- Email: [email protected]
Long-Range Adaptive Fuzzy Control of Neuromuscular Blockade
Copyright © IF AC Modelling and Control in Biomedical Systems, Warwick, UK, 1997
LONG-RANGE ADAPTIVE FUZZY CONTROL OF NEUROMUSCULAR BLOCKADE J. Valente de Oliveira+, T. F. Mendonc;a*, J. M. Lemos+ + INESC Grupo de Controlo de Sistemas Dintimicos Apartado 13069, 1000 Lisboa, Portu9al, em ail: [email protected]
• Departamento de Matematica Aplicada, UniveTsidade do Porto Rua das Taipas 135, 4050 Porto, Portugal, email:[email protected]
Abstract: An application of adaptive predictive fuzzy control to the neuromuscular blockade using a continuous infusion is presented. The proposed controller attempts·to minimize a quadratic multi-step cost using a receding horizon strategy. Using a (nonlinear) fuzzy model based predictive controller one may hope to extend the operation range of linear predictor based controllers. The adaptive feature of the controller is expected to cope with the variability of patient responses. The reported examples using simulated patients reinforce the claim of the usefulness of this approach. Keyv.rords: Adaptive control, Fuzzy control, Biomedical control, Delivery system.
l. INTRODUCTION
bust, adaptive and fuzzy algorithms (Jacklitsch, et al. 1987; Lemos, et al. 1991; Linkens, 1994; Lago et al. , 1993, Mendonc;a, et aI., 1989).
The control of neuromuscular blockade is characterized by an high degree of uncertainty associated with the dynamic behavior of the system under control. The variability of patient responses implies the frequent adaptation of infusion rates, thus increasing the anesthetist workload. Automatic feedback control of infusion has been shmvn to outperform manual control (Mendonc;a, 1992; Linkens, 1994).
In this work, an adaptive predictive fuzzy model based control algorithm (Valente de Oliveira et al. , 1995) is used to control the neuromuscular blockade induced by short-acting relaxants in patients under surgery. The patient response is described over an extended time horizon by a fuzzy relational structure. A receding horizon strategy is used, and the control law attempts to minimize a quadratiC cost over this horizon. Using a (nonlinear) fuzzy predictive controller, one may hope to extend the operation range of linear predictor based controllers, namely by considering the dynamics of patient responses in the entire region of interest. The adaptive feature of the controller is expected to cope with the variability of patient responses. The reported examples of atracurium infusion control on simulated patients (Ward, et al., 1983; Weatherley, et al., 1983) reinforce the claim of the usefulness of this approach.
In this application, the advantages of a controller should be primarily related to its robustness, reliability and performance in a clinical environment. Moreover, the control system should allow the introduction of modifications in the controller in a way that is easily understood by the anesthetist, thus accommodating clinical experience and peculiar clinical situations. The search for better performance and reliability has led to a variety of different controllers of increased comple..x.ity. Several control strategies have been proposed for the control of neuromuscular blockade based upon ro-
183
2. FUZZY RELATIONAL MODELLING IDENTIFICATION
A~D
concerns convergence rates) one needs a linear parametrized model. This can be obtained from (1) using averaging-product composition followed by a linear defuzzification method. A structure under these conditions are named simplified fuzzy structures and are generically denoted as (Valente de Oliveira, et al., 1997):
A fuzzy relational model is characterized by a numeric-to-linguistic (NIL) interface, a linguistic processing stage, and a linguistic-to-numeric (L/N) interface. The N /L interface has n primary linguistic terms (fuzzy sets). The i-th of these fuzzy sets is represented by the membership function X;(x), defined in Xc R. The membership function X;(x) is parametrized in ef (Xi(X) = X; (x; ef)). Usually eX , is a two-element vector " eX = (eX ,1' eX)T ,2 , with being the center, and being the width of the i-th input membership function. Given a numeric datum, say x, aN IL interface converts it into a fuzzy representation X according to:
e;1
y[k+l] = Y[kl'" Y[k-p-+-llU[kl' " U[k-q+llD (2) 3. A LONG-RAl'IGE ADAPTIVE CONTROL ALGORITHM
ea
Consider a single-input single-output plant whose dynamics can be represented by a discrete time input-output model of the form of (2). Recalling . that U[kl = .cu(u[k]) and Y[kl = .cy(Y[k]), (2) can be rewritten as:
y[k+ll The output (or Linguistic-to-Numeric) interface converts the fuzzy set Y produced by the fuzzy processing stage into a numeric sample. Using for output membership functions the same notation as the one used for input membership functions, the j-th output membership function is denoted by 1j(y) = 1j(y; being a vector of parameters defined in the model output space Y. The output fuzzy set Y is thus represented by an array of membership degrees such that Y = (Y1 , .•• , Yj , ... , Ym ), m being the number of output membership functions. Using the centroide defuzzification, the numeric model output y is given by:
=
where"T is a continuous static non-linear mapping. A basic predictive control problem is to find the optimal control sequence U- = (u"[k], u"[k + 1], ... ,u" [k + H - 1])' where H refers to the chosen time horizon, such that the following multistep cost is minimized
en, Br
- = V (Y) = Y ) y
H
Q
L..,j=l
)
)
81
Y[k] U[k]
0···0 0'"
0
Y[k - p ...!..1] U[k - q + 1]
- 1])2](4)
where index 0 refers to the iterations of the optimization process in each sampling time instant k; 0 is an a priori fixed step size gain such that o < 0 ::; 1. The increments L':luo are computed according to what is believed to be a descent direction of the multi-step criterion, i.e., in the negative gradient direction:
0 0
+ il- y[k -+- i])2 + Pi (L':lu[k -+- i
(5)
The fuzzy processing stage of the model is giwn by: =
[(r[k
The minimization of Q over the horizon of H steps may be performed under several ty'Pes of strategies. One possible strategy is to assume that u is free (i.e. u[k + jl, (j = 0, ... , H - 1) are free). The minimization of (4) relatively to the control sequence U, is carried out at each sampling time instant according to:
where denotes the center of the j-th output membership function .
Y[k + 1]
~ ~L ;=1
"m Y8)1Y "m Y L..,)=1
"T(y[kj,y[k-l], ... ,y[k-p+1], u[k], u[k - 1], .. . , u[k - q + 1]) "Tk+l (3)
=
R (1)
where 0 is the fuzzy composition operator and represents a general e - 0 composition, e and 2 being a triangular co-norm and norm, repecti\"ely. U[k] = .cu(u[k]) stands for the linguistic representation of the model input at the discrete time instant k, and Y[k] = .cy(Ny(Y[kJ)).
L':luo
= =
-vuQo (1 + r)-1~eo
where 1 is the H x H identity matrix, by
For adaptive control applications, fast com'ergence rate identification algorithms for fuzzy relational models are desirable. For this, a suitable selection of the norms e and 0 is useful. For instance to apply the Recursive Least Squares (RLS) algorithm (one of the fastest algorithms in what
000
184
r
is given
11
1r---~r---~~--~----~----~
100
a)
---ref(t) r(t) neunnouscular blockade
80 60
0.4
40
0.2
20
O.OS 0.1 O.lS 0.2 0.25 UoD (normalized atracurium infusion)
time (minutes)
0.1
Fig. 1. The continuous infusion rate of atracurium is modelled considering three linguistic terms.
bl
--u(t) mg/kg/min
0.08 0.06
and
0.04-
aT" ... ! aU[k+l]
aT""'2 aulk] aT·""'2 au[k+l]
aT .... H ~ aT""'H
aT .... ! au[HH-I ]
aT .... 2 au[k+H-l]
aT""'H au[HH-lj
[ ~ aT.~,
au[HI]
I
After meeting a given stop criterion, the optimization process (5) yields the control sequence ii, such that ii ~ i? However the control action u[k] actually applied to the plant is: u[k] = (1 , 0, · ·· ,0 )u+(
0.02-
rw
O~~r-,-~r-,-~~~~r-'Ir--r-'I
o
20
40
60 time (minutes)
80
100
Fig. 2. A typical result when unmodelled dynamics is considered and a zero initialization of the model parameters is assumed. a) Reference signal and neuromuscular activity; b) Control signal u(t) .
(6)
'-v--' H elements
This imposition is a major source of difficulty for the start-up of adaptive algorithms. The initial bolus leads the output variable r(t) to a very low value (usually zero) where it remains for some time (usually more than 15 minutes) therefore conveying very little information on the patient dynamic characteristics.
This means that only the first control action is actually used, all the other are discarded. This is known as the receding horizon strategy. The whole optimization process is then repeated at time k+ 1 (details are given in (Valente de Oliveira, et al., 1995; Valente de Oliveira et al., 1997)) . 4. CONTROL RESULTS BASED ON SIMULATED PATIENTS
To evaluate the performance of this particular controller, worst case experiments were conducted. These are characterized by assuming unmodelled dynamics in the fuzzy model with respect to the pharmacokinetic model. Therefore, the assumed fuzzy model ,vas:
The dynamic response of neuromuscular blockade may be modelled by a linear pharmacokinetic model relating the drug infusion rate u( t) with the plasma concentration ep(t) and a non-linear model relating ep(t) with the induced pharmacodynamic response r(t) (Ward, et al., 1983; Weatherley, et al., 1983) . The variable r(t) normalized between o and 100 measures the level of muscular activity, with 0 corresponding to full paralysis and 100 to full muscular activity (no paralysis at all) .
y[k + 1]
= Y[k]Y[k -
l]Y[k - 2]U[k - , ]n (7)
where J is a positive delay set to 2 after some preliminary tests. Three linguistic terms were used for both the input and output signals. Figure 1 shows the linguistic terms for the normalized atracurium infusion (after the initial bolus).
For clinical reasons the patient must undergo an initial bolus in order to induce rela.'(ation in a short period of time (smaller than five minutes).
Figure 2 shows a typical result obtained when n in (7) is initialized as a zero matrix. The con-
185
REFERENCES
100
Jaklitsch, R. and D.R. Westenskow(1987). A model based self-adjusting two-phase controller for vecuronium induced muscle relaxation during anesthesia, IEEE Trans. on Biomedical Engineering, 34, 583-594. Lago, P., T. Mendon<;a, J . Lemos, 11. Seabra, S. Esteves and M.S. Araujo (1993 ). A {3Robust controller for closed loop drug delivery systems: application to the infusion of atracurium in general anesthesia. Proceedings of the IFAC Symposium on Modelling and Control in Biomedical Systems (ed. Patterson B. W.), 175-176. Lemos, J., T. Mendon<;a, and E. Mosca (1991) . Long-range adaptive control with input constraints, Int. J . Control, 54, 289-306. Linkens, D.A.(ed.) (1994). Intelligent Control in Biomedicine. Taylor and Francis. Mendon<;a, T. (1992). Metodos e algoritmos para o controlo de sistemas biol6gicos: aplica<;a.o ao controlo do bloqueio neuromuscular, PhD Thesis (portuguese), University of Oporto, Portugal. Mendon<;a, T., J . Lemos and P. Lago (1989). Long-range adaptive control of muscle relaxation. Proceedings of the IFA.C-BME Workshop Decision Support for Patient Jfanage· ment: Measurement, Modelling and Control, British ~'ledical Informatics, London, 23124l. Valente de Oliveira, J. and J. 1\1. Lemos (1995) . Long-range predictive adaptive fuzzy relational control, Fuzzy Sets and Systems, special issue on modern fuzzy controL 70 , 337357. Valente de Oliveira, J. (1995) . A design methodology for fuzzy system interfaces,lEEE Trans . on Fuzzy Systems, 3, 4, 404-414. Valente de Oliveira, J . and J. M. Lemo:; (1997) . Improving adaptive fuzzy control performance by speeding up identification, Journal of Intelligent €3 Fuzzy Systems, accepted for publication. Ward, S., A. Neil, B. \Veatherley and ~I. Corall (1983). Pharmacokinetics of atracurium besylate in healthy patients (after a single Lv. bolus dose), British Journal of Anaesthesia, 55 , 113-116. Weatherley, B., S. Williams and E. Neill (1983) . Pharmacokinetics, pharmcodynamics and dose-response relationships of atracurium administered i.v., British Journal of A,naesthesia, 55, 39s-45s.
a)
--ref(t) - - r(t) neuromuscular blockade
80 60 40
20 0 0
20
40 60 time (minutes)
80
0.1
100 b)
--u(t) mg/kg/min 0.08 0.06 0.04 0.02
O~~-~~~~~~~~-T-~~
o
20
40 60 time (minutes)
80
100
Fig. 3. A typical result for a controller "'ith unmodelled dynamics but with accumulated clinical experience: a) Reference and neuromuscular activity: b) Control signal u(t).
troller goes on-line when neuromuscular acti\;ty· r (t) reaches the value 4 for the first time after the bolus. The control horizon used was H = 6 (sampling period of Ts = 20 s) . Furthermore, in (-!) Pi = 0 was used. Figure 3 shows a typical simulation when the controller starts with tuned parameters, hereby using the published mean parameters, thus simulating some previously accumulated information. 5. CONCLUSIONS This \"ork summarizes the application of a longrange adaptive fuzzy control algorithm to the control of neuromuscular blockade. Other references should be consulted for further details. The simulated results obtained with the control of neuromuscular blockade in patients under surgery reinforce the usefulness of the proposed approach. Furthermore, since the adaptive fuzzy controller is kept semantically valid (Valente de Oliveira, 1995), this control system should allow the introduction of modifications in the controller, thus improving the anesthetist interface.
186
LONG-RANGE ADAPTIVE FUZZY CONTROL OF NEUROMUSCULAR BLOCKADE J. Valente de Oliveira+, T. F. Mendonc;a*, J. M. Lemos+ + INESC Grupo de Controlo de Sistemas Dintimicos Apartado 13069, 1000 Lisboa, Portu9al, em ail: [email protected]
• Departamento de Matematica Aplicada, UniveTsidade do Porto Rua das Taipas 135, 4050 Porto, Portugal, email:[email protected]
Abstract: An application of adaptive predictive fuzzy control to the neuromuscular blockade using a continuous infusion is presented. The proposed controller attempts·to minimize a quadratic multi-step cost using a receding horizon strategy. Using a (nonlinear) fuzzy model based predictive controller one may hope to extend the operation range of linear predictor based controllers. The adaptive feature of the controller is expected to cope with the variability of patient responses. The reported examples using simulated patients reinforce the claim of the usefulness of this approach. Keyv.rords: Adaptive control, Fuzzy control, Biomedical control, Delivery system.
l. INTRODUCTION
bust, adaptive and fuzzy algorithms (Jacklitsch, et al. 1987; Lemos, et al. 1991; Linkens, 1994; Lago et al. , 1993, Mendonc;a, et aI., 1989).
The control of neuromuscular blockade is characterized by an high degree of uncertainty associated with the dynamic behavior of the system under control. The variability of patient responses implies the frequent adaptation of infusion rates, thus increasing the anesthetist workload. Automatic feedback control of infusion has been shmvn to outperform manual control (Mendonc;a, 1992; Linkens, 1994).
In this work, an adaptive predictive fuzzy model based control algorithm (Valente de Oliveira et al. , 1995) is used to control the neuromuscular blockade induced by short-acting relaxants in patients under surgery. The patient response is described over an extended time horizon by a fuzzy relational structure. A receding horizon strategy is used, and the control law attempts to minimize a quadratiC cost over this horizon. Using a (nonlinear) fuzzy predictive controller, one may hope to extend the operation range of linear predictor based controllers, namely by considering the dynamics of patient responses in the entire region of interest. The adaptive feature of the controller is expected to cope with the variability of patient responses. The reported examples of atracurium infusion control on simulated patients (Ward, et al., 1983; Weatherley, et al., 1983) reinforce the claim of the usefulness of this approach.
In this application, the advantages of a controller should be primarily related to its robustness, reliability and performance in a clinical environment. Moreover, the control system should allow the introduction of modifications in the controller in a way that is easily understood by the anesthetist, thus accommodating clinical experience and peculiar clinical situations. The search for better performance and reliability has led to a variety of different controllers of increased comple..x.ity. Several control strategies have been proposed for the control of neuromuscular blockade based upon ro-
183
2. FUZZY RELATIONAL MODELLING IDENTIFICATION
A~D
concerns convergence rates) one needs a linear parametrized model. This can be obtained from (1) using averaging-product composition followed by a linear defuzzification method. A structure under these conditions are named simplified fuzzy structures and are generically denoted as (Valente de Oliveira, et al., 1997):
A fuzzy relational model is characterized by a numeric-to-linguistic (NIL) interface, a linguistic processing stage, and a linguistic-to-numeric (L/N) interface. The N /L interface has n primary linguistic terms (fuzzy sets). The i-th of these fuzzy sets is represented by the membership function X;(x), defined in Xc R. The membership function X;(x) is parametrized in ef (Xi(X) = X; (x; ef)). Usually eX , is a two-element vector " eX = (eX ,1' eX)T ,2 , with being the center, and being the width of the i-th input membership function. Given a numeric datum, say x, aN IL interface converts it into a fuzzy representation X according to:
e;1
y[k+l] = Y[kl'" Y[k-p-+-llU[kl' " U[k-q+llD (2) 3. A LONG-RAl'IGE ADAPTIVE CONTROL ALGORITHM
ea
Consider a single-input single-output plant whose dynamics can be represented by a discrete time input-output model of the form of (2). Recalling . that U[kl = .cu(u[k]) and Y[kl = .cy(Y[k]), (2) can be rewritten as:
y[k+ll The output (or Linguistic-to-Numeric) interface converts the fuzzy set Y produced by the fuzzy processing stage into a numeric sample. Using for output membership functions the same notation as the one used for input membership functions, the j-th output membership function is denoted by 1j(y) = 1j(y; being a vector of parameters defined in the model output space Y. The output fuzzy set Y is thus represented by an array of membership degrees such that Y = (Y1 , .•• , Yj , ... , Ym ), m being the number of output membership functions. Using the centroide defuzzification, the numeric model output y is given by:
=
where"T is a continuous static non-linear mapping. A basic predictive control problem is to find the optimal control sequence U- = (u"[k], u"[k + 1], ... ,u" [k + H - 1])' where H refers to the chosen time horizon, such that the following multistep cost is minimized
en, Br
- = V (Y) = Y ) y
H
Q
L..,j=l
)
)
81
Y[k] U[k]
0···0 0'"
0
Y[k - p ...!..1] U[k - q + 1]
- 1])2](4)
where index 0 refers to the iterations of the optimization process in each sampling time instant k; 0 is an a priori fixed step size gain such that o < 0 ::; 1. The increments L':luo are computed according to what is believed to be a descent direction of the multi-step criterion, i.e., in the negative gradient direction:
0 0
+ il- y[k -+- i])2 + Pi (L':lu[k -+- i
(5)
The fuzzy processing stage of the model is giwn by: =
[(r[k
The minimization of Q over the horizon of H steps may be performed under several ty'Pes of strategies. One possible strategy is to assume that u is free (i.e. u[k + jl, (j = 0, ... , H - 1) are free). The minimization of (4) relatively to the control sequence U, is carried out at each sampling time instant according to:
where denotes the center of the j-th output membership function .
Y[k + 1]
~ ~L ;=1
"m Y8)1Y "m Y L..,)=1
"T(y[kj,y[k-l], ... ,y[k-p+1], u[k], u[k - 1], .. . , u[k - q + 1]) "Tk+l (3)
=
R (1)
where 0 is the fuzzy composition operator and represents a general e - 0 composition, e and 2 being a triangular co-norm and norm, repecti\"ely. U[k] = .cu(u[k]) stands for the linguistic representation of the model input at the discrete time instant k, and Y[k] = .cy(Ny(Y[kJ)).
L':luo
= =
-vuQo (1 + r)-1~eo
where 1 is the H x H identity matrix, by
For adaptive control applications, fast com'ergence rate identification algorithms for fuzzy relational models are desirable. For this, a suitable selection of the norms e and 0 is useful. For instance to apply the Recursive Least Squares (RLS) algorithm (one of the fastest algorithms in what
000
184
r
is given
11
1r---~r---~~--~----~----~
100
a)
---ref(t) r(t) neunnouscular blockade
80 60
0.4
40
0.2
20
O.OS 0.1 O.lS 0.2 0.25 UoD (normalized atracurium infusion)
time (minutes)
0.1
Fig. 1. The continuous infusion rate of atracurium is modelled considering three linguistic terms.
bl
--u(t) mg/kg/min
0.08 0.06
and
0.04-
aT" ... ! aU[k+l]
aT""'2 aulk] aT·""'2 au[k+l]
aT .... H ~ aT""'H
aT .... ! au[HH-I ]
aT .... 2 au[k+H-l]
aT""'H au[HH-lj
[ ~ aT.~,
au[HI]
I
After meeting a given stop criterion, the optimization process (5) yields the control sequence ii, such that ii ~ i? However the control action u[k] actually applied to the plant is: u[k] = (1 , 0, · ·· ,0 )u+(
0.02-
rw
O~~r-,-~r-,-~~~~r-'Ir--r-'I
o
20
40
60 time (minutes)
80
100
Fig. 2. A typical result when unmodelled dynamics is considered and a zero initialization of the model parameters is assumed. a) Reference signal and neuromuscular activity; b) Control signal u(t) .
(6)
'-v--' H elements
This imposition is a major source of difficulty for the start-up of adaptive algorithms. The initial bolus leads the output variable r(t) to a very low value (usually zero) where it remains for some time (usually more than 15 minutes) therefore conveying very little information on the patient dynamic characteristics.
This means that only the first control action is actually used, all the other are discarded. This is known as the receding horizon strategy. The whole optimization process is then repeated at time k+ 1 (details are given in (Valente de Oliveira, et al., 1995; Valente de Oliveira et al., 1997)) . 4. CONTROL RESULTS BASED ON SIMULATED PATIENTS
To evaluate the performance of this particular controller, worst case experiments were conducted. These are characterized by assuming unmodelled dynamics in the fuzzy model with respect to the pharmacokinetic model. Therefore, the assumed fuzzy model ,vas:
The dynamic response of neuromuscular blockade may be modelled by a linear pharmacokinetic model relating the drug infusion rate u( t) with the plasma concentration ep(t) and a non-linear model relating ep(t) with the induced pharmacodynamic response r(t) (Ward, et al., 1983; Weatherley, et al., 1983) . The variable r(t) normalized between o and 100 measures the level of muscular activity, with 0 corresponding to full paralysis and 100 to full muscular activity (no paralysis at all) .
y[k + 1]
= Y[k]Y[k -
l]Y[k - 2]U[k - , ]n (7)
where J is a positive delay set to 2 after some preliminary tests. Three linguistic terms were used for both the input and output signals. Figure 1 shows the linguistic terms for the normalized atracurium infusion (after the initial bolus).
For clinical reasons the patient must undergo an initial bolus in order to induce rela.'(ation in a short period of time (smaller than five minutes).
Figure 2 shows a typical result obtained when n in (7) is initialized as a zero matrix. The con-
185
REFERENCES
100
Jaklitsch, R. and D.R. Westenskow(1987). A model based self-adjusting two-phase controller for vecuronium induced muscle relaxation during anesthesia, IEEE Trans. on Biomedical Engineering, 34, 583-594. Lago, P., T. Mendon<;a, J . Lemos, 11. Seabra, S. Esteves and M.S. Araujo (1993 ). A {3Robust controller for closed loop drug delivery systems: application to the infusion of atracurium in general anesthesia. Proceedings of the IFAC Symposium on Modelling and Control in Biomedical Systems (ed. Patterson B. W.), 175-176. Lemos, J., T. Mendon<;a, and E. Mosca (1991) . Long-range adaptive control with input constraints, Int. J . Control, 54, 289-306. Linkens, D.A.(ed.) (1994). Intelligent Control in Biomedicine. Taylor and Francis. Mendon<;a, T. (1992). Metodos e algoritmos para o controlo de sistemas biol6gicos: aplica<;a.o ao controlo do bloqueio neuromuscular, PhD Thesis (portuguese), University of Oporto, Portugal. Mendon<;a, T., J . Lemos and P. Lago (1989). Long-range adaptive control of muscle relaxation. Proceedings of the IFA.C-BME Workshop Decision Support for Patient Jfanage· ment: Measurement, Modelling and Control, British ~'ledical Informatics, London, 23124l. Valente de Oliveira, J. and J. 1\1. Lemos (1995) . Long-range predictive adaptive fuzzy relational control, Fuzzy Sets and Systems, special issue on modern fuzzy controL 70 , 337357. Valente de Oliveira, J. (1995) . A design methodology for fuzzy system interfaces,lEEE Trans . on Fuzzy Systems, 3, 4, 404-414. Valente de Oliveira, J . and J. M. Lemo:; (1997) . Improving adaptive fuzzy control performance by speeding up identification, Journal of Intelligent €3 Fuzzy Systems, accepted for publication. Ward, S., A. Neil, B. \Veatherley and ~I. Corall (1983). Pharmacokinetics of atracurium besylate in healthy patients (after a single Lv. bolus dose), British Journal of Anaesthesia, 55 , 113-116. Weatherley, B., S. Williams and E. Neill (1983) . Pharmacokinetics, pharmcodynamics and dose-response relationships of atracurium administered i.v., British Journal of A,naesthesia, 55, 39s-45s.
a)
--ref(t) - - r(t) neuromuscular blockade
80 60 40
20 0 0
20
40 60 time (minutes)
80
0.1
100 b)
--u(t) mg/kg/min 0.08 0.06 0.04 0.02
O~~-~~~~~~~~-T-~~
o
20
40 60 time (minutes)
80
100
Fig. 3. A typical result for a controller "'ith unmodelled dynamics but with accumulated clinical experience: a) Reference and neuromuscular activity: b) Control signal u(t).
troller goes on-line when neuromuscular acti\;ty· r (t) reaches the value 4 for the first time after the bolus. The control horizon used was H = 6 (sampling period of Ts = 20 s) . Furthermore, in (-!) Pi = 0 was used. Figure 3 shows a typical simulation when the controller starts with tuned parameters, hereby using the published mean parameters, thus simulating some previously accumulated information. 5. CONCLUSIONS This \"ork summarizes the application of a longrange adaptive fuzzy control algorithm to the control of neuromuscular blockade. Other references should be consulted for further details. The simulated results obtained with the control of neuromuscular blockade in patients under surgery reinforce the usefulness of the proposed approach. Furthermore, since the adaptive fuzzy controller is kept semantically valid (Valente de Oliveira, 1995), this control system should allow the introduction of modifications in the controller, thus improving the anesthetist interface.
186