L1 adaptive control of end-tidal CO2 by optimizing ... · L 1 adaptive control has been successfully applied in flight control for NASA AirSTAR aircraft (Gregory et al., 2009), - [PDF Document] (2023)

L1 adaptive control of end-tidal CO2 by optimizing ...· L 1 adaptive control has been successfully applied in flight control for NASA AirSTAR aircraft (Gregory et al., 2009), - [PDF Document] (1)

L1 adaptive control of end-tidal CO2 by optimizing muscle force to

mechanically ventilated patients

Anake Pomprapa * Marian Walter * Christof Goebel ** Berno Misgeld * Steffen Leonhardt *

*Philips Chair for Medical Information Technology, RWTH Aachen University, Aachen, Tyskland

(e-mail:[email protected]).

**Weinmann Devices for Medicine GmbH, Hamborg, Tyskland

Abstract: This article presents a new approach to control end-tidal CO2 in mechanically ventilated patients. Assuming a homogeneous lung model, a regulation of arterial CO2 tension in blood can be

is achieved non-invasively using L1 adaptive control using an extremum search method to set

correct respiratory rate. By using these integrated approaches, not only end-water CO2 is regulated

specific level, but also muscle power for breathing is optimized to comfort the muscles involved

the respiratory organs. The simulation of the control algorithms shows the characteristic results based on linear

and non-linear Hammerstein models of the process. These were obtained from measurement data from a

human volunteer. The algorithm is applicable during pressure-controlled ventilation and provides a practical solution in various clinical situations.

Keywords: non-linear control systems, adaptive control, biomedical systems

1. INTRODUCTION

Carbon dioxide (CO2) is one of the by-products of metabolism in a living cell. In the human respiratory tract

system, the produced CO2 is transported through the blood

circulation and removed by the lung to the air below

expired. End-tidal CO2 (etCO2) is defined as CO2

pressure (in mmHg) at the outlet. If a homogeneous

lung is assumed without lung disease, etCO2 can be

used to estimate CO2 partial pressure in arterial blood (PaCO2) at steady state (Benallal and Busso, 2000).

Therefore, the control of etCO2 provides a regulation of PaCO2

and pH balance in the blood. By keeping its value in the normal range

range, avoiding hypercapnia or hypocapnia can be

noninvasively obtained for patients undergoing mechanical

ventilation procedures. The application of closed loop

ventilation can be used in various clinical situations, e.g

eg intensive medicine, anesthesia and ventilation

support during sleep.

To comfort the muscles involved in the respiratory system, the extremum search method is primarily used for

minimize the force of the breath, so that the optimum

respiratory rate (RR) is determined (Otis et al., 1950). That

patient model is subsequently identified using linear and

non-linear Hammerstein models for evaluating the model structure and model parameters. The simplified single-

input single-output (SISO) model is used for an operating system

design in this complex patient-in-the-loop system. It is quite

obvious that we are dealing with a non-linear time-varying

system (Pomprapa et al., 2013). It is therefore straightforward

use an adaptive controller for this system where challenges

for feedback control is the nonlinear, time-varying system

with uncertainties depending on the patient's age, size and lung

condition.

Adaptive control has attracted the attention of many researchers because it requires less a priori knowledge about

the limits of the uncertain system (Feng and Lozano, 1999).

Its principle is to adapt the control law to cope with the time-

varying system. The foundation is based on parameter

estimation and guaranteed stability to synthesize a

control law for the converged and bounded results. Many

adaptive control schemes have been developed viz

model reference adaptive control (MRAC), self-tuning regulator, extremum seeking control, iterative learning

control, gain planning or L1 adaptive control. Measured with

this paper will present an end-tidal control system design

carbon dioxide (etCO2) in mechanically ventilated patients

using the advanced L1 adaptive control of output

feedback.

L1 adaptive control has been successfully used in flight control for NASA AirSTAR aircraft (Gregory et al., 2009),

in a flight simulator for SIMONA 6DOF motion based

control (Stroosma et al., 2011), or in biomedical systems to

anesthesia control (Ralph et al., 2011 and Kharisov et al.,

2012). The structure of an L1 adaptive controller is similar

MRAC, but it contains an additional low-pass filter. That

mathematical proof of L1 adaptive controller

(Hovakimyan et al., 2011) clearly shows that the error norm is inversely proportional to the square root of

adaptation gain. By introducing the high adaptation gain,

asymptotic tracking can be achieved (Cao and Hovakimyan,

2007b). The key feature of this method is to guarantee

∞L -norms limited transient response for the faults in the model

modes and the control signals. A low-pass filter is used to get

get rid of the unwanted high frequencies in the control signals and the bandwidth of this filter is determined by using L1

small gain theorem (Cao and Hovakimyan, 2006) to stabilize

the whole system.

9th IFAC Symposium on Nonlinear Control Systems, Toulouse, France, 4-6 September 2013

WeC2.3

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The subsequent sections of this contribution are organized as follows. It starts with the physiological description in section

2 to provide the background for this particular process.

System identification is introduced in section 3 for

evaluation of the model structure, followed by the problem

declaration (section 4). L1 adaptive control design is

presented in section 5. A discussion follows in section 6 and

the article ends with the conclusion.

2. PHYSIOLOGICAL DESCRIPTION

The complex physiological system of a patient undergoing

mechanical ventilation can be simplified as a single entry

single-output (SISO) system vist i fig. 1. Minut

ventilation (MV) denotes the volume given into the lung i

a minute of a mechanical ventilator, which is calculated by

multiplying tidal volume (VT) and respiratory rate (RR). MV is applied to the system and is considered input, while etCO2

considered as the output of the system.

Fig. 1. SISO open-loop system for typical etCO2 control.

In fig. 2, the static nonlinearity of etCO2 is presented based

in an experiment with a male volunteer with a normal body

mass index (BMI = 21.5 kg/m2) at steady state. A ventilator

(VENTIlogic LS, Weinmann devices for medicine GmbH,

Hamburg, Germany) was set in the pressure control

ventilation mode with a fixed positive end-expiratory

pressure (PEEP) = 5 hPa and I:E ratio = 50%. Two variables viz. peak inspiratory pressure (PIP) and RR, were adjusted

incrementally to change the MV. EtCO2 was measured at a

capnography system with integrated pulse oximetry for

monitoring of peripheral oxygen saturation (SpO2) (CO2SMO+,

Philips Respironics, Pittsburgh, USA).

0 5 10 15 20 250

5

10

15

20

25

30

35

MV [L/min]

etc

O2 [

mm

Hg]

measured data from a male volunteer

estimated curve for the relationship

Fig. 2. Static non-linearity between MV and etCO2.

The response of etCO2 shown in Fig. 2 represents a non-linear

function corresponding to the input MV. The output of

system (etCO2) is inversely proportional to the input. You others

words, an increase in MV leads to a decrease in etCO2.

For simplicity, we consider the case of a homogeneous

lung model where PEEP and I:E ratio are set as indicated.

Otherwise, it would result in much more complicated

modeling of multivariate inputs. Nevertheless, ours is simplistic

The SISO model can be used in real clinical practice for support

or assist ventilation in intensive care or for home care.

The extremum search method (Tan et al., 2010) is primarily performed to identify the optimal RR. That

calculation of the respiratory force is provided in Eq. (1)

and is calculated from each breath.

∫ ⋅⋅= RR dttVtPRR

Current

60

0)()(

60& (1)

where Power represents the power of one breath (Watt),

)(tP symbolizes airway pressure (Nm-2) and )(tV& indicates

airway flow (m3sec-1). Conversion of the units is required

from hecto Pascal (hPa) to Pa or Nm-2 (1 hPa = 100 Pa) and from L/min to m3/sec.

12 14 16 18 20 22 24 26 28 300,05

0,055

0,06

0,065

0,07

0,075

0,08

Respiratory rate [bpm]

Pow

is

of

bre

ath

ing [

W]

Fig. 3. A relationship between the power of the breath and

respiratory rate (RR).

In fig. 3 is an initialization of the ventilation procedure

is performed to seek the optimal RR that optimizes the effect

of a respiratory cycle. By stepwise variation of RR, the effect

breathing is calculated and averaged for 5 consecutive

breathing cycles at rest. The extremum search method is

is used to find the global minima for the power of a

breathing. Based on data from the volunteer, an RR of 14

bpm is identified and it will be used for further processes i

system identification, simulation and control throughout this article.

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The formulation of the mathematical model is shifted from a

consideration of MV input to pressure difference ( P∆ = PIP-

PEEP). Since RR is predetermined to optimize the muscular

respiratory force and PEEP are also fixed, P∆ has a direct

impact on tidal volume. Therefore, P∆ is considered to

be an equivalent (except for a non-linear gain factor) input

into this system.

3. SYSTEMIDENTIFIKATION

To extract the dynamics of the cardiopulmonary system, a

step change of the pressure difference ( PEEPPIPP −=∆ )

was introduced to the mechanically ventilated patient. The range of pressure difference (P∆) was set between 2 and 10

hPa with PEEP of 5 hPa, I:E ratio of 50%, RR of 14 bpm and

oxygen concentration (FiO2) of 0.21 or 21%. Using these

settings, different minute ventilation steps were given i

system and it resulted in output end-tidal CO2 (etCO2).

Fig. 4. Input-output measurements for system identification.

The model that describes this system is identified using different

model structures of both linear and non-linear models (Pottmann and Pearson, 1998). The results of parameter

estimation is shown in fig. 4 with an overview of

performance results listed in Table 1. The evaluation of

different model structures are indicated for 2 data sets which are

estimation and validation data. The mathematical forms of

each individual model structure and the parameter estimation

technique is given in Appendix A. Based on a validation

dataset, a 1st order Hammerstein model gives the best result

among all listed models. The 1st order linear model as well

offers the best RMS error among all linear models. Controller

design and simulation are performed with 1st order linear

model for the full range of nonlinear operation i

the following section.

Table 1. Evaluation of model structure

RMS error off

estimation data

RMS error off

validation set

1st order model

2,2475 2,2880

2

order model 2.2116 2.2988

2

order with a zero 2.1597 2.4093

1st order Hammerstein 2.1988 1.6709

2

order Hammerstein 2.1680 1.7804

2

order Hammerstein

with a zero 2.1351 1.8085

Regarding the capnography for etCO2 measurement, its accuracy is ±2 mmHg within the range of 0 - 40 mmHg, 5%

of the reading for 41 - 70 mmHg and 8% of the reading for

71 - 150 mmHg. Considering this, the results of parameter

estimate is within an acceptable range for the description of

this system.

4. STATEMENT OF THE PROBLEM

The system to be managed can be described as a SISO system.

))()()(()( sdsusAsy += (2)

,where )(sy is the Laplace transform of the measured etCO2,

)(sA represents a strictly correct transfer function, )(su is

The Laplace transform of the control input or P∆ in this

system and )(sd is the Laplace transform of the time-varying

non-linear uncertainties and disturbances )(td and in general

assumed that ))(,()( tytftd = , where ))(,( tytf satisfies

Lipschitz continuity expressed in Eq. (3) with Lipschtiz

constant L > 0 and 0L > 0.

2121 ),(),( yyLytfytf −≤− , 0),( LyLytf +≤ (3)

The control objective is to design a low-frequency adaptive

controller )(tu uses output feedback in a way that

system output )(ty traces the given reference input )(tr .

Using a first-order reference model ms

msM

+=)( for

0>m, the output provided in Eq. (4) can be estimated by a

multiplication between reference model and reference

signal.

)()()( srsMsy ≈ (4)

Paraphrase eg. (4) using Eq. (2), we get

))()()(()( ssusMsy σ+= (5)

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,where)(

)()()()()()()(

sM

susMsdsAsusAs

−+=σ .

Subsequently, the closed-loop adaptive control system can be

formed based on the model reference )(sM .

5. L1 ADAPTIVE CONTROLLER

L1 adaptive controller consists of 3 main components,

namely an output predictor, an adaptive algorithm and a low-

pass filter. Its performance is expected to be accurate, adaptive and robust for controlling etCO2 in a wide range

of P∆ inputs. The closed loop structure of L1 adaptive

control chart is shown in fig. 5.

Fig. 5. Patient-in-the-loop-konfiguration med L1 adaptiv

controller.

Output predictor: Output predictor is designed for

observe the predicted output )(ˆ ty with an adaptive

mechanism from )(ˆ tσ , where )(ˆ tσ is the adaptive estimator.

))(ˆ)(()()(ˆ ttumtmyty σ++−=& , 0)0(ˆ =y (6)

Eq. 6 corresponds to the desired stable model reference

system )(sM , which is designed using a first order

differential equation.

Adaptive Algorithm: The adaptive algorithm is used to adapt

the reference signal for eliminating the output error and is

defined by

))(~),(ˆ(Pr)(ˆ tymPtojt −⋅Γ= σσ& , 0)0(ˆ =σ (7)

where +∈Γ R is the matching gain corresponding to

lower limit

->C

20

4

42

23,

)1(maks

c

ab

wow

ab

PP med 1>a

(Hovakimyan and Cao, 2010), ojPr denotes the projection

operator, which ensures that the signal )(ˆ tσ is bounded in a

compact convex set with a smooth edge (Cao and

Hovakimyan, 2007a), )()(ˆ)(~ tytyty −= , and P is obtained

by solving the well-known Lyapunov equation.

A low-pass filter is introduced to eliminate high-frequency components in the control signal. An abrupt change of

pressure difference will be avoided with this filter. The control

law is calculated by Eq. (8).

))(ˆ)()(()( ssrsCsu σ−= (8)

where ω

oh+

=s

sC )( and is subject to the L1 gain stability

requirements (Cao and Hovakimyan, 2007a). Therefore, our

choices for designing )(sM and )(sC are limited by

)())(1()()(

)()()(

sMsCsAsC

sMsAsH

−+= (9)

is stable and

1)(1

(10)

where ))(1)(()(sCsHsG −= .

The evidence (Hovayakim et al., 2011) shows that the error

norm is inversely proportional to the square root of

adaptation gain. Therefore, the design of high customization gain

Γ will minimize the error norm )(~ ty . A high Γ will be used

in the design of our control system. However, it is not possible to

introduce an extremely high adaptation gain due to

computational limitation of the processor used for

controller.

6. SIMULATION RESULTS AND DISCUSSION

The models from system identification obtained from Section 3 are analyzed for the control system design using linear and

Hammerstein models. A limitation of P∆ between 2 and 40

hPa is introduced for safety reasons. The parameters of L1

adaptive controllers are designed by 40000=Γ and various

low-pass filters at 03.0=ω, 0.05 and 0.1 rad/sec.

assessed in our study. The desired etCO2 is set to 35 mmHg

and the results of the control signal P∆ and the output signal etCO2 are shown in fig. 6.

Fig. 6. Simulation results of control input and etCO2 output based on a 1st order linear model with different cut-off

the low-pass filter frequencies.

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The higher the bandwidth of the low pass filter, the faster

response. The bandwidth at 10.0=ω rad/sec. Gives us

processing time of 90 sec. without steady state errors. There is no

chatter effect on the control channel for all the selected

bandwidths in the simulation. Furthermore, Gaussian white noise with a standard deviation of 1 mmHg was introduced

into the system to observe control performance and

disturbance rejection of the L1 adaptive controller. Further

examination is carried out on the basis of disturbances with

different power and under different conditions with pole safety.

The simulation results are shown in Fig. 7. L1 adaptive

controller shows good robustness to disturbance effects up to

1.5 mmHg2sec/rad. The steering can withstand a rod

uncertainty between -28% and 23%. If the uncertainty

beyond this range is introduced, loss of control may occur.

Fig. 7. Simulated output response of etCO2 with perturbation and pole uncertainty using a 1st order linear process model.

As the pole moves further into the left half-plane (pole uncertainty changes from -28% to 23%), a faster output

response of etCO2 can be observed at a shorter settling time

without steady state errors. The success or failure of this

controller mainly depends on the polarity of the output

predictor.

Fig. 8. Block diagram of the simulation using a 1st order Hammerstein model as a plant with the designed L1 adaptive

controller using a 1st order linear model.

Further investigation is carried out with a 1st order Hammerstein plant based on the designed parameters using a

first-order linear model for the design of L1 adaptive

controller. The structure of this simulation is presented in

Fig. 8. It closely imitates the real use of this controller

for the non-linear time-varying plant or the mechanical one

ventilated patient. But in some cases a loss of control

in etCO2 can be observed in the simulation. The control

signal P∆ is delivered at the maximum of the saturated

safety area and it keeps the unsatisfactory value for a longer time

duration. Therefore, a readjustment is necessary if we use L1

adaptive controller under these realistic situations. Thus

initial state of σ̂ in the projection of the adaptive

the algorithm is adjusted as well as the cutoff frequency for

low pass filter is reduced. The simulation result with additive

Gaussian white noise with power 0.5 is shown in fig. 9. The

desired reference )(tr is set to 35 mmHg in the simulation

time 200

200≥t sek.

Fig. 9. Simulation result of the tracking performance of L1 adaptive controller for a nonlinear Hammerstein model with

)(tr = 35 mmHg i 200

200≥t sek.

Fig. 9 shows that the etCO2 response reaches the desired reference signal approx. 100 seconds after the step change.

The model's rod is located in the left half-plane close to it

the origin of the complex plane at -0.0334 and the answer is

relatively slow, but acceptable for the heart-lungs

system. With respect to the control signal P∆, the overshoot is

also within an acceptable range for implementation. That

the controller can successfully tolerate the disturbance introduced into the system. The adaptive L1 controller can be considered

as a promising solution for controlling etCO2 to

nonlinear time-varying plant. However, the disturbance may

cause a rapid change in P∆ and result in a frequent change in i

tidal volume. To test this controller with the patient, a fine

tuning may be necessary during the experiment.

In the future, a more generalized approach to the model formulation of different PEEP should be introduced

values. The model must describe patients with various

physiological characteristics (large vs. small, diseased lung vs.

healthy) and with different PEEP settings. Basically PEEP

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parameter affects functional residual capacity (FRC).

More PEEP will certainly give a larger lung volume by

the end of the expiration and it causes a change in etCO2. Here

study, a simplification is made for a fixed PEEP at 5 hPa.

Second, it should be noted that our control method can be

applied for inhomogeneous lung model, e.g. a lung with a

restrictive disease (acute respiratory distress syndrome -

ARDS). But it can cause overdistension of aerated alveoli

and volutrauma, just targeting etCO2 and not minimizing

shear stress in the alveoli. Also the control of etCO2 i

diseased lungs have an even more complicated relationship with the physiological target value for PaCO2 in the blood, which

depends on individuality and the severity of the disease.

7. CONCLUSION

This paper presents the design of L1 adaptive controller to control etCO2 for a patient undergoing mechanical ventilation

with a homogeneous lung model. Uses the pressure control

ventilation, is a patient model from a male volunteer

identified using linear and non-linear Hammerstein models.

Based on the obtained models, the tracking performance and

the controller's robustness is evaluated by a simulation

with dynamic disturbance injection and pole safety. That

nonlinear Hammerstein extension is made for feasibility

study of real clinical implementation. The controller showed

stability and good performance in terms of adaptation to

uncertain, disturbed system, so that good results can be expected in the clinical application scenario. L1 adaptive

controller provides a practical solution for controlling

etCO2 to handle the nonlinear time-varying system and as

a secondary effect optimizes muscle power

also the respiratory organs.

APPEALS

The authors acknowledge German financial support

Federal Ministry of Science and Education (BMBF) through

The OXIvent project under grant 16SV5605.

REFERENCES

Benallal, H. and Busso, T. (2000). Analysis of end-tidal and

arterial PCO2 gradients using a breathing model. Eur J

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Proceedings of the 2007 American Control Conference,

1191-1196.

Cao, C. and Havakimyan, N. (2006). Design and Analysis of a New L1 Adaptive Controller, Part I: Control Signal and

asymptotisk stabilitet. Proceedings of the 2006 American

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Feng, G. and Lozano, R. (1999). Adaptive control systems.

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Gregory , I.M. , Cao , C. , Xargay , E. , Hovakimyan , N. , et al

Zou, X. (2009). L1 adaptive control design for NASA

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Appendix A. MODEL STRUCTURE

The model structures in this paper are given in this section for

a 1st order linear model, 2nd order linear model, 2nd order

linear model with zero, 1st order Hammerstein model, 2nd

order Hammerstein model and 2nd order Hammerstein model

with zero as given in (11) - (16) respectively. The model

parameters can be estimated from data by least squares

algorithm.

)()()( tbutayty +=& (11)

)()()()( 21 tbutyatyaty ++= &&&& (12)

)()()()()( 2121 tubtubtyatyaty +++= &&&&& (13)

)]([)()( tubNtata +=& (14)

)]([)()()( 21 tubNtyatyaty ++= &&& (15)

)]([)]([)()()( 2121 tuNbtuNbtyatyatyaty +++= &&&& (16)

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