the surgical stimulation and the success of the IT2FNN control system to reduce the ISE, ITAE and RMSE, resulting in smaller values than those obtained using the T1FNN controller and the AIT2FLC. References [1] Fung RF, Lin FJ, Wai RJ, Lu PY. Fuzzy neural network control of a motor-quickreturn servomechanism. Mechatronics 2000;10:145–67.
Please cite this article in press as: El-Nagar AM, El-Bardini M. Interval type-2 fuzzy neural network controller for a multivariable anesthesia system based on a hardware-in-the-loop simulation. Artif Intell Med (2014), http://dx.doi.org/10.1016/j.artmed.2014.03.002
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Please cite this article in press as: El-Nagar AM, El-Bardini M. Interval type-2 fuzzy neural network controller for a multivariable anesthesia system based on a hardware-in-the-loop simulation. Artif Intell Med (2014), http://dx.doi.org/10.1016/j.artmed.2014.03.002
