APPLICATION OF A NOVEL PID AUTO-TUNER

TO A LUNG FUNCTION TESTING DEVICE

Andres Hernandez, Robin De Keyser and Clara Ionescu

Department of Electrical energy, Systems and Automation, Ghent University

Technologiepark 913, B9052 Gent, Ghent, Belgium

Keywords: Respiratory impedance, Closed loop control, PID, Auto-tuner, Frequency response.

Abstract: The paper presents a closed loop approach for the lung function tests of forced oscillation technique. In this

method it is important to ensure that the desired excitation signal to be applied at the patient’s mouth will be

delivered by the lung function testing device, without introducing distortions and nonlinear effects. A novel

PID auto-tuner is applied in an initial phase of the investigation, without the patient, to verify whether the

closed loop control can be implemented. The results are promising, showing that the auto-tuner is able to

perform well and ensure the desired signal at the output of the device, within safety limits for the control

effort (air flow).

1 INTRODUCTION

Non-invasive lung function tests are broadly used

for assessing respiratory mechanics (Northrop, 2002;

Oostveen et al., 2003). Contrary to the forced

maneuvers from patient side and special training for

the technical medical staff necessary in spirometry

and in body plethysmography (Pellegrinno et al.,

2005; Miller et al., 2005), the technique of

superimposing air pressure oscillations is simple and

requires minimal cooperation from the patient,

during tidal breathing (Oostveen et al., 2003).

Among the air pressure oscillation techniques for

lung function testing, the most popular one is that of

Forced Oscillation Technique (FOT). FOT uses a

multisine signal to excite the respiratory mechanical

properties over a wide range of frequencies, usually

between 4-48Hz (Oostveen et al., 2003).

Using measurements of air pressure and air flow,

it is possible to extract information regarding the

human respiratory input impedance. However this is

a linear approximation of a nonlinear system, hence

the output will depend on the input’s amplitude and

frequency (Schoukens & Pintelon, 2001). It is

therefore important to ensure that the desired signal

to be applied at the patient’s mouth will be delivered

by the lung function testing device, without

introducing distortions and nonlinear effects. Hence,

a closed loop control system is necessary, to

continuously monitor and correct the errors between

the desired input signal and the one delivered by the

device at the patient’s mouth.

PID controllers can incorporate auto-tuning

capabilities (Åström & Hägglund, 1995). The auto-

tuners are equipped with a mechanism capable of

automatically computing a reasonable set of

parameters when the regulator is connected to the

process. Auto-tuning is a very desirable feature

because it does not require a-priori identification of

the system to be controlled. The auto-tuning features

provide easy-to-use controller tuning and have

proven to be well accepted among process engineers

(Leva et al. 2002).

The aim of this study is to apply a PID auto-tuner

to the FOT device and test whether the controller

can follow a multisine reference input. The objective

is that the nonlinear effects and distortions coming

from the FOT device itself are corrected by the

control action, such that the excitation signal of

interest is delivered to the patient. In this incipient

phase, the closed loop control will be designed for a

hypothetical patient: a respiratory tube and a rubber

balloon. The underlying reason is that we need to

ensure repeatability of our experiments, in order to

check the feasibility of implementing a closed loop

control strategy in the lung function device. The

final aim is to develop the closed loop control for the

case when the patient is breathing (i.e. in presence of

disturbance).

Hernandez A., De Keyser R. and Ionescu C..

APPLICATION OF A NOVEL PID AUTO-TUNER TO A LUNG FUNCTION TESTING DEVICE.

DOI: 10.5220/0003154500550061

In Proceedings of the International Conference on Biomedical Electronics and Devices (BIODEVICES-2011), pages 55-61

ISBN: 978-989-8425-37-9

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

The paper is organized as follows: the device and

test setup are described in the next section. The

underlying principles of the auto-tuner algorithm

and the PID parameters are given in the third

section, followed by a section describing the

validation of the closed loop control for multisine

setpoint and comparison to open loop performance.

A conclusion section summarizes the main outcome

of this investigation and points towards further

developments.

2 IMPEDANCE MEASUREMENT

The impedance was measured using a modified FOT

setup, able to assess the respiratory mechanics from

4-50 Hz. The specifications of the device are: 11kg,

50x50x60 cm, 40 seconds measurement time,

European Directive 93/42 on Medical devices and

safety standards EN60601-1.

Figure 1: A schematic overview (A) and an electrical

analogy of the FOT setup (B).

Typically for lung function testing purposes, the

subject is connected to the setup from figure 1 via a

mouthpiece, suitably designed to avoid flow leakage

at the mouth and dental resistance artefact. The

oscillation pressure is generated by a loudspeaker

(LS) connected to a chamber. The LS is driven by a

power amplifier fed with the oscillating signal

generated by a computer. The movement of the LS

cone generates a pressure oscillation inside the

chamber, which is applied to the patient's respiratory

system by means of a tube connecting the LS

chamber and the bacterial filter (bf). A side opening

of the main tubing (BT) allows the patient to have

fresh air circulation. Ideally, this pipeline will have

high impedance at the excitation frequencies to

avoid the loss of power from the LS pressure

chamber. It is advisable that during the

measurements, the patient wears a nose clip and

keeps the cheeks firmly supported. Before starting

the measurements, the frequency response of the

pressure transducers (PT) and of the

pneumotachograph (PN) are calibrated. The PN is a

meter for measuring gas flow rates during breathing

by recording pressure differences across a device of

fixed-flow resistance that has known pressure and

flow characteristics. The measurements of air-

pressure P and air-flow Q during the FOT lung

function test are done at the mouth of the patient.

The FOT excitation signal was kept within a

range of a peak-to-peak range of 0.1-0.3 kPa, in

order to ensure patient comfort and linearity

(Oostveen et al., 2003). From these signals, the non-

parametric representation of the patient’s lung

impedance Z

is obtained assuming a linear

dependence between the breathing and

superimposed oscillations at the mouth of the patient

(Ionescu & De Keyser, 2008). The algorithm for

estimating Z

can be summarized from the electrical

analogue in figure 1-B:

(1)

where s denotes the Laplace operator. Since the

excitation signal is designed such that it is not

correlated with the breathing of the patient,

correlation analysis can be applied to the measured

signals. Therefore, one can estimate the respiratory

impedance as the ratio:

()

(2)

whereas the P corresponds to pressure (its electrical

equivalent is voltage) and Q corresponds to air-flow

(its electrical equivalent is current), U the excitation

signal,

()

the cross-correlation spectra

between the various input-output signals, ω is the

angular frequency and

1/2

(1)j =−

, resulting in the

complex variable Z

. From the point of view of the

forced oscillatory experiment, the signal components

of respiratory origin (U

) have to be regarded as pure

noise for the identification task (Ljung, 1999). In

this application, the patient is replaced by a system

without disturbance U

=0 (i.e. a respiratory tube

with a rubber balloon attached at the end) in order to

ensure repeatability and test the feasibility of

implementing a closed loop control algorithm in the

FOT setup.



() () () ()

Ps Z sQs U s=+

BIODEVICES 2011 - International Conference on Biomedical Electronics and Devices

3 UNDERLYING PRINCIPLES OF

KCR PID AUTO-TUNING

The development of this auto-tuning algorithm is

based on the prior art where two relay-based PID

auto-tuners have been presented: the Kaiser-Chiara

auto-tuner and the Kaiser-Rajka auto-tuner. Hence

the proposed algorithm is an extended combination

of the two: the Kaiser-Chiara-Rajka auto-tuner

algorithm (KCR) (De Keyser & Ionescu, 2010).

Notice that the development of the PID controller

does not require a-priori knowledge of the system.

The closed loop transfer function of a second

order system with complex conjugate poles, and

static gain one is given by:

()

ωω

(3)

with

the natural frequency and

the damping

factor. From (3) one has the relationship between the

closed loop percent overshoot (OS%) and the peak

magnitude M

in frequency domain (Nise, 2007):

% 100 ,

OS e M

πζ

ζζ

−−

−

(4)

By specifying the allowed overshoot in the

closed loop, it follows that the closed loop transfer

function must fulfill the condition:

()

1()

Tj M

(5)

with

()Gj

the open loop transfer function of both

the process and controller. Re-writing (5) in its

complex form:

[]

() ()

()

1() ()

RjI

ωω

(6)

with R the real part and I the imaginary part, and

taking

()Tj

results that:

()

cIr++=

(7)

where

−−

, which is nothing

else than the equation of a (Hall-)circle with radius r

and center in

{

}

,0c−

(Nise, 2007). In order to have a

peak magnitude, only those circles with M>1 are of

interest. Intersection with the unit circle is achieved

by using (7) and the condition:

I+=, hence

solving for R and I yields:

0.5

−

and

0.25

−

=−

(8)

The phase margin is given by

tan

thus:

0.25

tan

0.5

−

(9)

We stated in (De Keyser & Ionescu, 2010) that

specifying PM does not suffice to guarantee a good

closed loop performance in all situations. Therefore,

the next step is to determine the cross-over

frequency; i.e. the frequency where the process and

controller crosses the 0 dB line (open loop).

If the settling time of the closed loop is specified,

then using

and (4) we can obtain the

bandwidth frequency:

()

242

1442

BW n

ωω ζ ζζ

−+ −+

(10)

From (De Keyser & Ionescu, 2010), we use that

1.5

BW c

≈

and the generalization to higher order

systems which gives

cBW c

ωω

≤

≤ . By having the

cross-over frequency

, a sinusoid with period

can be applied to the process and obtain

the output:

(

)

() cos sin

Gj Me M j

ϕϕ

== +

(11)

using the transfer function analyzer algorithm

(Ionescu et al, 2010). The task is now to find the

controller parameters such that the specification for

phase margin is fulfilled, by giving

%OS ,

T , M

and

The controller is derived in its textbook form,

which for the critical frequency becomes:

() 1

cp d

Rj K jT

⎡

⎤

⎛⎞

⎢

⎥

⎜⎟

⎢

⎥

⎜⎟

=+ −

⎢

⎥

⎜⎟

⎢

⎥

⎝⎠

⎣

⎦

(12)

Starting from the controller frequency response

in (12), the loop frequency response is given by:

APPLICATION OF A NOVEL PID AUTO-TUNER TO A LUNG FUNCTION TESTING DEVICE

( 180 )

()()1

cos( 180 ) sin( 180 )

jPM

Rj Gj e

Mj PM

ajb

ωω

−°+

=⋅

=−°+ +⋅−°+

=− −

(13)

with cosaPM= and sinbPM

, schematically

shown in figure 2.

Figure 2: Schematic of the KCR tuning principle.

Based on (12), the controller is given by:

()1 (1 )

cdc p

jjT Kj

ωα

⎡⎤

⎛⎞

=+ − = +

⎢⎥

⎜⎟

⎝⎠

⎣⎦

(14)

Equivalence of (13) with (14) gives that

[]

(cos sin ) (sin cos )

(cos sin )

KM j

PM j PM

αϕ ϕα ϕ

−++

=− +

(15)

From the real and imaginary parts of (15), we have

that:

tan tan

tan( )

1tan tan

−

==−

=−

(16)

and using T

=4T

, (16) becomes:

tan( )

TPM

−= −

(17)

from where

sin( ) 1

cos( )

−

(18)

which gives only one positive result. From (15) and

(16) we have that

()

*1 1,

(19)

With

11tan( )

cos (

)

αϕ

+=+ −=

−

(20)

which gives the K

controller parameter:

cos( )

−

=±

(21)

with only one positive result.

4 CONTROLLER VALIDATION

4.1 Open Loop Identification

In order to verify the performance of the controller,

we shall identify the open loop frequency response

of the system. We shall make use of the Chirp-TFA

(Chirp Transfer Function Analyzer) technique

proposed in (Ionescu et al., 2010). In short, the

frequency of such a sinusoidal test signal varies

from a minimum frequency (

) until a maximum

frequency (

) in a certain time (

t ), known as a

chirp signal. In the Chirp-TFA framework, the

sampling period varies such that a fixed number of

samples per period are ensured (

N ), independent of

the increasing frequency.

An example of a chirp signal with fixed number

of samples per period is given in figure 3. Notice

that the sampling period will be adjusted at every

sampling instant, because the frequency is varying

continuously. After the measurement is performed,

in order to process the data, the chirp signal is

divided into sections, such that sub-sequent sections

which have approximately the same frequency. Each

of these sections will be used to obtain one point for

gain and one point for the phase in the Bode diagram

of the system. The schematic flowchart of the Chirp-

TFA discrete-time implementation is depicted in

figure 4.

Figure 3:

Chirp signal from 1 to 10 Hz in 5 seconds, 20

samples per

erio

ωω

-1

M=1

-a

-b

ωω

-1

M=1

-a

-b

0 1 2 3 4 5

-1

-0.5

0.5

Seconds

BIODEVICES 2011 - International Conference on Biomedical Electronics and Devices

Figure 4: Scheme of the Chirp-TFA discrete

implementation.

The form of the chirp signal is given by

sin(2 ) sin(2 ( ) ( ))

tftkTt

⋅⋅= ⋅ ⋅ . Hence, at

every time instant t, a variable sampling period

()

is calculated, such that one period contains

samples. The relation is given by

() ()

ft Tt

⋅=.

The

N samples are then given by

sin( )

, with

k=0, 1…

N -1. The sinusoidal output of a system in

terms of its magnitude, phase and noise, can be

written as:

() sin(2 / ) ()

yk b k N nk

=++

(22)

where n is the noise, b is the amplitude, ω is the

angular frequency and φ is the phase shift. For

example, at the n

interval, we have that:

(1)

() ()sin( ) ()

kN n

data n y k T k

−

=−

∑

(23)

(1)

() ()cos( ) ()

kN n

data n y k T k

−

=−

∑

(24)

where

()

Tkrepresents the sampling period at the

sample in the data vector and:

()

Nfk

(25)

where

()

k denotes the frequency at the

sample. Considering that we obtain one point in a

Bode plot for each period on which we integrate, it

makes sense to increase the frequency exponentially

with time, in order to get the same resolution (points

per decade) for all frequency intervals in the plot.

Therefore, the frequency points are calculated from:

() ( )

ft f

(26)

which is then a function of the design parameters.

As the measurement time T

increases, (24) and (25)

can be reduced to the approximations:

() cos

sm m

data T T

≈

() sin

cm m

data T T

≈

(27)

from where it follows that

() ()

mcm

b data T data T

(28)

()

arctan

()

data T

(29)

Plotting the b/a and

values for a range of

frequencies provides the Bode diagram for the

observed system.

4.2 Real-time Implementation

In practice, in order to send a sinusoidal signal of 50

Hz, is necessary to have a sample rate for about 500

Hz, which means about 10 samples per sinusoid

period. The sampling time obtained was 0.002

seconds. In this particular example, it is not possible

to work with Matlab, because the delay for

calculations in the closed loop is about 14ms, much

higher than the desired sample rate. A solution to

overcome this limitation consists in using Real Time

Windows Target (RTWT) Toolbox in Matlab. This

toolbox assigns some resources of the system

exclusively for this task, ensuring the desired

sampling time. A corresponding Simulink model

was developed in order to send and receive signals

to/from the real FOT system, as depicted in figure 5.

Next, the Simulink model is automatically compiled

in C-language, making in this way a direct

communication using the National Instruments

DAQCard 6024E (which is recognized by Matlab

and supported for real time applications).

Figure 5: Real Time Simulink model used in closed loop.

APPLICATION OF A NOVEL PID AUTO-TUNER TO A LUNG FUNCTION TESTING DEVICE

4.3 Open Loop versus Closed Loop

Performance

The open loop and closed loop identification using

the Chirp-TFA algorithm is given by means of Bode

plots in figure 6. It can be observed that the

bandwidth (frequency at -3dB) of the system is

about 45Hz.

Figure 6: Bode characteristic of the open and closed loop.

For this preliminary study an OS%=20 and

Ts=0.041 seconds are defined as design

specifications. The KCR experiment is based on

obtaining the magnitude and phase of the system for

(crossover frequency), or represented in time as

. This value is obtained from the settling time as

<2*T

it follows that T

=0.082 seconds, with

=76.62 rad/s or 12.19 Hz. A sinusoidal signal to

this frequency ω

was applied to the system, from

where the magnitude (M) and phase (φ) were 10.48

dB, and 11.21° respectively.

Once the values for OS%, T

are defined and M

and φ are experimentally obtained, then by using

(4,9, 18, 21) as presented on the KCR algorithm in

section 3, it follows that the PID parameters are:

=0.4775; T

=0.0141; T

=0.0035 (30)

The designed PID controller (30), was tested by

applying a multisine setpoint with frequencies of

3Hz and 20Hz. The response is presented in figure 7.

In order to be able to follow a reference signal in

a closed loop is necessary that the magnitude of the

closed loop remains in 0dB and the phase in 0º into

the frequencies of interest. From the Bode plot in

figure 6 for the closed loop, we can observe that the

results are in agreement with the expected

bandwidth, and that the controller performs

satisfactorily. This result is also visible when a

comparison in time domain between open loop and

Figure 7: Comparison test between Open loop and PID

controller performance for a multisine setpoint at 3 Hz and

20 Hz.

closed loop is done. The controller avoids distortions

and nonlinear effects at the output of the lung

function device; the desired signal will be

successfully delivered at the patient’s mouth as

depicted in figure 7.

5 CONCLUSIONS

In this paper, the problem of closed loop control of a

medical device for lung function testing was

initiated. Preliminary results show that a proposed

PID autotuner can be applied and developed with

desired closed loop performance specifications for

settling time and overshoot. The next step is to

develop the method in the presence of noise, i.e.

interference with the breathing signal coming from

the patient.

REFERENCES

Åström, K. J., & Hägglund, T., 1995. PID Controllers:

Theory, Design and Tuning. Instrument Society of

America, Research Triangle Park, NC, USA,

De Keyser R., Ionescu C., 2010. “A generally valid

algorithm for PID auto-tuners”, accepted for

presentation at the Int Conf of Mathematical Methods

in Engineering, Coimbra, Portugal, to be held 21-24

October.

De Keyser R., Ionescu C., 2006. “FRtool: a frequency

response tool for CACSD in MatLab”, IEEE Conf on

Computer Aided Control Systems Design (CACSD-

CCA-ISIC), Munchen, Germany, pp. 2276-2280

0 5 10 15 20 25 30 35 40 45 50

-15

-10

-5

Bode characteristic Closed and Open loop

Magnitude [dB]

Open loop

Closed loop

0 5 10 15 20 25 30 35 40 45 50

-100

-50

100

150

Phase [Degree]

Frequency [Hz]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2

-1

Time [sec]

Voltage [V]

Reference

Out openloop

Out closedloop

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2

-1

Time [sec]

Voltage [V]

Ctrl effort

BIODEVICES 2011 - International Conference on Biomedical Electronics and Devices

Ionescu, C. & De Keyser, R., 2008. Parametric models for

characterizing the respiratory input impedance.

Journal of Medical Engineering & Technology, Taylor

& Francis, 32(4), pp. 315-324

Ionescu C., Robayo F., De Keyser R., Naumovic M.,

2010. “The frequency response analysis revisited”, in

Proc. the IEEE 18

Mediterranean Conference on

Control and Automation, 23-25 June, Marrakesh,

Marocco, pp. 1441-1446

Leva, A., Cox C. & Ruano A., 2002. “Hands-on Pid

autotuning: a guide to better utilisation”. IFAC

Professional Brief

Ljung, L., 1999, System Identification, NJ: Prentice Hall

Northrop R., 2002. Non-invasive measurements and

devices for diagnosis, CRC Press

Miller, M., Hankinson, J., Brusasco, V., Burgos, F.,

Casaburi, R., Coates, A., Crapo, R., Enright, P., van

der Grinten, C., Gustafsson, P., Jensen, R., Johnson,

D., MacIntyre, N., McKay, R., Navajas, D., Pedersen,

O., Pellegrino, R., Viegi, G., Wanger, J., 2005,

Standardization of spirometry, Eur Resp J, 26, 319-

338

Nise, N. S, 2007. Control systems engineering, Wiley

India Pvt. Ltd., 4th edition

Oostveen, E., Macleod, D., Lorino, H., Farré, R., Hantos,

Z., Desager, K., Marchal, F, 2003, The forced

oscillation technique in clinical practice: methodology,

recommendations and future developments, Eur

Respir J, 22, 1026-1041

Pellegrino, R., Viegi, G., Brusasco, V., Crapo, R., Burgos,

F., Casaburi, R., Coates, A., van der Grinten, C. P. M.,

Gustafsson, P., Hankinson, J., Jensen, R., Johnson, D.

C., McKay, R., Miller, M. R., Navajas, D., Pedersen,

O. F., Wanger, J., 2005, Interpretative Strategies for

Lung Function Tests. European Respiratory Journal,

26: 948-968

Schoukens J., and Pintelon, R., System Identification, a

frequency-domain approach, (IEEE Press, 2001)

APPLICATION OF A NOVEL PID AUTO-TUNER TO A LUNG FUNCTION TESTING DEVICE