IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL 39, NO
IO, OCTOBER 1992
1071
Adaptive Control of Multiplexed Closed-circuit Anesthesia Gyu-In Jee, Member, IEEE, and Rob J. Roy, Senior Member, IEEE
Abstract-This paper describes the design of an adaptive closed-circuitanesthesia controller based on a multiplexed mass spectrometer system. The controller deals with measurement deterioration caused by measurement delay and rise time through a long catheter as well as long sampling times due to the multiplexed measurements. Measurement data are extrapolated between sampling periods to increase the estimation convergence rate. A multiple-step-ahead predictive control algorithm is used to calculate intermediate control inputs between sampling intervals. Simulations are used to validate the designed controller.
INTRODUCTION HE anesthetic process typically consists of the following: The patient is given an intravenous bolus of a short acting barbiturate, followed by a short acting muscle relaxant. The patient is then orally intubated with an endotracheal tube to provide control of the patient’s airway. Inhalation agents, typically oxygen, nitrous oxide, and other anesthetic gas such as halothane or forane, are administered to the patient through a semiclosed anesthetic circle. A semiclosed anesthetic circle is one in which fresh gases are administered to the patient in excess of their body uptake, the expired gases are passed through a cannister which removes the carbon dioxide and the remaining gases returned to the circle. There is a popoff valve which releases excess gases to a scavenging system and then to the atmosphere. This valve usually releases at a pressure between one and two centimeters of water. An inflatable bag is T-connected on the inspiratory side of the circle. By squeezing this bag the anesthesiologist can force gases into the patient’s lung if the patient is not breathing. These bags have a pressure-volume characteristic such that they are filled between 1-2 cm of H 2 0 . If the fresh gas flow is far in excess of the patient’s requirements then most of the fresh gas flow is wasted, being passed to the atmosphere through the scavenging system. However, the concentration of gases that the patient is receiving is easily calculated, depending mostly on the fresh gas flow.
T
Manuscript received November 8, 1990; revised March 9, 1992. This work was supported by NIGMS Grant 1-ROI-GM38191. G.-I. Jee was with the Department of Biomedical Engineering, Rensselaer Polytechnic University, Troy, NY 12180. He is now with the Department of Electronics Engineering, Konkuk University, Seoul 133-701, Korea. R. J. Roy is with the Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180. IEEE Log Number 9202580.
In a closed-circuit the popoff valve is closed and the fresh gas flow is then set equal to the patient’s uptake plus any leak from the circuit. The early use of closed-circuit anesthesia [l] required a great deal of assumptions and estimates concerning the patient since accurate monitoring equipment was not available. Oxygen consumption was estimated by an empirical formula which used the three-quarter power of the patient’s weight. Liquid anesthetic was injected into the circuit and allowed to vaporize. It was assumed that anesthetic uptake was inversely proportional to the square root of time. Consequently, a unit liquid dose of anesthetic agent, based on an empirical formula for cardiac output, was injected into the circuit at 1, 4,9, 16, 25, 36, etc., min. In the hands of an experienced anesthesiologist this system provided excellent low cost anesthesia. By observing the volume in the breathing bag the patient’s uptake could be estimated. This gave a “feel” for the patient’s anesthetic state that was not obtainable in a semiclosed system. However, this was not a safe anesthetic regime for the less experienced. With the advent of reliable operating room mass spectrometers the monitoring required for such an anesthetic was available, and closed circuit anesthesia could be administered in a safe, reliable manner. Unfortunately, by this time the high flow, wasteful anesthetic administration had become well established and closed-circuit anesthesia became a curiousity. Recently with the increasing cost of anesthetics and decreasing government and other third party payments there has been renewed interest in providing a safe, low cost anesthetic. The coupling of low cost computers and increasingly sophisticated monitoring has brought about a reintroduction of closed circuit anesthetic regimes into operating suites. A recent publication by Vishnoi and Roy [2] addresses the coupling of computer and mass spectrometer to provide adaptive control of closed circuit anesthesia (CCA). By considering flow and mass balances around the closed circuit, bilinear state equations are obtained for alveolar oxygen, circuit volume, and alveolar halothane concentration. A recursive least-squares algorithm is used to estimate oxygen consumption, nitrous oxide uptake, and halothane uptake from the four measured quantities: alveolar oxygen, halothane, nitrous oxide concentration, and circuit volume. The use of these estimators allows the control system to adapt to a variation of system parameters among patients as well as within a patient over time.
0018-9294/92$03.00 0 1992 IEEE
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 39. NO. IO. OCTOBER 1992
1072
Since the state equations for the CCA system are nonlinear, a predictive or "receding horizon" control system was used. This paper examines the problems raised when the control system of [2] is used with a multiplexed mass spectrometer. Since a single mass spectrometer in each operating room is too costly, the usual arrangement is to have a central mass spectrometer service a number of operating rooms, typically ten. In this situation the gas profiles from the patients at ten current sites are stored in long sampling catheters and directed to the centrally located mass spectrometer [ 3 ] . The spectrometer switches between the catheters, providing end tidal and inspired gas concentrations which are displayed at each site. Since the sampling catheters are long (typically 30 m) the gas concentration measurements are degraded [4]. It has been shown [ 5 ] that the gas delivery characteristics of the catheter can be modeled as a first order system with a transport delay. The process of multiplexing increases the sampling interval of an individual site. A mass spectrometer dedicated to a single site will have a sampling interval of 6 s. If the mass spectrometer services ten locations then each location will be sampled every 60 s. If one of these locations is using the adaptive closed circuit anesthesia delivery system of [2], then a modification of the control law must be made to account for the increased sampling interval, rise time, and transport delay. A further understanding of a closed-circuit anesthesia delivery system can be obtained by following the path of gases around the circuit shown in Fig. 1. Flow from the ventilator is divided into the inspired minute volume VI and the circuit leak VL which is not recovered. The inspired minute volume vaporizes the liquid anesthetic input into U,. The inspired gases oxygen (0,) nitrous oxide (N,O) and halothane (H) are delivered to the patient as the inspired alveolar ventilation ( Both halothane and nitrous oxide are anesthetic agents while oxygen is necessary for body metabolism. In the lungs the blood takes up these three gases and delivers them to the body tissues. The rate of uptake of these gases into the body is given ~ . body produces carbon dioxide by VH, Vo2,and V N ~The (COz) which is released through the lungs back into the airway. The expired gases (V,,) pass through the carbon dioxide absorber where all the carbon dioxide is removed. Two mass flow controllers (MFC) deliver additional oxygen U,, and nitrous oxide (UNzo)into the circuit mixing with the expired gases to form the input to the ventilator. The position of the standing bellows ventilator at the end of expiration is determined by an infrared sensor and indicates changes in the circuit volume Vs. The rate of change of circuit volume Vs is a rate balance between body uptake, fresh gas inflow, and circuit leak. The mass spectrometer detects the fractional inspired and expired gases from the lung (FH,FN20, and FoJ. From a knowledge of the body uptake of gases, the circuit leak, and the change in circuit volume, the computer then calculates the required fresh gas flows (UH,UN20,and U02)to achieve a desired alveolar fractional concentration of these gases.
vA,).
tt
1 -1 s
K
~
BODY
Fig. 1. Closed-circuit anesthesia delivery system.
MODELEQUATIONS In order to understand the equations presented in this paper, a few key equations from (2) will be presented here. These are the equations for a single closed circuit delivery system. In examining these equations note that there is a mixed nomenclature. A flow is indicated by a dot over a variable. However the system is also discrete, as events occur breath by breath. There is a continuous inspired flow VI during the nth breath as well as a continuous expired flow during the nth breath. The computer of course uses only difference equations and the flows are considered constant during a given breath. The model of the closed circuit is derived by setting up the gas mass flow rate equation across the components of the circuit shown in Fig. 1. The result is a single inputsingle output (SISO) system for the halothane concentration and a multiple input-multiple output (MIMO) system for oxygen and circuit volume with oxygen and nitrous oxide as inputs. The effect of halothane on the MIMO system is assumed to be negligible. By considering the total inflows to and outflows from the circuit, the state equation for the circuit volume is obtained as
vE
JEE AND ROY: ADAPTIVE CONTROL OF CLOSED-CIRCUIT ANESTHESIA
Considering the mass balance for oxygen at the ventilator and completing the circuit we obtain a bilinear state equation for the alveolar oxygen Concentration.
+
FAO2 =
+ Uo, -
VNIO)
- FA02(U02
~O2I/VF.
+
UNzO)
(2)
Thus we obtain a MIMO nonlinear (bilinear) state space model for the alveolar oxygen concentration and circuit volume. Following the same procedure, we can obtain a similar equation for alveolar halothane concentration:
1073
control [7]. One common assumption on the future inputs is that the control input will remain constant. This constant future control input assumption is used here. The k-step-ahead predictive controller is designed to minimize a quadratic cost function consisting of the control effect and predicted output error variance from the desired output. The incorporation of the estimator with this controller results in an indirect adaptive predictive control scheme [2].
DESIGNCONSIDERATIONS The manner in which the gases are drawn has a direct -k V N 2 0 ) - FAH(UOz + U N 2 0 + U H ) FAH = bearing on the design of the control system. A 20 s profile of airway gas concentrations is stored in each sampling + - VH]/I/F. (3) catheter. This profile is sent down a long (30 m), thin In the derivation it is assumed that the inspired and al- (1.07 mm int. diam) catheter to the central mass specveolar concentrations of oxygen and halothane are equal. trometer. The gas flow rate is low due to a small inlet (760 This assumption, though not valid for open-circuit, gen- torr) to outlet (500 torr) pressure gradient. This results in erally holds in the closed-circuit in steady state. a transit time of approximately 21 seconds. When a particular catheter is switched to the mass spectrometer for gas analysis the outlet pressure drops to 80 torr, and the ESTIMATOR gas flow is rapidly increased. The high flow and sudden The estimated parameters are oxygen consumption, nitrous oxide and halothane uptakes. The controlled vari- decompression allows the 20 s gas profile to be analyzed ables are alveolar oxygen concentration (FAO?), circuit in approximately 6 s. The analysis provides for a single volume (V,) and alveolar halothane concentration. Four breath to be validated. If the same catheter remains conmeasurements are available, alveolar oxygen, halothane nected (single room analysis) the transport delay is reand nitrous oxide concentrations, and circuit volume. This duced to 6.5 s. Consequently the range of transport delay allows the estimator to be decoupled into the following is from 6.5 s to 21 s. This sampling, compression, and expansion of the airways provides a single breath analysis three equations: for each of 10 operating rooms every 60 s even though the data from each operating room may have undergone a V02 = Uo2 - v F F A O 2 - V , F ~ @- v s F 1 o 2 - PLF,,, 20 s transport delay. Furthermore, since the sampling (4) catheters have a small internal diameter the presence of airway moisture and mucus causes a degradation in the v N 2 0 = uN20 - I/FFAN20 - V S F A N 2 0 (5) gas delivery characteristic which can be modeled by the previously discussed transport delay and a first order lin- V LFAN20 - ‘ S F I N 2 0 ear system. However, the worst case time constant of the V,y = U, - vFFAH - V s ~ A H - VLFAH - v S F ~ H first order system is of the order of 50 ms which is negligible with respect to the transport delay. Therefore the (6) current measurement vector from the mass spectrometer The estimation of these parameters is performed by Z ( n ) is assumed equal to the true value Y ( n ) of the gas using recursive least squares estimation [6], [9]. It is as- concentration Nd samples in the past: sumed that the circuit leak (V,) and the lung functional z ( n ) = Y ( n - Nd). (7) residual capacity (VF) are known. CONTROLLER To control the bilinear system a predictive control algorithm [6] is used. It has the advantage of being able to utilize the true nonlinear model of the process. Thus it may be better able to capture the intrinsic features of the control problem than a control law based on a linear approximation. Predictive control is based on an assumed model of the process and on an assumed scenario for the future control inputs. This gives a sequence of control inputs. Only the first one is applied to the process, and a new sequence of control signals is calculated at the next sampling time. This strategy is called a receding horizon
The problems caused by the increased sampling time and transport lag lie in the controller design and the estimator equations. The original state equations of the closed circuit system are in continuous time. The conversion to a sampled data system involves calculating the derivatives as backward differences. As the sampling interval increases these approximations to the derivatives become less accurate. One of the assumptions used in the derivation of the state equations is that the alveolar and inspired concentrations are equal. This is certainly not true during transient disturbances, and as the sampling interval increases becomes less true during transients, causing large errors.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 10, OCTOBER 1992
1074
The state equations are bilinear, requiring a k-stepahead predictive control. The bilinearity becomes more pronounced as the sampling interval lengthens, increasing the difficulty of providing smooth control. Although all of the preceding problems with increased sampling interval cause a degradation of system performance the principal problem is with the recursive least squares estimator. The estimates of oxygen uptake, nitrous oxide uptake, and halothane uptake are necessary to provide the proper fresh gas flow to maintain the circuit volume and the proper concentrations. The number of samples required for convergence is the same regardless of the sampling interval. If the sampling interval is increased by a factor of ten, then the estimator takes ten times longer for the parameter values to converge. This is clearly too long for adequate control, and simulations have shown that instabilities can arise with erroneous estimation. Faster convergence is achieved by using extrapolated data between samples. The extrapolation used is based on the assumption that uptake is proportional to the inverse of the square root of time. With this type of extrapolation the estimator is reinitialized at each true data point. This method produces a smooth and rapid convergence.
In discrete time form it is rewritten as -
+ 1) =
FAH(~
[
1
h
.
+ VF (Vo2(n -
-
Nd)
+ VN20(n - N d )
h
-V F V,(n - Nd)
.
.
Now we need to calculate the gas uptakes Vo2,VNz0. From (4) O2 consumption at time t - Td is given by
V02(t - Td) = Uo2(t - T ~ -) VFFA02(t- T ~ ) - VS(t
-
Td)FAOz(t
- Td)
- VLFAO>(t - Td) - Vs(t - T d ) 6 0 2 ( t -
Td)
MODIFICATION OF CONTROL ALGORITHM The previous changes are incorporated in the design of a multiplexed adaptive controller. Predictor, estimator, and controller are modified to compensate for the delay in the measurements. First consider the equation for the halothane concentration. = [FAH(t)(VOz(t)
+
VN,O(t)
-
oO,(t)
- ONzO(t)
- o,> + QY(4 - KY0)1/VF. (8) Define F A H ( t ) as a current halothane concentration measurement through the mass spectrometer, which is delayed due to a long sampling catheter. FAH(~)
6 FAH(~
- ~d).
- VLFAo2(n>-
h .
- - VH(t-
VF
Nd
I n).
(15)
We define the prediction of O2 consumption at step n , Vo2(nI n) as Vo2(nI4
A oO2(4 -
equation (8) can be written as
NdF,02(4
(14)
a Vo2(n =
(9)
Define a basic time base h as the interval at which the controller and estimator are calculated and updated. This will be set to 6 s. Using a backward approximation for the derivative term,
VS(n -
- V F F A 0 2 ( 4 - VSs~)FAo*(n)
V&02(n>
- VS(4&02(4.
(16)
Following the same procedure N 2 0 and halothane consumption are defined as
Td).
1075
JEE AND ROY: ADAPTIVE CONTROL OF CLOSED-CIRCUIT ANESTHESIA
Using the same reasoning the coupled O2 and Vs prediction can be given by -
1
F A O ~ (N ~d
L
+ k I n)
Vs(n + k ( n )
1
where
1
k- 1
G(n)
E(n) 4
Note that the above equation is a time-varying bilinear equation (i.e., AH(n 1 n ) , BH(nI n) and EH(n 1 n) are timevarying). Now we consider the prediction of halothane concentration. Note that A H ( n 1 In), * * * , E H ( n 1 In), . . . , are unknown future variables at time n. We will consider them as constant during the derivation of the prediction equation. Then Nd-step-ahead prediction of halothane concentration at time n, F A H ( n + Nd1 n) which corresponds to F A H ( n 1 n) can be obtained by
+
pAH
IO, OCTOBER 1992
1071
Adaptive Control of Multiplexed Closed-circuit Anesthesia Gyu-In Jee, Member, IEEE, and Rob J. Roy, Senior Member, IEEE
Abstract-This paper describes the design of an adaptive closed-circuitanesthesia controller based on a multiplexed mass spectrometer system. The controller deals with measurement deterioration caused by measurement delay and rise time through a long catheter as well as long sampling times due to the multiplexed measurements. Measurement data are extrapolated between sampling periods to increase the estimation convergence rate. A multiple-step-ahead predictive control algorithm is used to calculate intermediate control inputs between sampling intervals. Simulations are used to validate the designed controller.
INTRODUCTION HE anesthetic process typically consists of the following: The patient is given an intravenous bolus of a short acting barbiturate, followed by a short acting muscle relaxant. The patient is then orally intubated with an endotracheal tube to provide control of the patient’s airway. Inhalation agents, typically oxygen, nitrous oxide, and other anesthetic gas such as halothane or forane, are administered to the patient through a semiclosed anesthetic circle. A semiclosed anesthetic circle is one in which fresh gases are administered to the patient in excess of their body uptake, the expired gases are passed through a cannister which removes the carbon dioxide and the remaining gases returned to the circle. There is a popoff valve which releases excess gases to a scavenging system and then to the atmosphere. This valve usually releases at a pressure between one and two centimeters of water. An inflatable bag is T-connected on the inspiratory side of the circle. By squeezing this bag the anesthesiologist can force gases into the patient’s lung if the patient is not breathing. These bags have a pressure-volume characteristic such that they are filled between 1-2 cm of H 2 0 . If the fresh gas flow is far in excess of the patient’s requirements then most of the fresh gas flow is wasted, being passed to the atmosphere through the scavenging system. However, the concentration of gases that the patient is receiving is easily calculated, depending mostly on the fresh gas flow.
T
Manuscript received November 8, 1990; revised March 9, 1992. This work was supported by NIGMS Grant 1-ROI-GM38191. G.-I. Jee was with the Department of Biomedical Engineering, Rensselaer Polytechnic University, Troy, NY 12180. He is now with the Department of Electronics Engineering, Konkuk University, Seoul 133-701, Korea. R. J. Roy is with the Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180. IEEE Log Number 9202580.
In a closed-circuit the popoff valve is closed and the fresh gas flow is then set equal to the patient’s uptake plus any leak from the circuit. The early use of closed-circuit anesthesia [l] required a great deal of assumptions and estimates concerning the patient since accurate monitoring equipment was not available. Oxygen consumption was estimated by an empirical formula which used the three-quarter power of the patient’s weight. Liquid anesthetic was injected into the circuit and allowed to vaporize. It was assumed that anesthetic uptake was inversely proportional to the square root of time. Consequently, a unit liquid dose of anesthetic agent, based on an empirical formula for cardiac output, was injected into the circuit at 1, 4,9, 16, 25, 36, etc., min. In the hands of an experienced anesthesiologist this system provided excellent low cost anesthesia. By observing the volume in the breathing bag the patient’s uptake could be estimated. This gave a “feel” for the patient’s anesthetic state that was not obtainable in a semiclosed system. However, this was not a safe anesthetic regime for the less experienced. With the advent of reliable operating room mass spectrometers the monitoring required for such an anesthetic was available, and closed circuit anesthesia could be administered in a safe, reliable manner. Unfortunately, by this time the high flow, wasteful anesthetic administration had become well established and closed-circuit anesthesia became a curiousity. Recently with the increasing cost of anesthetics and decreasing government and other third party payments there has been renewed interest in providing a safe, low cost anesthetic. The coupling of low cost computers and increasingly sophisticated monitoring has brought about a reintroduction of closed circuit anesthetic regimes into operating suites. A recent publication by Vishnoi and Roy [2] addresses the coupling of computer and mass spectrometer to provide adaptive control of closed circuit anesthesia (CCA). By considering flow and mass balances around the closed circuit, bilinear state equations are obtained for alveolar oxygen, circuit volume, and alveolar halothane concentration. A recursive least-squares algorithm is used to estimate oxygen consumption, nitrous oxide uptake, and halothane uptake from the four measured quantities: alveolar oxygen, halothane, nitrous oxide concentration, and circuit volume. The use of these estimators allows the control system to adapt to a variation of system parameters among patients as well as within a patient over time.
0018-9294/92$03.00 0 1992 IEEE
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 39. NO. IO. OCTOBER 1992
1072
Since the state equations for the CCA system are nonlinear, a predictive or "receding horizon" control system was used. This paper examines the problems raised when the control system of [2] is used with a multiplexed mass spectrometer. Since a single mass spectrometer in each operating room is too costly, the usual arrangement is to have a central mass spectrometer service a number of operating rooms, typically ten. In this situation the gas profiles from the patients at ten current sites are stored in long sampling catheters and directed to the centrally located mass spectrometer [ 3 ] . The spectrometer switches between the catheters, providing end tidal and inspired gas concentrations which are displayed at each site. Since the sampling catheters are long (typically 30 m) the gas concentration measurements are degraded [4]. It has been shown [ 5 ] that the gas delivery characteristics of the catheter can be modeled as a first order system with a transport delay. The process of multiplexing increases the sampling interval of an individual site. A mass spectrometer dedicated to a single site will have a sampling interval of 6 s. If the mass spectrometer services ten locations then each location will be sampled every 60 s. If one of these locations is using the adaptive closed circuit anesthesia delivery system of [2], then a modification of the control law must be made to account for the increased sampling interval, rise time, and transport delay. A further understanding of a closed-circuit anesthesia delivery system can be obtained by following the path of gases around the circuit shown in Fig. 1. Flow from the ventilator is divided into the inspired minute volume VI and the circuit leak VL which is not recovered. The inspired minute volume vaporizes the liquid anesthetic input into U,. The inspired gases oxygen (0,) nitrous oxide (N,O) and halothane (H) are delivered to the patient as the inspired alveolar ventilation ( Both halothane and nitrous oxide are anesthetic agents while oxygen is necessary for body metabolism. In the lungs the blood takes up these three gases and delivers them to the body tissues. The rate of uptake of these gases into the body is given ~ . body produces carbon dioxide by VH, Vo2,and V N ~The (COz) which is released through the lungs back into the airway. The expired gases (V,,) pass through the carbon dioxide absorber where all the carbon dioxide is removed. Two mass flow controllers (MFC) deliver additional oxygen U,, and nitrous oxide (UNzo)into the circuit mixing with the expired gases to form the input to the ventilator. The position of the standing bellows ventilator at the end of expiration is determined by an infrared sensor and indicates changes in the circuit volume Vs. The rate of change of circuit volume Vs is a rate balance between body uptake, fresh gas inflow, and circuit leak. The mass spectrometer detects the fractional inspired and expired gases from the lung (FH,FN20, and FoJ. From a knowledge of the body uptake of gases, the circuit leak, and the change in circuit volume, the computer then calculates the required fresh gas flows (UH,UN20,and U02)to achieve a desired alveolar fractional concentration of these gases.
vA,).
tt
1 -1 s
K
~
BODY
Fig. 1. Closed-circuit anesthesia delivery system.
MODELEQUATIONS In order to understand the equations presented in this paper, a few key equations from (2) will be presented here. These are the equations for a single closed circuit delivery system. In examining these equations note that there is a mixed nomenclature. A flow is indicated by a dot over a variable. However the system is also discrete, as events occur breath by breath. There is a continuous inspired flow VI during the nth breath as well as a continuous expired flow during the nth breath. The computer of course uses only difference equations and the flows are considered constant during a given breath. The model of the closed circuit is derived by setting up the gas mass flow rate equation across the components of the circuit shown in Fig. 1. The result is a single inputsingle output (SISO) system for the halothane concentration and a multiple input-multiple output (MIMO) system for oxygen and circuit volume with oxygen and nitrous oxide as inputs. The effect of halothane on the MIMO system is assumed to be negligible. By considering the total inflows to and outflows from the circuit, the state equation for the circuit volume is obtained as
vE
JEE AND ROY: ADAPTIVE CONTROL OF CLOSED-CIRCUIT ANESTHESIA
Considering the mass balance for oxygen at the ventilator and completing the circuit we obtain a bilinear state equation for the alveolar oxygen Concentration.
+
FAO2 =
+ Uo, -
VNIO)
- FA02(U02
~O2I/VF.
+
UNzO)
(2)
Thus we obtain a MIMO nonlinear (bilinear) state space model for the alveolar oxygen concentration and circuit volume. Following the same procedure, we can obtain a similar equation for alveolar halothane concentration:
1073
control [7]. One common assumption on the future inputs is that the control input will remain constant. This constant future control input assumption is used here. The k-step-ahead predictive controller is designed to minimize a quadratic cost function consisting of the control effect and predicted output error variance from the desired output. The incorporation of the estimator with this controller results in an indirect adaptive predictive control scheme [2].
DESIGNCONSIDERATIONS The manner in which the gases are drawn has a direct -k V N 2 0 ) - FAH(UOz + U N 2 0 + U H ) FAH = bearing on the design of the control system. A 20 s profile of airway gas concentrations is stored in each sampling + - VH]/I/F. (3) catheter. This profile is sent down a long (30 m), thin In the derivation it is assumed that the inspired and al- (1.07 mm int. diam) catheter to the central mass specveolar concentrations of oxygen and halothane are equal. trometer. The gas flow rate is low due to a small inlet (760 This assumption, though not valid for open-circuit, gen- torr) to outlet (500 torr) pressure gradient. This results in erally holds in the closed-circuit in steady state. a transit time of approximately 21 seconds. When a particular catheter is switched to the mass spectrometer for gas analysis the outlet pressure drops to 80 torr, and the ESTIMATOR gas flow is rapidly increased. The high flow and sudden The estimated parameters are oxygen consumption, nitrous oxide and halothane uptakes. The controlled vari- decompression allows the 20 s gas profile to be analyzed ables are alveolar oxygen concentration (FAO?), circuit in approximately 6 s. The analysis provides for a single volume (V,) and alveolar halothane concentration. Four breath to be validated. If the same catheter remains conmeasurements are available, alveolar oxygen, halothane nected (single room analysis) the transport delay is reand nitrous oxide concentrations, and circuit volume. This duced to 6.5 s. Consequently the range of transport delay allows the estimator to be decoupled into the following is from 6.5 s to 21 s. This sampling, compression, and expansion of the airways provides a single breath analysis three equations: for each of 10 operating rooms every 60 s even though the data from each operating room may have undergone a V02 = Uo2 - v F F A O 2 - V , F ~ @- v s F 1 o 2 - PLF,,, 20 s transport delay. Furthermore, since the sampling (4) catheters have a small internal diameter the presence of airway moisture and mucus causes a degradation in the v N 2 0 = uN20 - I/FFAN20 - V S F A N 2 0 (5) gas delivery characteristic which can be modeled by the previously discussed transport delay and a first order lin- V LFAN20 - ‘ S F I N 2 0 ear system. However, the worst case time constant of the V,y = U, - vFFAH - V s ~ A H - VLFAH - v S F ~ H first order system is of the order of 50 ms which is negligible with respect to the transport delay. Therefore the (6) current measurement vector from the mass spectrometer The estimation of these parameters is performed by Z ( n ) is assumed equal to the true value Y ( n ) of the gas using recursive least squares estimation [6], [9]. It is as- concentration Nd samples in the past: sumed that the circuit leak (V,) and the lung functional z ( n ) = Y ( n - Nd). (7) residual capacity (VF) are known. CONTROLLER To control the bilinear system a predictive control algorithm [6] is used. It has the advantage of being able to utilize the true nonlinear model of the process. Thus it may be better able to capture the intrinsic features of the control problem than a control law based on a linear approximation. Predictive control is based on an assumed model of the process and on an assumed scenario for the future control inputs. This gives a sequence of control inputs. Only the first one is applied to the process, and a new sequence of control signals is calculated at the next sampling time. This strategy is called a receding horizon
The problems caused by the increased sampling time and transport lag lie in the controller design and the estimator equations. The original state equations of the closed circuit system are in continuous time. The conversion to a sampled data system involves calculating the derivatives as backward differences. As the sampling interval increases these approximations to the derivatives become less accurate. One of the assumptions used in the derivation of the state equations is that the alveolar and inspired concentrations are equal. This is certainly not true during transient disturbances, and as the sampling interval increases becomes less true during transients, causing large errors.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 10, OCTOBER 1992
1074
The state equations are bilinear, requiring a k-stepahead predictive control. The bilinearity becomes more pronounced as the sampling interval lengthens, increasing the difficulty of providing smooth control. Although all of the preceding problems with increased sampling interval cause a degradation of system performance the principal problem is with the recursive least squares estimator. The estimates of oxygen uptake, nitrous oxide uptake, and halothane uptake are necessary to provide the proper fresh gas flow to maintain the circuit volume and the proper concentrations. The number of samples required for convergence is the same regardless of the sampling interval. If the sampling interval is increased by a factor of ten, then the estimator takes ten times longer for the parameter values to converge. This is clearly too long for adequate control, and simulations have shown that instabilities can arise with erroneous estimation. Faster convergence is achieved by using extrapolated data between samples. The extrapolation used is based on the assumption that uptake is proportional to the inverse of the square root of time. With this type of extrapolation the estimator is reinitialized at each true data point. This method produces a smooth and rapid convergence.
In discrete time form it is rewritten as -
+ 1) =
FAH(~
[
1
h
.
+ VF (Vo2(n -
-
Nd)
+ VN20(n - N d )
h
-V F V,(n - Nd)
.
.
Now we need to calculate the gas uptakes Vo2,VNz0. From (4) O2 consumption at time t - Td is given by
V02(t - Td) = Uo2(t - T ~ -) VFFA02(t- T ~ ) - VS(t
-
Td)FAOz(t
- Td)
- VLFAO>(t - Td) - Vs(t - T d ) 6 0 2 ( t -
Td)
MODIFICATION OF CONTROL ALGORITHM The previous changes are incorporated in the design of a multiplexed adaptive controller. Predictor, estimator, and controller are modified to compensate for the delay in the measurements. First consider the equation for the halothane concentration. = [FAH(t)(VOz(t)
+
VN,O(t)
-
oO,(t)
- ONzO(t)
- o,> + QY(4 - KY0)1/VF. (8) Define F A H ( t ) as a current halothane concentration measurement through the mass spectrometer, which is delayed due to a long sampling catheter. FAH(~)
6 FAH(~
- ~d).
- VLFAo2(n>-
h .
- - VH(t-
VF
Nd
I n).
(15)
We define the prediction of O2 consumption at step n , Vo2(nI n) as Vo2(nI4
A oO2(4 -
equation (8) can be written as
NdF,02(4
(14)
a Vo2(n =
(9)
Define a basic time base h as the interval at which the controller and estimator are calculated and updated. This will be set to 6 s. Using a backward approximation for the derivative term,
VS(n -
- V F F A 0 2 ( 4 - VSs~)FAo*(n)
V&02(n>
- VS(4&02(4.
(16)
Following the same procedure N 2 0 and halothane consumption are defined as
Td).
1075
JEE AND ROY: ADAPTIVE CONTROL OF CLOSED-CIRCUIT ANESTHESIA
Using the same reasoning the coupled O2 and Vs prediction can be given by -
1
F A O ~ (N ~d
L
+ k I n)
Vs(n + k ( n )
1
where
1
k- 1
G(n)
E(n) 4
Note that the above equation is a time-varying bilinear equation (i.e., AH(n 1 n ) , BH(nI n) and EH(n 1 n) are timevarying). Now we consider the prediction of halothane concentration. Note that A H ( n 1 In), * * * , E H ( n 1 In), . . . , are unknown future variables at time n. We will consider them as constant during the derivation of the prediction equation. Then Nd-step-ahead prediction of halothane concentration at time n, F A H ( n + Nd1 n) which corresponds to F A H ( n 1 n) can be obtained by
+
pAH
