A Barrier Function-Based Integral Sliding Mode Control of Heart Rate During Treadmill Exercise Taghreed MohammadRidha * , Shibly Ahmed Al-Samarraie Control and Systems Engineering Department, University of Technology – Iraq, Baghdad 10066, Iraq Corresponding Author Email: taghreed.m.ridha@uotechnology.edu.iq https://doi.org/10.18280/mmep.100120 ABSTRACT Received: 31 July 2022 Accepted: 20 January 2023 The objective of this work is to design an Integral Sliding Mode Controller based on barrier function (ISMCbf) for a human Heart Rate (HR) during a treadmill exercise. ISMCbf commands the speed of the treadmill such that the individual HR follows a time-varying profile. This profile is pre specified as part of rehabilitation exercises for patients with cardiovascular diseases. ISMCbf is chosen due to its well-known robustness properties as well as to its simple design procedure as compared to classic SMC and ISMC. It does not require the upper bounds of the uncertainties and perturbations in its design. Moreover, it does not have discontinuous function, hence it is a chattering-free controller. ISMCbf designed in this work for the first time for this system and its performance is compared to Quasi SMC (QSMC) and Super Twisting SMC (STSMC) from previous studies. The simulated exercises were conducted on a nonlinear model describing HR response to the walking speed of a treadmill. For ISMCbf, the model parameters and their upper bound of uncertainties are considered unknown. During two different exercise scenarios, the three controllers guided HR to follow the time-varying reference profile. However, ISMCbf showed higher quantitative performance by recording less Integral Squared Error (ISE) and Integral Time Absolute Error (ITAE) indices as compared to the other controllers. Keywords: heart rate control, cardiovascular system, exercise, integral sliding mode control, barrier function, nonlinear systems, reference tracking 1. INTRODUCTION Cardiovascular diseases (CVDs) are the first main cause of death representing 32% of all global deaths in 2019 according to the World Health Organization (WHO) [1]. The study underlined that cardiovascular events could be prevented by addressing behavioral risk factors like physical inactivity. Many studies showed that Cardiac Rehabilitation (CR) represents an important secondary prevention model that can contribute favourably to the reduction of mortality and disability [2]. A CR program includes many elements like educational sessions on different topics such as risk factors. An important element of CR programs is the physical exercise that help improving physical capacity and fitness. During dynamic exercise, the metabolic demand increases, which in turn increases the Heart Rate (HR) and stroke volume. Different mathematical models were developed in the literature [3-5] to predict HR response during exercises. Modelling HR kinetics helps understanding the system behavior and provides some useful means for the prevention of cardiac failure [3]. Ludwig et al. [6] presented an overview of measurement, prediction, and control of individual HR responses. Available sensor technologies measuring HR are analyzed and the feasibility for wearables is analyzed as well. Wang and Hunt [7] investigated the linear second-order models of HR response to the available linear first-order models. They performed experimental tests on eleven participants each performed two open-loop identification tests while running at moderate-to-vigorous intensity on a treadmill. The authors concluded that second-order models give significantly better fitting performance than first-order models. A similar comparison of first and second-order models of HR of cycle-ergometer was performed in the study of Spörri et al. [8]. However, physiological systems like HR response to exercise are known to exhibit nonlinear behaviors. One of the nonlinear models that can capture HR dynamic behavior using a reduced number of parameters is the one of [3] that is frequently used in the literature for control design. This model that describes HR response to the speed of a treadmill is also applicable to represent HR response to the speed of cycle-ergometer [9]. Moreover, it was also shown in the study of Liu et al. [10] that it can also well describe HR response in the outdoor running exercise by re identifying its parameters. The main purpose of HR models is to precisely control HR response to follow a pre-specified profile prescribed for healthy subjects like athletes or for patients of cardiovascular problems. A nonlinear controller was designed in the study of Scalzi et al. [11] for the nonlinear HR model given in the study of Cheng et al. [3] during a treadmill exercise. The authors considered it as a generalization of the classical proportional- integral controller that was designed in the literature for linear HR models. This nonlinear controller is also used in the study of Paradiso et al. [9] to control HR during cycle-ergometer exercises at constant cycling speed. Adaptive  ∞ controller is deigned in the study of Baig et al. [12] to control HR response during aerobic activities of unknown type. The  ∞ controller is designed based on Linear Time-Varying (LTV) model. The controller is adaptively re-designed after three sampling intervals based on the estimation of the LTV model parameters Mathematical Modelling of Engineering Problems Vol. 10, No. 1, February, 2023, pp. 179-188 Journal homepage: http://iieta.org/journals/mmep 179