IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

765

Multiple-Model Adaptive Predictive Control of Mean Arterial Pressure and Cardiac Output Clement Yu, Member, IEEE, Rob J . Roy, Senior Member, IEEE, Howard Kaufman, Senior Member, IEEE, and B. Wayne Bequette

Abstmct-A multiplemodel adaptive predictive controller has been designed to simultaneously regulate mean arterial pressure and cardiac output in congestive heart failure subjects by adjusting the infusion rates of nitroprusside and dopamine. The

had been designed a priori to give a specified performance. In a Bayesian approach, the current probability of each model representing the actual patient response is calalgorithm is based on the multiple-model adaptive controller culated, and the results are used to determine the new inand utilizes model predictive controliers to provide reliable fusion rate. As the response of the subject changes, the control in each model subspace. A total of 36 linear small-signal probabilities associated with each model will also adjust models were needed to span the entire space of anticipated re- themselves such that the model closest to the current subsponses. To reduce computation time, only the six models with ject response will have the greatest probability. The conthe highest probabilities were used in the control calculations. The controller was evaluated on laboratory animals that were cept of the MMAC was originally introduced by Lainiotis either surgically or pharmacologically altered to exhibit symp- [ 121, and it had been used in the control of aircraft [ 131, toms of congestive heart failure. During trials, the controller the management of arterial oxygen saturation levels [14), performance was robust with respect to excessive switching be- and in the regulation of mean arterial pressure [8], [9]. tween models and nonconvergence to a single dominant model. A comparison is also made with a previous multiple-drug con- The attraction of this controller, especially in the medical field, is the absence of any system identification maneuver troller design. prior to initiating control. It also provides explicit bounds on the parameters, preventing the estimation of physiologically unrealistic values. I. INTRODUCTION In the clinical setting, it is often desirable to simultaHE closed-loop control of mean arterial pressure neously regulate multiple variables by infusing more than (MAP) by adjusting the rate of infusion of a drug, usually sodium nitroprusside (NP), has enjoyed an enor- one drug. The concurrent infusion of a vasodilator and an inotropic agent has been shown to benefit a subject sufmous amount of interest in the past ten years. These studfering from congestive heart failure [15], [16]. Several ies culminated with the approval by the Food and Drug studies have explored the automation of such a multiple Administration for the marketing of the MAP-NP condrug-delivery application [ 171-[ 191. Serna et al. [17] troller [11. Initially, nonadaptive methods, usually industry-proven proportional-integral (PI) and proportional-in- studied the simultaneous control of cardiac output (CO) tegral-derivative (PID) controllers, were used to regulate and MAP by adjusting the infusion rates of dopamine arterial pressure about a set level [2], [3]. Since then, more (DP), an inotropic agent, and NP, a vasodilator. They complex adaptive control schemes were utilized to pro- separated the NP-MAP loop from the DP-CO loop, using vide consistent performance despite interpatient and time- an adaptive pole-placement self-tuning controller to regdependent variability in patient sensitivity to the drug [3]- ulate MAP and a heuristic control law to maintain CO. McInnis et al. [ 181 used a one-stepahead controller with 1113. One such adaptive algorithm is the multiple-model a bilinear model and recursive least squares identification adaptive controller (MMAC) [8], [9] which assumes that scheme to regulate MAP and central venous pressure. It the patient response to the drug can be represented by one was assumed that CO can be adjusted indirectly by reguof a finite number of models. For each model, a controller lating the two pressures. However, the relationship between the two blood pressures and CO depends on the vascular resistance and this may change with time. NeiManuscript received October 25, 1991; revised February 28, 1992. This ther Serna et al. nor M c h N s et al. had rigorously tested work was supported by the National Science Foundation under Award EETtheir designs in the laboratory. Voss et al. [19] applied a 8620246. C. Yu is with the BOC Group, Group Technical Center, Murray Hill, control advance moving average controller (CAMAC) to NJ 07974. the drug infusion problem and verified his design with anR. J. Roy is with the Department of Biomedical Engineering, Rensselaer imal experiments. However, large fluctuations in the drug Polytechnic Institute, Troy, NY 12180. H.Kaufman is with the Department of Electrical, Computer and Systems infusion levels were observed and identification of the paEngineering, Rensselaer Polytechnic Institute, Troy, NY 12180. tient parameters was needed prior to closing the control B. W.Bequette is with the Department of Chemical Engineering, Rensloop. Also,the controller was not applied on subjects with selaer Polytechnic Institute, Troy, NY 12180. IEEE Log Number 9201480. symptoms of heart failure.

T

0018-9294/92$03.00 0 1992 IEEE

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I

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

This paper describes the application of a multiple-model adaptive predictive control strategy for the simultaneous regulation of MAP and CO in heart failure subjects by adjusting the infusion rates of DP and NP. Instead of the conventional PI and PID controllers utilized in previous single-input, single-output MMAC designs [8], [9], a model predictive controller (MPC) was used for each model subspace. Hence, the term “predictive” has been added to the controller name. The MPC is comparatively simpler to tune than the PI and PID controllers in a multivariable environment [20]. The MPC, which is closely related to the Smith predictor in structure [21], has been shown to be robust with respect to uncertainty in the response delay time [22], and infusion rate constraints can be directly handled in the control computation [21], [23]. These properties make the MPC ideal for biomedical control applications. Adaptive versions of the MPC [21], [22] have generally utilized recursive least squares estimators to track changes in subject response. The controller described here uses the novel combination of the MMAC and the MPC to adapt to the nonlinear and time-varying nature of the patient response. No a priori identification period is needed for the MMAC. To alleviate the enormous amount of computations associated with this design, only the six models with the highest probabilities were considered in the calculation of the drug infusion rates. The multiplemodel adaptive predictive controller can be categorized as an adaptive model predictive controller as described by Garcia et al. [21]. The controller had been previously tested by simulating the patient response with an elaborate model [24], [25]. The results showed that the multiple-model adaptive predictive design is a feasible control algorithm for the management of multiple-drug infusion. The success of the computer simulations prompted further evaluation of the design with laboratory animals whose cardiovascular systems were pharmacologically or surgically altered to simulate congestive heart failure. To the authors’ knowledge, this is also the first time that an automated multidrug infusion controller, using a positive inotropic agent and a vasodilator, had been tested on such animals. This paper describes the result of those experiments. A comparison is also made with the CAMAC design.

11. MODELING THE PATIENT RESPONSE The multiple-model adaptive predictive controller requires a set of models to predict the patient response for a finite time into the future. Although extremely accurate models are desired, complex models are impractical due to the large amount of parameters that have to be estimated [26]. This is why relatively simple but insightful models are used in control designs. It must be stressed that the primary goal of the models in this application is reliable control, not the determination of some subtle patient characteristic. The controller assumes that the patient response can be

represented by a linear small-signal first-order transfer function matrix (1). The first order behavior associated with each element of the matrix was derived from past studies by different investigators. Using cross-correlation techniques, Slate [3] had shown that the MAP response to NP can be represented by a first-order system with two time delays. The second delay was omitted in this design. Sheppard et al. [27] also demonstrated that the NP action on aortic blood flow was inversely correlated with that on MAP when the patient exhibited a state of congestion. Similarly, Gingrich et al. [28] documented a first-order dynamic response for both MAP and CO with stepwise changes in infusion rates of DP in laboratory animals with acute heart failure. Packer et al. [29] used a first-order model to control MAP with DP in hypotensive patients.

:[

K1 e

(1

Eo:s?l=

K12e

+ 711s) -T~Is

(1

K e -T22s 22

721s) (1

1I:zl*

+ 712s) + 722s)

(1)

The nominal values and ranges of each parameter in the model are given in Table I [30]. These were derived from previous clinical and laboratory studies on the hemodynamic effects of NP and DP. In this design, the hypotensive effects of dopamine at small infusion rates ( e 2 pg/kg/min) were ignored. Likewise, the congestive state in heart failure subjects minimizes the preload effects of NP, resulting in increases in CO when the drug is infused. Therefore, the values for K I 2 and KZl were restricted to positive values. Since the model operates on changes in the output variables about the baseline level, it is assumed the baseline values of CO and MAP do not change. Although changes in MAP can affect CO and alterations in CO can modify MAP levels, (1) implies that MAP and CO are independent. The coupling between CO and MAP, while not explicitly stated in the model, is instead manifested by changes in parameter values, particularly gain. For example, if an increase in NP infusion decreased the MAP, the concurrent augmentation of CO by a reduction in afterload will tend to restore MAP back up again. This effect is perceived as a lower gain associated between NP and MAP. If the outputs were sampled, and changes in infusion rates were implemented at discrete times, the patient response can be expressed in the form of an controlled-autoregressive-integrated-moving-average(CARIMA) model. A(q-’)y(k) - = B(q-’) &(k

where A(q-’) and B(q-’) are 2

X

- 1)

2 polynomial matrices,

y(k) - is the 2 x 1 incremental output vector [AMAP (k) ACO @)IT, &(k) is the 2 x 1 incremental input vector [ANP(k) ADP(k)lT, and 4-l is the backward shift operator. Voss et al. [19] used a similar representation for pa-

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YU et al.: PREDICTIVE CONTROL OF ARTERIAL PRESSURE AND CARDIAC OUTPUT

TABLE I SMALL SIGNAL MODELPARAMETER VALUES Range

Nominal Value

Parameter

-6 0.667 0.75 12 0.667 0.75

3 2.0

1 .o 5 5.0 1 .o

-1.5 to -54

0.5 to 1 0.333 to 1 - 15 to 30 0.5 to 1 0.333 to 1 - 3 to 9 1.5 to 3 . 0 0.333 to 1.5 1 to 10

3.0 to 9.0 0.333 to 1.5

"Values are in terms of mm Hg (pg/kg/min)-'. bVaIues are in terms of mL/pg. 'Values are in terms of min.

._______________. c&lmmu Bak

I

Fig. 1. System structure. The plant is the subject in the experiments. The MPC's correspond to the individual controllers in the controller bank. The reference model is described by ( 5 ) and the models of the response constitute the model bank.

tient response but had a second-order behavior for each input-output pair. 111. DESIGN AND IMPLEMENTATION The multiple-model adaptive predictive controller combines the control strategies of the MMAC and MPC. The MMAC design here is based on that documented in [8], [9], and [14], and the MPC is similar to the controller described by Clarke et al. [22]. The overall schematic is displayed in Fig. 1. The control strategy of the MPC is to find a sequence of control signals [&(k), &(k l), &(k Nu l)] that will bring the predicted outputs [ y (k + NI( k ) ,y_ ( k + NI + 1 (k),* y-(k N,(k)],which is based on an

+

--

+

---

+

assumed model of the drug response, to some desired vec, ~ ( +k N2)]. The tor [ r ( k NI), _r(k N I l), span of the control sequence, 0 to Nu,is called the control horizon, while that of the predicted output, NI to N2,is called the output costing horizon. Although the computed control values are based on Nu time steps, only the first move is implemented and a new sequence of control inputs is computed at the next sampling instance. The computation of a control trajectory at each sample time accounts for model uncertainty and disturbances, and gives the controller integral action. The minimum value of the output costing horizon, NI, should be longer than output response time delays to maintain stability. If N I = N2and Nu = 1, the algorithm represents a weighted k-step ahead controller [31] with k = NI.

+

+

+

--

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

768

ply with the specified settling times while the time delays equal the maximum expected time delay of the response. The time constants and time delays in ( 5 ) are expressed min J in terms of minutes. The difference in desired settling (AM@).' . & ( k + N u - l ) ) times also led to the use of a longer output costing horizon N2 for CO (NI = 5 min, N2 = 15 min) than for MAP (NI = =E Ily(k + j ( k ) - r ( k + j ) h 1.5 min, N2 = 2.5 min). However, the same number of J=NI samples (samples = 3) was used in evaluating the cost N" function for both outputs to facilitate design computa( 3 ) tions. In each of the outputs, the errors were equally + J = I I I W k + j - 1)ll6] spaced throughout the output costing horizons. The dewhere [ ( k ) is the 2 x 1 desired output vector, y ( k j Ik) viation between predicted and desired values of MAP were is the j-step ahead predicted output given the present out- evaluated at 1.5, 2, and 2.5 min into the future, while put measurements, r is the 2 x 2 positive definite output those of CO were determined at 5 , 10, and 15 rnin ahead. error weighting matrix, /3 is the 2 X 2 positive semidefi- A sampling time of 30 s was specified. nite input weighting matrix, and E [ -1 is the mathematical A nonzero 0 matrix in (3) may also be used to shape expectation operator. The term inside the double vertical the closed-loop response of the system [20]. By penalizlines in (3) represent a quadratic product, that is: 1) x_ (I9 = ing changes in the input variables, a slower but smoother x_ TQx_.The weighting matrices I? and 0, the output costing servo trajectory can be achieved. A similar effect can be horizon parameters NI and N 2 , and the control horizon produced by restricting the number of moves in the conlength Nu are tuning parameters which can be used to trol horizon. In this design, the control horizon length was shape the closed-loop response of the system. set to one (Nu= 1) to minimize computations, and the Although the design calls for the regulation of both out- elements of the 0 matrix were chosen primarily to stabiputs, a tighter regulation of MAP is sought (*10 mm Hg). lize the system in the presence of noise. However, the To achieve this, the weight associated with the square of elements of 0 should also be small enough to enable the the deviation between predicted and desired MAP values system outputs to track the reference model despite the in the cost function was set to be 50 times greater than restrictions. The matrix /3 is of the form that for the square of the CO error. The larger penalty assures that the MAP steady-state error will remain smaller than that of CO even if one or both of the drug inputs are constrained. Part of the large contrast in error weights reflect the disparity in units for both output er- The selection of values for and Pz2 was found to be rors. If none of the drug infusion rates are at their con- dependent on the steady-state gains of the model. And strained values, then the expected values of the errors for since the MMAC, which will be coupled to the MPC in MAP and CO are zero due to the integral action of the this application, will utilize a number of models, the subMPC. The absolute values of the error weights were set script i had been added in (6). Increasing the values of PI] as follows and p22 reduces the controller gain and, by adjusting their values, the steady-state gain of the forward transfer func(4) tion of the controlled system can remain relatively fixed despite changes in response parameters. Effectively, this is what the adaptive controller seeks to achieve. An anaThe transient response should have a settling time of lytical approach to the selection of the elements of 0 has less than 10 min for MAP and 20-25 rnin for CO. The yet to be outlined. In this design, PI I and p2* were chosen difference in settling times stem from the disparity in time to be a function of the steady-state gains associated with constants for the two drugs. The closed-loop dynamics of the diagonal elements of the small signal transfer function the system were shaped primarily by filtering the setpoints matrix (1). Their values were determined by simulation with a reference model. The reference model, which was and were adjusted during the animal studies to further tune derived from the design specifications, has the following the controller. The values for the PI and p2* are given in transfer function matrix: Table 11. As in any practical application, constraints due to physe -Is ical limitations may exist. The drug infusion rates were (1 2s) limited to 10 pg/kg/min for NP and 7 pg/kg/min for ?-(s) = e -Is DP. The limit for NP is the maximum recommended in0 fusion rate to prevent cyanide toxicity. The level set for (1 + 5 s ) J DP is to keep the drug in its inotropic range and prevent In (3,f ( s ) is the command input and ~ ( s is ) the reference its alpha adrenergic properties from countering any benused when minimizing the cost function used in the MPC eficial action already achieved. Infusion rates below zero algorithm. The time constants in ( 5 ) were chosen to com- are physically unachievable. These constraints were ex-

The following cost function is chosen to implement such a strategy.

[ .x .z

+

[

+

O 1

~

769

YU er al.: PREDlCTIVE CONTROL OF ARTERIAL PRESSURE A N D CARDIAC OUTPUT TABLE I1 MODELPARAMETERS

K , ,a

Klz'

-2.7778 -2.7778 -2 .I778 -2.7778 -2.7778 -2.7778 -2.7718 -2.7118 -2 .I778 -2.7778 -2.7778 -8.3333 -8.3333 -8.3333 -8.3333 -8.3333 -8.3333 -8.3333 -8.3333 -8.3333 -8.3333 -8.3333 - 8.3333 -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO -25.oooO

9 . m 9 . m 9 . m 9.oooo 9.0000 9 . m 3 . m 3.oooo 3 .m 3.oooo 3 . m 9 . m 9.0000 9 . m 9 . m 9 . m 9 . m 9.oooo 3.0000 3.oooo 3.oooo 3.oooo 3.oooo 3.m 3 . m 3.oooo 3 . m 3 . m 3.oooo 3 . m 9 . m 9.oooo 9.oooo 9 . m 9 . m 9 . m

Model 1 2 3 4

5 6

I 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

K2,h

PI1

022

3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 10.0000 3.3333 3.3333 3.3333 10.0000 10.0000 10.0000 3.3333 3.3333 3.3333 10.0000 10.0000 10.0000

150.0 150.0 150.0 150.0 150.0 150.0 150.0 150.0 150.0 150.0 150.0 150.0 400.0 400.0 400.0 400.0 400.0 400.0 400.0 400.0 400.0 400.0 400.0 400.0 1200.0 1200.0 1200.0 1200.0 1200.0 1200.0 1200.0 1200.0 1200.0 1200.0 1200.0 1200.0

30.0 70.0 30.0 70.0 30.0 70.0 30.0 70.0 30.0 70.0 30.0 70.0 30.0 70.0 30.0 70.0 30.0 70.0 30.0 30.0 30.0 70.0 70.0 70.0 30.0 30.0 30.0 70.0 70.0 70.0 30.0 30.0 30.0 70.0 70.0 70.0

K2Ib

1,6667 1.6667

5.oooo 5.oooo 15.oooO 15.oooO 1.6661 1.6667 5 . m

5.oooo 15.oooO 15.0000 1.6667 1.6667 5 . m 5 . m

15.oooO 15.oooO 1.6661 1.6667 5 .OOOO 5 . m 15.0000 15.oooO 1.6667 5 . m 15.oooO 1.6667

5.oooo 15.0000 1.6667 5 . m

15.oooO I .6667 5 . m 15.oooO

"Values are in terms of mm Hg (pg/kg/min)-'. bValues are in terms of mL/pg.

pressed in the form of linear inequalities. 1

0

NP,,,

-

NP(k - 1)

DP,,,

-

DP(k

-

1)

P(k - 1)

0 -1

(7)

where NP,,, and DP,,, are the maximum allowable infusion rates, and NP(k - 1) and DP(k - 1) are the current infusion levels for NP and DP, respectively. By using the CARIMA model in (2), the cost function (3) can be expressed in the form of a quadratic programming (QP) problem [23], [32]. min J &(k)

=

;Au(k)% &(k)

+ A(l~)~&(k)

(8)

subject to: (7) where Q is the 2 X 2 Hessian matrix and A ( k ) is the 2 x 1 QP gradient vector. The derivation for (8) can be found in [30], in [22] for the single-input single-output case, and

in [ 2 3 ] for the truncated step response model. Note that limits on the maximum allowable changes in infusion rates can be directly handled by this algorithm by adding the proper inequalities to (7). Because of interpatient and intrapatient variations in the response of the subject, the model used by the MPC has to be updated to track these changes. Generally, a recursive least squares algorithm is used to estimate the model parameters. However, in the MMAC procedure, a finite number of models are used, each predicting the response of the subject. For each model, an MPC was designed a priori so as to meet the closed-loop specifications. At each control interval, computations are performed for all controllers, each assuming that its corresponding model accurately describes the subject. The actual incremental control signal to the patient is a weighted sum of these controller outputs: rn

Au ( k ) = -

.zpi (4 4( k )

r=l

(9)

where m represents the total number of models, and p i (k) (0 Ip,(k) 5 1) is a scalar weighting variable associated with the i t h model. By adjusting the value of p i @ ) for each model, the controller can adapt to any variations in the subject response. The identification problem here is determining the model weights. Values of p i @ ) close to one indicate that the corresponding model can accurately predict the drug response. The models used by the MMAC are the same models used by each MPC to compute the control sequences. Simulations showed that variations in the response time constants and dead times do not adversely affect the closed-loop response if the model time constants and model dead times were frozen at their nominal and maximum values respectively. The parameter space was then divided on the basis of response steady-state gains alone. A total of 36 models were needed to span the entire parameter space. The gains and input weights for each model are shown in Table 11. To obtain a measure of matching between the response and the models, a two-dimensional residual vector, gi(k), associated with the ith model is defined as the difference between the observed output, y ( k ) , and the predicted values, y-, ( k ( k - 1):

The identification algorithm then proceeds as follows: at every sampling interval, the residual vector ~ ( kand ) the weighting variables p i (k) are computed. This computation involved several stages wherein the weights are recursively updated using Bayes' theorem:

I ' 1 I

1)

I

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 8, AUGUST 1992

770

bounded away from zero to facilitate transition between models p/(k) = pi(k) =6

>6 p [ ( k ) I6

forp[(k)

(12)

and then normalized:

The threshold 6 is used to limit past information, while pi (k) and p/ (k) represent intermediate variables. The residual covariance matrix, Ci(k),used in computing the weighting variablepj(k)in (1 l), was assumed to be a constant diagonal matrix whose diagonal elements are equivalent to the variance of noise associated with each output:

Note that since Ci(k)is assumed constant, the weighting variables p i (k)do not exactly correspond to the true conditional probabilities. The values of the weights were initialized to be uniform. i=1 pi(0) = 36. (15)

---

No identification was needed prior to initiating control. The active set method [33] was used to solve the quadratic programming problem (8) for each MPC. To circumvent the immense computational overhead associated with the use of so many models, only the control inputs from the MPC's corresponding to the six models with the highest weights were generated. The weights for these six models were subsequently normalized before determining the actual control input from the weighted sum of their computed control moves. In the case of equal weights, the higher model number took precedence. During start-up, the control input &(k) will be in error, and a large undershoot for MAP may result from a high gain subject. To prevent such an occurrence, the models with larger MAP gains were assigned to higher model numbers, and therefore took precedence. Also, both elements of &(k) was limited to a maximum of 0.2 pg/kg/min during the first 5 min of control. Because of the presence of a dead zone in the output response to DP at low infusion rates [28], DP(0) was initialized to 1 CLg/kg/min. IV. METHODS The controller was evaluated with animal experiments. Mongrel dogs (n = 6), weighing 18-25 kg, were anesthetized with sodium pentobarbital (25-30 mg/kg), restrained in a supine position, intubated with a endotracheal tube, and mechanically ventilated (Siemens-Elema 9OOC Servo-Ventilator) at 20 mL/kg and 14 breaths/min.

The inhaled gas was initially comprised of room air. Minute ventilation was adjusted to keep end-tidal percent carbon dioxide (%CO2) levels near normal (4.5-5.5%). A continuous drip of sodium pentobarbital (2-3 mg /kg /min) maintained anesthesia, and supplemental boluses of sodium pentobarbital (2-3 mg/kg) were given when necessary. A catheter was inserted into the left femoral artery to provide continuous arterial pressure tracings (Mennen Horizon Monitor) and arterial blood samples for gas analysis (Radiometer ABL-2) which was performed every hour. If the gas analysis indicated metabolic acidosis (base excess < -0.5), sodium bicarbonate (base excess X 0.2 meq/kg) was administered to correct the condition. Likewise, O2 concentration of the inspired gas was increased if arterial oxygen saturation fell below 95%. A SwanGanz catheter (American Edwards, 7.5 F) was introduced through the left femoral vein and advanced to the pulmonary artery. Isotonic solutions were given intravenously to maintain central venous pressure between 0 and 4 mm Hg. In earlier experiments, an esophageal doppler was used to monitor CO. The probe (Datascope Accucom) was inserted into the esophagus and positioned to give an optimal velocity signal. Later experiments utilized electromagnetic flowmeters (Biotronex Laboratory Model BL160) for CO measurements. Electromagnetic flowmeters required that a midline sternotomy be performed, the pericardium opened, and the heart placed in a pericardial cradle. The ascending aorta was isolated and the flow probe (12-16 mm diameters) was then placed around it. The CO values obtained from either monitor were calibrated using triplicate thermodilution measurements (Sorenson CO Computer Model 03950). Control calculations were handled on a Zenith 386 16Mhz AT-compatible computer with the ASYST laboratory software (ver. 3.0). MAP values were sent to the computer through an RS232 link at every sampling instance. When using the esophageal doppler, CO values were likewise transmitted through an RS232 port. In the case of the electromagnetic flowmeter, the flow signal was sampled with an A/D converter (Data Translation DT2801) 25 times during a 5 s interval at every sample point. The average of these samples was taken as the CO measurement. The control loop was closed through the use of rotary infusion pumps (Critikon Simplicity 2100A) modified to accept digital outputs from the computer. The administration sets from these pumps were connected to a venous puncture. Infusion rates were automatically adjusted using a digital interface (Data Translation DT2801) on the computer. The rotary pumps possessed a resolution of 1 mL/h. Solutions of 200 pg/mL for NP and 320 pg/mL for DP were used. Initial experiments used phenylephrine (10-40 pg/min) to increase the load on the heart and depress CO. Large doses of phenylephrine had been shown to diminish the preload effects of NP [34], giving an afterload-sensitive preparation. Phenylephrine was infused in steps of 10

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TABLE 111 CONTROLLER PARAMETERS Parameter

Initial Value

Final Value

0.0001 4.0 100.0

0.0001 4.0 25.0

6 &P .TO

TABLE IV CONTROLLER PERFORMANCE MAP

CO

Step down in MAP (n = 13): Step size

-21.15 f 4.45 mm Hg -9.7 to -24.80 mm Hg

41.05 f 13.83 mL/min/kg 21.3 to 69.8 mL/min/kg

Undershoot/Overshoot

5.92 _+ 3.54 mm Hg 0 to 12 mm Hg

16.64 f 11.85 mL/min/kg 4.5 to 43.0 mL/min/kg

Response time

5.81 f 1.98 min 3 to 10.5 min

15.31 f 5.36 min 3.5 to 22 min

Step size

7.50 f 1.43 mm Hg 5.5 to 8.75 mm Hg

22.57 f 8.77 mL/min/kg 14.32 to 34.7 mL/min/kg

Overshoot

2.67 f 0.47 mm Hg 2 to 3 mm Hg

18.35 f 10.53 mL/min/kg 4.5 to 30.0 mL/min/kg

Response time

5.5 f 0.41 min 5 to 6 min

9.17 f 5.33 min 4 . 0 to 16.5 min

~~

~

Step up in MAP (n = 3):

Regulation after step (n = 16): ~~

Standard deviation

4.00 f 1.35 mm Hg 1.95 to 6.59 mm Hg

pglmin until CO had been depressed from its baseline value. Hypertension usually accompanies the introduction of phenylephrine. Later experiments were performed with the animal model of acute heart failure described by Smiseth [15]. Induction of failure was achieved by the repeated infusion of 50 pm microsphere (New England Nuclear, 005) solutions (1 mg microspheres per ml dextran) into the left coronary artery (LCA). Access to the coronary arteries was achieved with a #8F right angle coronary catheter (Cordis) introduced through the right common carotid artery and advanced to the root of the aorta. The microspheres was administered in 5-10 mL boluses every 5 min until the left ventricular end-diastolic pressure reached 15 mm Hg or CO was reduced 30% relative to control in the presence of S-T segment and T-wave changes consistent with myocardial ischemia. After the hemodynamic state of the animal had stabilized, the controller was activated by closing the loop. V. RESULTS The results from each experimental trial are depicted in four graphs. The top two plots show the response of the

T - I

-

t

10.36 f 7.99 mL/min/kg 3.28 to 29.67 mL/min/kg

controlled variables MAP and CO. In each of these diagrams, the broken lines represent the reference model while the solid lines portray the history of the actual outputs. The time course for the NP (solid) and DP (dashed) infusion rates are shown in the lower left graph. The lower right graph depicts the dynamic evolution of the six most dominant weights during the course of the experiment. The model number associated with the largest weight at any specific time is printed on top of the graph. Before the start of each run, baseline values for each output were obtained by collecting and averaging twelve MAP and CO samples in a 3-min span. Each trial started with a step command for both outputs and continued for 80 min. The initial step always asked for an increase in CO while MAP could either be increased or decreased, depending upon the baseline pressure. In the middle of each run, disturbances may be introduced by initiating or altering the infusion of a third drug (usually phenylephrine) or by requesting a setpoint change for one or both outputs. Controller performance to the initial step commands are summarized in Table IV. The response time is the time needed for the output to come within 5 mm Hg of the

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Fig. 2. Animal experiment where MAP was initially dropped from 149 to 125 mm Hg and CO raised from 120 to 150 mL/min/kg. At time = 40 min, the CO setpoint was increased to 175 mL/min/kg. Top graphs display actual (solid) outputs compared with their reference (dashed) levels. Bottom left show the infusion rates of NP (solid) and DP (dashed) versus time. The bottom right displays the time course of the six most dominant models. The model number corresponding to the plot are shown above the corresponding curve.

steady state setpoint pressure for MAP and the time taken to climb within 5 mL/min/kg of the steady-state setpoint flow for CO. A total of 16 runs were performed but one run was omitted from the computations because the setpoint was never reached despite maximum infusion levels. The standard deviation was taken 10 min after the step for MAP and 20 min after for CO. The response times of the outputs are within the settling times specified in the design. The mean MAP undershoot is less than 10 mm Hg and the standard deviation during a constant setpoint is 4.00 mm Hg. Assuming a normal distribution, this implies that 95% of the outputs will lie within +8.00 mm Hg of the setpoint. The numerical value of the standard deviation for CO (190 f 0.14mL/min) is higher than that of MAP. This was expected since MAP errors have larger penalties. Large overshoots were sometimes observed with a step command increase in CO, but there is no design constraint on this value. Fig. 2 shows a typical experimental trial. After the initial step commands, the controller successfully regulated both outputs at their desired levels. The identification algorithm converged to Model 22 after 25 min. At time = 40 min, an increase in CO was requested and the controller responded by raising the DP infusion rate. Notice that DP had minimal effect on MAP during this step change as seen by the insignificant change in NP level after DP was raised. The switch from Model 22 to Model 23 at

time = 68 min may be due to the nonlinear response of the subject. In Fig. 3, the algorithm converged to Model 15 after 28 min. Setpoint changes in MAP were implemented in this trial. When an elevated blood pressure was requested at time = 41 min, the controller initially augmented the DP infusion level while lowering that of NP. These actions provoked CO to deviate from its setpoint, and the DP rate was then decreased to return CO back to its reference level. When a step down in MAP was specified at time = 60 min, the controller correspondingly raised the NP infusion rate. This feat allowed CO to stray once again from its setpoint. Since DP was constrained at the zero infusion level at this time, the controller had no option for lowering CO while maintaining MAP at its present level, and the error in CO was allowed to persist. Convergence of the weights seemed elusive for the run depicted in Fig. 4. This was caused by selecting an exceedingly large value for the variance of the CO residual (E = diag [4,1001).In computing the weights (ll),the residuals are weighted by the inverse of the covariance matrix. Thus, a large variance can blur small differences in the residuals and prevent a dominant model from evolving. At the start of this particular run, phenylephrine was infused at a rate of 10 pglmin, but this was halted at time = 40 min. Without the vasoconstrictor, smaller doses of

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Fig. 3. Animal experiment where MAP was initially lowered from 155 to 130 mm Hg and CO raised from 118 to 145 mL/min/kg. MAP setpoint was increased to 145 mm Hg at time = 41 min and returned to 130 mm Hg at time = 60 min. Top graphs display actual (solid) outputs compared with their reference (dashed) levels. Bottom left show the infusion rates of NP (solid) and DP (dashed) versus time. The bottom right displays the time course of the six most dominant models. The model number corresponding to the plot are shown above the corresponding curve.

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NP were needed to maintain the MAP at its setpoint. A drop in NP infusion rate was observed closely after the phenylephrine infusion was terminated. A small positive hump in CO was also attributed to the disturbance, and this was eliminated after 20 min. In Fig. 5 , the outputs were corrupted by frequent premature ventricular contractions (PVC) caused by the embolization of the coronaries. The PVC’s caused large swings in both pressure and flow signals which are unaffected by the infused drugs. Since these large perturbations are uncorrelated with the inputs, they can be considered as increased noise and they make it more difficult for the controller to regulate both outputs. Lidocaine (1 mg/min) was infused in an attempt to control the PVC’s, but the PVC’s persisted. The increased amplitude of the noise caused a large amount of switching between models. Fig. 6 shows another run where the coronaries were embolized. This time, PVC’s were successfully controlled with lidocaine. The controller converged to Model 16 after 15 min. VI. DISCUSSION During the animal trials, the magnitudes of a step up in pressure are significantly smaller than those of a step down. This is relevant because a considerable increase in pressure could augment the load on the heart. Also, the alpha adrenergic properties of DP, which can bring a

strong vasopressor response, only appear at dosages higher than the maximum infusion level imposed for the drug. A decrease in MAP of over 50 mm Hg from baseline causes a loss of contact between the aorta wall and the electromagnetic flowmeters, which in turn produces a degradation of the flow signal. When the transesophageal doppler was used, the pressure drop may likewise cause erroneous CO measurements by altering the cross-sectional area of the descending aorta. Because of this limitation in the flow sensors and for the reason stated in the previous paragraph, a wide range of MAP values was not tested for each trial. Although this limited the evaluation of the controllers, the limited range is consistent with the use of small-signal linear models. The residual associated with the model closest to the subject’s response is expected to be smaller that those of the mismatched models. If this situation persists over several measurements, then from ( 1 1)-( 13), the value of pi(k) should approach 1 where i indexes the model closest to the actual behavior of the system. This behavior has been defined as “regular” [13j. It may be possible to find a model, not in the original 36, that will yield a smaller residual error and, because the uncertainty is less, tighter control. But an increase in number of available models creates a greater computational burden for the controller. Besides, as previously mentioned, the simple linear models used in the control design will never exactly co-

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.

incide with the actual drug response. This performance versus number of models tradeoff is one of the fundamental issues in the design. The basic idea is to have enough models spanning the space of expected responses such that the actual system behavior is close enough to any one of the models for the corresponding controller to provide satisfactory control. Unfortunately, there is no known general solution to this model-selection problem. In the current design, extensive simulations were used to select the models. Circumstances can exist when the residuals of two or more models are approximately equal for an extended sequence of samples and convergence to one dominant model may be hampered. One such condition is when the one of the inputs is constrained at zero, and it will be difficult to distinguish between two models whose only differences are in the gains associated with the input maintained at zero. This situation occurred in the latter portion of Fig. 3. With DP constrained at zero (time > 60 min), the identification routine was unable to select between Models 9 and 10. However, there is no need to resolve the situation because, so long as the DP infusion rate remains at zero, Models 9 and 10, with their identical NP responses, will prescribe identical NP dosage levels. If the DP infusion rates become nonzero, the identification routine should once again be able to distinguish between the two models.

1

The controller seems to be robust with respect to problems in the identification algorithm. This property was also evident in the MMAC design for the control of the F-8C aircraft [13]. In Fig. 4, the large assumed values for the variances prevented the weights from converging to a single dominant model, and yet exemplary control was executed. At the other end of the spectrum, the algorithm was still able to regulate the outputs despite the increased switching brought about by an increase in the noise amplitude during the run in Fig. 5. The behavior of the model weights can benefit from an accurate estimate of the variance, and since it is difficult to predict the amount of noise that may be encountered in an experiment, an on-line estimator for the output noise variance can enhance the identification portion of the algorithm. However, this can only be achieved at the expense of added computational resources. And if the controller remains robust with respect to the nonconvergence of the model weights, the advantages of the on-line variance estimator appear to be small. The conflicting results from the identification and controller portions of the algorithm show that the design should not be evaluated based on the dynamic evolution of the model weights alone. Evaluation should instead focus on whether the controller meets the design specifications or on how the multiple model predictive controller compares against other adaptive algorithms. This opinion

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had also been voiced by Athans et al. [13]. They argue that, due to the constant covariance matrix, the model weights do not correspond to the true conditional probabilities and therefore, have no physical meaning. Besides, a value of pj(k)equal to one implies that the corresponding model can precisely predict the behavior of the patient, which is not true. The use of only six models, instead of the entire 36, to compute the changes in drug infusion rates provided a controller that met the design specifications with a decrease in computational requirements. However, there is no current theoretical guideline that shows the use of six models, or any number of models, for the control calculations is optimal. The control configuration presented here belongs to the same class of adaptive predictive controller as the CAMAC used by Voss et al. [19]. However, there are significant differences between the two. The most prominent of these is the dissimilar methods both controllers use to estimate the parameters of the patient response. The CAMAC uses a recursive least squares (RLS) algorithm while the multiple-model adaptive predictive design adheres to the MMAC approach. One advantage of MMAC method over the RLS used in the CAMAC is the elimination of any parameter identification maneuver prior to closing the control loop. The MMAC also intrinsically provides bounds on the values of the estimated parameters. On the other hand, the CAMAC is more general in its application. A bank of models and their corresponding controllers do not have to be created before using the algorithm. Other differences include the lack of any penalties on changes in drug infusion rates and the absence of a reference model in the CAMAC. These gave the CAMAC a more aggressive control strategy which resulted in shorter response times, tighter regulation, and larger undershoots when compared with the multiple-model adaptive predictive controller. Large fluctuations in the infusion rates were also observed due to the lack of restraints. Despite a more cautious approach, the multiple-model adaptive predictive controller was still able to meet the specifications of the design. However, the restrictions on the control input by weighting matrix & cause coupling between the outputs [20]. Coupling becomes apparent when a setpoint change is requested on one output while the other is asked to remain at its present value. For the current design, setpoint changes in MAP seem to have a greater effect on CO (Fig. 3) than the other way around (Fig. 2). This may be due to the larger penalty on MAP. Although the CAMAC initially used DP [36] as its positive inotropic agent, a switch was made to dobutamine due to problems encountered by the estimator on the dose dependent response of DP. Dobutamine is also less vasoactive than DP. Both controllers are effective in regulating MAP and CO. However, further laboratory and clinical trials will be needed to evaluate both methods. The CAMAC has yet to be tested on subjects exhibiting symptoms of congestion.

VII. FUTURECONSIDERATIONS Animal studies have shown the feasibility of simultaneously controlling CO and MAP by adjusting the infusion rates of NP and DP with a multiple-model adaptive predictive control scheme. The controller was robust with respect to excessive switching between models and nonconvergence to a single dominant model. At several instances during the animal experiments, the increased noise due to PVC’s seemed to pose significant challenges to the algorithm. A “smart” sensor that will filter these unwanted artifacts may be needed to improve the regulatory performance of the controller. The small-signal models used in this design were obtained by linearizing around the operating points, usually the baseline values of the outputs. However, if the baseline state of the subject changes or if large steps in the outputs are requested, the linear models may be unable to accurately represent the patient response. The use of a conservative control policy, such as increasing the values in the p matrix, can help prevent unwanted behavior. Another consideration is to move the operating points to the new steady-state values. Perhaps the best option is to utilize nonlinear models which take into account absolute levels of infusion rates in future designs. A model with nonlinear gain such as that proposed by Gingrich [28] could be used. With most design issues being resolved through extensive simulations, theoretical studies could be pursued to provide insight on convergence properties, optimal selection of models and system stability. Another future consideration is the use of a third drug, like phenylephrine, to raise MAP in hypotensive situations. Such a modification will result in a nonsquare control system. Cardiac outputs are generally measured using the thermodilution method. Estimates of CO can be made available every 5- 10 min with thermodilution measurements, but to incorporate this measurement technique into the design, a multirate sampling scheme may have to be employed to handle the faster settling time for MAP. However, the introduction of new methods for measuring continuous CO, such as the transesophageal [37] or the transtracheal doppler [38], can bring the application of the MAP-CO controller closer to reality. REFERENCES C. F. Walker, “Biomedical engineering R&D picks up,” Spectrum, vol. 27, pp. 52-53, 1990. L. C. Sheppard, “Computer control of the infusion of vasoactive drugs,” Ann. Biorned. Eng., vol. 8, pp. 431-444, 1980. J . B. Slate, “Model-based design of a controller for infusing sodium nitroprusside during postsurgical hypertension,” Doctoral Thesis, Univ. Wisconsin-Madison, Wisconsin, 1980. K. S. Stem, B. K. Walker, and P. G . Katona, “Automated blood pressure control using a self-tuning regulator,” IEEE Frontiers Eng. Health Care, 1981, pp. 255-258. J. M. Amsparger, B. C. McInnis, J . R. Glover Jr., and N. A. Normann, “Adaptive control of blood pressure,” IEEE Trans. Biorned. Eng., vol. BME-30, pp. 168-176, 1983. H. Kaufman, R. Roy, and X. Xu, “Model reference adaptive control of drug infusion rate,” Auromarica, vol. 20, pp. 205-209, 1984.

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L. J. Meline, D. R. Westenskow, A. Somerville, R. T. Wernick, J. Jacobs, and N. L. Pace, “Evaluation of two adaptive sodium nitroprusside control algorithms,” J. Clin. Monir., vol. 2, pp. 79-86, 1986. W. G. He, H. Kaufman, and R. Roy, “Multiple-model adaptive control procedure for blood pressure control,” IEEE Trans. Biomed. Eng., vol. BME-33, pp. 10-19, 1986. 1. F. Martin, A. M. Schneider, and N. T. Smith, “Multiple-model adaptive control of blood pressure using sodium nitroprusside,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 603-611, 1987. G. A. Pajunen, M. Steinmetz, and R. Shankar, “Model reference adaptive control with constraints for postoperative blood pressure management,” IEEE Trans. Biomed. Eng., vol. BME-37, pp. 679687, 1990. B. Widrow and S. D. Steams, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1985, pp. 270-279. D. G. Lainiotis, “Partitioning: A unifying framework for adaptive systems, 11: Control,” Proc. IEEE, vol. 64, pp. 1182-1198, 1976. M. Athans, D. Castanon, K.-P. Dunn, C. S. Greene, W. H. Lee, N. S. Sandell Jr., and A. S. Willsky, “The stochastic control of the F-8C aircraft using a multiple model adaptive control (MMAC) method-Part I: Equilibrium flight,” IEEE Trans. Automat. Contr., vol. AC-22, pp. 768-780, 1977. C. Yu, W. G. He, J. M. So, R. Roy, H. Kaufman, and J. Newell, “Improvement in arterial oxygen control using multiple-model adaptive control procedures,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 567-574, 1987. R. R. Miller, N. A. Awan, J. A. Joye, K. S . Maxwell, A. N. DeMaria, E. A. Amsterdam, and D. T. Mason, “Combined dopamine and nitroprusside therapy in congestive heart failure,” Circularion, vol. 5 5 , pp. 881-884, 1977. D. R. Stemple, J. H. Kleiman, and D. C. Harrison, “Combined nitroprusside-dopamine therapy in severe chronic congestive heart failure,” Amer. J. Cardiol., vol. 42, pp. 267-275, 1978. V. Serna, R. Roy, and H. Kaufman, “Adaptive control of multiple drug infusions,” presented at the Amer. Contr. Conf., San Francisco, CA, June 22-24, 1983. B. C. McInnis and L. Z. Deng, “Automatic control of blood pressures with multiple drug inputs,” Ann. Biomed. Eng., vol. 13, pp. 217-225, 1985. G. I. Voss, P. G. Katona, and H. J. Chizeck, “Adaptive multivariable drug delivery: Control of arterial pressure and cardiac output in anesthetized dogs,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 617-623, 1987. C. E. Garcia and M. Morari, “Internal model control: 3. Multivariable control law computation and tuning guidelines,” Ind. Eng. Chem. Process Des. Dev., vol. 24, pp. 484-494, 1985. C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive control: Theory and practice-a survey,” Auromatica, vol. 25, pp. 335348, 1989. D. W. Clarke, C. Mohtadi, and P. S. Tuffs, “Generalized predictive control-Part 1. The basic algorithm,” Autornatica, vol. 23, pp. 137148, 1987. C. E. Garcia and A. M. Morshedi, “Quadratic programming solution of dynamic matrix control (QDMC),” Chem. Eng. Commun., vol. 46, pp, 73-87, 1986. C. Yu, R. J. Roy, H. Kaufman, and B . W. Bequette, “Multiplemodel adaptive control applied to the multiple-drug infusion problem,” in Proc. I990 Con$ Inform. Sci. Syst., Princeton University, Princeton, NJ, Mar. 21-23, 1990, pp. 274-279. C. Yu, R. J. Roy, and H. Kaufman, “A circulatory model for combined nitroprusside-dopamine therapy in acute heart failure,” Med. Prog. Tech., vol. 16, pp. 77-88, 1990. A. D. Forbes, “Modeling and control,” J. Clin. Monit., vol. 6, pp. 227-235, 1990. L. C. Sheppard, W. F. Holdefer, N. T. Kouchoukos, and J. W. Kirklin, “Analysis of multiple effects of vasoactive and positive inotropic agents on cardiovascular system variables, ’’ in Cardiovascular System Dynamics: Models and Measurements. New York: Plenum, 1982, pp. 647-656. K. J. Gingrich and R. J. Roy, “Modeling the hemodynamic response to dopamine in acute heart failure,” IEEE Trans. Biomed. Eng., vol. BME-38, pp. 267-272, 1991. J. S. Packer, D. G. Mason, J. F. Cade, and S. M. McKinley, “Adaptive closed-loop control of dopamine infusion in seriously ill hypotensive patients,” in IFAC Symp. Model. Conrr. Biomed. Sysr., Venice, Italy, April 6-8, 1988, pp. 141-145.

[30] C. Yu, “Adaptive control of cardiac output and mean arterial pressure using multiple drug infusions,” Doctoral Thesis, Rensselaer Polytech. Inst., Troy, NY, 1990. [31] G. C. Goodwin and K. S. Sin, Adaptive Filtering, Predicrion and Control. Englewood Cliffs, NJ: Prentice Hall, 1984, pp. 120-128. [32] N . L. Ricker, “Use of quadratic programming for constrained internal model control,” Ind. Eng. Chem. Process Des. Dev., vol. 24, pp. 925-936, 1985. [33] D. G. Luenberger, Linear and Nonlinear Programming. Reading, MA: Addison Wesley, 1984, pp. 423-427. [34] G. I. Voss, P. G. Katona, and P. J. Dauchot, “Effectiveness of sodium nitroprusside as a function of total peripheral resistance in the anesthetized dog,” Anesrhesiol., vol. 62, pp. 130-134, 1985. [35] 0. A. Smiseth and 0. D. Mjos, “A reproducible and stable model of acute ischaemic left ventricular failure in dogs,” Clin. Physiol., vol. 2, pp. 225-239, 1982. [36] G. I. Voss, “A self-tuning controller for drug delivery,” Doctoral Thesis, Case Western Reserve University, Cleveland, OH, 1986. [37] A. C. Pemno Jr., J. Fleming, and K. R. LaMantia, “Transesophageal doppler ultrasonography: Evidence for improved cardiac output monitoring,” Anes. Analg, vol. 71, pp. 651-657, 1990. [38] J. H. Abrams, R. E. Weber, and K. D. Holmen, “Transtracheal doppler: A new procedure for continuous cardiac output measurement,” Anesthesiol., vol. 70, pp. 134-138, 1989.

Clement Yu (S’83-M’89) received the Bachelor degree in electrical engineering from the University of the Philippines, Diliman, in 1981, two M.E. degrees in biomedical engineering and in electrical engineering in 1986, and the Ph.D. degree in biomedical engineering in 1990 from the Rensselaer Polytechnic Institute, Troy, NY. After completing his doctorate degree, he served as a visiting assistant professor at the Department of Biomedical Engineering at Rensselaer Polytechnic Institute. He is cumntlv a senior engineer with The BOC Group, Technical Center, Murray Hili, NJ, where-he is engaged in bioinstrumentation design and development. His interests include biological systems modelling, biosensor design, digital signal processing and adaptive control systems.

Rob J. Roy (S’56-M’57-SM’73) received the B.S.E.E. degree from Cooper Union, New York City, NY, the M.S.E.E. degree from Columbia University, New York City, NY, and the D.Eng.Sc. degree from Rensselaer Polytechnic Institute, Troy, NY. He was an NIH Career Development Fellow in Pulmonary Physiology and later received the M.D. Degree from Albany Medical College, Albany, NY. He is currently the Department Head of the Department of Biomedical Engineering, Rensselaer Polytechnic Institute. He is a Board certified anesthesiologist and Fellow of the American College of Anesthesiology. He is a Professor and Attending anesthesiologist at Albany Medical Center Hospital, Albany, NY. His research interests are in biological signal processing and adaptive cardiorespiratory control systems. He has published extensively in the areas of pattern recognition, control systems, radar processing, noninvasive cardiac output monitoring, and closed-circuit anesthesia.

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Howard Kauhnan(S’64-M’66-SM’75) received the B.E.E., M.E.E., and Ph.D. degrees from the Department of Electrical Engineering, Rensselaer Polytechnic Institute, Troy, NY in 1%2, 1963, and 1965, respectively. Since 1969 he has been a member of the Department of Electrical, Computer and System Engineering, Rensselaer Polytechnic Institute, Troy, NY where he is a Professor teaching collrses in systems analysis, optimal, and adaptive control theory and digital systems. His research intmsts are in the are-aaof adaptive and digital systems. He has written many papers in these areas and has served as a consultant in related projects. Fmm 19651968 he was with the Computer Research Department of Cornel1 Aeronautical Laboratory, Buffalo, NY in 1965, where he was engaged in the development of procedures for applying digital computers to process estimation, identification, and control. In 1968 he joined the General Electric Research and Development Center as a System Engineer and developed computer simulations of large scale industrial processes. During the Summer of 1972 he was awarded a NASA Summer Faculty Fellowship at NASA-Langley Research Center where he conducted research in the development of digital adaptive flight control systems, and during the Summer of 1982, he was an NSF Industrial Reswch Participant at General

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Electric Corporate Research and Development where he was involved in computer aided control systems design. Dr.Kaufman is a member of Sigma Xi, Eta Kappa Nu, and Tau Beta Pi honor societies.

B. Wayne Jkquette received the B.S. degree in chemical engineering from the University of Arkansas, Fayetteville, in 1980 and the M.S.E. and W.D. degrees in chemical engineering from the University of Texas at Austin in 1985 and 1986, respectively. He worked for three years as a process engineer for American Petrofina between his undergraduate and graduate studies. From 1986 to 1987 he was B postdoctoral research associate at the University of Texas and from 1987 to 1988 he was a Visiting Lecturer with the Department of Chemical Engineering at the University of California at Davis. Since 1988 he has been an Assistant Professor with the Depamnent of Chemical Engineering at Rensselaer Polytechnic Institute. His research interests include nonlinear dynamics and control, chemical process modeling and optimization.