2. Experiments

the possibility of both type 2 Diabetes and CVD [1]. The development of automated exercise assisted equipment can greatly enhance the efficiency of exercises and reduce the requirement of supervision. An easy-to-measure indication of exercise strength is heart rate (HR). One of the most efficient ways for the manipulation of exercise intensity for either the training of athletes or rehabilitation patients is to simulate the HR in order to follow a pre-designed HR profile. As a result, an automated control system can offer numerous benefits for different groups of users. For instance, it will provide assistance for patients with cardiac diseases who might be prescribed treadmill exercise rehabilitation. It also can be used for training the athletes and safely regulating the exercise intensity within a suitable profile in order to achieve the predefined HR ranges. It has been well documented that by determining one's maximum and minimum values of HR responses the exercise profile can be individually designed [2]. Specifically, rehabilitation patients will be guided to perform the exercise in terms of 50–60% of the maximum HR, 60–70% of the maximum HR zone then suitable for subjects who target weight control, and the range between 70 and 90% is preferred for the cardio-endurance exercise. In the study, a two-input single-output (2ISO) control system is proposed, which can specially further improve the efficiency of treadmill exercises for different subject groups as

Recently, most studies [3–6] only consider using one manipulate variable (treadmill speed or gradient) to regulate HR responses. In [7, 8], for exercise testing and rehabilitation of subjects with impaired exercise tolerance, ramp type protocols were proposed by simultaneously manipulating both speed and gradient (without feedback), which could produce a low initial metabolic rate that then increases the work rate linearly to reach the subject's limit of tolerance in approximately 10 min. In [9], a multi-loop proportion-integral (PI) controller based HR tracking system has been presented, which independently tuned both treadmill speed and gradient in closed loop, and achieved good performance. However, in paper [9], it is assumed that the HR response to treadmill exercise is in linear range, and only linear modeling and control approaches have been presented. The experimental evidences of the advantages of using both speed and gradient

This study introduces the 2ISO HR process which employs two actuators, treadmill speed and gradient, to regulate the HR response. Such process control has the following merits. First of all, it can increase the non-saturation range. For example, practical systems always have physical limitations and therefore have limited non-saturation range. If simultaneously executing multiple actuators, the output range can be extended. On the other hand, it can increase the maximum gain of the actuator so that the fast tracking or regulation of manipulated variable can be achieved [9]. Also, redundancy of actuators can facilitate fault accommodation for the implementation of faulttolerant control strategies. As the stability of the closed loop system can be practically achieved by adding suitable constraints in control inputs and their derivatives, the achievable performance

For the regulation of HR, multi-loop PI controllers were developed [9] in order to achieve zero steady-state tracking error. For multi-loop PI control of square process, Skogestad and Morari [10] introduced the concept of decentralized integral controllability (DIC). DIC analysis [11–13] determines the stability, integral control ability, and faulty-tolerant control of the multi-loop system. This study extends the DIC analysis for nonlinear and non-square processes especially

well as diversify more reliable and safe treadmill exercise protocols.

276 Adaptive Robust Control Systems

to regulate HR are therefore only valid in a certain linear response range.

(especially in steady state) is more important for most industrial control processes.
