Comparison of linear and nonlinear feedback control of heart rate for treadmill running

Hunt, K.J.; Maurer, Roman R.

doi:10.1080/21642583.2016.1179139

Cited by 10 publications

(23 citation statements)

References 10 publications

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“…Since, nominally, higher control signal activity would be expected to drive down RMSE, this result may indicate that, at this very high level of tracking precision, an empirical lower bound on the achievable RMSE is being approached. This observation is supported by a previous study where RMSE was similar to that reported here (mean RMSE of 2.29 bpm over 32 tests), and where even higher levels of average control signal power did not lead to any observable reduction in RMSE …”

Section: Discussioncontrasting

confidence: 87%

“…In the sequel, the plant transfer function P d = B / A is taken to be a first‐order LTI system; this assumption follows previous observations that such a model gives a good representation of HR dynamics during moderate‐to‐vigorous treadmill exercise, and, further, that feedback control design based on this model structure can be highly accurate and robust. () Thus, with steady‐state gain k and time constant τ, the transfer function for HR dynamics is represented in continuous and discrete forms as

u \to y : P_{c} false(s false) = \frac{k}{normalτ s + 1} \overset{⟷}{true} P_{d} false(z^{- 1} false) = \frac{B false(z^{- 1} false)}{A false(z^{- 1} false)} = \frac{b_{0} z^{- 1}}{1 + a_{1} z^{- 1}},

where the double arrow denotes transformation between the continuous and discrete domains using sample period T s . The discrete model parameters, expressed in k , τ, and T s , are

a_{1} = - e^{\frac{- T_{s}}{τ}}, b_{0} = k (() 1 - e^{\frac{- T_{s}}{normalτ}}) .

…”

Section: Control Design Methodsmentioning

confidence: 99%

“…Recent studies on feedback control of HR have used quantitative measures of closed‐loop control performance() and these measures are adopted in the present work. Tracking accuracy is assessed using the RMSE on an interval [ t 0 , t 1 ]

RMSE = \sqrt{\frac{1}{N} \sum_{t = t_{0}}^{t_{1}} {((), y_{nom} false(t false) - y false(t false))}^{2}},

where N = t 1 − t 0 +1.…”

Section: Control Design Methodsmentioning

confidence: 99%

“…For comparative purposes, a feedback compensator was also designed using a pole assignment (PA) method where the closed‐loop poles were specified using the closed‐loop rise time (this approach was also taken in previous works()). The feedback compensator transfer function was taken to be identical to that used in the optimal design, ie, C ( q −1 )=( g 0 + g 1 q −1 )/(1− q −1 ), Equation , but instead of using the optimal characteristic polynomial D c obtained by spectral factorisation , D c in Equation was calculated using a desired 10%‐90% closed‐loop rise time t r and critical relative damping ζ=1 to give (see the work of Hunt and Hunt)

d_{c 1} = - 2 e^{- 3.35 T_{s} false/ t_{r}}, d_{c 2} = e^{- 6.7 T_{s} false/ t_{r}} .

All other details of the algebraic problem solution are then identical to the optimal case, ie, only step 2 of the 4‐step algorithm summary in Section 2.4 differs for the PA design; the parameters d c 1 and d c 2 in Equation are obtained from Equation instead of from Equation .…”

Section: Control Design Methodsmentioning

confidence: 99%

“…In contrast, the primacy of broad‐spectrum physiological HR variability (HRV) as a disturbance source and, therefore, its role as the principal feedback design challenge have been asserted elsewhere: the observations therein were supported by empirical results, which used quantitative performance outcome measures, control designs based on simple and approximate linear time‐invariant (LTI) plant models, and subject cohorts of sufficient size to facilitate comparative statistical analysis. Furthermore, a direct head‐to‐head comparative study of linear and nonlinear controllers did not reveal any significant differences in performance outcomes at moderate exercise intensity . These results have demonstrated that accurate, stable, and robust HR control performance can be obtained using simple models and LTI design methods, provided that the HRV disturbance spectrum is appropriately dealt with.…”

Section: Introductionmentioning

confidence: 99%

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Optimal control of heart rate during treadmill exercise

Hunt

Liu

2017

Optim Control Appl Methods

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SummaryFeedback control of heart rate (HR) for treadmills is important for exercise intensity specification and prescription. This work aimed to formulate HR control within a stochastic optimal control framework and to experimentally evaluate controller performance. A quadratic cost function is developed and linked to quantitative performance outcome measures, namely, root-mean-square tracking error and average control signal power. An optimal polynomial systems design is combined with frequency-domain analysis of feedback loop properties, with focus on the input sensitivity function, which governs the response to broad-spectrum HR variability disturbances. These, in turn, are modelled using stochastic process theory. A simple and approximate model of HR dynamics was used for the linear time-invariant controller design. Twelve healthy male subjects were recruited for comparative experimental evaluation of 3 controllers, giving 36 tests in total. The mean root-mean-square tracking error for the optimal controllers was around 2.2 beats per minute. Significant differences were observed in average control signal power for 2 different settings of the control weighting (mean power 22.6 vs 62.5 × 10 −4 m 2 /s 2 , high vs low setting, p = 2.3 × 10 −5 ). The stochastic optimal control framework provides a suitable method for attainment of high-precision, stable, and robust control of HR during treadmill exercise. The control weighting can be used to set the balance between regulation accuracy and control signal intensity, and it has a clear and systematic influence on the shape of the input sensitivity function. Future work should extend the problem formulation to encompass low-pass compensator and input sensitivity characteristics.

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Section: Discussioncontrasting

confidence: 87%

u \to y : P_{c} false(s false) = \frac{k}{normalτ s + 1} \overset{⟷}{true} P_{d} false(z^{- 1} false) = \frac{B false(z^{- 1} false)}{A false(z^{- 1} false)} = \frac{b_{0} z^{- 1}}{1 + a_{1} z^{- 1}},

where the double arrow denotes transformation between the continuous and discrete domains using sample period T s . The discrete model parameters, expressed in k , τ, and T s , are

a_{1} = - e^{\frac{- T_{s}}{τ}}, b_{0} = k (() 1 - e^{\frac{- T_{s}}{normalτ}}) .

…”

Section: Control Design Methodsmentioning

confidence: 99%

RMSE = \sqrt{\frac{1}{N} \sum_{t = t_{0}}^{t_{1}} {((), y_{nom} false(t false) - y false(t false))}^{2}},

where N = t 1 − t 0 +1.…”

Section: Control Design Methodsmentioning

confidence: 99%

d_{c 1} = - 2 e^{- 3.35 T_{s} false/ t_{r}}, d_{c 2} = e^{- 6.7 T_{s} false/ t_{r}} .

Section: Control Design Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal control of heart rate during treadmill exercise

Hunt

Liu

2017

Optim Control Appl Methods

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Robust control of heart rate for cycle ergometer exercise

Hunt

Hurni

2019

Med Biol Eng Comput

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The objective was to assess the performance and robustness of a novel strategy for automatic control of heart rate (HR) during cycle ergometry. Control design used a linear plant model and direct shaping of the closed-loop input-sensitivity function to achieve an appropriate response to disturbances attributable to broad-spectrum heart rate variability (HRV). The controller was evaluated in 73 feedback control experiments involving 49 participants. Performance and stability robustness were analysed using a separately identified family of 73 plant models. The controller gave highly accurate and stable HR tracking performance with mean root-mean-square tracking error between 2.5 beats/min (bpm) and 3.1 bpm, and with low average control signal power. Although plant parameters varied over a very wide range, key closed-loop transfer functions remained invariant to plant uncertainty in important frequency bands, while infinite gain margins and large phase margins (> 62∘) were preserved across the whole plant model family. Highly accurate, stable and robust HR control can be achieved using LTI controllers of remarkably simple structure. The results highlight that HR control design must focus on disturbances caused by HRV. The input-sensitivity approach evaluated in this work provides a transparent method of addressing this challenge. Graphical AbstractHeart rate control using a cycle ergometer

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