4.1. MATLAB®/Simulink® simulation verification

In order to identify the coefficients of PI controllers and verify the proposed MIC conditions for 2ISO HR regulation, the SISO and 2ISO control loops are designed and implemented through MATLAB®/Simulink® simulations. The schematic diagram for simulations is illustrated in Figure 3, where the constructed HR model is regulated by either treadmill speed/gradient (the SISO loops) or both of those actuators (the 2ISO loop).

Figure 5 shows the simulation result for which only speed has been manipulated. This indicates another advantage for the using of two control inputs and fault tolerance. It can be proved based on Theorem 1 and also by simulation that if the open loop gain for healthy actuator is significantly

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Figure 4. Steady-state HR response to speed (gradient is zero).

Figure 5. HR regulation results when only speed is manipulated.

A two-input single-output Hammerstein model is used for modeling of HR-based open-loop characteristics. The linear dynamic part of the model is noted as G sð Þ¼ <sup>K</sup> Tsþ1, where <sup>K</sup> and <sup>T</sup> are the steady state gain and the time constant, respectively [24, 25]. The static nonlinear part is modeled by a cubic polynomial function. The input–output relationship between treadmill speed and heart rate is shown in Figure 4 when the gradient is zero.

Based on the previous experimental survey in [9], the multi-loop integral controllers for the single loops of speed-HR and gradient-HR are developed respectively. The experimental results confirm the static nonlinearity of HR responses to treadmill walking/running exercises which has been shown in Figure 8. As a result, the passive sector condition can be found in the zones of 0–6.8 km/h (indicating the walking condition), and 7.2–7.8 km/h (indicating the running condition). This means when treadmill speed is within the walking zone, for instance, the subject merely needs to undergo a walking motion to follow the treadmill protocol. Moreover, if the treadmill speed reaches the running zone, the running motion has to be taken by all of subjects. However, it also could be observed from Figure 4 that the passivity sector condition in the transition zone from 6.8 to 7.2 km/h is not valid. This means that if the reference HR variation is selected as 49.5374 (HR is 124.5374 bpm) which is located in the transition zone, the regulation of HR by only adjusting speed would be problematic even under small perturbations in the measurement. This is because the subject will frequently switch his/her motion actions between walking and running in order to stabilize his/her desired HR level.

Figure 3. The schematic diagram of 2ISO PI control structure for MATLAB®/Simulink®-based HR tracking simulation. a. HR tracking of 2ISO control loop and b. SISO control loop of speed-HR or gradient-HR.

Figure 5 shows the simulation result for which only speed has been manipulated. This indicates another advantage for the using of two control inputs and fault tolerance. It can be proved based on Theorem 1 and also by simulation that if the open loop gain for healthy actuator is significantly

Figure 4. Steady-state HR response to speed (gradient is zero).

Figure 3, where the constructed HR model is regulated by either treadmill speed/gradient (the

A two-input single-output Hammerstein model is used for modeling of HR-based open-loop

the steady state gain and the time constant, respectively [24, 25]. The static nonlinear part is modeled by a cubic polynomial function. The input–output relationship between treadmill

Based on the previous experimental survey in [9], the multi-loop integral controllers for the single loops of speed-HR and gradient-HR are developed respectively. The experimental results confirm the static nonlinearity of HR responses to treadmill walking/running exercises which has been shown in Figure 8. As a result, the passive sector condition can be found in the zones of 0–6.8 km/h (indicating the walking condition), and 7.2–7.8 km/h (indicating the running condition). This means when treadmill speed is within the walking zone, for instance, the subject merely needs to undergo a walking motion to follow the treadmill protocol. Moreover, if the treadmill speed reaches the running zone, the running motion has to be taken by all of subjects. However, it also could be observed from Figure 4 that the passivity sector condition in the transition zone from 6.8 to 7.2 km/h is not valid. This means that if the reference HR variation is selected as 49.5374 (HR is 124.5374 bpm) which is located in the transition zone, the regulation of HR by only adjusting speed would be problematic even under small perturbations in the measurement. This is because the subject will frequently switch his/her motion actions between walking and running in order

Figure 3. The schematic diagram of 2ISO PI control structure for MATLAB®/Simulink®-based HR tracking simulation. a.

HR tracking of 2ISO control loop and b. SISO control loop of speed-HR or gradient-HR.

Tsþ1, where <sup>K</sup> and <sup>T</sup> are

characteristics. The linear dynamic part of the model is noted as G sð Þ¼ <sup>K</sup>

speed and heart rate is shown in Figure 4 when the gradient is zero.

SISO loops) or both of those actuators (the 2ISO loop).

282 Adaptive Robust Control Systems

to stabilize his/her desired HR level.

Figure 5. HR regulation results when only speed is manipulated.

bigger than that for faulty actuator, then the offset free tracking is still possible. Although the regulated HR is quite close to reference HR, treadmill speed swings between 6.5 and 7.5 km/h.

In order to identify the comfort HR zones of each individual for treadmill experiments, moderate exercise intensity was adopted which offers safe treadmill intensity (speed ranged from 3 to 7 km/h, and gradient from 2 and 15%). This guarantees the HR operating zone for the subject to be located between 90 and 170 bpm. In addition, the HR level at 135 bpm was

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Considering the time-delay situation between control inputs (such as treadmill speed and gradient) and system output (instantaneous HR) during the practical experiment validation, a stack buffer with a 5-s timer was used to obtain the instantaneous HR values. The latest HR value was stored from the top of the stack buffer. The control input commands will be sent to the treadmill every 5 s based on the up-to-date stack buffer stores. For de-noising the raw instantaneous HR values measured by the Alive Technologies HR sensor, an improved exponential, weighted, moving, average filter together with a simple outlier detection algorithm

In the first SISO open loop test, the speed of the treadmill is employed to be the system input, and HR is considered as the system output. Moreover, the gradient of treadmill is fixed at 2% and the input is set to be adjustable between 2 and 5 km/h. The PI parameters of speed-HR loop with a stable operation range 0.01–1.05 for kp, 0.001–0.075 for ki were determined. Based on the open loop experimental results, for the SISO controller with speed as the input, PI controllers, kpand ki, are set to 0.7 and 0.05, respectively, in order to achieve acceptable control characteristics. The HR response is shown in Figure 7, in which the HR model parameters K and T values were determined to be equal to 15.91 and 37.44, respectively. The results show

In addition, based on the gradient-HR test, a fixed speed of 4 km/h is maintained while the gradient is changed from 2 to 12%. The stable operation range (kp:0.01–0.4665, ki:0.001) for gradient-HR loop is observed from the experimental results. As a result, for the SISO controller with gradient as the input, kp and ki values are adjusted to 0.311 and 0.001 respectively. The coefficients for the speed-input controller slightly varied from the theoretical values, while in the case of the gradient-input controller, the theoretical values were acceptable. The response

Figure 7. Experimental data for HR step responses to either treadmill speed or gradient (open-loop test with speed and HR).

selected to be a reference input for the setpoint value of both SISO and 2ISO tests.

was adopted for the estimation of the HR stored in the stack buffer [4].

the HR increasing quickly and a small reaction delay.

Due to the discomfort evoked by the speed swinging, we can simultaneously manipulate both treadmill speed and gradient and adjust the gradient regulation loop to avoid the swinging. Simulation results in Figure 6 prove the effectiveness of the simultaneous manipulation strategy.

The simulation results from PI control loops of SISO (speed-HR and gradient-HR) and 2ISO indicate that those three structures can well achieve the HR tracking performance merely by tuning the PI parameters of multiple integral controllers with quite small values; the simultaneous manipulation strategy for the regulation of HR responses to treadmill exercises is effective if MIC conditions are satisfied; the 2ISO closed loop can provide the ability of fault tolerance which means that especially in the case of one of the actuators (either speed or gradient) being out of service, the offset free tracking is still achievable.

#### 4.2. Experimental validation

In order to evaluate real-time HR tracking performance obtained from the proposed 2ISO control loop, the experimental verification is also used in this study, and a comparative study is made by comparing the tracking performance of the 2ISO loop with that of both SISO loops of speed-HR and gradient-HR.

Figure 6. HR regulation results when both speed and gradient are manipulated.

In order to identify the comfort HR zones of each individual for treadmill experiments, moderate exercise intensity was adopted which offers safe treadmill intensity (speed ranged from 3 to 7 km/h, and gradient from 2 and 15%). This guarantees the HR operating zone for the subject to be located between 90 and 170 bpm. In addition, the HR level at 135 bpm was selected to be a reference input for the setpoint value of both SISO and 2ISO tests.

bigger than that for faulty actuator, then the offset free tracking is still possible. Although the regulated HR is quite close to reference HR, treadmill speed swings between 6.5 and 7.5 km/h.

Due to the discomfort evoked by the speed swinging, we can simultaneously manipulate both treadmill speed and gradient and adjust the gradient regulation loop to avoid the swinging. Simulation results in Figure 6 prove the effectiveness of the simultaneous manipulation strategy. The simulation results from PI control loops of SISO (speed-HR and gradient-HR) and 2ISO indicate that those three structures can well achieve the HR tracking performance merely by tuning the PI parameters of multiple integral controllers with quite small values; the simultaneous manipulation strategy for the regulation of HR responses to treadmill exercises is effective if MIC conditions are satisfied; the 2ISO closed loop can provide the ability of fault tolerance which means that especially in the case of one of the actuators (either speed or

In order to evaluate real-time HR tracking performance obtained from the proposed 2ISO control loop, the experimental verification is also used in this study, and a comparative study is made by comparing the tracking performance of the 2ISO loop with that of both SISO loops

gradient) being out of service, the offset free tracking is still achievable.

Figure 6. HR regulation results when both speed and gradient are manipulated.

4.2. Experimental validation

284 Adaptive Robust Control Systems

of speed-HR and gradient-HR.

Considering the time-delay situation between control inputs (such as treadmill speed and gradient) and system output (instantaneous HR) during the practical experiment validation, a stack buffer with a 5-s timer was used to obtain the instantaneous HR values. The latest HR value was stored from the top of the stack buffer. The control input commands will be sent to the treadmill every 5 s based on the up-to-date stack buffer stores. For de-noising the raw instantaneous HR values measured by the Alive Technologies HR sensor, an improved exponential, weighted, moving, average filter together with a simple outlier detection algorithm was adopted for the estimation of the HR stored in the stack buffer [4].

In the first SISO open loop test, the speed of the treadmill is employed to be the system input, and HR is considered as the system output. Moreover, the gradient of treadmill is fixed at 2% and the input is set to be adjustable between 2 and 5 km/h. The PI parameters of speed-HR loop with a stable operation range 0.01–1.05 for kp, 0.001–0.075 for ki were determined. Based on the open loop experimental results, for the SISO controller with speed as the input, PI controllers, kpand ki, are set to 0.7 and 0.05, respectively, in order to achieve acceptable control characteristics. The HR response is shown in Figure 7, in which the HR model parameters K and T values were determined to be equal to 15.91 and 37.44, respectively. The results show the HR increasing quickly and a small reaction delay.

In addition, based on the gradient-HR test, a fixed speed of 4 km/h is maintained while the gradient is changed from 2 to 12%. The stable operation range (kp:0.01–0.4665, ki:0.001) for gradient-HR loop is observed from the experimental results. As a result, for the SISO controller with gradient as the input, kp and ki values are adjusted to 0.311 and 0.001 respectively. The coefficients for the speed-input controller slightly varied from the theoretical values, while in the case of the gradient-input controller, the theoretical values were acceptable. The response

Figure 7. Experimental data for HR step responses to either treadmill speed or gradient (open-loop test with speed and HR).

graph is shown in Figure 8, where K and T values obtained are 2 and 26, respectively. A reaction delay can be observed due to the mechanical time that the treadmill needs to reach

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Using experimental verification results with the determined PI coefficients, two closed loop SISO controllers and one 2ISO controller were implemented. Figures 9 and 10 provide a clear view of both SISO control with speed-input and gradient-input compared with 2ISO. For the SISO speed-input controller, it demonstrates that the system outputs have a slight overshoot followed by a fast rise to track the setpoint. However, for the SISO gradient-input controller, it shows a more stable performance compared to that of the SISO speed-input controller. The comparative results shown in Figure 10 demonstrate that 2ISO control loop can achieve the fastest HR tracking performance and stay close to the reference HR during steady state, while

The main advantage of using 2ISO control in treadmill exercises is to improve the HR tracking performance. It can be concluded that the 2ISO controller outperforms both SISO controllers and can provide shorter rise time, best steady-state stability, as well as the lowest steady-state error. In addition, the 2ISO automatic treadmill exercise system also offers more comfortable

This study investigates the benefits of using two controller inputs, the speed and gradient, for the regulation of HR during treadmill exercises. The main goal of HR control in treadmill exercises is to ensure a reliable, fast, and offset-free tracking, as well as to offer the faulty tolerance ability in the case of one of actuators (either treadmill speed or gradient) being out of service. For this purpose, we extended the concept of nonlinear decentralized integral controllability (DIC) to nonlinear 2ISO processes and presented a sufficient condition which only needs checking the steady state input-output relationship of controlled processes. Based on the proposed condition, we investigate the new defined multi-loop integral controllability (MIC) for walking, running, and walking-running zones. The experimental validation presents that by simultaneously using two control inputs, the automated system can achieve the fastest HR-tracking performance and stay close to the reference HR during steady state, while comparing with two SISO structures and offer the fault-tolerant ability if the gains of the two multiloop integral controllers are well tuned. It has a vital implication for the applications of exercise

This work is supported by the Fundamental Research Funds for the Central Universities, China (grant#ZYGX2015J118), the Sichuan Provincial Program on Key R & D projects (grant#2017GZ0162), National Natural Science Foundation of China (grant No. 51675087, 61522105), and the China Postdoctoral Science Foundation funded project (2017M612950).

rehabilitation and fitness in relation to the automated control system.

the desired gradient.

comparing with two SISO structures.

and safer exercise conditions for users.

5. Conclusion

Acknowledgements

Figure 8. Experimental data for HR step responses to either treadmill speed or gradient (open-loop test with gradient and HR).

Figure 9. HR tracking performance comparison of SISO (speed) test with 2ISO.

Figure 10. HR tracking performance comparison of SISO (gradient) test with 2ISO.

graph is shown in Figure 8, where K and T values obtained are 2 and 26, respectively. A reaction delay can be observed due to the mechanical time that the treadmill needs to reach the desired gradient.

Using experimental verification results with the determined PI coefficients, two closed loop SISO controllers and one 2ISO controller were implemented. Figures 9 and 10 provide a clear view of both SISO control with speed-input and gradient-input compared with 2ISO. For the SISO speed-input controller, it demonstrates that the system outputs have a slight overshoot followed by a fast rise to track the setpoint. However, for the SISO gradient-input controller, it shows a more stable performance compared to that of the SISO speed-input controller. The comparative results shown in Figure 10 demonstrate that 2ISO control loop can achieve the fastest HR tracking performance and stay close to the reference HR during steady state, while comparing with two SISO structures.

The main advantage of using 2ISO control in treadmill exercises is to improve the HR tracking performance. It can be concluded that the 2ISO controller outperforms both SISO controllers and can provide shorter rise time, best steady-state stability, as well as the lowest steady-state error. In addition, the 2ISO automatic treadmill exercise system also offers more comfortable and safer exercise conditions for users.
