Identification of heart rate dynamics 
during treadmill exercise: comparison of first‑ 
and second‑order models
Hanjie Wang*  and Kenneth J. Hunt 
Background
Characterisation of heart rate (HR) dynamics with respect to changes in exer-
cise intensity provides models that can be used to synthesise control algorithms to 
maintain target HR levels [1]. The control of HR is important in the design of train-
ing protocols that aim both to maintain and to improve cardiorespiratory fitness; 
this applies to healthy individuals [2] and also in different patient populations [3, 
Abstract 
Background: Characterisation of heart rate (HR) dynamics and their dependence on 
exercise intensity provides a basis for feedback design of automatic HR control systems. 
This work aimed to investigate whether the second-order models with separate Phase 
I and Phase II components of HR response can achieve better fitting performance com-
pared to the first-order models that do not delineate the two phases.
Methods: Eleven participants each performed two open-loop identification tests 
while running at moderate-to-vigorous intensity on a treadmill. Treadmill speed was 
changed as a pseudo-random binary sequence (PRBS) to excite both the Phase I and 
Phase II components. A counterbalanced cross-validation approach was implemented 
for model parameter estimation and validation.
Results: Comparison of validation outcomes for 22 pairs of first- and second-order 
models showed that root-mean-square error (RMSE) was significantly lower and 
fit (normalised RMSE) significantly higher for the second-order models: RMSE was 
2.07 bpm ± 0.36 bpm vs. 2.27 bpm ± 0.36 bpm (bpm = beats per min), second 
order vs. first order, with p = 2.8 × 10−10 ; fit was 54.5% ± 5.2 % vs. 50.2% ± 4.8 %, 
p = 6.8 × 10−10.
Conclusion: Second-order models give significantly better goodness-of-fit than first-
order models, likely due to the inclusion of both Phase I and Phase II components of 
heart rate response. Future work should investigate alternative parameterisations of 
the PRBS excitation, and whether feedback controllers calculated using second-order 
models give better performance than those based on first-order models.
Keywords: Heart rate dynamics, System identification, Treadmills
Open Access
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RESEARCH
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37  
https://doi.org/10.1186/s12938‑021‑00875‑7
BioMedical Engineering
OnLine
*Correspondence:   
hanjie.wang@bfh.ch 
Department of Engineering 
and Information Technology, 
Division of Mechanical 
Engineering, Institute 
for Rehabilitation 
and Performance Technology, 
Bern University of Applied 
Sciences, 3400 Burgdorf, 
Switzerland
Page 2 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
4]. Target heart rate profiles come in various forms such as high-intensity inter-
val training (HIIT) that repeats high-intensity exercise connected by low-intensity 
recovery intervals; HIIT has potential to enhance cardiovascular health and fitness 
when compared to training at constant work rates (systematic reviews: [5, 6]).
Recent work investigated the effect of exercise intensity and time on HR dynamics 
using first-order models [7], but it may be beneficial to include higher order effects: 
based on physiological study, the dynamics of both oxygen uptake and HR responses 
to changes in exercise intensity are known to have three distinct phases [8]. These 
are: (i) a Phase I component lasting ∼ 15 s with a relatively small-magnitude venti-
latory response, but where HR can increase by about 50% of its total response [9]; 
(ii) a Phase II component between around 15 s and 3 min contributing the further 
increase of cardiopulmonary response; and then, (iii) if the applied exercise inten-
sity exceeds the anaerobic threshold, a Phase III component is prolonged and rises 
slowly. The three components can each be modelled as single exponentials (first-
order systems) each with their own time delay, gain, and time constant [10]. In addi-
tion to these primary dynamic responses, the phenomenon of heart rate variability 
(HRV) can be added to the model to represent the regulatory activities of the auto-
nomic nervous system; in the context of feedback control of HR, HRV represents a 
broad-spectrum disturbance term [1].
Because it can be challenging to estimate the separate Phase I and II components 
using data which is noisy, those two phases have often been identified as a com-
bined single exponential model with a time constant termed the mean response time 
(MRT, [8]), which is effectively the first-order approach taken in the previous studies 
that focused on system identification [7] and feedback control [1]. In feedback con-
trol, the slow Phase III component can readily be neglected as it is compensated by 
inclusion of an integrator in the controller. The focus of the present work is there-
fore the investigation of whether the separate identification of Phase I and II compo-
nents, i.e., the employment of a second-order model, can give better model fidelity.
Other recent approaches to HR dynamics identification focused mainly on the 
modelling of the Phase II and III components of the HR response. Several studies 
employed a non-linear state-space model structure comprising two different states 
( x1 and x2 ) to separately describe the Phase II and Phase III dynamics [11–15]. Other 
work used linear time-varying systems to model the slow Phase III dynamic [16, 
17]. While inclusion of Phase III may improve overall model fidelity, it will, as noted 
above, have negligible impact on feedback-control performance as it will be elimi-
nated by the integral action. In contrast, it can be anticipated that separate mod-
elling of the Phase I and II components might lead to better control performance 
when the model is used as the basis of an analytical, model-based feedback design.
To this end, this work aimed to investigate whether second-order models with sep-
arate Phase I and Phase II components of HR response can achieve better fitting 
performance compared to first-order models that do not delineate the two phases. 
Here, an input signal of PRBS (pseudo-random binary sequence) form was designed 
to excite both the Phase I and Phase II components.
Page 3 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
 
Results
To illustrate the procedures of data preprocessing and model validation, an exemplary 
result from participant P04 is shown (Fig. 1); the raw data for the same participant are 
shown above below in the section ‘Method’. For this example, the second-order model P2 
gave better performance than the first-order model P1 : fit was 51.9% vs. 50.9% ( P2 vs. P1 ) 
and RMSE was 2.01 bpm vs. 2.05 bpm.
The overall statistical comparison of validation outcomes for the 22 pairs of first- and 
second-order models showed that RMSE was significantly lower and fit significantly 
higher for the second-order models: RMSE was 2.07 bpm ± 0.36 bpm vs. 2.27 bpm ± 
0.36  bpm, P2 vs. P1 , with p = 2.8 × 10−10 (Table  1; Fig.  2a); fit was 54.5% ± 5.2 % vs. 
50.2% ± 4.8 %, p = 6.8 × 10−10 (Table  1; Fig.  2b). The graphical illustration of overall 
outcomes (Fig. 2) shows how widely individual samples and their differences are dis-
persed, together with means and their 95% confidence intervals (CIs). These plots allow 
visual determination of significant differences, if they exist: whenever there is a signifi-
cant difference, the value 0 will not be contained within the corresponding CI.
The sample size was estimated a priori by a statistical power calculation that used esti-
mates of expected effect sizes and sample standard deviations, with significance level set 
to 5% ( α = 0.05) and with a statistical power of 80% ( 1 − β = 0.8).
The observed outcomes show large effect sizes (approximately 9% for both outcomes) 
and extremely low p values (on the order of 10−10 ), thus pointing to a well-powered sta-
tistical analysis. In fact, post hoc statistical power analysis based on observed effect sizes 
and sample dispersions gives an observed power of 100% for both outcomes.
A graphical illustration of the dispersion of estimated model parameters for the 22 
first- and 22 second-order models is provided (Fig.  3). The overall first- and second-
order models were obtained by averaging the individual gains and time constants. For 
the first-order models, the overall gain was k1 = 28.57  bpm/(m/s) ± 5.27  bpm/(m/s) 
time/s
300
600
900
1200
1500
1800
-20
-10
0
10
20
heart rate/bpm
300
600
900
1200
1500
1800
-0.5
-0.25
0
0.25
0.5
speed/(m/s)
Fig. 1 Data preprocessing and model validation: exemplary data for participant P04 (the raw data for this 
test are shown in section ‘Method’). Upper plot: HR measurement from validation data set after detrending 
(solid black line), simulated HR response of first-order model ( P1sim , blue dashed line), and simulated HR 
response of second-order model ( P2sim , green dashed line). Lower plot: treadmill speed from validation data 
set after mean removal
Page 4 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
(mean ± standard deviation) while the time constant was τ1 = 70.56 s ± 16.84 s. For the 
second-order models, the overall gain was k2 = 24.70 bpm/(m/s) ± 5.07 bpm/(m/s) and 
the overall time constants were τ21 = 18.60 s ± 7.88 s and τ22 = 37.95 s ± 16.01 s. This 
gives the average transfer functions for first- and second-order models as follows:
Discussion
This study aimed to investigate whether second-order models with separate Phase 
I and Phase II components of heart rate response can achieve better fitting perfor-
mance compared to first-order models that do not delineate the two components. 
(1)
u  → y: P1(s) =
28.57
70.56s + 1,
(2)
u  → y: P2(s) =
24.70
(18.60s + 1)(37.95s + 1).
Table 1 Overall outcomes for first- and second-order models and comparison of outcome 
differences (see also Fig. 2)
n = 22
P1 first‑order models, P2 second‑order models, SD standard deviation, MD mean difference, 95% CI confidence interval for 
the mean difference , p-value paired one‑sided t tests,  RMSE root‑mean‑square error, fit normalised root‑mean‑square error, 
bpm beats per min
Mean ± SD
MD (95% CI)
p-value
P1
P2
P2 − P1
RMSE/bpm
2.27 ± 0.36
2.07 ± 0.36
−0.19 ( −∞ , −0.16)
2.8 × 10−10
fit/%
50.2 ± 4.8
54.5 ± 5.2
4.3 (3.6, +∞)
6.8 × 10−10
****
a Root-mean-square error, RMSE.
****
b Normalised root-mean-square error, ﬁt.
Fig. 2 Primary outcomes: data samples and differences for RMSE and fit between 22 first-order models, P1 , 
and 22 second-order models, P2 (see also Table 1). Sample pairs for each participant are connected by green 
lines; mean values are shown as red horizontal bars (with numerical values given in Table 1). Sample-pair 
differences are shown as D ( P2 − P1 ). The mean difference (MD) is depicted as a red bar and the blue arrow 
is the corresponding 95% confidence interval (CI). For both RMSE and fit, the 95% CI does not contain the 
value 0, thus showing a significant improvement for P2 vs. P1 ( p < 0.05 , Table 1; the notation **** denotes 
p < 0.0001)
Page 5 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
 
The results clearly demonstrate that second-order models give significantly better 
goodness-of-fit, in terms of both RMSE and fit (NRMSE): RMSE was on average 
10
15
20
25
30
35
40
45
50
0
20
40
60
80
100
120
first-order models
average model
a First-order models.
10
15
20
25
30
35
40
45
50
0
20
40
60
80
second-order models
average model
10
15
20
25
30
35
40
45
50
0
20
40
60
80
second-order models
average model
b Second-order models.
Fig. 3 Dispersion of estimated model parameters for 22 first- and 22 second-order models. The stars depict 
the average models. The 95% confidence intervals for the mean gains and time constants are shown as 
rectangular boxes
-25
-10
speed/(m/s)
evaluation period (290 s to 2085 s )
0.5 m/s
time/min
PRBS (schematic)
warm up
rest
formal measurement phase
cool down
0
10
20
30
40
50
-
5
2.0
+
5
2.0
a
0
300
600
900
1200
1500
1800
2100
100
150
200
heart rate/bpm
0
300
600
900
1200
1500
1800
2100
time/s
1
1.5
2
2.5
3
speed/(m/s)
b
Fig. 4 Identification test protocol. a Test phases and treadmill speed. b Original data record from one 
participant (P04; upper plot—HR measurement; lower plot—speed of the treadmill); the evaluation period is 
depicted by the red horizontal bar
Page 6 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
0.19 bpm lower and fit 4.3% higher for the second-order model structure (p values 
were on the order of 10−10 in both cases); that these significance levels were achieved 
with a small sample size of only 11 participants underline the difference.
The approach taken here focused on control-orientated model structures, in the 
sense that the estimated models would be intended to be used for analytical (model-
based) design of heart rate control systems. For this reason, slow Phase III com-
ponents in the data were eliminated by detrending prior to parameter estimation. 
This is consistent with feedback-control scenarios where slowly drifting Phase III 
variations in heart rate are automatically compensated using integral action in the 
controller.
A further difference between the methodology employed here and heart rate mod-
elling approaches taken in the physiological literature, [8], is that a nominal operat-
ing point was assumed, and small deviations around this point were considered (in 
this case, the operating point was set at the transition between exercise levels con-
sidered to be moderate and vigorous). This is consistent with linear feedback design 
approaches, which are implicitly based on models that are small-signal linearisations 
around an operating point; the purpose of feedback control is indeed to maintain the 
controlled variable, viz., heart rate, close to a target level.
For these reasons, it is not possible to compare the overall estimated model param-
eters (gains and time constants, Eqs. (1) and (2) with values given in the physiologi-
cal literature (e.g., [9, 10]), because, there, responses are usually recorded using large 
steps from a resting or low-intensity baseline.
A consequence of the control-orientated methodology followed here is that the 
design of the PRBS input signal becomes important. For non-linear systems, it is 
known that the parameters of linear approximations are input dependent [18], which 
motivates further work to explore the effect of PRBS amplitude and frequency con-
tent on model fidelity; in particular, it is important to focus the information content 
on frequencies around the intended crossover band of the closed-loop system [19].
Future work should investigate whether the observed improvement in model fidel-
ity translates into better feedback-control performance, i.e., whether controllers 
designed on the basis of second-order models perform better, in some sense, than 
those designed using first-order models. Because of the fundamental property of 
feedback that plant uncertainty (including modelling error) is reduced, the answer 
to this question will likely not be as clear cut as in the open-loop identification case.
Conclusions
Second-order models give significantly better goodness-of-fit than first-order mod-
els, likely due to the inclusion of both Phase I and Phase II components of heart rate 
response. Future work should investigate alternative parameterisations of the PRBS 
excitation, and whether feedback controllers calculated using second-order models 
give better performance than those based on first-order models.
Page 7 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
 
Methods
Participants
Eleven healthy participants were recruited (8 males, 3 females) with age 32.5 years 
± 12.3 years (mean ± standard deviation), body mass 75.5 kg ± 14.4 kg, and height 
179 cm ± 12 cm. For inclusion, each participant was required to be a regular exerciser 
(30-min bouts, 3 times per week) and non-smoker, and to be free of injury and illness.
Test protocols
To generate separate estimation and validation data sets, each participant took part 
in two identification tests; there was an interval of at least 48 h between the two tests. 
Before each test, participants were asked to meet the following requirements: refrain 
from strenuous activity for 24 h, caffeine for 12 h, avoid large meals for 3 h. Each test 
session had four phases: a 15 min warm up, a 10 min rest, a 36 min formal measure-
ment, and a 10 min cool down (Fig. 4a).
In the warm up, a feedback-control system was employed to automatically regulate 
the speed of the treadmill to maintain a constant target HR. The target HR, denoted 
HRref , was computed individually for each participant and corresponded to the HR 
at the transition between intensity levels considered to be moderate or vigorous [3], 
as follows: HRref = 0.765 × (220 − age) [beats/min, bpm] (except for participant P03, 
for whom the factor 0.7 was used, because 0.765 led to HR remaining in the vigor-
ous-intensity regime). The mean speed of the treadmill during the final 2 min of the 
warm up phase was subsequently used as the mid-level speed, denoted vm , for the 
next phase.
In the formal measurement phase, the speed of the treadmill, denoted v, was 
designed as a fifth-order PRBS with mean speed vm and amplitude 0.25  m/s, i.e., 
v = vm ± 0.25 m/s (to illustrate, a single original data record is provided; Fig.  4b). 
Model parameter estimation and validation was performed over a full cycle of the 
PRBS using an evaluation period from 290 s to 2085 s (Fig. 4); the first 5 min were 
excluded to eliminate the initial transient. During the cool down phase, the speed of 
the treadmill was kept constant at v = vm − 0.5 m/s.
Equipment
All tests were carried out using a treadmill (model Venus, h/p/cosmos Sports & Med-
ical GmbH, Germany) controlled by a PC running real-time Matlab/Simulink (The 
MathWorks, Inc., USA). HR recording was performed with a chest strap (H10, Polar 
Electro Oy, Finland) and a wireless receiver (Heart rate Monitor Interface, Spark-
fun Electronics, USA) connected to the Simulink model via a serial port. HR meas-
urements were received at a rate of 1 Hz and then downsampled to a sample rate of 
0.2 Hz (sample period 5 s) by averaging consecutive batches of five individual samples.
Data preprocessing, model identification, and outcome measures
As noted above, each participant completed two identification tests, thus gener-
ating individual data sets (I and II) for model parameter estimation and validation. 
To prevent over-fitting and to eliminate potential order-of-presentation effects, a 
Page 8 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
counterbalanced cross-validation approach was implemented: for each participant, 
data set I was used to estimate model parameters and data set II was used as valida-
tion data for the estimated models; then, for the same participant, data set II was used 
for model estimation and data set I for validation. Thus, for the 11 participants, a total 
of 22 estimation data sets and 22 validation data sets were obtained.
According to the test protocol (Sect. 5.2, Fig. 4a), an evaluation interval from 290 s to 
2085 s was used to estimate and validate model parameters. This interval, within one 
single PRBS period, was selected, such that the number of samples where the input 
was high ( v = vm + 0.25m/s) equalled the number of samples where the input was low 
( v = vm − 0.25m/s). Here, on the evaluation period from 290 s to 2085 s and with a sam-
ple period of 5 s, the total number of samples was N = 360, thus giving 180 low samples 
and 180 high samples.
To remove any potential drifting Phase III dynamic of the HR response, the mean 
value and any trend were removed (Matlab “detrend” function) prior to estimation and 
validation; the mean value of the input signal was also removed. An exemplary data set 
following this preprocessing procedure is provided (Fig.  1), with raw data are shown 
above (Fig. 4b).
For each estimation data set, two linear time-invariant transfer functions were 
employed to model the dynamic response from treadmill speed to HR: a first-order 
transfer function (Eq. 3) which combined Phases I and II into a single time constant, and 
a second-order transfer function (Eq. 4) with separate time constants for Phases I and 
II. Hence, for the 11 participants, a total of 22 first-order models and 22 second-order 
models were estimated:
Here, k1 and k2 are steady-state gains, and τ1 , τ21 , and τ22 are time constants. Model 
parameters were calculated from the estimation data sets using a least-squares optimisa-
tion procedure (“procest” function from the Matlab System Identification Toolbox; The 
Mathworks, Inc., USA).
After model estimation, the corresponding validation data sets were used to compute 
goodness-of-fit measures for the resulting first- and second-order models. Two outcome 
measures were used: the normalised root-mean-square error [denoted fit, Eq. (5)], and 
the root-mean-square error [denoted RMSE, Eq. (6)], as follows:
(3)
u  → y: P1(s) =
k1
τ1s + 1,
(4)
u  → y: P2(s) =
k2
(τ21s + 1)(τ22s + 1).
(5)
ﬁt (NRMSE) [%] =

1 −




N
i=1(HR(i) − HRsim(i))2
N
i=1(HR(i) − HR)2

 × 100 %,
(6)
RMSE [bpm] =



 1
N
N

i=1
(HRsim(i) − HR(i))2.
Page 9 of 10
Wang and Hunt  BioMed Eng OnLine           (2021) 20:37 
 
Here, HRsim is the simulated HR response obtained using the estimated models and the 
input signal, and HR is the measured HR from the validation data. ¯
HR is the mean value 
of HR . i is the discrete time index and N is the number of discrete samples considered 
(as described above, N = 360 ). Both of the above outcomes were calculated using the 
“compare” function from the Matlab System Identification Toolbox.
Statistics
Statistical analysis was performed to test the hypothesis that the goodness-of-fit out-
comes of second-order models are better (higher fit and lower RMSE) compared to first-
order models. Prior to analysis, normality of differences between the goodness-of-fit 
outcomes was formally assessed using the Matlab “lilliefors” function (this implements 
a Kolmogorov–Smirnov test with correction according to the Lillifors method). As it 
transpired that all differences were not significantly different from a normal distribution, 
paired one-sided t tests were employed for hypothesis testing. Hypothesis testing used 
a significance threshold of 5% ( α = 0.05 ). The Matlab Statistics and Machine Learning 
Toolbox (The Mathworks, Inc., USA) was employed.
Abbreviations
bpm: Beats/min; CI: Confidence interval; fit/NRMSE: Normalised root-mean-square error; HR: Heart rate; HRV: Heart rate 
variability; MD: Mean difference; PRBS: Pseudo-random binary sequence; RMSE: Root-mean-square error; SD: Standard 
deviation; k: Steady-state gain; τ: Time constant.
Acknowledgements
Lars Brockmann (Institute for Rehabilitation and Performance Technology, Bern University of Applied Sciences) critically 
reviewed the manuscript for important intellectual content.
Authors’ contributions
KH and HW designed the study. HW did the data acquisition. HW and KH contributed to the analysis and interpretation 
of the data. HW wrote the manuscript; KH revised it critically for important intellectual content. Both authors read and 
approved the final manuscript.
Funding
This study was funded by the Swiss National Science Foundation as part of the project “Heart Rate Variability, Dynamics 
and Control During Exercise” (Ref. 320030-185351).
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable 
request.
Declarations
 Ethics approval and consent to participate
This research was performed in accordance with the Declaration of Helsinki. The study was reviewed and approved by 
the Ethics Committee of the Swiss Canton of Bern (Ref. 2019-02184). All participants provided written, informed consent.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Received: 13 January 2021   Accepted: 28 March 2021
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Identification of heart rate dynamics during treadmill exercise: comparison of first- and second-order models.
Wang, Hanjie,Hunt, Kenneth J