IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 16 ...IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 16, NO. 2, MARCH 2005 387 Passivity-Based Neural Network Adaptive Output Feedback Control

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 16, NO. 2, MARCH 2005 387

Passivity-Based Neural Network AdaptiveOutput Feedback Control for Nonlinear

Nonnegative Dynamical SystemsTomohisa Hayakawa, Member, IEEE, Wassim M. Haddad, Senior Member, IEEE, James M. Bailey, and

Naira Hovakimyan, Senior Member, IEEE

Abstract—The potential clinical applications of adaptive neuralnetwork control for pharmacology in general, and anesthesia andcritical care unit medicine in particular, are clearly apparent.Specifically, monitoring and controlling the depth of anesthesiain surgery is of particular importance. Nonnegative and com-partmental models provide a broad framework for biological andphysiological systems, including clinical pharmacology, and arewell suited for developing models for closed-loop control of drugadministration. In this paper, we develop a neural adaptive outputfeedback control framework for adaptive set-point regulation ofnonlinear uncertain nonnegative and compartmental systems.The proposed framework is Lyapunov-based and guaranteesultimate boundedness of the error signals corresponding to thephysical system states and the neural network weighting gains.The approach is applicable to nonlinear nonnegative systems withunmodeled dynamics of unknown dimension and guarantees thatthe physical system states remain in the nonnegative orthant of thestate–space for nonnegative initial conditions. Finally, a numericalexample involving the infusion of the anesthetic drug midazolamfor maintaining a desired constant level of depth of anesthesia fornoncardiac surgery is provided to demonstrate the efficacy of theproposed approach.

Index Terms—Adaptive control, automated anesthesia, bispec-tral index (BIS), electroencephalography, exponential passivity,neural networks, nonlinear nonnegative systems, nonnegativecontrol, output feedback.

I. INTRODUCTION

ADMINISTRATION of drugs to produce general anes-thesia has traditionally been guided by clinical evaluation.

However, the clinical measures of depth of anesthesia are im-perfect, primarily since the most reliable, purposeful movementin response to noxious stimulus, is masked by the concomitant

Manuscript received March 14, 2003; revised November 12, 2003. This workwas supported in part by the Air Force Office of Scientific Research under GrantsF49620-03-1-0178 and F49620-03-1-0443.

T. Hayakawa was with the School of Aerospace Engineering, GeorgiaInstitute of Technology, Atlanta, GA 30332 USA. He is now with the JapanScience and Technology Agency, Saitama 332-0012, Japan (e-mail: [email protected]).

W. M. Haddad is with the School of Aerospace Engineering,Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail:[email protected]).

J. M. Bailey is with the Department of Anesthesiology, North-east Georgia Medical Center, Gainesville, GA 30503 USA (e-mail:[email protected]).

N. Hovakimyan is with the Department of Aerospace and Ocean Engineering,Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA(e-mail: [email protected]).

Digital Object Identifier 10.1109/TNN.2004.841782

administration of paralytic agents, given to improve operatingconditions for the surgeon. There has been a long-standinginterest in the use of the electroencephalogram (EEG) as an ob-jective, quantitative measure of consciousness. Recent work hasdemonstrated that a derivative of the EEG signal, the bispectralindex (BIS), correlates with changes in consciousness [1]–[3].The BIS is a scalar measure ranging from 0 to 100, with theupper value of 100 corresponding to the awake state and thelower limit of 0 corresponding to an isoelectrical EEG signal.The ease of BIS monitoring and its ready availability for use inthe operating room, opens the possibility of closed-loop controlof anesthetic drug administration, using the BIS as the perfor-mance and measurement variable. Current standard practice,open-loop control (manual control) by clinical personnel, canbe tedious, imprecise, time-consuming, and sometimes of poorquality, depending on the skills and judgment of the clinician.Underdosing can result in patients psychologically traumatizedby pain and awareness during surgery, while overdosing, at thevery least, may result in delayed recovery from anesthesia and,in the worst case, may result in respiratory and cardiovascularcollapse. Closed-loop control may improve the quality of drugadministration, lessening the dependence of patient outcomeon the skills of the clinician.

There has been a great deal of interest in the development ofalgorithms for closed-loop control of intravenous anesthesia.Algorithms for closed-loop control of inhalation anesthesia,using anesthetic concentration as the performance variable,have been developed. However, since it is not possible withcurrent sensor technology to rapidly measure the plasma con-centration of intravenously-administered drugs (in contrast toinhalation agents), these algorithms are not useful for intra-venous agents. Furthermore, drug concentration, even if it couldbe measured rapidly, is not the best measurement variable. Weare far more interested in drug effect than concentration. Morerelevant are recently described algorithms for the control ofintravenous anesthesia using a processed electroencephalo-graph as the control variable. Building on pioneering work byBickford [4], Absalom et al. [5] developed a proportional-in-tegral-derivative controller using the BIS, a processed EEGsignal, as the performance variable to control the infusion ofthe hypnotic, propofol. While the median performance of thesystem was good, in three out of ten patients oscillations of theBIS signal around the set point were observed and anesthesiawas deemed clinically inadequate in one out of the ten patients.This would not be acceptable for clinical practice.

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388 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 16, NO. 2, MARCH 2005

Alternative algorithms have been devised by both Schwildenet al. [6], [7] and Struys et al. [8]. Both groups have devel-oped and clinically tested closed-loop, model-based adaptivecontrollers for the delivery of intravenous anesthesia usinga processed EEG signal as the measurement variable. Thealgorithms are based on a pharmacokinetic model predictingthe drug concentration as a function of infusion rate and timeand a pharmacodynamic model relating the processed EEGsignal to concentration. The pharmacokinetic and pharma-codynamic models are characterized by specific parameters.The two algorithms are similar in assuming that certain modelparameters are equal to the mean values from previous pharma-cokinetic/pharmacodynamic studies while varying a few selectparameters of the models to minimize the difference betweenthe desired and observed processed EEG signal. The primarydifference between the two algorithms is in the parameterswhich are fixed to the mean values from previous studies andthe parameters that are chosen for variation. Schwilden et al.[6], [7] assumed that the pharmacodynamic parameters maybe fixed to mean values taken from the literature and varypharmacokinetic parameters to minimize bias from the targetsignal.

In contrast, Struys et al. [8] assumed that the pharmacokineticparameters are always correct and that any variability in indi-vidual patient response is due to pharmacodynamic variability.Thus, they vary pharmacodynamic parameters to minimize thedifference between the observed and target processed EEGsignal. Both algorithms have been implemented in the operatingroom with clinically acceptable performance in small numbersof patients. However, as pointed out by Glass and Rampil [9]in an analysis of the algorithm of Struys et al. [8], the systemsmay not have been fully stressed. For example, in their study,Struys et al. [8] administered a relatively high fixed dose of theopioid remifentanil, in conjunction with closed-loop controlof the hypnotic, propofol. This blunted the patient responseto surgical stimuli and meant that the propofol was neededonly to produce unconsciousness in patients who were pro-foundly analgesic. The result was that only small adjustmentsin propofol concentrations were necessary. Whether eithersystem would have been robust in less controlled situations isan open question. And it should be noted that both algorithmsare model dependent and only partially adaptive, in the sensethat only select pharmacokinetic/pharmacodynamic parametersare varied to minimize the signal bias from the target.

Given the uncertainties in both pharmacokinetic and phar-macodynamic models, and the magnitude of interpatientvariability, in this paper we present a neural network adaptivecontrol framework that accounts for combined interpatientpharmacokinetic and pharmacodynamic variability. In partic-ular, we develop a neural adaptive output feedback controlframework for adaptive set-point regulation of nonlinear uncer-tain nonnegative and compartmental systems. Neural networkadaptive controllers are ideal for controlling nonlinear uncertaindynamical systems due to their ability to approximate a largeclass of continuous nonlinear functions [10]–[16]. However,with the notable exception of [17], neuro adaptive controllersfor nonlinear uncertain nonnegative and compartmental sys-tems have not been developed in the literature. Nonnegative

and compartmental models provide a broad framework forbiological and physiological systems, including clinical phar-macology, and are well suited for the problem of closed-loopcontrol of drug administration. Specifically, nonnegative andcompartmental dynamical systems [18]–[24] are composed ofhomogeneous interconnected subsystems (or compartments)which exchange variable nonnegative quantities of materialwith conservation laws describing transfer, accumulation, andelimination between the compartments and the environment. Itthus follows from physical considerations that the state trajec-tory of such systems remains in the nonnegative orthant of thestate–space for nonnegative initial conditions. Using nonneg-ative and compartmental model structures, a Lyapunov-basedneural adaptive control framework is developed that guaranteesultimate boundedness of the error signals corresponding to thephysical system states as well as the neural network weightinggains.

The neuro adaptive controllers are constructed without re-quiring knowledge of the system dynamics while guaranteeingthat the physical system states remain in the nonnegative orthantof the state–space. Furthermore, since in pharmacological ap-plications involving active drug administration control (source)inputs as well as the system states need to be nonnegative,the proposed neuro adaptive controller also guarantees thatthe control signal remains nonnegative. We emphasize thateven though neuro adaptive full-state feedback controllers fornonnegative systems have been recently addressed in [17],our present formulation addresses adaptive output feedbackcontrollers for nonlinear systems with unmodeled dynamics ofunknown dimension using the exponential passivity, feedbackequivalence, and stabilizability of exponentially minimumphase notions developed in [25] and [26]. The frameworkdeveloped in [17] is limited to full-state feedback controllersand does not address the problem of unmodeled dynamics ofunknown dimension. Output feedback controllers are crucial inclinical pharmacology since key physiological (state) variablescannot be measured in practice. Furthermore, the results in [17]are based on the new notions of partial boundedness and partialultimate boundedness as opposed to the approach of this paperwhich imposes passivity and positive real requirements on thesystem dynamics. Thus, the approach of the present paper isrelated to the neuro adaptive control methods developed in [27]and [28].

II. MATHEMATICAL PRELIMINARIES

In this section, we introduce notation, several definitions, andsome key results concerning nonlinear nonnegative dynamicalsystems [24], [29] and exponentially passive systems [25], [26]that are necessary for developing the main results of this paper.Specifically, for we write (resp., ) toindicate that every component of is nonnegative (resp., posi-tive). In this case, we say that is nonnegative or positive, re-spectively. Likewise, is nonnegative1 or positive ifevery entry of is nonnegative or positive, respectively, which

1In this paper, it is important to distinguish between a square nonnegative(resp., positive) matrix and a nonnegative–definite (resp., positive–definite)matrix.

HAYAKAWA et al.: PASSIVITY-BASED NEURAL NETWORK ADAPTIVE OUTPUT FEEDBACK CONTROL 389

is written as or , respectively. Let anddenote the nonnegative and positive orthants of ; that is, if

, then and are equivalent, respectively,to and . Finally, we write to denote trans-pose, for the trace operator, for the Euclidean vectornorm, for the Frobenius matrix norm, for the interior ofthe set , and for the Fréchet derivative of at . The fol-lowing definition introduces the notion of a nonnegative (resp.,positive) function.

Definition 2.1: Let . A real functionis a nonnegative (resp., positive) function if (resp.,

) on the interval .The following definition introduces the notion of essentially

nonnegative vector fields [24], [30].Definition 2.2: Let , where

is an open subset of that contains . Then is essentiallynonnegative with respect to , , if

for all , and such that, , where denotes the th element of . is

essentially nonnegative if for all , andsuch that .

In this paper, we consider controlled nonlinear dynamicalsystems of the form

(1)

(2)

where , , , , ,, is locally Lipschitz continuous and satisfies

, is continuous, andis continuous. We assume that has at least one equilibriumso that, without loss of generality, and . Fur-thermore, for the nonlinear dynamical system we assume thatthe required properties for the existence and uniqueness of solu-tions are satisfied; that is, , , and satisfy sufficientregularity conditions such that the system (1) has a unique solu-tion forward in time. The following definition and propositionare needed for the main results of the paper.

Definition 2.3: The nonlinear dynamical system given by (1)is nonnegative if for every and , ,the solution , , to (1) is nonnegative.

Proposition 2.1 ([24]): The nonlinear dynamical systemgiven by (1) is nonnegative if is essentiallynonnegative and , .

It follows from Proposition 2.1 that a nonnegative input signal, , is sufficient to guarantee the nonnegativity

of the state of (1).Next, we introduce the notion of exponential passivity.Definition 2.4 ([26]): A nonlinear dynamical system of the

form (1), (2) is exponentially passive if there exists a constantsuch that the dissipation inequality

(3)

is satisfied for all with . A nonlinear dynamicalsystem of the form (1), (2) is passive if the dissipation inequality(3) is satisfied with .

For the statement of the following result, recall the definitionsof zero-state observability and complete reachability given in[31].

Theorem 2.1 ([26]): Let be zero-state observable and com-pletely reachable. is exponentially passive if and only if thereexist functions and , and ascalar such that is continuously differentiable, pos-itive–definite, , is continuous, , and, forall

(4)

(5)

As shown in [26], an equivalent statement for exponentialpassivity of using (4), (5) is given by

(6)

Hence, if is exponentially passive (resp., passive), thenthe undisturbed ( ) nonlinear dynamical system(1) is asymptotically (resp., Lyapunov) stable. If, in addi-tion, there exist scalars and such that

, , then the undisturbed( ) nonlinear dynamical system (1) is exponentiallystable. This leads to the following stronger notion of exponen-tial passivity [25].

Definition 2.5: A nonlinear dynamical system of the form(1), (2) is strongly exponentially passive if is exponentiallypassive and there exist a continuously differentiable function

and positive scalars such that

(7)

Since, in this paper, we consider nonlinear dynamical systemsin normal form, for the remainder of this section we restate someof the key results of [25] in a concise and unified format that sup-ports the developments in Section III. Specifically, we considerthe normal form characterization of (1), (2) given by

(8)

(9)

(10)

where , , and , , are thestate vectors, , , is the control input,

, , is the system output,and satisfies , ,

and satisfies , , andwith , .

The following definition introduces the notion of exponentiallyminimum phase.

Definition 2.6: A nonlinear dynamical system of the form(8)–(10) is exponentially minimum phase if there exist a con-tinuously differentiable function and positiveconstants , , , and such that

(11)

(12)

(13)


It follows from converse Lyapunov theory that if the zero so-lution to , , , isexponentially stable and is continuously differentiable,then there exists a continuously differentiable function

such that (11)–(13) hold. Finally, the following def-inition and theorem are needed for the main results of this paper.For the statement of this definition let ,

, and .Definition 2.7 ([25]): A nonlinear dynamical system of

the form (8)–(10) is semiglobally output feedback exponentiallypassive if, for any compact set , there exists a contin-uous feedback of the form

(14)

where , , such that the closed-loopsystem given by (8)–(10) and (14), or, equivalently

(15)

(16)

where and, is strongly exponentially passive from to for

all .Theorem 2.2 ([25]): Consider the nonlinear dynamical

system given by (8)–(10). Assume that the input matrixfunction , , can be factored as

(17)

where and arecontinuously differentiable matrix functions such that

, , and , . Thenis semiglobally output feedback exponentially passive if and

only if is exponentially minimum phase.Remark 2.1: As noted in [25], if is globally Lipschitz

continuous in , is uniformly positive–definite;that is, there exists such that ,

, and the zero solution to ,, , is globally exponentially stable, then the

previous result holds globally.Remark 2.2: It is important to note that if the conditions

in Theorem 2.2 are satisfied, then there exists an output feed-back control law of the form (14) which renders the closed-loopsystem exponentially passive from to . Specifically, as shownin [25], the output feedback controller achieving exponentialpassivity is given by

(18)

where is a positive constant. Finally, it is important tonote that in the case where , in (14) takesthe form

(19)

This fact will be used for our main result presented in the fol-lowing section.

III. NEURAL OUTPUT FEEDBACK ADAPTIVE CONTROL FOR

NONLINEAR NONNEGATIVE UNCERTAIN SYSTEMS

In this section, we consider the problem of character-izing neural adaptive output feedback control laws for non-linear nonnegative and compartmental uncertain dynamicalsystems to achieve set-point regulation in the nonnega-tive orthant. Specifically, consider the controlled nonlinearuncertain dynamical system given by (8)–(10) where

is essentially nonnegative withrespect to but otherwise unknown and satisfies ,

, is essentially non-negative with respect to but otherwise unknown and satisfies

, , and is an unknownnonnegative input matrix function. Furthermore, the systemdimension need not be known. The control input in (8)is restricted to the class of admissible controls consisting ofmeasurable functions such that , .

As discussed in the Introduction, control (source) inputs ofdrug delivery systems for physiological and pharmacologicalprocesses are usually constrained to be nonnegative as are thesystem states. Hence, in this paper we develop neuro adaptiveoutput feedback control laws for nonnegative systems with non-negative control inputs. Specifically, for a given desired set point

and for given , our aim isto design a nonnegative control input , , such that

and for all , where, and and , , for all

. However, since in many applications ofnonnegative systems and in particular, compartmental systems,it is often necessary to regulate a subset of the nonnegative statevariables which usually include a central compartment, here weonly require that , . Furthermore, we as-sume that we have independent control inputs so that the inputmatrix function is given by ,where , . For compartmental sys-tems, this assumption is not restrictive since control inputs cor-respond to control inflows to each individual compartment.

In this paper, we assume that for a given set pointthere exist and such that

(20)

(21)

and the solution to (9) with is globallyexponentially stable so that given by (8)–(10) is exponentiallyminimum phase at with constant control input . Notethat is an equilibrium point of (8), (9)if and only if there exists such that (20), (21) hold.Next, defining , , and

, and using (20), (21), it follows that:

(22)


and

(23)

where and. Since, by as-

sumption, the solution to (9) with is glob-ally exponentially stable, it follows from Definition 2.6 and con-verse Lyapunov theory that is exponentially minimum phaseand, hence, it further follows from Theorem 2.2 and Remark 2.2that for any compact set , whereand , and for all , there existcontinuous functions and

with , , such that, with, (22), (23) is strongly exponentially

passive from to . Next, adding and sub-tracting to and from (22), it follows that (22) canbe rewritten as:

(24)

Now, we assume that for a given , the th componentof the vector function can beapproximated over a compact set

by a linear in parameters neural network up to a desiredaccuracy so that for , there exists such that

, , and

(25)where , , are optimal unknown (con-stant) weights that minimize the approximation error over ,

, , are a set of basis functionssuch that each component of takes values between 0 and1, , , are the modeling errors, and

, where , , are bounds for the op-timal weights , . Since and arecontinuous functions, we can choose , , froma linear space of continuous functions that forms an algebraand separates points in . In this case, it follows from theStone–Weierstrass theorem [32, p. 212] that is a dense subsetof the set of continuous functions on . Hence, as is the casein the standard neuro adaptive control literature [13], we canconstruct the signal involving the estimatesof the optimal weights as our adaptive control signal. For the fol-lowing theorem let denote the total numberof basis functions or, equivalently, the number of nodes of theneural network.

Theorem 3.1: Consider the nonlinear uncertain systemgiven by (8)–(10) where is essentially nonnegative withrespect to , is essentially nonnegative with respectto , and is nonnegative and given by

. For a given , assumethere exist positive vectors and such that(20) and (21) hold and the equilibrium point of (8), (9)is globally asymptotically stable with . In addition,

assume that is exponentially minimum phase at .Finally, let and , , be positive constants. Thenthe neural adaptive output feedback control law

(26)

where

(27)

and , , , with update law

(28)

guarantees that there exists a compact positively invariant setsuch that , where

, and the solution , , of theclosed-loop system given by (8), (9), (26), and (28) is ultimatelybounded for all with ultimate bound

, , where

(29)

(30)

block (31)

, ,

and , are positive constants. Furthermore, ,, and for all and.

Proof: First, since, by assumption, (8), (9) is exponen-tially minimum phase at , it follows from Theorem 2.2and Remark 2.2 that for any compact set and for all

, there exist continuous functionsand with ,

, such that, with , (22), (23)is strongly exponentially passive from to . Hence, itfollows from Theorem 2.1 that there exist a continuously dif-ferentiable function , a continuous function

, and positive constants , , such that is pos-itive–definite, , , and, for all ,

(32)

(33)

and (7) hold, where and.

Next, define

block (34)


where

ifotherwise

(35)

and note that with , , given by (26) it follows that (23)and (24) become:

(36)

and

(37)

or, equivalently

(38)

To show ultimate boundedness of the closed-loop system (28)and (38), consider the Lyapunov-like function

(39)

where , block ,and . Note that and, since

and are positive–definite, for all. Next, letting , , denote the solutions to (38) and

using (25), (28), (32) and (33), it follows that the time derivativeof along the closed-loop system trajectories is givenby (40), shown at the bottom of the page. Now, for each

and for the two cases given in (35), the last term onthe right-hand side of (40), gives the following.

1) If , then and, hence

2) Otherwise, and, hence

Hence, it follows from (40) that in either case:

(41)

Next, completing squares yields

(42)

(40)


Fig. 1. Visualization of sets used in the proof of Theorem 3.1.

where is given by (30). Now, for

(43)

or

(44)

it follows that for all ; that is,for all and

, where

(45)

(46)

Next, define

(47)

where is the maximum value such that , anddefine

(48)

where

(49)To show ultimate boundedness of the closed-loop system(28) and (38), assume2 that (see Fig. 1). Now,since for all and

, it follows that is positively invariant.Hence, if , then the solution

, , to (28) and (38) is ultimately

2This assumption is standard in the neural network literature and ensures thatin the error space ~D there exists at least one Lyapunov level set ~D � ~D . Inthe case where the neural network approximation holds in � , thisassumption is automatically satisfied. See Remark 3.1 for further details.

bounded. Furthermore, since is positively invariant, itfollows that:

(50)

is also positively invariant. Now, to show that ,, suppose there exists such

that and . In this case, andfor all and, hence, is

trivially satisfied for all . Alternatively, suppose theredoes not exist such that and .In this case, consider the Lyapunov-like function

(51)where and

Note that is continuous on andis positively invariant. Furthermore, note that

(52)

Now, it follows from the generalized Krasovskii–LaSalle in-variant set theorem [33, Th. 2.3] that

as , where denotes the largest in-variant set contained in

. Hence, since , , and, there exists

such that for all and, hence

(53)

Since , (53) implies , .Finally, , , is a restatement of (26). Now,

since is essentially nonnegative,, , and , , it follows from

Proposition 2.1 that and for all and.

Remark 3.1: It follows from Theorem 2.2 that if givenby (8)–(10) is exponentially minimum phase, then is semi-globally output feedback exponentially passive. Hence, for anyarbitrarily large compact set there exists an output feed-back control law of the form (14) that renders the closed-loopsystem (8)–(10) exponentially passive. For this compact set ,as is common in the neural network literature, we assume thatthere exists an approximator for the unknown nonlinear map

up to a desired accuracy. Fur-thermore, we assume that in the error space there exists atleast one Lyapunov level set such that .

A block diagram showing the neuro adaptive control architec-ture given in Theorem 3.1 is shown in Fig. 2. In Theorem 3.1, weassumed that the equilibrium point of (8), (9) is globallyasymptotically stable with . In general, however, un-like linear nonnegative systems with asymptotically stable plant


Fig. 2. Block diagram of the closed-loop system.

dynamics, a given set point for the non-linear nonnegative dynamical system (8), (9) may not be asymp-totically stabilizable with a constant control .However, if is homogeneous,cooperative; that is, the Jacobian matrix is essen-tially nonnegative for all , the Jacobianmatrix is irreducible for all [29], and thezero solution of the undisturbed ( )system (8), (9) is globally asymptotically stable, then the setpoint satisfying (20) and (21) is a uniqueequilibrium point with and is also asymptoticallystable for all [34]. This implies that thesolution to (8), (9) with isasymptotically stable for all .

IV. NEURAL ADAPTIVE CONTROL FOR GENERAL ANESTHESIA

Almost all anesthetics are myocardial depressants whichlower cardiac output (i.e., the amount of blood pumped bythe heart per unit time). As a consequence, decreased cardiacoutput slows down redistribution kinetics; that is, the transferof blood from the central compartments (heart, brain, kidney,and liver) to the peripheral compartments (muscle and fat).In addition, decreased cardiac output could increase drugconcentrations in the central compartments causing even moremyocardial depression and further decrease in cardiac output.To study the effects of pharmacological agents and anestheticswe propose the nonlinear two-compartment model shownin Fig. 3, where denotes the mass of drug in the centralcompartment, which is the site for drug administration and isgenerally thought to be comprised of the intravascular bloodvolume as well as highly perfused organs such as the heart,brain, kidney, and liver. These organs receive a large fraction ofthe cardiac output. Alternatively, is the mass of drug in theperipheral compartment, comprised of muscle and fat whichreceive a smaller proportion of the cardiac output.

A mass balance of the two-state compartment model yields

(54)

(55)

where is the drug concentration in the central com-partment, is the volume of the central compartment,is the rate of transfer of drug from Compartment I to Compart-

Fig. 3. Pharmacokinetic model for drug distribution during anesthesia.

ment II, is the rate of transfer of drug from Compart-ment II to Compartment I, is the rate of drug metabo-lism and elimination (metabolism typically occurs in the liver),and , , is the infusion rate of an anesthetic drug. Inorder to formulate a physiologically realistic nonlinear model,we assume that the rate transfers are proportional to the car-diac output. This reflects the fact that the drug transfer fromthe central compartment to the peripheral compartment (or viceversa) requires physical transport via the blood stream fromthe heart, brain, etc., to muscle and fat (or vice versa). It isgenerally assumed that this transport in the vascular tree willbe proportional to the cardiac output . Furthermore, formany drugs the rate of metabolism (i.e., ) will be propor-tional to the rate of transport of drug to the liver and, hence,we assume that is also proportional to the cardiac output.Thus, we assume , , and

, where , , and are positive con-stants. Many anesthetics depress the heart, decreasing the car-diac output. Furthermore, the transfer coefficients are functionsof the concentration in the central compartment. Thus, to de-velop a physiologically plausible model we assume a sigmoidrelationship between drug concentration in the central compart-ment and effect so that , where theeffect is related to (since that is the presumed concentrationin the highly perfused myocardium), is a constant,

is the drug concentration associated with a 50% de-crease in the cardiac output, and determines the steep-ness of this curve (that is, how rapidly the cardiac output de-creases with increasing drug concentration). Furthermore, thismodel assumes instantaneous mixing and as increases, therate coefficients decrease through their dependence on the car-diac output. Even though the transfer and loss coefficients ,

, and are nonnegative, and , , and ,these parameters can be uncertain due to patient gender, weight,pre-existing disease, age, and concomitant medication. Hence,the need for adaptive control to regulate intravenous anestheticsduring surgery is crucial.

Midazolam is an intravenous anesthetic that has been used forboth induction and maintenance of general anesthesia. A simpleyet effective patient model for the disposition of midazolam isbased on the two-compartment model shown in Fig. 3 with thefirst compartment acting as the central compartment. Here, weuse the BIS as a measure of anesthetic effect. The BIS signalis a nonlinear monotonically decreasing function of the level ofconsciousness and is given by

BIS BIS (56)


Fig. 4. BIS index versus effect site concentration.

where BIS denotes the baseline (awake state) value and, byconvention, is typically assigned a value of 100, is the mi-dazolam concentration in nanograms/liter in the effect site com-partment (brain), is the concentration at half maximal ef-fect and represents the patient’s sensitivity to the drug, anddetermines the degree of nonlinearity in (56). Here, the effectsite compartment is introduced as a correlate between the centralcompartment concentration and the central nervous system con-centration. The effect site compartment concentration is relatedto the concentration in the central compartment by the first-orderdelay model

(57)

where in is a positive time constant. Assuming, it follows that:

(58)

In reality, the effect site compartment equilibrates with the cen-tral compartment in a matter of a few minutes. The parameters

, , and are determined by data fitting and vary frompatient to patient. BIS index values of 0 and 100 correspond, re-spectively, to an isoelectric EEG signal and an EEG signal of afully conscious patient; the range between 40 and 60 indicatesa moderate hypnotic state.

In the following numerical simulation we set, , and BIS , so that the BIS signal

is shown in Fig. 4. The target (desired) BIS value, BIS ,is set at 50. Furthermore, for simplicity of exposition, weassume that the effect site compartment equilibrates instan-taneously with the central compartment; that is, we assumethat , so that (57) gives , . Now,defining BIS BIS and , where

, (54), (55) can be written in form of(8)–(10) with

(59)

(60)

(61)

Note that is essentially nonnegative with respect to ,is essentially nonnegative with respect to , and is

nonnegative. In addition, note that since is a monotonicallyincreasing function, the mapping is diffeo-morphic. Furthermore, note that since

(62)

where

(63)

and , it follows that the system zero dynamicsare exponentially stable and, hence, the system given by (54),(55) is exponentially minimum phase at . Thus, sincethe input matrix function satisfies (17), it follows from Theorem2.2 that (54), (55) is semiglobally output feedback exponentiallypassive. Now, using the adaptive output feedback controller

(64)

where

BIS (65)

, , and is a given basis function,with update law

BIS BIS BIS

BIS BIS (66)

where is an arbitrary positive constant, it follows fromTheorem 3.1 that the control input (anesthetic infusion rate)

is nonnegative for all and there exist positiveconstants and such that BIS BIS , ,for all (uncertain) positive values of the transfer and loss co-efficients as well as all (uncertain) nonnegativecoefficients , , and . It is important to note that duringactual surgery the BIS signal is obtained directly from the EEGand not (56). For our simulation we assume ,

, ,, , and . Note

that these parameter values for and probably exaggeratethe effect of midazolam on cardiac output. They have been


Fig. 5. Compartmental concentrations versus time.

Fig. 6. BIS index versus time and control signal (infusion rate) versus time.

selected to accentuate nonlinearity but they are not biolog-ically unrealistic. To illustrate the efficacy of the proposedneuro adaptive controller we switch the pharmacodynamicparameters and , respectively, from 150 and 3 to170 and 2 at and back to 150 and 3 at

. Furthermore, here we consider noncardiac surgerysince cardiac surgery often utilizes hypothermia which itselfchanges the BIS signal. With , ,

BIS

BIS BIS BIS BIS

, and initial conditions , ,, Fig. 5 shows the concentrations of

midazolam in the two compartments versus time. Fig. 6 shows

the BIS index and the control signal (midazolam infusion rate)versus time.

Even though we did not calculate the analytical boundsgiven by (29) due to the fact that one has to solve an opti-mization problem with respect to (25) to obtain and ,

, the closed-loop BIS signal response shown inFig. 6 is clearly acceptable. Furthermore, the basis functionsfor BIS are chosen to cover the domain of interest of ourpharmacokinetic/pharmacodynamic problem since we knowthat the BIS index varies from 0 to 100. Hence, the basis func-tions are distributed over that domain. The number of basisfunctions, however, is based on trial and error. This goes backto the Stone–Weierstrass theorem, which only provides anexistence result without any constructive guidelines. Finally,we note that simulations using a larger number of neuronsresulted in imperceptible differences in the closed-loop systemperformance.

V. CONCLUSION

Nonnegative and compartmental systems are widely usedto capture system dynamics involving the interchange of massand energy between homogenous subsystems or compartments.Thus, it is not surprising that nonnegative and compartmentalmodels are remarkably effective in describing the dynam-ical behavior of biological systems, physiological systems,and pharmacological systems. In this paper, we developeda neural adaptive output feedback control framework foradaptive set-point regulation of nonlinear uncertain nonneg-ative and compartmental systems. Using Lyapunov methods,the proposed framework was shown to guarantee ultimateboundedness of the error signals corresponding to the physicalsystem states and the neural network weighting gains whileadditionally guaranteeing the nonnegativity of the closed-loopsystem states associated with the plant dynamics. Finally, usinga nonlinear two-compartment patient model for the dispositionof anesthetic drug midazolam, the proposed adaptive controlframework was used to monitor and control a desired constantlevel of consciousness for noncardiac surgery. Even thoughmeasurement noise was not addressed in our framework, itshould be noted that EEG signals may have as much as 10%variation due to noise. In particular, the BIS signal may becorrupted by electromyographic noise; that is, signals ema-nating from muscle rather than the central nervous system.Clinical implementation of the proposed algorithm would thushave to include muscle paralysis to minimize the effects ofelectromyographic noise.

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Tomohisa Hayakawa (S’00–M’04) received theB.Eng. degree in aeronautical engineering fromKyoto University, Kyoto, Japan, in 1997, the M.S.degree in aerospace engineering from the StateUniversity of New York (SUNY), Buffalo, in 1999,and the M.S. degree in applied mathematics and thePh.D. degree in aerospace engineering, both fromthe Georgia Institute of Technology, Atlanta, in 2001and 2003, respectively.

After working in the Department of Aeronauticsand Astronautics, Kyoto University, he is currently a

Research Fellow in the Department of Information Physics and Computing, theUniversity of Tokyo, Tokyo, Japan, which is affiliated with the Japan Scienceand Technology Agency (JST). His research interests include stability of non-linear systems, nonnegative and compartmental systems, hybrid systems, non-linear adaptive control, neural networks and intelligent control, adaptive dis-crete-time and sampled-data control, and applications to aerospace vehicles,robotic systems, and biological/biomedical systems.

Wassim M. Haddad (S’87–M’87–SM’01) receivedthe B.S., M.S., and Ph.D. degrees in mechanical en-gineering from Florida Institute of Technology, Mel-bourne, in 1983, 1984, and 1987, respectively, withspecialization in dynamical systems and control.

From 1987 to 1994, he served as a Consultant forthe Structural Controls Group of the GovernmentAerospace Systems Division, Harris Corporation,Melbourne, FL. In 1988, he joined the faculty of theMechanical and Aerospace Engineering Department,Florida Institute of Technology, where he founded

and developed the Systems and Control Option within the graduate program.Since 1994, he has been a Member of the faculty in the School of AerospaceEngineering, Georgia Institute of Technology, where he holds the rank ofProfessor. His research contributions in linear and nonlinear dynamical systemsand control are documented in over 400 archival journal and conferencepublications. He is the coauthor of the books Hierarchical Nonlinear SwitchingControl Design with Applications to Propulsion Systems (Springer-Verlag,2000) and Thermodynamics: A Dynamical Systems Approach (PrincetonUniversity Press, 2005). His recent research is concentrated on nonlinear robustand adaptive control, nonlinear dynamical system theory, large-scale systems,hierarchical nonlinear switching control, hybrid and impulsive control fornonlinear systems, system thermodynamics, thermodynamic modeling of me-chanical and aerospace systems, nonlinear analysis and control for biologicaland physiological systems, and active control for clinical pharmacology.

Dr. Haddad is a National Science Foundation Presidential Faculty Fellow anda Member of the Academy of Nonlinear Sciences.


James M. Bailey received the B.S. degree fromDavidson College, Davidson, NC, in 1969, the Ph.D.degree in chemistry (physical) from the University ofNorth Carolina, Chapel Hill, in 1973, and the M.D.degree from Southern Illinois University School ofMedicine, Carbondale, in 1982.

He was a Helen Hay Whitney Fellow at theCalifornia Institute of Technology, Pasadena, from1973 to 1975 and an Assistant Professor of chemistryand biochemistry at Southern Illinois Universityfrom 1975 to 1979. After receiving the M.D. degree,

he completed a residency in anesthesiology and then a fellowship in cardiacanesthesiology at the Emory University School of Medicine, Atlanta, GA,affiliated hospitals. From 1986 to 2002, he was an Assistant Professor ofanesthesiology and then an Associate Professor of anesthesiology at Emory,where he also served as Director of Critical Care Service. In 2002, he movedhis clinical practice to the Northeast Georgia Medical Center, Gainesville, asDirector of Cardiac Anesthesia and Consultant in Critical Care Medicine. Heremains affiliated with the Emory University School of Medicine Departmentof Anesthesiology as a Clinical Associate Professor. He is board certified inanesthesiology, critical care medicine, and transesophageal echocardiography.His research interests have focused on pharmacokinetic and pharmacodynamicmodeling of anesthetic and vasoactive drugs and, more recently, applicationsof dynamical system theory in medicine. He is the author or coauthor of 98journal articles, conference publications, or book chapters.

Naira Hovakimyan (M’01–SM’02) received thePh.D. degree in physics and mathematics from theInstitute of Applied Mathematics of the RussianAcademy of Sciences, Moscow, in 1992.

After receiving the Ph.D. degree, she joinedthe Institute of Mechanics, Armenian Academy ofSciences, as a Research Scientist, where she workeduntil 1997. In 1997, she was awarded a governmentalpostdoctoral scholarship to work at INRIA, France.The subject areas in which she has published includedifferential pursuit-evasion games, optimal control

of robotic manipulators, robust control, adaptive estimation, and control. In1998, she was invited to the School of Aerospace Engineering, Georgia Instituteof Technology, Atlanta, where she worked as a Research Faculty Memberuntil 2003. In 2003, she joined the Department of Aerospace and OceanEngineering, Virginia Tech University, Blacksburg, as an Associate Professor.She has authored over 90 refereed publications. Her current interests are in thetheory of adaptive control and estimation, neural networks, and stability theory.

Dr. Hovakimyan received the SICE International Scholarship for the bestpaper of a young investigator in the VII ISDG Symposium, Japan, 1996. She isalso the recipient of the 2004 Pride @ Boeing Award. She is a Senior Memberof AIAA, a Member of AMS and ISDG, and is an Associate Editor for the IEEEControl Systems Society, the IEEE TRANSACTIONS ON NEURAL NETWORKS, andthe International Journal of Control Systems and Automation.

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