Robust control of heart rate for cycle ergometer exercise

Well-established guidelines exist for the definition of cardiopulmonary exercise testing protocols and for the prescription of training regimes. Specific testing and prescription guidelines are available for healthy individuals and for patients across a diversity of health conditions [1]; the most common exercise modalities are treadmill walking/running and cycle ergometry, while exercise intensity can be characterised using such variables as heart rate (HR), oxygen uptake or a subjective rating of perceived exertion, RPE [2].

HR is a quantitative variable that can be easily measured, and several approaches have been investigated for automatic control of HR during both treadmill and cycle ergometer exercise ([3‐5] and [6‐8], respectively). These feedback systems facilitate tracking of arbitrary HR profiles by automatically and continuously adjusting a manipulated variable, which for treadmills can be speed or slope, or both, and which for cycling is usually work rate.

Since cycle ergometers provide a stable-seated position, they are the preferred modality for exercise testing and prescription in cardiac rehabilitation; HR controllers have long been investigated in this context [6, 9, 10], but also, subsequently, for healthy persons [7, 8, 11].

A most elegant treatment of HR control for cycle ergometers was provided by Kawada et al. [6]. In that work, a single linear transfer function model of HR response to changes in work rate was obtained as an average from open-loop system identification experiments with 10 individual participants (8 men, 2 women). The model was then used in simulation to tune the two free parameters of a linear proportional-integral (PI) controller. The single time-invariant PI controller thus obtained was then tested in HR control experiments with 55 healthy participants (45 men, 10 women) and with 12 patients with cardiac disease (10 men, 2 women). In the healthy participants (n = 55), the mean root-mean-square tracking error (RMSE) for a constant HR target of 60 % of maximal HR (HRmax) was 2.5 beats/min (bpm); when the HR target was 75 % of HRmax, mean RMSE was 3.8 bpm. For the cardiac patients (n = 12) exercising at a constant target HR of 20 bpm above resting HR, mean RMSE was 3.0 bpm. This work, which reported HR control data from 122 individual HR control tests with 67 participants in the two experimental cohorts, thus provides strong empirical evidence that a single linear, time-invariant (LTI) controller of very simple structure can provide accurate and robust HR control.

A variety of nonlinear approaches to HR modelling and control for both treadmills and cycle ergometers have been proposed. Nonlinear models have been used to represent the different gains and time constants that exist for positive and negative step changes in speed [12]; asymmetry has also been observed and modelled during moderate-intensity treadmill running [13]. For the purpose of control design, a nonlinear state-space model, where the control signal appears in quadratic form, was employed and combined with linear-quadratic and H-infinity optimisation [14]; the same model structure was used, but with a nonlinearity-cancellation strategy, for HR control using a treadmill [4] or cycle ergometer [7]. A related approach using a Hammerstein model structure and a compensator with cancellation of the nonlinear model term was combined with model-predictive control [15]. Other approaches include linear H-infinity control with static nonlinearity compensation [16] and a nonlinear neural network approach [17]. A limitation common to most of these reports is that quantitative measures of controller performance (i.e. RMSE and control signal intensity) were not employed and that very small numbers of participants were included in experimental evaluations, thus making it difficult to objectively gauge their utility.

A recent study of HR control during cycle ergometry combined an LTI proportional-integral-derivative (PID) controller with an auditory biofeedback signal [8]. Despite the human-in-the-loop nature of this approach, quite accurate tracking was achieved with mean RMSE on the range 3.7 bpm to 5.0 bpm (various experiments with 24 healthy male participants).

In concordance with some of the above observations, a growing body of evidence has emerged from treadmill studies that points towards heart rate variability (HRV, [18]) as the principal challenge in the design of HR control systems, in contradistinction to parametric and/or structural sources of plant uncertainty. From a control-theoretical perspective, HRV presents as a broad-spectrum disturbance signal [19]; care must therefore be taken to ensure that the control signal is not unduly excited at frequencies that might disturb the exercising subject. In short, the said studies have demonstrated that simple approximate linear models, [20], can be employed to design LTI controllers that give highly accurate, stable and robust HR control performance, e.g. [5, 21] (20 to 30 participants, mean RMSE below 3 bpm).

To directly address the HRV disturbance, a HR control approach was developed that allows the frequency-domain characteristics of the closed-loop input-sensitivity function, which is the transfer function from the HRV disturbance to the control signal, to be appropriately shaped [5]; for treadmill exercise, HR control was accurate (mean RMSE of 3.0 bpm, n = 30) and the control signal was smooth and stable (average power of changes in the control signal was low). Using this design approach as a foundation, and based on the observation that HR dynamics are not significantly different between treadmills and cycle ergometers, [22], a common control strategy was derived and experimentally tested with these two exercise modalities; it was found to give accurate tracking (mean RMSE of 3.1 bpm vs. 2.8 bpm, cycle ergometer vs. treadmill; n = 25) and low control signal intensity [11].

The primary contribution of the present work is, for the first time, the application of the input-sensitivity-shaping approach for feedback control of HR to cycle ergometer exercise and the systematic analysis of its performance and robustness in a large experimental test series. A secondary contribution is a comparison with alternative linear and nonlinear controllers based upon data available in the literature. A single LTI feedback compensator was calculated using a linear first-order plant model. The aim of the work was to assess controller performance in several experimental scenarios using quantitative measures of tracking accuracy and control signal intensity (a total of 73 feedback control experiments involving 49 individual participants were performed), and to analyse performance and stability robustness properties of the compensator using a large family of empirically derived plant models (73 individual plant models were used for the robustness analysis).

The single linear compensator was found to give highly accurate HR tracking performance in both experimental cohorts and under the different experimental conditions: mean RMSE was on the range 2.5 bpm to 3.1 bpm. Due to the dynamic nature of square-wave reference tracking, mean RMSE for this condition (3.1 bpm) was higher than for the two constant target regulation series (2.5 bpm and 2.6 bpm).

The input-sensitivity-shaping control design approach gives a simple, closed-form analytical procedure that allows the closed-loop bandwidth to be set in consideration of the broad-spectrum HRV disturbance. In the present set of experiments, this gave a stable and smooth control signal whose changes had low average power P_∇u (mean of 2.4 W² for constant HR regulation and 10.3 W² for square-wave tracking).

Although the HR response was represented using the approximation of a simple linear model of the form y = P_o(s)u + d (Eq. 1, Fig. 1), where the term d represents the lumped effects of the HRV disturbance at a nominal operating point, it should be emphasised that human heart rate variability arises from complex interactions between the sympathetic and parasympathetic divisions of the autonomic nervous system [18]. These divisions are continuously engaged in regulation of cardiac output by adjustment of stroke volume and heart rate, thus leading to the observed variations in the time between individual beats. This HRV depends on many factors that are not dependent upon the control signal u (target work rate) including hydration level, ambient temperature and health status. Thus, it is not the purpose of the feedback control loop and, in particular, the control signal u, to directly influence the level of HRV. Rather, HRV is treated as a lumped, unmeasurable output disturbance d; the task of the controller is then, in the face of the unknown HRV disturbance d, to achieve a sufficient level of accuracy in the tracking of the target HR profile while maintaining an acceptable intensity of the control signal u. This amounts to the classical trade-off between tracking accuracy and control signal intensity: choice of a higher closed-loop bandwidth will tend to give a more dynamic controller resulting in lower tracking error but higher control signal intensity, and vice versa.

In comparison with the study of Kawada et al. [6], which employed a PI controller and evaluated only constant HR regulation, the RMSE values for regulation in the present work are slightly lower (\(\sim \)2.5 bpm here vs. 2.5 bpm to 3.8 bpm in [6]); but this comparison should be interpreted with caution since RMSE will also have been affected by the differing experimental conditions and the respective methods for controller-parameter tuning.

A direct comparison of the intensity of control signal activity between the two studies is not possible: here, this was evaluated using the average power of sample-to-sample changes in the control signal P_∇u; but in [6], no quantitative assessment of control signal intensity was performed. It can be conjectured, however, that the control signal intensity when using a PI controller (as in [6]) would be higher. This is because, in the present work, the compensator, Eq. 2, was constrained at the outset to be strictly proper (low pass). This in turn gives a strictly proper, low-pass input-sensitivity function U_o because, from Eq. 6, U_o = C_fb/(1 + C_fbP_o). Thus, when \(\lim _{\omega \rightarrow \infty } |C_{\text {fb}}| = 0\), it follows that \(\lim _{\omega \rightarrow \infty } |U_{o}| = \lim _{\omega \rightarrow \infty } |C_{\text {fb}}| = 0\). Thus, the control signal will not respond to disturbances at frequencies above the specified input-sensitivity bandwidth p (set here as frequency f = 0.01 Hz; see |U_o| in Fig. 2).

In contrast, for a PI controller C_fb(s) = k_p + k_i/s with proportional gain k_p and integrator gain k_i (this is the exact structure employed in [6]), the magnitude of C_fb tends to the value k_p at high frequency. Consequently, |U_o| also tends to the value k_p because, employing the condition that P_o is strictly proper (low pass), \(\lim _{\omega \rightarrow \infty } |U_{o}| = \lim _{\omega \rightarrow \infty } |C_{\text {fb}}| = k_{p}\). This shows that, for a PI controller, the control signal will react to disturbance and noise inputs across the whole frequency spectrum.

Finally, in comparison with the study in [6], it is noted that the nominal plant gain used here for controller calculation (k = 0.39, mean from 25 participants) was very close to the value estimated in [6] (k = 0.42, mean from 10 participants). The nominal time constant used here cannot be compared because a non-parametric model was estimated in [6].

The performance of the controller proposed and tested in the present work can be compared with nonlinear strategies that have previously been applied to HR control. One nonlinear approach has been applied to HR control during both treadmill [4] and cycle ergometer [7] exercise. This nonlinear method is based upon a plant model where the control signal u appears in quadratic form, and where the controller cancels this term using the inverse nonlinearity, viz. the square-root function. This approach has the important theoretical property that global convergence of regulation errors is guaranteed for the class of nonlinear models considered. However, the experimental evidence provided in [4] and [7] is weak because no quantitative measures of controller performance were employed, and because short-duration tests were performed with only two (treadmill, [4]) or three (cycle, [7]) participants. Furthermore, a later independent study systematically compared this nonlinear approach to a linear PI controller using quantitative outcome measures and a cohort of 16 healthy male participants during treadmill exercise [26]. Using formal statistical analysis methods, this study found no significant difference between the linear and nonlinear controllers in HR tracking accuracy (for both controllers, RMSE was approximately 2.3 bpm) and in average control signal power. Moreover, the nonlinear controller was found to be highly sensitive at low control signal levels, which was attributed to the fact that the square-root function, which is included in the compensator, has a gain that tends to infinity as the control signal tends to zero.

The HR tracking accuracy reported in [26] for both the linear and nonlinear controllers, i.e. RMSE of approximately 2.3 bpm, is slightly lower than the range of 2.5 bpm to 3.1 bpm observed in the present work. This can likely be attributed to the non-strictly-proper nature of the linear/nonlinear controllers implemented in [26], in contrast to the strictly proper constraint applied here (Eq. 2): when the controller is not strictly proper, its gain does not roll off with frequency, thus making it more dynamic across the whole frequency range, which tends to drive down the RMSE; the price to be paid for this improved HR tracking accuracy, however, is an increased sensitivity to higher frequency HRV disturbances and consequent higher average control signal power.

Notwithstanding this critical analysis of nonlinear control strategies, further work is recommended to investigate appropriate nonlinear plant model and controller structures, while experimental evaluations are recommended that comprise quantitative performance-outcome measures and participant cohorts with sufficient sample size to allow formal statistical comparison with other linear/nonlinear approaches.

Within the present work, the quantification of parametric plant uncertainty showed that steady-state gains and time constants vary over a very wide range; overall, k was on the range 0.180 bpm/W to 0.796 bpm/W and τ ranged from 26.5 to 133.2 s (Section 3.2, Fig. 6). Despite this high level of plant dispersion, the controller was accurate and stable in all 73 experiments involving a total of 49 individual participants.

This empirically observed, high degree of controller robustness is underscored by the performance and stability robustness analysis (Section 3.3):

The single-linear, time-invariant controller was found to give accurate and stable performance with low values of the quantitative outcomes root-mean-square tracking error RMSE and average control signal power P_∇u. The empirical evidence of controller robustness was corroborated by numerical analysis of key closed-loop transfer functions and stability margins across the available plant model family.

These results, taken together with data reported in Kawada et al. [6], demonstrate that highly accurate, stable and robust heart rate control can be achieved using LTI controllers of remarkably simple structure. Furthermore, the results highlight that design methods for HR control must give adequate attention to plant disturbances caused by physiological heart rate variability. The input-sensitivity approach evaluated in this work provides a direct and transparent method of addressing this challenge.