# Pressure- and work-limited neuroadaptive control for mechanical ventilation of critical care patients

**ABSTRACT** In this paper, we develop a neuroadaptive control architecture to control lung volume and minute ventilation with input pressure constraints that also accounts for spontaneous breathing by the patient. Specifically, we develop a pressure-and work-limited neuroadaptive controller for mechanical ventilation based on a nonlinear multi-compartmental lung model. The control framework does not rely on any averaged data and is designed to automatically adjust the input pressure to the patient's physiological characteristics capturing lung resistance and compliance modeling uncertainty. Moreover, the controller accounts for input pressure constraints as well as work of breathing constraints. Finally, the effect of spontaneous breathing is incorporated within the lung model and the control framework.

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**ABSTRACT:**We develop optimal respiratory airflow patterns using a nonlinear multicompartment model for a lung mechanics system. Specifically, we use classical calculus of variations minimization techniques to derive an optimal airflow pattern for inspiratory and expiratory breathing cycles. The physiological interpretation of the optimality criteria used involves the minimization of work of breathing and lung volume acceleration for the inspiratory phase, and the minimization of the elastic potential energy and rapid airflow rate changes for the expiratory phase. Finally, we numerically integrate the resulting nonlinear two-point boundary value problems to determine the optimal airflow patterns over the inspiratory and expiratory breathing cycles.Computational and Mathematical Methods in Medicine 01/2012; 2012:165946. · 0.79 Impact Factor

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614IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 4, APRIL 2011

Pressure- and Work-Limited Neuroadaptive Control

for Mechanical Ventilation of Critical Care Patients

Konstantin Y. Volyanskyy, Member, IEEE, Wassim M. Haddad, Fellow, IEEE, and James M. Bailey

Abstract—In this paper, we develop a neuroadaptive control

architecture to control lung volume and minute ventilation with

input pressure constraints that also accounts for spontaneous

breathing by the patient. Specifically, we develop a pressure- and

work-limited neuroadaptive controller for mechanical ventilation

based on a nonlinear multicompartmental lung model. The

control framework does not rely on any averaged data and

is designed to automatically adjust the input pressure to the

patient’s physiological characteristics capturing lung resistance

and compliance modeling uncertainty. Moreover, the controller

accounts for input pressure constraints as well as work of

breathing constraints. Finally, the effect of spontaneous breathing

is incorporated within the lung model and the control framework.

Index Terms—Compartmental systems, mechanical ventila-

tion, multicompartment lung model, neuroadaptive control,

pressure-limited ventilation, work-limited ventilation.

I. INTRODUCTION

T

primary lung pathology, such as pneumonia, but also as

a secondary consequence of heart failure or inflammatory

illness, such as sepsis or trauma. When this occurs, it is

essential to support patients while the fundamental disease

process is addressed. For example, a patient with pneumonia

may require mechanical ventilation while the pneumonia is

being treated with antibiotics, which will eventually effectively

“cure” the disease. Since the lungs are vulnerable to critical

illness and respiratory failure is common, support of patients

with mechanical ventilation is very common in the intensive

care unit.

The goal of mechanical ventilation is to ensure adequate

ventilation, which involves a magnitude of gas exchange that

leads to the desired blood level of carbon dioxide (CO2), and

adequate oxygenation, which involves a blood concentration

of oxygen that will ensure organ function. Achieving these

goals is complicated by the fact that mechanical ventilation

can actually cause acute lung injury, either by inflating the

HE lungs are particularly vulnerable to acute critical

illness. Respiratory failure can result not only from

Manuscript received February 14, 2010; revised October 13, 2010 and

January 20, 2011; accepted January 23, 2011. Date of publication March

14, 2011; date of current version April 6, 2011. This work was supported

in part by the U.S. Army Medical Research and Material Command, under

Grant 08108002.

K. Y. Volyanskyy and W. M. Haddad are with the School of Aerospace

Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA

(e-mail: gtg891s@mail.gatech.edu; wm.haddad@aerospace.gatech.edu).

J. M. Baileyis withthe Department

east Georgia Medical Center, Gainesville, GA 30503 USA (e-mail:

james.bailey@nghs.com).

Digital Object Identifier 10.1109/TNN.2011.2109963

ofAnesthesiology,North-

lungs to excessive volumes or by using excessive pressures to

inflate the lungs. The challenge to mechanical ventilation is to

produce the desired blood levels of CO2and oxygen without

causing further acute lung injury.

The earliest primary modes of ventilation can be classified,

approximately, as volume-controlled or pressure-controlled

[1]. In volume-controlled ventilation, the lungs are inflated (by

the mechanical ventilator) to a specified volume and then al-

lowed to passively deflate to the baseline volume. The mechan-

ical ventilator controls the volume of each breath and the num-

ber of breaths per minute. In pressure-controlled ventilation,

the lungs are inflated to a given peak pressure. The ventilator

controls this peak pressure as well as the number of breaths per

minute. In early ventilation technology, negative pressure ven-

tilation was employed, wherein a patient’s thoracic area is en-

closed in an airtight chamber and the volume of the chamber is

expanded, inflating the patient’s lungs. Such ventilator devices

include tank ventilators, jacket ventilators, and cuirassess [2].

The primary determinant of the level of CO2in the blood

is minute ventilation, which is defined as the tidal volume (the

volume of each breath) multiplied by the number of breaths

per minute [3], [4]. With volume-controlled ventilation, both

tidal volume and the number of breaths are determined by

the machine (the ventilator), and typically, the tidal volumes

and breaths per minute are selected by the clinician caring

for the patient. In pressure-controlled ventilation, the tidal

volume is not directly controlled. The ventilator determines

the pressure that inflates the lungs and the tidal volume is

proportional to this driving pressure and the compliance or

“stiffness,” of the lungs. Consequently, the minute ventilation

is not directly controlled by the ventilator and any change

in lung compliance (such as improvement or deterioration

in the underlying lung pathology) can result in changes in

tidal volume, minute ventilation, and ultimately, the blood

concentration of CO2.

In respiratory management, the goal is to control arterial

partial pressure of CO2 in the blood denoted by PaCO2(t).

The means to do this is reflected in the equation relating

PaCO2(t) to the volume of gas exchange in the lungs in a

given unit of time, the alveolar ventilation. The relationship

between PaCO2(t) and ventilation is given by [3]

PaCO2(t) = 0.863VCO2

where VCO2is the total body production of CO2per minute,

which is approximately 259 m?/min in healthy subjects, 0.863

is a constant to reconcile units, and Va(t) is alveolar venti-

Va(t),

t ≥ 0

1045–9227/$26.00 © 2011 IEEE

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VOLYANSKYY et al.: NEUROADAPTIVE CONTROL FOR PRESSURE-LIMITED VENTILATION 615

lation. In patients who are totally dependent on mechanical

ventilation (and not taking any independent breaths), Va(t) is

given by [3]

Va(t) = (TV(t) − Vd)RR(t),

where TV(t) denotes the volume of each breath set on the

ventilator (i.e., tidal volume), RR(t) denotes the respiratory

rate set on the ventilator, and Vd denotes the dead space of

the lungs. The product TV(t)RR(t) is referred to as minute

ventilation [3] and Vd is approximately one-third of minute

ventilation in healthy subjects. During mechanical ventilation

TV(t) ∈ [400, 700] m? and RR(t) ∈ [12, 25]. The tidal

volume is the difference between the lung volume at the start

of expiration and the lung volume at the end of expiration.

In this paper, we will denote the lung volume by V(t)

and the inspiration time and expiration time by Tin and Tex,

respectively, over a single breathing cycle T = Tin+Tex. Fur-

thermore, we assume that the inspiration process starts from

a given initial state V(0) and is followed by the expiration

process where its initial state is the final state of inspiration.

Hence, the explicit relationship between the delivered air

volume and the tidal volume over a ventilatory cycle is given

by TV = V(Tin) − V(Tin+ Tex) = V(Tin) − V(0).

The concentration of oxygen in the blood is determined by

the underlying lung pathology, the concentration of oxygen in

the gas delivered by the mechanical ventilator, and also by the

pressure that is used to inflate the lungs. In very general terms,

oxygenation can be improved by higher mean pressures in the

lungs, although higher peak pressures during the inflation–

deflation cycle are associated with lung injury [5], [6].

With the increasing availability of microchip technology,

it has been possible to design mechanical ventilators that

have control algorithms which are more sophisticated than

simple volume or pressure control. Examples are proportional-

assist ventilation [7], [8], adaptive support ventilation [9],

SmartCare ventilation [10], and neurally adjusted ventilation

[11]. In proportional-assist ventilation, the ventilator measures

the patient’s volume and rate of inspiratory gas flow, and

then applies pressure support in proportion to the patient’s

inspiratory effort [12]. In this mode of ventilation, inspired

oxygen and positive end-expiratory pressure are manually

adjusted by the clinician.

In adaptive support ventilation, tidal volume and respiratory

rate are automatically adjusted [13]. In particular, minute

ventilation (TV(t)RR(t)) is calculated from a % MinVol

parameter and the patient’s ideal body weight. The patient’s

respiratory pattern is measured pointwise in time and fed back

to the controller to provide the required (target) tidal volume

and patient respiratory rate. Adaptive support ventilation does

not provide continuous control of minute ventilation, positive

end-expiratorypressure, and inspired oxygen, these parameters

need to be adjusted manually.

SmartCare ventilation monitors tidal volume, respiratory

rate, and end-tidal pressure of CO2to maintain the patient in a

respiratory “comfort” zone by automatically adjusting the level

of pressure support [14], [15]. SmartCare ventilators do not

account for patient respiratory variations and do not generally

guarantee adequate minute ventilation during weaning. In

t ≥ 0

addition, positive end-expiratory pressure and inspired oxygen

need to be manually adjusted.

Neurally adjusted ventilation is fundamentally different

from the aforementioned automatic ventilation technologies in

the sense that it uses the patient’s respiratory neural drive as

a measurement signal to the ventilator [16]. In this mode of

ventilation, rather than controlling pressure, the patient’s respi-

ratory neural drive signal to the diaphragmatic electromyogram

is controlled using electrodes placed on an esophageal catheter

[17]. Even though this approach has been shown to be effective

in some recent clinical studies [18], [19], its effectiveness

is affected if the patient is highly sedated. In addition, as

in the aforementioned ventilator technologies, positive end-

expiratory pressure and inspired oxygen need to be manually

controlled.

The common theme in modern ventilation control algo-

rithms is the use of pressure-limited ventilation while also

guaranteeing adequate minute ventilation. One of the chal-

lenges in the design of efficient control algorithms is that the

fundamental physiological variables defining lung function,

i.e., the resistance to gas flow and the compliance of the lung

units, are not constant but rather vary with lung volume. This

is particularly true for compliance, strictly defined as dV/dP,

where V is the lung unit volume and P is the pressure driving

inflation. More simply, lung volume is a nonlinear function

of driving pressure. In addition, these physiological variables

vary from patient to patient, as well as within the same

patient under different conditions, making it very challenging

to develop models and effective control law architectures for

active mechanical ventilation.

In this paper, we develop an adaptive control architecture

to control lung volume and minute ventilation with input

pressure constraints that also accounts for spontaneous work of

breathing by the patient. Specifically, we develop a pressure-

and work-limited neuroadaptive controller for mechanical ven-

tilation based on a nonlinear multicompartmental lung model.

The control framework does not rely on any averaged data

and is designed to automatically adjust the input pressure

to the patient’s physiological characteristics, capturing lung

resistance and compliance modeling uncertainty.Moreover,the

controller accounts for input pressure constraints as well as

work of breathing constraints. Finally, the effect of sponta-

neous breathing is incorporated within the lung model and the

control framework.

The contents of this paper are as follows. In Section II,

we provide definitions and mathematical preliminaries on

nonlinear nonnegative dynamical systems that are necessary

for developing the main results of this paper. In Section III,

we develop a neuroadaptive control framework for nonnega-

tive dynamical systems with actuator amplitude and control

integral constraints. It is important to note here that, even

though adaptive and neuroadaptive controllers for nonnegative

dynamical systems have been developed in the literature

[20]–[24], neuroadaptive control with actuator saturation and

integral constraints is virtually nonexistent. Notable exceptions

include [25] (see also [26]). Then, in Section IV, we extend the

linear multicompartment lung model given in [27] to address

system model nonlinearities and spontaneous patient work of

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616IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 4, APRIL 2011

breathing effects. To demonstrate the efficacy of the proposed

neuroadaptive control framework, in Section V we apply our

framework to control the ventilatory drive of a pressure-

and work-limited respirator in the face of lung modeling

uncertainty. Finally, in Section VI we draw conclusions.

II. MATHEMATICAL PRELIMINARIES

In this section, we introduce notation, several definitions,

and some key results concerning nonlinear nonnegative dy-

namical systems [28], [29] that are necessary for developing

the main results of this paper. Specifically, for x ∈ Rnwe write

x ≥≥ 0 (resp., x >> 0) to indicate that every component of

x is nonnegative (resp., positive). In this case, we say that x is

nonnegative or positive, respectively. Likewise, A ∈ Rn×mis

nonnegative1or positive if every entry of A is nonnegative

or positive, respectively, which is written as A ≥≥ 0 or

A >> 0, respectively. Furthermore, let Rn

nonnegative and positive orthants of Rn: that is, if x ∈ Rn,

then x ∈ Rn

x ≥≥ 0 and x >> 0. Finally, we write (·)Tto denote

transpose, (·)?to denote Fréchet derivative, tr(·) for the trace

operator, λmin(·) (resp., λmax(·)) to denote the minimum (resp.,

maximum) eigenvalue of a Hermitian matrix, and ? · ? for a

vector norm in Rn.

Definition 1: Let T > 0. A real function u : [0,T] → Rm

is a nonnegative (resp., positive) function if u(t) ≥≥ 0 (resp.,

u(t) >> 0) on the interval [0,T].

The following definition introduces the notion of essentially

nonnegative vector fields [28], [29].

Definition 2: Let f = [ f1,..., fn]T: D ⊆ Rn

Then f is essentially nonnegative if fi(x) ≥ 0 for all i =

1,...,n and x ∈ Rn

xi denotes the ith component of x.

It follows from Definition 2 that, if xi is an element of the

boundary of Rn

note that, if f (x) = Ax, where A ∈ Rn×n, then f is essentially

nonnegative if and only if A is essentially nonnegative or

Metzler: that is, A(i,j) ≥ 0, i, j = 1,...,n, i ?= j, where

A(i,j)denotes the (i, j)th entry of A.

In this paper, we consider controlled nonlinear dynamical

systems of the form

+and Rn

+denote the

+and x ∈ Rn

+are equivalent, respectively, to

+→ Rn.

+such that xi = 0, i = 1,...,n, where

+, then fi(·) is directed toward Rn

+. In addition,

˙ x(t) = f (x(t)) + G(x(t))u(t), x(0) = x0, t ≥ 0

where x(t) ∈ Rn, t ≥ 0, u(t) ∈ Rm, t ≥ 0, f : Rn→ Rn

is continuous and satisfies f (0) = 0, G : Rn→ Rn×mis

continuous, and u : [0,∞) → Rmis measurable and locally

bounded. For the nonnegative system (1), we assume that f (·),

G(·), and u(·) satisfy sufficient regularity conditions such that

(1) has a unique solution forward in time.

The following definition and proposition are needed for the

main results of this paper.

Definition 3: The nonlinear dynamical system given by (1)

is nonnegative if, for every x(0) ∈ Rn

the solution x(t), t ≥ 0, to (1) is nonnegative.

1In this paper, it is important to distinguish between a square nonnegative

(resp., positive) matrix and a nonnegative-definite (resp., positive-definite)

matrix.

(1)

+and u(t) ≥≥ 0, t ≥ 0,

Proposition 1 ([28], [29]): The nonlinear dynamical sys-

tem given by (1) is nonnegative if f : Rn→ Rnis essentially

nonnegative and G(x) ≥≥ 0, x ∈ Rn

It follows from Proposition 1 that, if f (·) is essentially

nonnegative, then a nonnegative input signal G(x(t))u(t),

t ≥ 0, is sufficient to guarantee the nonnegativity of the state

of (1).

+.

III. NEUROADAPTIVE OUTPUT FEEDBACK CONTROL WITH

ACTUATOR CONSTRAINTS

In this section, we consider the problem of characterizing

neuroadaptive dynamic output feedback control laws for non-

linear uncertain dynamical systems with actuator amplitude

and integral constraints to achieve reference model output

tracking. While our framework is applicable to general non-

negative and compartmental dynamical systems [29] with ac-

tuator amplitude and integral constraints, the main focus of this

paper is the application of this framework to pressure-limited

and work-limited control of mechanical ventilation. In this

section, however, we present a general neuroadaptive control

framework for nonlinear nonnegative dynamical systems with

actuator amplitude and integral constraints.

Consider the controlled nonlinear uncertain dynamical sys-

tem G given by

˙ x(t) = A0x(t) + B?f (x(t),h(u(t)),θ(t)) + B?h(u(t)),

x(0) = x0,

y(t) = Cx(t)

where x(t) ∈ Rn, t ≥ 0, is the state vector, u(t) ∈ Rm, t ≥ 0,

is the control input, y(t) ∈ Rm, t ≥ 0, is the system output,

A0 ∈ Rn×nis a nominal known Hurwitz and essentially

nonnegative matrix, B ∈ Rr×mis a known nonnegative input

matrix, ? ∈ Rm×mis an unknown nonnegative and positive-

definite matrix, h(u(t)) = [h1(u1(t),...,hm(um(t))]Tis the

constrained control input given by

⎧

⎩

where u∗

Dθ is a known bounded continuous function, where Dθ⊂ R

is a compact set, f : Rn× Rm× Dθ → Rmis Lipschitz

continuous and essentially nonnegative for all u ∈ Rmand

θ ∈ Dθ but otherwise unknown (that is, f (·,·,·) is such that

fi(x,h(u),θ) ≥ 0 if xi = 0, i = 1, ..., n, for all u ∈ Rm

and θ ∈ Dθ), and C ∈ Rm×nis a known output matrix.

For the mechanical ventilation problem, the control input

u(t), t ≥ 0, represents the pressure input to the ventilator and

the control input constraint (4) captures pressure amplitude

limitations. Furthermore, as we see in Section V, the function

θ(t), t≥0, is introduced to account for a continuous transition

of the respiratory parameters (e.g., lung resistance and com-

pliance) from inspiration to expiration. Finally, note that the

system structure given by (2) involves a non-affine system in

the control input. For details of such systems, see [30], [31].

t ≥ 0(2)

(3)

hi(ui) ?

⎨

0,

u∗

ui,

if ui≤ 0,

if ui≥ u∗

otherwise

i,

i,

(4)

i> 0, i = 1, ..., m, are given constants, θ : R →

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VOLYANSKYY et al.: NEUROADAPTIVE CONTROL FOR PRESSURE-LIMITED VENTILATION617

In order to achieve output tracking, we construct a reference

nonnegative dynamical system Gref given by

˙ xref(t) = Arefxref(t) + Brefr(t), xref(0) = xref0, t ≥ 0

yref(t) = Cxref(t)

where xref(t) ∈ Rn, t ≥ 0, is the reference state vector, r(t) ∈

Rd, t ≥ 0, is a bounded piecewise continuous nonnegative

reference input, Aref ∈ Rn×nis a Hurwitz and essentially

nonnegative matrix, and Bref∈ Rn×dis a nonnegative matrix.

Control (source) inputs for mechanical ventilation involving

pressure control are usually constrained to be nonnegative as

are the system states, which typically correspond to compart-

mental volumes. Hence, in this paper we develop neuroad-

aptive dynamic output feedback control laws for nonnegative

systems with nonnegative control inputs. Specifically, for the

reference model output tracking problem our goal is to design

a nonnegative control input u(t), t ≥ 0, predicated on the

system measurement y(t), t ≥ 0, such that ?y(t)−yref(t)? < γ

for all t ≥ T, where ?·? denotes the Euclidean vector norm on

Rm, γ > 0 is sufficiently small, and T ∈ [0,∞), x(t) ≥≥ 0,

t ≥ 0, for all x0 ∈ Rn

is restricted to the class of admissible controls consisting of

measurable functions u(t) = [u1(t), ..., um(t)]T, t ≥ 0, such

that (4) holds and

?t

where τs> 0 and η∗

and ui(t) ≡ 0 for all t ∈ [−τs, 0] and i = 1, ..., m. Note

that ηi(t), i = 1, ..., m, t ≥ 0, given by (7), satisfies

˙ ηi(t) = hi(ui(t)) − hi(ui(t − τs)), ηi(0) = 0, t ≥ 0.

For the mechanical ventilation problem, the pressure control

integral constraint (7) enforces an upper bound on the amount

of work performed by the ventilator.

Here, we assume that the function f (x,h(u),θ) can be

approximated over a compact set Dx× Du× Dθ by a linear

in parameters neural network up to a desired accuracy. In

this case, there exists ˆ ε : Rn× Rm× Dθ → Rmsuch that

?ˆ ε(x,h(u),θ)? < ˆ ε∗for all (x,h(u),θ) ∈ Dx× Du× Dθ,

where ˆ ε∗> 0, and

f (x,h(u),θ) = WT

(x,u,θ) ∈ Dx× Du× Dθ

where Wf ∈ Rs×mis an optimal unknown (constant) weight

that minimizes the approximation error over Dx× Du× Dθ,

ˆ σ : Rn× Rm× Dθ→ Rsis a vector of basis functions such

that each component of ˆ σ(·,·,·) takes values between 0 and

1, and ˆ ε(·,·,·) is the modeling error. Note that s denotes the

number of basis functions or equivalently, the number of nodes

of the neural network approximating the nonlinear function

f (x,h(u),θ) on a compact set (x,u,θ) ∈ Dx× Du× Dθ.

Since f (·,·,·) is continuous on Rn× Rm× Dθ, we can

choose ˆ σ(·,·,·) from a linear space X of continuous functions

that forms an algebra and separates points in Dx× Du× Dθ.

In this case, it follows from the Stone–Weierstrass theorem

[32, p. 212] that X is a dense subset of the set of continuous

(5)

(6)

+, and the control input u(·) in (2)

ηi(t) ?

t−τs

hi(ui(s))ds ≤ η∗

i, i = 1, ..., m, t ≥ 0(7)

i> 0, i = 1, ..., m, are given constants

(8)

fˆ σ(x,u,θ) + ˆ ε(x,u,θ),

(9)

functions on Dx×Du×Dθ. Now, as is the case in the standard

neuroadaptive control literature [33], we can construct a signal

involving the estimates of the optimal weights and basis

functions as our adaptive control signal. It is important to note

here that we assume that we know the structure and the size of

the approximator. This is a standard assumption in the neural

network adaptive control literature. In online neural network

training, the size and the structure of the optimal approximator

are not known and are often chosen by the rule that the larger

the size of the neural network and the richer the distribution

class of the basis functions over a compact domain, the tighter

the resulting approximation error bound ˆ ε(·,·,·). This goes

back to the Stone–Weierstrass theorem which only provides

an existence result without any constructive guidelines.

In order to develop an output feedback neuroadaptive con-

troller, we use the approach developed in [34] for recon-

structing the system states via the system delayed inputs and

outputs. Specifically, we use a memory unit as a particular

form of a tapped delay line (TDL), which takes a scalar time

series input and provides an (2mn − r)-dimensional vector

output consisting of the present values of the system outputs

and system inputs, and their 2(n − 1)m − r delayed values

given by

κ(t) ? [y1(t), y1(t − d),..., y1(t − (n − 1)d),...,

ym(t), ym(t − d),..., ym(t − (n − 1)d);

u1(t),u1(t − d),...,u1(t − (n −r1− 1)d),

...,um(t),um(t − d),...,

um(t − (n −rm− 1)d)]T

where ri denotes the relative degree of G with respect to the

output yi, i = 1,...,m, and r ? r1+ ··· + rm denotes the

(vector) relative degree of G.

The following matching conditions are needed for the main

result of this paper.

Assumption 1: There exist Ky ∈ Rm×mand Kr ∈ Rm×d

such that A0+ BKyC = Aref and BKr= Bref.

Assumption 1 involves standard matching conditions for

model reference adaptive control appearing in the literature,

see, for example [35, Ch. 5].

Using the parameterization ? = ˆ? + ??, where ?? ∈

Rm×mis an unknown symmetric matrix, the dynamics in (2)

can be rewritten as

˙ x(t) = A0x(t) + Bˆ?h(u(t)) + B[??h(u(t))

+?f (x(t),h(u(t)),θ(t))], x(0) = x0, t ≥ 0. (11)

Define W ??WT

(11) can be rewritten as

˙ x(t) = A0x(t) + Bˆ?u(t) + BWTσ(ζ(t),h(u(t)))

+ B ?ˆ ε(x(t),u(t),θ(t)) + B ˆ??h(t)

+ B WT

t ≥ 0(10)

1,WT

2

?T∈ R(s+m)×m, where W1? Wf?

and W2? ??T, and ζ(t) ??κT(t), θ(t)?T, t ≥ 0. Using (9),

1

?ˆ σ(x(t),u(t),θ(t)) − σζ(ζ(t))?,

x(0) = x0,

t ≥ 0(12)

where

σ(ζ(t),h(u(t))) ?

?

σT

ζ(ζ(t)),hT(u(t))

?T

(13)

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618IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 4, APRIL 2011

σζ : R2mn−r+1→ Rsis a vector of basis functions such that

each component of σζ(·) takes values between 0 and 1, and

?h(u(t)) ? h(u(t)) − u(t), t ≥ 0.

Next, consider a sequence of positive numbers {ρi}∞

that limi→∞ρi = 0 and define the time-dependent set ?t,i

and saturation impact times τ∗

?t,i??τ ≥ 0 : ηi(τ) = η∗

t ≥ 0,

⎧

⎩

where θi> 0, i = 1, ..., m, are design parameters.

Now, consider the control input u(t), t ≥ 0, given by

u(t) = ?(η(t))ψ(t),

where ?(η(t)) ? diag[φ1(η1(t)),...,φm(ηm(t))], t ≥ 0

φi(ηi(t)) ?

⎧

⎪⎪⎪⎪⎩

0 < δi < η∗

(chosen to be sufficiently small), and ψ(t) ∈ Rm, t ≥ 0, is

given by

i=1such

i(t) by

iand there exists N > 0

such that for all i ≥ N, ηi(τ − ρi) < η∗

i = 1, ..., m

θi+ max?τ : τ ∈ ?t,i

0, otherwise,

t ≥ 0,

i

?,

(14)

τ∗

i(t) ?

⎨

?, if ?t,i?= ∅,

i = 1, ..., m

(15)

t ≥ 0 (16)

⎪⎪⎪⎪⎨

1, if 0 ≤ ηi(t) ≤ η∗

i− δi and t ≥ τ∗

i(t),

1

δi(η∗

i− ηi(t)), if η∗

i− δi≤ ηi(t) ≤ η∗

and t ≥ τ∗

i,

i(t)

0, otherwise

t ≥ 0,

i = 1, ..., m.

(17)

i, i = 1, ..., m, are design constant parameters

ψ(t)

=

ψn(t) − ψad(t),

t ≥ 0(18)

where

ψn(t) = ˆ?−1?Kyy(t) + Krr(t)?,

and ˆ W(t) ∈ R(s+m)×m, t ≥ 0, is an update weight. Note

that, for all t ≥ 0 and i = 1, ..., m, 0 ≤ φi(ηi(t)) ≤ 1.

Furthermore, if ηi(ˆ t) = η∗

Now, it follows from (8) that ˙ η(ˆ t) = −hi(ui(ˆ t − τs)) ≤ 0,

ˆ t ≥ 0, and hence, ηi(ˆ t) is upper bounded by η∗

integral constraint (7) is satisfied. Fig. 1 shows the interplay

between ηi(t) and φi(ηi(t)), i = 1, ..., m.

Remark 1: The choice of φi(ηi), i = 1, ..., m, is not

limited to the piecewise linear continuous function given by

(17). In particular, on the interval η∗

can be chosen as any decreasing continuous function such that

φi(η∗

Defining the tracking error state e(t) ? x(t) − xref(t),

t ≥ 0, and using (16), (18)–(20), and Assumption 1, the error

dynamics is given by

t ≥ 0 (19)

(20)

ψad(t) = ˆ?−1ˆ WT(t)σ(ζ(t),h(u(t))),

t ≥ 0

ifor every ˆ t ≥ 0, then hi(ui(ˆ t)) = 0.

i. Thus, the

i− δi ≤ ηi ≤ η∗

i, φi(ηi)

i− δi) = 1 and φi(η∗

i) = 0.

˙ e(t) = Arefe(t) + B˜ WT(t)σ(ζ(t),h(u(t))) + Bˆ??h(u(t))

+ ε(t), e(0) = x0− xref0, t ≥ 0(21)

θi

τ∗

i

t

t

φ(ηi(t))

0

0

1

ηi(t)

η∗

i − δi

η∗

i

Fig. 1. Visualization of the effect of φi(ηi(t)) for a given function ηi(t).

where

ε(t) ? Bˆ?(?(t) − Im)ψ(t) + BWT

−σζ(ζ(t))] + B?ˆ ε(x(t),u(t),θ(t))

and ˜ W(t) ? W −ˆ W(t), t ≥ 0.

Next, to account for the effects of saturation (pressure lim-

itation) on the error state e(t), t ≥ 0, consider the dynamical

system given by

1[ˆ σ(x(t),u(t),θ(t))

˙ es(t) = Arefes(t) + Bˆ??h(u(t)),es(0) = es0, t ≥ 0

ys(t) = Ces(t)

where es(t) ∈ Rn, t ≥ 0, and define the shifted error state

˜ e(t) ? e(t)−es(t), t ≥ 0. Now, it follows from (21) and (22)

that

(22)

(23)

˙˜ e(t) = Aref˜ e(t) + B˜ WT(t)σ(ζ(t),h(u(t))) + ε(t),

˜ e(0) = 0,

t ≥ 0.

(24)

For the statement of the main result, define the projection

operator Proj(˜ W,Y) by

⎧

⎪⎪⎪⎩

where ˜ W ∈ Rs×m, Y ∈ Rn×m, μ(˜ W) ? (tr˜ WT˜ W − ˜ w2

ε˜ W, ˜ wmax∈ R is the norm bound imposed on˜ W, and ε˜ W> 0.

Note that for a given matrix ˜ W ∈ Rs×mand Y ∈ Rn×m, it

follows that

Proj(˜ W,Y) ?

⎪⎪⎪⎨

Y,

if μ(˜ W) < 0,

if μ(˜ W) ≥ 0 and μ?(˜ W)Y ≤ 0,

Y −μ?T(˜ W)μ?(˜ W)Y

Y,

μ?(˜ W)μ?T(˜ W)μ(˜ W), otherwise

max)/

tr[(˜ W − W)T(Proj(˜ W,Y) − Y)]

=

i=1

≤ 0

where coli(X) denotes the ith column of the matrix X.

n

?

[coli(˜ W − W)]T[Proj(coli(˜ W),coli(Y)) − coli(Y)]

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VOLYANSKYY et al.: NEUROADAPTIVE CONTROL FOR PRESSURE-LIMITED VENTILATION619

Consider the update law given by

˙ˆ W(t) = ?WProj[ˆ W(t),σ(ζ(t),h(u(t)))ξT

c(t)PB],

ˆ W(0) =ˆ W0, t ≥ 0(25)

where ?W ∈ R(s+m)×(s+m)is a positive definite matrix, P ∈

Rn×nis a positive-definite solution of the Lyapunov equation

0 = AT

where R > 0, and ξc(t) ∈ Rnξ, t ≥ 0, is the solution to the

estimator dynamics

˙ξc(t) =ˆAξc(t) + L [y(t) − yref(t) − yc(t) − ys(t)],

refP + PAref+ R

(26)

ξc(0) = ξc0,

t ≥ 0 (27)

(28)

yc(t) =ˆCξc(t)

where ˆA ∈ Rnξ×nξis Hurwitz, L ∈ Rnξ×m, andˆC ∈ Rm×nξ.

In addition, let ˜P ∈ Rnξ×nξbe a positive-definite solution of

the Lyapunov equation

0 = (ˆA − LˆC)T˜P +˜P(ˆA − LˆC) +˜R

where˜R > 0.

Now, since the projection operator used in the update law

(25) guarantees the boundness of the update weight ˆ W(t),

t ≥ 0, it follows that there exist u∗> 0 and δ∗> 0 such that

?u(t)? ≤ u∗and ??h(u(t))? ≤ δ∗for all t ≥ 0. Furthermore,

note that there exists ε∗> 0 such that ?ε(t)? ≤ ε∗for all t ≥ 0

and (x(t),u(t),θ(t)) ∈ Dx× Du× Dθ. Finally, there exists

α1> 0 such that ?˜ WT(t)σ(ζ(t),h(u(t)))? ≤ α1for all t ≥ 0.

For the statement of the main result of this paper, let ?·??:

Rn×n→ R be the matrix norm equi-induced by the vector

norm ? · ???: Rn→ R, let ? · ????: Rn×m→ R be the matrix

norm induced by the vector norms ? · ???: Rn→ R and

? · ?????: Rm→ R, and let ? · ?∗: Rnξ×nξ→ R be the matrix

norm equi-induced by the vector norm ? · ?∗∗: Rnξ→ R.

Furthermore, recall the definition of ultimate boundness of a

state trajectory given in [36, p. 241].

Theorem 1: Consider the nonlinear uncertain dynamical

system G given by (2) and (3) with u(t), t ≥ 0, given by

(16)–(20) and reference model Gref given by (5) and (6) with

tracking error dynamics given by (21). Assume Assumption 1

holds, λmin(R) > 1, and λmin(˜R) > ?˜PLˆC?∗2. Then there

exists a compact positively invariant set Dα⊂ Rn×Rn×Rnξ×

R(s+m)×msuch that (0,0,0,W) ∈ Dα, where W ∈ R(s+m)×m,

and the solution (e(t),es(t),ξc(t),ˆ W(t)), t ≥ 0, of the closed-

loop system given by (2), (3), (16), (22), (23), (25), (27), and

(28) is ultimately bounded for all (e(0),es(0),ξc(0),ˆ W(0)) ∈

Dαwith ultimate bound ?y(t) − yref(t)? < γ, t ≥ T, where

γ >(λmin(R) − 1)−1

+

+ λmax(?−1

ν ? (λmin(R) − 1)α2

αe?

λmin(R) − 1

αξ?

(29)

??

2√ν + αe

?2

?

(λmin(˜R) − ?˜PLˆC?∗2)−1

?1

e+ (λmin(˜R) − ?˜PLˆC?∗2)α2

?

1

λmin(˜R) − ?˜PLˆC?∗2?PB????α1

2√ν + αξ

?2

W) ˆ w2max

2

(30)

ξ(31)

1

?P??ε∗+ ?PB????α1

?

(32)

(33)

and ˆ wmaxis a norm bound imposed on ˆ W. Furthermore, u(t)

satisfies (7) for all t ≥ 0, h(u(t)) ≥≥ 0, t ≥ 0, and x(t) ≥≥ 0,

t ≥ 0, for all x0∈ Rn

Proof:

Ultimate boundness of the closed-loop system

follows by considering the Lyapunov-like function candidate

V(˜ e,ξc,˜ W) = ˜ eTP˜ e + ξT

where P > 0 and ˜P > 0 satisfy, respectively, (26) and

(29). Note that (34) satisfies α(?z?) ≤ V(z) ≤ β(?z?) with

z = [˜ eT,ξT

where ?z?2? ˜ eTP˜ e + ξT

denotes the column stacking operator. Furthermore, note that

α(?z?) is a class K∞ function [36]. Now, using (25), and

after considerable, albeit standard, algebraic manipulations

(see [24], [37] for similar details), the time derivative of

V(˜ e,ξc,˜ W) along the closed-loop system trajectories satisfies

˙V(˜ e(t),ξc(t),˜ W(t)) ≤ −

+ ν −

Now, for

?

or

?

it follows that ˙V(˜ e(t),ξc(t),˜ W(t)) ≤ 0 for all t ≥ 0, that is,

˙V(˜ e(t),ξc(t),˜ W(t)) ≤ 0 for all (˜ e(t),ξc(t),˜ W(t)) ∈ ˜De\˜Dr

and t ≥ 0, where

˜De?

(x, ˆ u,θ) ∈ Dx× Dˆ u× Dθ

˜Dr?

or ?ξc? ≤ αξc

Next, define

?

V(˜ e,ξc,˜ W) ≤ α

where α is the maximum value such that˜Dα⊆˜De, and define

˜Dη?

V(˜ e,ξc,˜ W) ≤ η

where

η > β(μ) = μ = α2

To show ultimate boundedness of the closed-loop system

(2)–(3), (16), (22)–(25), (27), and (28) assume2that˜Dη⊂˜Dα.

2This assumption is standard in the neural network literature and ensures

that in the error space ˜Dethere exists at least one Lyapunov level set ˜ Dη⊂

˜Dα. Equivalently, imposing bounds on the adaptation gains ensures˜Dη⊂˜Dα

[38]. In the case where the neural network approximation holds in Rnwith

delayed values, this assumption is automatically satisfied.

+.

c˜Pξc+ tr˜ WT?−1

W

˜ W

(34)

c,(vec˜ W)T]Tand α(?z?) = β(?z?) = ?z?2,

c˜Pξc + tr˜ WT?−1

W

˜ W and vec(·)

?

λmin(RP−1) − 1

?

(?˜ e(t)? − αe)2

?2, t ≥ 0.

?

λmin(˜R) − ?˜PLˆC?∗2???ξc(t)? − αξ

(35)

?˜ e? ≥ α˜ e?

ν

λmin(RP−1) − 1+ αe

(36)

?ξc? ≥ αξc?

ν

λmin(˜R) − ?˜PLˆC?∗2+ αξ

(37)

?

?

(˜ e,ξc,˜ W) ∈ Rn× Rnξ× R(s+m)×m:

?

(38)

(˜ e,ξc,˜ W) ∈ Rn× Rnξ× R(s+m)×m: ?˜ e? ≤ α˜ e

?

.

(39)

˜Dα?

(˜ e,ξc,˜ W) ∈ Rn× Rnξ× R(s+m)×m:

?

(40)

?

(˜ e,ξc,˜ W) ∈ Rn× Rnξ× R(s+m)×m:

?

(41)

˜ e+ α2

ξc+ λmax(?−1

W) ˆ w2max.

(42)

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620IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 4, APRIL 2011

1

Plant (2), (3)

−

Observer(27), (28)

y

NN Controller

(16)–(20)

TDL

ξc

ζ

h(u)

u

r

Ref. model (5), (6)

yZ

System (22), (23)

ys

Neuroadaptive controller

η

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

Now, since ˙V(˜ e,ξc,˜ W) ≤ 0 for all (˜ e,ξc,˜ W) ∈˜De\˜Dr and

˜Dr⊂˜Dα, it follows that ˜Dαis positively invariant. Hence, if

(˜ e(0),ξc(0), ˜ W(0)) ∈ ˜Dα, then it follows from [36, Corr. 4]

that the solution (˜ e(t),ξc(t),˜ W(t)), t ≥ 0, to (24), (25), (27),

and (28) is ultimately bounded with ultimate bound given by

γ = α−1(η) =√η, which yields (30).

The nonnegativity of h(u(t)), t ≥ 0, is immediate from (4).

The fact that u(t), t ≥ 0, satisfies (7) follows from (16), (17),

and the fact that h(u) ≥≥ 0 for all u ∈ Rm. Since A0 is

essentially nonnegative, B? ≥≥ 0, h(u(t)) ≥≥ 0, t ≥ 0, and

f (x,h(u),θ) is essentially nonnegative for all u ∈ Rmand

θ ∈ Dθ, it follows from (2) and Proposition 1 that x(t) ≥≥ 0,

t ≥ 0, for all x0∈ Rn

A block diagram showing the neuroadaptive control archi-

tecture given in Theorem 1 is shown in Fig. 2.

Remark 2: To apply Theorem 1 to a set-point regulation

problem, let xe∈ Rn

Brefr∗and yref(t) ≡ yd = Cxe, where yd ∈ Rm

desired set-point. In this case, the control signal u(t) is given

by (16) and (18) with ψn(t) ≡ 0.

+. This completes the proof.

+and r(t) ≡ r∗be such that 0 = Arefxe+

+is a given

IV. NONLINEAR MULTICOMPARTMENT MODEL FOR A

PRESSURE-LIMITED RESPIRATOR

In this section, we extend the linear multicompartment lung

model of [27] to develop a nonlinear model for the dynamic

behavior of a multicompartmentrespiratory system in response

to an arbitrary applied inspiratory pressure. Here we assume

that the bronchial tree has a dichotomy architecture [39], that

is, in every generation each airway unit branches in two airway

units of the subsequent generation. In addition, we assume that

lung compliance is a nonlinear function of lung volume. First,

for simplicity of exposition, we consider a single-compartment

lung model as shown in Fig. 3.

In this model, the lungs are represented as a single lung

unit with nonlinear compliance c(x) connected to a pressure

source by an airway unit with resistance (to air flow) of R.

At time t = 0, an arbitrary pressure pin(t) is applied to the

opening of the parent airway, where pin(t) is determinedby the

mechanical ventilator. This pressure is applied to the airway

opening over the time interval 0 ≤ t ≤ Tin, which is the

inspiratory part of the breathing cycle. At time t = Tin, the

applied airway pressure is released and expiration takes place

c(x)

R

papp

Fig. 3.Single-compartment lung model.

passively, that is, the external pressure is only the atmospheric

pressure pex(t) during the time interval Tin≤ t ≤ Tin+ Tex,

where Texis the duration of expiration.

The state equation for inspiration (inflation of lung) is

given by

Rin˙ x(t)+

1

cin(x)x(t) = pin(t),x(0) = xin

0, 0 ≤ t ≤ Tin

(43)

where x(t) ∈ R, t ≥ 0, is the lung volume, Rin ∈ R is

the resistance to air flow during the inspiration period, cin :

R → R+is a nonlinear function defining the lung compliance

at inspiration, xin

0

∈ R+ is the lung volume at the start

of the inspiration and serves as the system initial condition.

Equation (43) is simply a pressure balance equation where

the total pressure pin(t),t ≥ 0, applied to the compartment

is proportional to the volume of the compartment via the

compliance and the rate of change of the compartmental

volume via the resistance. We assume that expiration is passive

(due to elastic stretch of lung unit). During the expiration

process, the state equation is given by

Rex˙ x(t) +

1

cex(x)x(t) = pex(t), x(Tin) = xex

Tin≤ t ≤ Tin+ Tex

where x(t) ∈ R, t ≥ 0, is the lung volume, Rex ∈ R is the

resistance to air flow during the expiration period, cex: R →

R+ is a nonlinear function defining the lung compliance at

expiration, and xex

0

∈ R+ is the lung volume at the start of

expiration.

Next, we develop the state equations for inspiration and

expiration for a 2n-compartment model, where n ≥ 0. In

this model, the lungs are represented as 2nlung units which

are connected to the pressure source by n generations of

airway units, where each airway is divided into two airways

of the subsequent generation leading to 2ncompartments

(see Fig. 4 for a four-compartment model).

Let xi, i = 1,2,...,2n, denote the lung volume in

the ith compartment, cin

compliance of each compartment as a nonlinear function of

the volume of ith compartment, and let Rin

i = 1,2,...,2j, j = 0,...,n, denote the resistance (to

air flow) of the ith airway in the jth generation during

the inspiration (resp., expiration) period with Rin

Rex

01) denoting the inspiration (resp., expiration) of the parent

(i.e., 0th generation) airway. As in the single-compartment

model, we assume that a pressure of pin(t),t ≥ 0, is applied

during inspiration.

0,

(44)

i(xi), i = 1,2,...,2n, denote the

j,i(resp., Rex

j,i),

01(resp.,

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VOLYANSKYY et al.: NEUROADAPTIVE CONTROL FOR PRESSURE-LIMITED VENTILATION 621

c2(x2)

c1(x1)

c3(x3)

c4(x4)

x1

x4

x2

x3

Rin

2, 1

Rin

2,2

Rin

2,3

Rin

2, 4

Rin

1, 1

Rin

1, 2

Rin

0, 1

papp

Fig. 4.Four-compartment lung model.

xi

xi

xi2

in

xi1

in

xi2

ex

xi2

ex

cin

i (xi)

cex

i (xi)

Fig. 5.

of compartmental volumes.

Typical inspiration and expiration compliance functions as function

Now, the state equations for inspiration are given by

Rin

n,i˙ xi(t) +

1

cin

i(xi(t))xi(t) +

n−1

?

j=0

Rin

j,kj

×

kj2n−j

?

l=(kj−1)2n−j+1

˙ xl(t) = pin(t),

xi(0) = xin

i0,

0 ≤ t ≤ Tin,

i = 1,2,...,2n

(45)

where cin

i = 1,2,...,2n, given by [40]

i(xi), i = 1,2,...,2n, are nonlinear functions of xi,

cin

i(xi) ?

⎧

⎪⎪⎪⎪⎩

⎪⎪⎪⎪⎨

ain

i1+ bin

i1xi, if 0 ≤ xi≤ xin

i1,

ain

i2, if xin

i1≤ xi≤ xin

i2,

ain

i3+ bin

i3xi, if xin

i2≤ xi≤ TVi,

i = 1, ..., 2n

ij, j = 1,3, are unknown para-

i3< 0, xin

(46)

where ain

meters with bin

volume ranges wherein the compliance is constant, TVi de-

notes tidal volume, and

?kj+1− 1

ij, j = 1,2,3, and bin

i1> 0 and bin

ij, j = 1,2, are unknown

kj=

2

?

+ 1,

j = 0,...,n − 1,

kn= i

(47)

where ?q? denotes the floor function which gives the largest

integer less than or equal to the positive number q. Fig. 5

shows a typical piecewise linear compliance function for

inspiration. A similar compliance representation holds for

expiration, which is also shown in Fig. 5.

To further elucidate the inspiration state equation for a

2n-compartment model, consider the four-compartment model

shown in Fig. 4 correspondingto a two-generation lung model.

Let xi, i = 1,2,3,4, denote the compartmental volumes.

Now, the pressure (1/cin

i(xi(t))xi(t)) due to the compliance in

the ith compartment will be equal to the difference between

the external pressure applied and the resistance to air flow at

every airway in the path leading from the pressure source to

the ith compartment. In particular, for i = 3 (see Fig. 4)

1

cin

+ ˙ x4(t)] − Rin

−Rin

or, equivalently

3(x3(t))x3(t) = pin(t) − Rin

0,1[˙ x1(t) + ˙ x2(t) + ˙ x3(t)

1,2[˙ x3(t) + ˙ x4(t)]

2,3˙ x3(t)

Rin

2,3˙ x3(t) + Rin

+ ˙ x3(t) + ˙ x4(t)] +

1,2[˙ x3(t) + ˙ x4(t)] + Rin

3(x3(t))x3(t) = pin(t).

0,1[˙ x1(t) + ˙ x2(t)

1

cin

Next, we consider the state equation for the expiration

process. As in the single-compartment model, we assume that

the expiration process is passive and the external pressure

applied is pex(t),t ≥ 0. Following an identical procedure

as in the inspiration case, we obtain the state equation for

expiration as

Rex

n,i˙ xi(t) +

n−1

?

j=0

Rex

j,kj

kj2n−j

?

l=(kj−1)2n−j+1

˙ xl(t)

+

1

cex

i(xi(t))xi(t) = pex(t),

Tin≤ t ≤ Tex+ Tin,

xi(Tin) = xex

i = 1,2,...,2n

i0,

(48)

where

cex

i(xi) ?

⎧

⎪⎪⎪⎪⎩

⎪⎪⎪⎪⎨

aex

i1+ bex

aex

i1xi, if 0 ≤ xi≤ xex

i1,

i2, if xex

i1≤ xi≤ xex

i2,

aex

i3+ bex

i3xi, if xex

i2≤ xi≤ TVi,

i = 1, ..., 2n

(49)

aex

with bex

ranges wherein the compliance is constant, and kj is given

by (47).

ij, j = 1,2,3, and bex

i1> 0 and bex

ij, j = 1,3, are unknown parameters

i3< 0, xex

ij, j = 1,2, are unknown volume

V. NEUROADAPTIVE CONTROL FOR PRESSURE- AND

WORK-LIMITED MECHANICAL VENTILATION

In this section, we illustrate the efficacy of the neuroadaptive

control framework of Section III on the nonlinear multi-

compartmental lung model developed in Section IV. First,

however, we rewrite the state equations (45) and (48) for

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622IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 4, APRIL 2011

inspiration and expiration, respectively, in the form of (2).

Specifically, define the state vector x ? [x1,x2,...,x2n]T,

where xi denotes the lung volume of the ith compartment.

Now, the state equation (45) for inspiration can be rewritten as

Rin˙ x(t) + Cin(x(t))x(t) = pin(t)e,

x(0) = xin

0,

0 ≤ t ≤ Tin

(50)

where e ? [1,...,1]Tdenotes the ones vector of order 2n,

Cin(x) is a diagonal matrix function given by

?

Cin(x) ? diag

1

cin

1(x1),··· ,

1

cin

2n(x2n)

?

(51)

and

Rin?

n

?

j=0

2j

?

k=1

Rin

j,kZj,k ZT

j,k

(52)

where Zj,k ∈ R2nis such that the l-th element of Zj,k is 1

for all l = (k −1)2n−j+1, (k −1)2n−j+2,...,k2n−j,k =

1,...,2j, j = 0,1,...,n, and zero elsewhere.

Similarly, the state equation (48) for expiration can be

rewritten as

Rex˙ x(t) + Cex(x(t))x(t) = pex(t)e,

x(Tin) = xex

0,

Tin≤ t ≤ Tex+ Tin

(53)

where

Cex(x) ? diag

?

1

cex

1(x1),··· ,

1

cex

2n(x2n)

?

(54)

and

Rex?

n

?

j=0

2j

?

k=1

Rex

j,kZj,k ZT

j,k.

(55)

Now, since, by Proposition 4.1 of [27], Rin and Rex are

invertible, it follows that (50) and (53) can be equivalently

written as

˙ x(t) = Ain(x(t))x(t) + Binpin(t),

x(0) = xin

0

0 ≤ t ≤ Tin

(56)

˙ x(t) = Aex(x(t))x(t) + Bexpex(t),

x(Tin) = xex

0,

Tin≤ t ≤ Tex+ Tin

(57)

where Ain(x) ? −R−1

−R−1

To account for a continuous transition of the lung resistance

and compliance parameters between the inspiration and expira-

tion phase, consider the bounded continuous periodic function

θ(t) ∈ R, t ≥ 0, given by

⎧

1

εin(Tin− t), if Tin− εin≤ t ≤ Tin

inCin(x), Bin ? R−1

exe.

ine, Aex(x) ?

exCex(x), and Bex? R−1

θ(t) ?

⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, if 0 ≤ t ≤ Tin− εin,

0, if Tin≤ t ≤ Tin+ Tex− εex

1

εex(t + εex− Tin− Tex), otherwise

(58)

where εin > 0 and εex > 0 are sufficiently small constants

representing the transition times form inspiration to expiration

and vice versa, respectively, and θ(t) = θ(t + Tin+ Tex) for

all t ≥ 0. It is important to note that small variations in the

parameters εin and εex result in imperceptible differences in

the closed-loop system performance. Now, (56) and (57) can

be written as

˙ x(t) =?θ(t)Ain(x(t)) + (1 − θ(t))Aex(x(t))?x(t)

+ Pmusc(eTx(t))?, x(0) = xin(0), t ≥ 0

where u(t)

?

pk(t), k

∈

?t

⎧

⎩

remaining in the lung after the completion of each breath

[40], Pmax denotes the peak pressure of the ventilator, and

Pmusc(eTx(t),z(t)), t ≥ 0, introduced in (59) represents a

nonnegative pressure term due to the lung muscle activity of

a patient and accounts for the effect of spontaneous breath-

ing of a patient in the lung model. Here, we assume that

Pmusc(eTx(t),z(t)), t ≥ 0, is a nonlinear function given by

Pmusc(eTx(t),z(t)) = e−αptκ(t)WT

+ [θ(t)Bin+ (1 − θ(t))Bex][h(u(t)) + Pex

(59)

{in,ex}, t

≥

0, z(t)

?

t−τmeTx(s)ds, τm> 0, t ≥ 0, h(u(t)), t ≥ 0, is a saturation

constraint on the applied airway pressure given by

0, if u ≤ 0,

Pmax, if u ≥ Pmax,

u, otherwise.

Pex ∈ R2n

h(u) ?

⎨

(60)

denotes the end-expiratory pressure due to air

mσm(eTx(t),z(t)),

t ≥ 0(61)

where αp ≥ 0, κ : R → R is a given bounded function,

Wm∈ Rlm×2n≥≥ 0 is an unknown matrix, and σm(eTx,z) :

R×R → Rlm

volume as well as past system volume over a moving time

window. The scaling factor e−αptκ(t) in (61) is introduced to

account for the effect of the anesthetic on the spontaneous

breathing of the patient. Specifically, if αpis large, indicating

a heavily sedated patient, then the lung muscle activity of

the patient is negligible, whereas if αp is small, indicating a

moderately sedated or agitated patient, then the lung muscle

activity of the patient is accounted for by (61).

Note that since, by Proposition 4.1 of [27], −R−1

are essentially nonnegative, Cin(x) and Cex(x) are diagonal,

and θ(t) ≥ 0, t ≥ 0, it follows that Ain(x) and Aex(x) in

(59) are essentially nonnegative. Hence, since h(u(t)) ≥≥ 0,

t ≥ 0, Pmusc(eTx(t),z(t)) ≥≥ 0, t ≥ 0, and Pex ≥≥ 0, it

follows from Proposition 1 that x(t) ≥≥ 0, t ≥ 0, for all

xin(0) ∈ R2n

Next, we rewrite (59) in the form of (2) and (3) as

+is a known function of the instantaneous system

inand −R−1

ex

+.

˙ x(t) = A0x(t) + B0h(u(t)) + f (x(t),h(u(t)),θ(t)),

x(0) = xin(0),

y(t) = Cx(t)

where A0 = −R−1

and Cavare nominal parameter matrices given by

t ≥ 0(62)

(63)

avCav, B0 = R−1

ave, and C = eT, and Rav

Rav?

n

?

j=0

2j

?

k=1

Rav

j,kZj,k ZT

j,k,

Cav? diag

?1

cav

1

,··· ,

1

cav

2n

?

Page 10

VOLYANSKYY et al.: NEUROADAPTIVE CONTROL FOR PRESSURE-LIMITED VENTILATION 623

05 101520 2530

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [sec]

V(t)[liters]

Delivered air volume: Without adaptation

V(t)

Vref(t)

05 101520 2530

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [sec]

V(t)[liters]

Delivered air volume: With adaptation

V(t)

Vref(t)

Fig. 6. Delivered air volume V(t) = eTx(t) versus time with pressure-limited

input h(u(t)).

05 101520 25 30

0

2

4

6

8

10

12

14

16

18

20

Time [sec]

P(t) [cm H2O]

P(t) [cm H2O]

Constrained pressure: Without adaptation

05 101520 2530

0

2

4

6

8

10

12

14

16

18

20

Time [sec]

Constrained pressure: With adaptation

P(t)

Pmax

P(t)

Pmax

Fig. 7.Constrained pressure P(t) = h(u(t)) versus time.

where Rav

nal resistance (to air flow) of the ith airway in the jth genera-

tion, and cav

of each compartment. Now, the nonlinear unknown function

f (x,h(u),θ) capturing resistance and compliance uncertainty

in (62) during the inspiration and expiration phases is given by

j,i, i = 1,2,...,2j, j = 0,...,n, denote the nomi-

i, i = 1,2,...,2n, denote the nominal compliance

f (x,h(u),θ) = [θ(Ain(x) − A0) + (1 − θ)(Aex(x) − A0)]x

+ [θ(Bin− B0) + (1 − θ)(Bex− B0)]

× [h(u) + Pmusc(eTx) + Pex].

Finally, to account for work limitation constraints by the

mechanical ventilator over an inspiration-expiration cycle, we

assume that the constraint (7) holds and is given by η(t) ?

?t

fying the aforementioned input constraints while guaranteeing

output tracking of a clinically plausible reference model in the

face of physiological parameter uncertainty. For the system

given by (62) and (63), which is a special case of (2) and (3),

we consider an output tracking problem with a reference model

of the form given by (5) and (6), and design a neuroadaptive

controller using Theorem 1.

For our simulation, we consider a two-compartment lung

model and use the values for lung resistance and compliance

found in [40]. In particular, we set cav

cav

ain

i2

i3

−0.0067, xin

aex

i1

i1

H2O, aex

xex

i2

i3

H2O/?/sec, Rav

(64)

t−τlh(u(s))ds ≤ η∗, t ≥ 0, τl> 0, where η∗> 0.

Our goal here is to design a neuroadaptive controller satis-

1

= 0.022?/cm H2O,

i1= 0.0233,

= 0.2532?/cm H2O, bin

i2= 0.48?, xin

= 0.078, aex

i3= 0.1025?/cm H2O, bex

= 0.43?, xex

1,1= 30.67 cm H2O/?/sec, Rav

2= 0.03?/cm H2O, ain

= 0.025?/cm H2O, ain

i1= 0.3?, xin

= 0.02?/cm H2O, bex

i1= 0.018?/cm H2O, bin

i3

=

i3= 0.63?, i = 1, 2,

i2

= 0.038?/cm

i3= −0.15, xex

= 0.63?, i = 1, 2, Rav

i1= 0.23?,

0,1= 6.29 cm

1,2= 13 cm

05 101520 25 30

0

5

10

15

20

25

30

35

40

45

Time [sec]

0510 152025 30

0

Time [sec]

η(t)

5

10

15

20

25

30

35

40

45

η(t)

Without adaptation

η(t)

η*

With adaptation

η(t)

η*

Fig. 8.

η(t) versus time.

05 101520 25 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [sec]

0510152025 30

0

Time [sec]

V(t)[liters]

Delivered air volume: Without adaptation

0.1

0.2

0.3

0.4

0.5

0.6

0.7

V(t)[liters]

Delivered air volume: With adaptation

V(t)

Vref(t)

V(t)

Vref(t)

Fig. 9.

pressure input u(t).

Delivered air volume V(t) = eTx(t) versus time with unconstrained

0510 15202530

0

5

10

15

20

25

Time [sec]

P(t) [cm H2O]

P(t) [cm H2O]

Unconstrained pressure: Without adaptation

P(t)

Pmax

051015202530

0

5

10

15

20

25

Time [sec]

Unconstrained pressure: With adaptation

P(t)

Pmax

Fig. 10. Unconstrained pressure P(t) = u(t) versus time.

TABLE I

MAE, MAXIMUM VOLUME UNDERSHOOT, AND MAXIMUM VOLUME

OVERSHOOT

Without adaptation

53.98

364.46

88.33

With adaptation

0.94

92.43

64.96

MAE (%)

Max. overshoot (%)

Max. undershoot (%)

H2O/?/sec, Rin

H2O/?/sec, Rin

Rex

1,1

Tin= 5 sec, Tex= 10 sec, εin= εex= 0.001 sec, Pex(t) =

θ(t)P1

H2O, P2

ex

=

[0.01, 0.03, 0.23, 0;0.02, 0.01, 0, 0.17]T, and σm(y,z) =

[1/(1

1/(1 + e−0.5z)]T.

For our first simulation, we set Aref = A0, Bref = 0.6B,

r(t) = 17θ(t) + 5 cm H2O, Kr = 0.6, σ(ζ(t),h(u(t))) =

[1/(1 + e−ay(t)), 1/(1 + e−ay(t−d)),1/(1 + e−aP(t)), θ(t),

σT

ˆ W0

0,1

1,2= 10 cm H2O/?/sec, Rex

= 40 cm H2O/?/sec, Rex

=

6 cm H2O/?/sec, Rin

1,1

=

25 cm

0,1= 6 cm H2O/?/sec,

= 20 cm H2O/?/sec,

1,2

ex+ (1 − θ(t))P2

ex, P1

ex= [−0.1105, −0.3113]Tcm

[−0.0894, −0.1964]Tcm H2O, Wm

e−0.2y), 1/(1

+

=

+

e−0.3y)1/(1

+

e−0.3z),

m(y(t),0)]T, t ≥ 0, a = 0.02, x0 = xref0 = [0, 0]T,

= 08×2, ?W

= 100I8, Wm

= [0.01, 0.03, 0, 0;

Page 11

624IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 4, APRIL 2011

0510 15 20 2530

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time [sec]

κ(t)

Fig. 11.

κ(t) versus time.

05 10152025 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [sec]

V(t)[liters]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

V(t)[liters]

Delivered air volume: Without adaptation

V(t)

Vref(t)

V(t)

Vref(t)

05 10 1520 2530

Time [sec]

Delivered air volume: With adaptation

Fig. 12.

limited input h(u(t)).

Delivered air volume V(t) = eTx(t) versus time with pressure-

0.02, 0.01, 0, 0]T, peak pressure limit Pmax = 19 cm H2O,

and η∗= 43 sec·cm H2O. From the structure of Wm and

σm(y,0) it follows that lung muscle activity of the patient

is not a function of the past system volume. In addition, we

set αp = 0 and κ(t) ≡ 1. Figs. 6–8 show the delivered air

volume V(t) = eTx(t) versus time, the constrained pressure

P(t) = h(u(t)) versus time, and the integrated constrained

pressure over the time interval τl = 5 s with and without

adaptation for the pressure-limited input h(u(t)), t ≥ 0. Figs. 9

and 10 show the delivered air volume versus time and the

unconstrained pressure input u(t), t ≥ 0, versus time with

and without adaptation. Here, “with adaptation” refers to the

control signal (16) with the adaptive signal ψad(t), t ≥ 0,

given by (20), and “without adaptation” refers to the control

signal (16) with ψad(t) ≡ 0. In addition, Table I summa-

rizes performance measures of the control algorithm with

and without adaptation for mean absolute error (MAE) (de-

fined as measured delivered volume minus the target volume,

normalized to the target), maximum volume overshoot, and

maximum volume undershoot. It can be seen from Table I that

adaptation provides significantly better tracking performance

of the reference model.

As can be seen from Fig. 6, the delivered air volume signifi-

cantly exceeds the desired values in the absence of adaptation,

whereas satisfactory tracking of the desired air volume is

achieved with adaptation. As discussed in the introduction,

failure to adequately regulate the mode and parameters of

ventilatory support can result in failure to oxygenate, failure to

achieve adequate lung expansion or overexpansion of the lung

resulting in lung tissue rupture. These problems oftentimes

occur when open-loop volume control or pressure control

is employed or when averaged respiratory data is used to

051015202530

0

2

4

6

8

10

12

14

16

18

20

Time [sec]

051015 2025 30

0

Time [sec]

P(t) [cm H2O]

2

4

6

8

10

12

14

16

18

20

P(t) [cm H2O]

Constrained pressure: Without adaptation

P(t)

Pmax

Constrained pressure: With adaptation

P(t)

Pmax

Fig. 13.Constrained pressure P(t) = h(u(t)) versus time.

05 1015 202530

0

5

10

15

20

25

30

35

40

45

Time [sec]

0510 15 202530

0

Time [sec]

Without adaptation

With adaptation

η(t)

5

10

15

20

25

30

35

40

45

η(t)

η(t)

η*

η(t)

η*

Fig. 14.

η(t) versus time.

TABLE II

MAE, MAXIMUM VOLUME UNDERSHOOT, AND MAXIMUM VOLUME

OVERSHOOT

Without Adaptation

29.86

539.52

71.47

With Adaptation

1.83

129.21

66.57

MAE (%)

Max Overshoot (%)

Max Undershoot (%)

choose the parameters for a closed-loop ventilation control

algorithm. In contrast, the proposed neuroadaptive control

algorithm avoids reliance on average respiratory data and

achieves system performance without excessive reliance on

system model parameters.

Finally, to account for the lung muscle activity being a

function of the instantaneous system volume as well as the

past system volume, we set τm= 3 s and σ(ζ(t),h(u(t))) =

[1/(1 + e−ay(t)), 1/(1 + e−ay(t−d)),1/(1 + e−aP(t)), θ(t),

σT

of the anesthetic agent on spontaneous breathing, we set

αp = 0.03 and κ(t) to be a saw-type function generated by

filtering a pulse function of period 3 s and amplitude 1 through

a first-order filter with a pole at −2 (see Fig. 11). Figs. 12–14

show the effect of the integral term z(t), t ≥ 0, in the muscle

activity model. As can be seen from Fig. 12, in the absence of

adaptation the delivered volume dynamics significantly differs

from the dynamics shown in Fig. 6. However, as can be seen

from Fig. 12, the neuroadaptive controller captures the effect

of the integral term z(t), t ≥ 0, and provides satisfactory

tracking of the reference model. Table II gives analogous

performance measures to Table I for the case where lung

muscle activity is accounted for in the simulations.

m(y(t),z(t))]T, t ≥ 0. In addition, to account for the effect

VI. CONCLUSION

Acute respiratory failure due to infection, trauma, and

major surgery is one of the most common problems en-

Page 12

VOLYANSKYY et al.: NEUROADAPTIVE CONTROL FOR PRESSURE-LIMITED VENTILATION625

countered in intensive care units and mechanical ventilation

is the mainstay of supportive therapy for such patients. In

particular, mechanical ventilation of a patient with respiratory

failure is a critical life-saving procedure performed in the

intensive care unit. Failure to adequately regulate the mode

and parameters of ventilatory support can result in failure

to oxygenate, failure to achieve adequate lung expansion,

or overexpansion of the lung resulting in lung tissue rup-

ture. In this paper, we developed a neuroadaptive control

algorithm for mechanical ventilation to control lung volume

and minute ventilation. The adaptive controller accounts for

input pressure constraints as well as work of breathing con-

straints in the face of lung resistance and compliance model

uncertainty.

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Kostyantyn Y. Volyanskyy (S’07–M’10) received

the B.S., M.S., and Ph.D. degrees in applied mathe-

matics from the National Taras Shevchenko Univer-

sity of Kyiv, Kiev, Ukraine, in 1998, 1999, and 2003,

respectively, with a specialization in modeling and

control of complex dynamical systems. He received

another Ph.D. degree in aerospace engineering from

the Georgia Institute of Technology, Atlanta, in

2010.

He is currently a Post-Doctoral Fellow at the

Georgia Institute of Technology. His current research

interests include nonlinear adaptive control and estimation, neural networks

and intelligent control, nonlinear analysis and control for biological and

physiological systems, and active control for clinical pharmacology.

Wassim M. Haddad (S’87–M’87–SM’01–F’09)

received the B.S., M.S., and Ph.D. degrees in me-

chanical engineering from the Florida Institute of

Technology, Melbourne, in 1983, 1984, and 1987,

respectively, with a specialization in dynamical sys-

tems and control.

He served as a Consultant for the Structural Con-

trols Group of the Government Aerospace Systems

Division, Harris Corporation, Melbourne, from 1987

to 1994. In 1988, he joined the faculty of the

Mechanical 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 Aerospace Engineering at the Georgia

Institute of Technology, Atlanta, where he holds the rank of Professor and

Chair of the Flight Dynamics and Control discipline. His research contribu-

tions in linear and nonlinear dynamical systems and control are documented in

over 520 archival journal and conference publications. He is a co-author of the

books Hierarchical Nonlinear Switching Control Design with Applications to

Propulsion Systems (Springer-Verlag, 2000), Thermodynamics: A Dynamical

Systems Approach (Princeton University Press, 2005), Impulsive and Hybrid

Dynamical Systems: Stability, Dissipativity, and Control (Princeton University

Press, 2006), Nonlinear Dynamical Systems and Control: A Lyapunaov-Based

Approach (Princeton University Press, 2008), Nonnegative and Compartmen-

tal Dynamical Systems (Princeton University Press, 2010), and Stability and

Control of Large-Scale Systems (Princeton University Press, 2011). His current

research interests include nonlinear robust and adaptive control, nonlinear

dynamical system theory, large-scale systems, hierarchical nonlinear switching

control, analysis and control of nonlinear impulsive and hybrid systems,

adaptive and neuroadaptive control, system thermodynamics, thermodynamic

modeling of mechanical and aerospace systems, network systems, expert

systems, nonlinear analysis and control for biological and physiological

systems, and active control for clinical pharmacology. His secondary interests

include the history of science and mathematics, as well as western philosophy.

Prof. Haddad is a National Science Foundation Presidential Faculty Fellow

and a member of the Academy of Nonlinear Sciences.

James M. Bailey received the B.S. degree from

Davidson College, Davidson, NC, in 1969, the Ph.D.

degree in chemistry (physical) from the University of

North Carolina, Chapel Hill, in 1973, and the M.D.

degree from Southern Illinois University School of

Medicine, Springfield, in 1982.

He was a Helen Hay Whitney Fellow at the Cali-

fornia Institute of Technology, Pasadena, from 1973

to 1975, and an Assistant Professor of chemistry and

biochemistry at Southern Illinois University from

1975 to 1979. After receiving the M.D. degree, he

completed a residency in anesthesiology and then a fellowship in cardiac

anesthesiology at the Emory University School of Medicine Affiliated Hos-

pitals, Atlanta, GA. From 1986 to 2002, he was an Assistant Professor of

anesthesiology and then Associate Professor of anesthesiology at the Emory

University School of Medicine Affiliated Hospitals, where he also served as

a Director of Critical Care Service. In September 2002, he moved his clinical

practice to Northeast Georgia Medical Center, Gainesville, as a Director of

Cardiac Anesthesia and Consultant in critical care medicine. He has served

as Chief Medical Officer of Northeast Georgia Health Systems, Gainesville,

since 2008. He is board certified in anesthesiology, critical care medicine,

and transesophageal echocardiography. He is the author or co-author of over

100 journal articles, conference publications, and book chapters. His current

research interests include pharmacokinetic and pharmacodynamic modeling of

anesthetic and vasoactive drugs and applications of dynamical system theory

in medicine.

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