Tracking Control and Robustness Analysis for a Nonlinear Model of Human Heart Rate During Exercise

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1 Tracking Control and Robustness Analysis for a Nonlinear Model of Human Heart Rate During Exercise Frédéric Mazenc a Michael Malisoff b,1 Marcio de Querioz c, a Projet INRIA DISCO, CNRS-Supélec, 3 rue Joliot Curie, 9119 Gif-sur-Yvette, France b Department of Mathematics, Louisiana State University, Baton Rouge, LA , USA c Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA , USA Abstract The control of human heart rate during exercise is an important problem that has implications for the development of protocols for athletics, assessing physical fitness, weight management, and the prevention of heart failure. Here we provide a new stabilization technique for a recently-proposed nonlinear model for human heart rate response that describes the central and peripheral local responses during and after treadmill exercise. The control input is the treadmill speed, and the control objective is to make the heart rate track a prescribed reference trajectory. We use a strict Lyapunov function analysis to design new state and output feedback tracking controllers that render the error dynamics globally exponentially stable. This allows us to prove input-to-state stability properties for our feedback stabilized systems under actuator errors. This robustness property quantifies the effects of variations of the treadmill speed from the controller values. We illustrate our feedback design through simulations. Key words: Cardiovascular system, feedback control, nonlinear systems 1 Introduction Human heart rate HR is a widely used indicator of exercise intensity, because it is easy and inexpensive to monitor Achten & Jeukendrup, 003. For example, HR can potentially play a role in detecting and preventing overtraining. Thus, as noted in Cheng, Savkin, Su, Celler, & Wang, 008, controlling HR in real time during exercise can be useful for developing training or rehabilitation exercise protocols for patients with cardiovascular problems, athletes, people with obesity problems, and patients recovering from cardiothoracic surgery. This has motivated considerable research on modeling HR during exercise Brodan, Hájek, & Kuhn, 1971; Coyle & Alonso, 001; Hájek, Potucek, & Brodan, 1980; Johnson & Rowell, 197; Rowell, 1993; Su, Wang, Celler, Savkin, & Guo, 007. The works Brodan et al., 1971 and Hájek et al., 1980 use feedforward and feedback components, and Su et al., 007 uses a Hammerstein system consisting of a static nonlinearity cascaded at the input of a linear system. See Cooper, Fletcher-Shaw, & Robertson, 1998 and Kawada, Sunagawa, Takaki, et al., 1999 for linear HR models using classical PID control and model reference control. However, these earlier works are mostly derived for explaining short duration exercise. HR is known to continue to increase during prolonged exercise Coyle Corresponding author: Tel.: Malisoff. addresses: Frederic.Mazenc@lss.supelec.fr Frédéric Mazenc, malisoff@lsu.edu Michael Malisoff, dequeiroz@me.lsu.edu Marcio de Querioz. 1 Supported by AFOSR Grant FA and NSF DMS Grant Supported by NSF/CAREER Grant & Alonso, 001. Moreover, it is natural to use nonlinear systems, because the increase in HR is not proportional to walking speed, and because the transient HR behavior depends on both the walking speed and the intensity of the exercise, which contrasts with typical step responses of stable LTI systems, whose transient behavior is invariant with respect to the magnitude of the input Cheng, Savkin, Celler, & Su, 008. These nonlinear effects can be significant for patients who have cardiovascular diseases or are recovering from cardiosurgery. A nonlinear dynamic model of HR variation during treadmill exercise was recently proposed and experimentally verified in Cheng et al., 008. The resulting second-order dynamics model both the central and local peripheral response effects on the HR. Further, it accounts for the effects of long-duration exercises on the HR. The model input is the speed of the treadmill. A feedback control law for the treadmill speed was also designed in Cheng et al., 008 by approximating the nonlinear model by a piecewise linear model. The proposed control switches between a set of basic controllers that are composed of a linear quadratic feedforward component and an H feedback component. The present paper presents an alternative nonlinear feedback controller for the HR model in Cheng et al., 008. The contribution of our work is that our control design does not rely on a piecewise linear approximation of the original nonlinear model, nor does it require a switching scheme. Rather, our results use Lyapunov-type stability arguments to design continuous, model-based, nonlinear feedback laws for the treadmill speed that ensure that a properly-chosen, time-varying HR profile is tracked exponentially fast. Our first result uses state feedback by assuming that the system state representing the local peripheral effects is zero Preprint submitted to Automatica 4 November 010

2 at rest and thus can be explicitly calculated from the system model. We remove this assumption in our second result by designing an output feedback controller in conjunction with a nonlinear observer for the aforementioned state. Our third result shows input-to-state stability ISS for the closed loop tracking dynamics with respect to actuator errors, provided that the state dynamics are restricted to an appropriate domain where the states are nonnegative. The ISS property is valid for all initial states when the output feedback and observer are used. This robustness property quantifies the effect of deviations of the treadmill speed from the controller values. Simulation results show the tracking performance of the proposed control for a typical warm-up/exercise/cool-down sequence for the exerciser. System Model and Problem Statement The nonlinear dynamics of the human heart rate during treadmill exercise is Cheng et al., 008 ẋ 1 = a 1 x 1 + a x + a 6 u 1a x 1 ẋ = a 3 x + a 4, 1b 1 + e cx1 a where x 1 represents the deviation of the HR from the at-rest heart rate, u is the treadmill speed and control input, and the a i s and c are positive parameters. The state x models the local peripheral effects on the HR. That is, the human HR is not only externally affected by the treadmill speed during exercise, but also by internal, slower-acting effects such as hormones, metabolism, body temperature, and loss of body fluid from sweating Cheng et al., 008. By 1a, x is an input or disturbance to the x 1 dynamics. Moreover, since x 0 and a > 0, these internal effects tend to further increase the HR during exercise. We introduced the constant c to render cx 1 dimensionless. The nonlinearity u is motivated by studies that show that the metabolic cost from walking on level ground is approximately proportional to the square of the walking speed Cheng et al., 008. Given any input u and initial state x0 0,, we have x 1 t 0 and x t 0 for all t 0 Cheng et al., 008, so we take the state space 0, for 1. Also, since a 6 > 0, we can assume in what follows that a = a 6, by scaling u by a 6 /a 1/ without relabeling. The model 1 then agrees with the one in Cheng et al., 008 when c = 1. The control problem for 1 is to design a feedback law u = ux, t such that x 1 tracks some desired HR profile t, representing a given exercise regime, while keeping x bounded. State feedback is possible since x 1 can be physically measured while x can be calculated by solving 1b, which is a stable linear system driven by x 1 /1 + exp{ cx 1 a }, if x 0 = 0. The main technical obstacle is the nonnegative input nonlinearity u. We overcome this obstacle using a strict Lyapunov function construction. 3 Main Result Consider the reference model ẋ 1r = a 1 + a x r + a u r a ẋ r = a 3 x r + a 4 b 1 + e cx1r a for 1, where, x r, u r : 0, 0, are bounded reference trajectories for x 1, x, and u, respectively. In Section 8, we discuss how to construct relevant reference trajectories from. Then the tracking error variable x = x 1, x = x 1, x x r has the dynamics x 1 = a 1 x 1 + a x + a u u r t x x = a 3 x + a 1 t e cx 1 a 1+e c t a Our main result is as follows, where q denotes the essential supremum of any function q defined on 0,, b = e a, and global exponential stability of x = 0 is the requirement that there exist positive constants c 1 and c so that xt c 1 e ct x0 4 for all trajectories xt = xt x r t for 3 and all trajectories x : 0, 0, of 1 but see Section 6 for results under less stringent conditions on, and see Section 7 for analogs for cases where x 0 is not known: Theorem 1 Provided that a 1a 3 a a 4 > 1 + c, the nonlinear controller u c x, t= where max R x 1, t = { 0, u r t } 1 + R x1,t x, 6 1+be c 1+ cx t 0 1rt e c x 1 m dm 1 1+be c x 1 + t 1+be c t renders x = 0 globally exponentially stable over all trajectories of 1 on the state space 0,, and therefore solves the aforementioned control problem. Remark 1 By our choice of c, both sides of are dimensionless, so the condition is physically meaningful. The feedback 6 does not depend on the parameters a i for i = 1,, 3, 4. Hence, the strictness of the inequality makes the controller robust to uncertainty in the a i s for i = 1,, 3, 4, because if the inequality holds for a fixed quadruple a 1, a, a 3, a 4 0, 4, then it still holds if we perturb the quadruple slightly, which means we can still apply 6 for perturbed values of a 1, a, a 3, a 4 0, 4 as long as continues to hold. The larger the gap a 1 a 3 /{a a 4 } {1 + c }, the greater the degree of robustness; see Section 9 for a specific example where both sides of were computed. 4 A Simple Lemma In some of what follows, we omit the arguments of our functions when they are clear. Also, all equalities and inequalities should be understood to hold globally, unless otherwise indicated. In Appendix A.1, we prove: Lemma 1 R x 1, t 1 + c everywhere. Proof of Theorem 1 First notice that x 1 1+be t cx 1 1+be c t = 1+be c t x 1+bte c t 1 e c x 1 1+be cx 1 1+be c t 1+be cx = 1rt +b cte c t 0 e c x 1 m dm 1 1+be cx 1 1+be c t, 7 x 1 = R x 1, t x 1.

3 Hence, we can rewrite the x dynamics from 3 as x = a 3 x + a 4 R x 1, t x 1. We deduce that along all trajectories of 3 in closed loop with the feedback 6, the time derivative of V x 1, x = 1 x 1 + k x, where k = satisfies V = a 1 x 1 + a x 1 x + a x 1 u c u r t +k a 3 x + a 4 R x 1, t x x 1 a a 4 = a 1 x 1 ka 3 x + x 1 a u c u r t ka4 a R x 1, t x. We next show that the right side of 9 is bounded above by a negative definite quadratic function of x. To this end, we distinguish between two cases. Case 1. u r t 1 + ka 4 R x 1, t/a x > 0. Then u c = u r t 1 + ka 4 R x 1, t/a x, so 9 gives 8 9 V a 1 x 1 ka 3 x. 10 Case. u r t 1+ ka4 a R x 1, t x 0. Then u c = 0, so V = a 1 x 1 ka 3 x + x 1 a u r t ka4 a R x 1, t x. 11 If x 1 0, then 11 again gives 10, since a > 0 and the term in brackets in 11 is nonnegative. If x 1 > 0, then Lemma 1 and 11 give V a 1 x 1 ka 3 x + x 1 a u r t + 1+ ka4 a 1+ c x a 1 x 1 ka 3 x + a x 1 x a 1 a x1 = x 1 x a ka 3 x 1 where the second inequality used our choice of k and nonnegativity of x 1 to drop the term u r t. Condition ensures that the right side of 1 is negative definite. Hence, in all cases, we have V x, where = min{ λ max, a 1, ka 3 } 13 and λ max < 0 is the maximum eigenvalue of the negative definite quadratic form on the right side of 1. It follows from 8 that V σv, where σ = max{k,1}, 14 which gives the desired global exponential stability estimate 4. This proves the theorem. Remark It is worth discussing the controller design technique from Theorem 1 from the viewpoint of control Lyapunov functions. The Lyapunov function candidate 8 we chose in our proof is a standard quadratic function. Then we designed our new controller so that V decays exponentially along all of the closed loop trajectories. Since V is quadratic, this gives the desired exponential decay estimate 4 for the tracking error variable x. 6 Conservativeness of Condition Condition is an easily checked sufficient condition that guarantees that a given trajectory component can be tracked using our controller 6. Theorem 1 remains true if we replace 1+ c in and 6 by any constant P > 0 satisfying P sup{r x 1, t : t 0, x 1 > 0}. 1 The proof is identical to the one we gave, except with 1 + c replaced by P throughout. For example, we can choose P according to the following proposition, which we prove in Appendix A.: Proposition 1 For any constant ε > 0, the choice P = P ε where P ε = max { 1+ε ε, C ε} and C ε = sup t 0 1+b1+ c{1+ε}te c t 1+be c{1+ε} t 1+be c t satisfies 1. With such a choice of P, the new condition 16 a 1a 3 a a 4 > P ε 17 is harder to check than, but it is physically meaningful because of our choice of c, and it can be checked numerically. In Section 9, we give an example of a reference trajectory component for which is violated, but for which 17 is satisfied with ε = 0., and which therefore can be tracked using our method. This will show the conservativeness of. An even less conservative condition can be obtained by simply replacing 1 + c on the right side of and in the controller with the value of the supremum from the right side of 1. However, 16 is easier to compute since it only takes a supremum in one scalar variable. 7 Observer Design If we assume that x 0 = 0, then we can use the original dynamics 1 to solve for x in terms of x 1. Knowing x allows us to implement the feedback u c from 6, because x r and x 1 are both known. When x 0 is unknown, then we can instead implement the feedback u c x 1, ˆx, t = max { 0, u r t 1 + R x1,t ˆx }, where ˆx is from the observer ˆx 1 = a 1ˆx 1 + a ˆx + k 1 x 1 ˆx 1 + a u cx 1, ˆx, t u r t ˆx = a 3ˆx + a 4 R x 1, t x 1 + k x 1 ˆx 1, where k 1 > 0 and k > 0 are appropriate tuning parameters. This follows from the following proposition, where we set x i = x i ˆx i for i = 1,, x = x 1, x, and x = x 1, x : Proposition The x, x dynamics in closed loop with 18 is globally exponentially stable to the origin. See Appendix A.3 for the proof of Proposition.

4 8 Constructing Reference Trajectories Because the dynamics in the equation 1 describes the behavior during exercise, the reference trajectory component can be viewed as a designed parameter to control human heart rate during exercise. For longer duration exercise, should typically increase during a warm-up, then stay near a certain higher level during exercise, and then decrease during a cool down period; see Section 9 for a specific example. The triplet, x r, u r used in 3 needs to be a trajectory of the reference model in. Then, gives a sufficient condition for the signal to be trackable by our controller. This section addresses the problem of how to select the triplet, x r, u r to represent a desired exercise regime while satisfying the above conditions. Method 1: One way to approach this problem is to first select a nonnegative C 1 function t representing a desired HR profile for the exerciser. Given, we can solve b for x r : τ x r t = x r 0e a3t t +a 4 0 ea3τ t dτ. 0 1+be c τ Then, we use a to calculate u r t = J x1r t, where J x1r t = 1 a ẋ1r t+a 1 t x r 0e a3t a 4 t 0 ea3τ t τ dτ. 1+be c τ 1 The condition J x1r t 0 is important for using our controller u r ; if the condition fails, then the controller cannot be applied. In fact, nonnegative valuedness of 1 is necessary and sufficient for to be the first component of a trajectory for. Hence, we can track if a condition holds or 17 holds for some constant ε > 0 and b J x1r t is nonnegative valued. We next give sufficient conditions for J x1r t to be nonnegative valued. Throughout this paragraph, we assume for simplicity that x r 0 = 0, but the general case follows by similar methods. One easily checks that if t is any constant c > 0, then J x1r t is positive for all t 0 if a 1 a 3 /{a a 4 } > 1 e a3t /1+be cc for all t 0, which is satisfied under or if 17 holds for some ε > 0. Hence, we can track any trajectory having t c 0 constant. It follows from a continuity argument that we can also track trajectories of the form t = c + t with positive constant c s, provided the functions and are sufficiently small. More generally, since τ/{1 + be cx1rτ } for all choices of nonnegative valued functions and all τ 0, the function J x1r t is everywhere nonnegative if ẋ 1r t + a 1 t aa4 a 3 t 0. Method : An alternative approach for constructing reference trajectories is to first select a nonnegative valued continuous function u r t representing a desired treadmill speed profile for the exerciser. The given u r is then input to which is numerically solved for the reference trajectory components and x r either offline or online. Note that this second approach bypasses the issue of the nonnegative valuedness of 1. In the next section, we use some of these observations to construct a time-varying reference trajectory and evaluate the proposed controls. 9 Simulations We simulated 1 using the parameter values c = 1, a 1 =., a = 19.96, a 3 = , a 4 = 0.006, a = 8.3, and a 6 = These values are in the 9% independent confidence intervals for the model parameters in Cheng et al., 008, and they give a 1 a 3 /{a a 4 } = With these parameters, time t is measured in minutes, and HR is related to the at-rest heart rate according to HR = 4x 1 + HR Rest and measured in beats per minute bpm. To validate Theorem 1 using these parameter values, we chose the reference profile u r shown in Fig. 1, which represents a typical warm-up/exercise/cool-down sequence. The corresponding reference functions for and x r were calculated as described in Method in the previous section with 0 = x r 0 = 0. The trajectory was multiplied by 4 so that it corresponds to actual deviation from the at-rest heart rate in bpm, and then plotted in Fig.. In this case,, so is violated, but the constant P ε from 16 with the choice ε = 0. can be computed numerically to be 3.4. Hence, 17 is satisfied with ε = 0., so our discussion in Section 6 shows that Theorem 1 still applies, except with 1 + c replaced by P 0. in the formula 6 for the control u c x, t. In the first simulation, the system initial conditions were set to x0 =, 0, and we chose the controller u c x, t from 6. Fig. shows the actual x 1 t tracking t. In the second simulation, we chose the system initial conditions to be x0 = 0.01, 0.0, but this time we used the output feedback controller u c x 1, ˆx, t from 18 with the initial values ˆx0 =, 0.3 for the observer 19 and k 1 = k = 0. In Fig. 3, we show the actual x 1 t tracking t for the second simulation. In both cases, the closed loop input u has a profile similar to Fig. 1. Reference treadmill speed km hr Fig. 1. Reference input u r in simulations of 1 scaled to km/h. Remark 3 Since we need a 1 a 3 /{a a 4 } > P ε, the gap a 1 a 3 /{a a 4 } P ε determines the robustness with respect to the a i s for i = 1,, 3, 4. Different u r s can lead to a bigger gap by reducing P ε. For example, if we scale u r in the previous example by replacing it with u r and keep the a i s and c the same but use ε = 1.6 instead of ε = 0., then the new input u r is such that P 1.6 = 1.6, so the left side a 1 a 3 /{a a 4 } is more than twice the right side P 1.6. This illustrates how we can have considerable robustness against individual differences or intraday variation in the parameter values, which is useful for clinical applications. 10 Robustness to Actuator Errors In Remark 1, we saw how our controller design was robust to uncertainty in the parameters a i for i = 1,, 3, 4. We

5 HR above rest bpm Fig.. State feedback controller: blue and dashed and state x 1 red and solid. HR aove rest bpm Fig. 3. Output feedback controller: blue and dashed and state x 1 red and solid. next consider robustness under small changes in the treadmill speed. Our feedback control design requires the treadmill speed to correspond to the controller values throughout the exercise cycle. This may be difficult to guarantee in practice. Moreover, significant deviations of the treadmill speed from the controller values could have undesirable consequences, such as heart failure during exercise. We can model this situation by replacing the control input u in 1 with u + d, where the actuator error d represents the uncertainty in the treadmill speed. Using the Lyapunov function V from our proof of Theorem 1, we can prove robustness properties for the feedback stabilization with respect to actuator errors. To explain this robustness property precisely, we first need the following definitions. Throughout this section, we fix a bounded reference triplet, x r, u r for 1, so a and b are satisfied. Let D denote the set of all measurable essentially bounded functions d : 0, R. Readers unfamiliar with measure theory can view D as the set of all bounded piecewise continuous functions d : 0, R. For each interval I 0, and each d D, we use d I to denote the essential supremum of the restriction of d to I, and d is the essential supremum of d on all of 0,. Viewing the elements of D as additive uncertainties acting on the controller u c gives the perturbed heart model tracking dynamics { x 1 = a 1 x 1 + a x + a uc + d u r t x x = a 3 x + a be t cx 1 1+be c t where the constants a i and the controller u c are as in Theorem 1 but see Corollary below for robustness results based on the output feedback u c x 1, ˆx, t from Section 7. As before, we only consider componentwise nonnegative solutions of the original dynamics 1. The perturbed system 3 models the realistic scenario where there are perturbations in the treadmill control mechanism. We maintain our convention that the initial times for our trajectories are always t 0 = 0. We say that 3 is inputto-state stable ISS with respect to d provided there are functions 3 β KL and γ K so that for each d D and each initial condition x0 = x0 x r 0 = x 0, the corresponding trajectory t xt, x 0, d of 3 satisfies xt, x 0, d β x 0, t + γ d 0,t 4 for all t 0. The ISS property first appeared in Sontag It is an important paradigm in nonlinear control that has been analyzed and applied in many engineering models Sontag, 008. Proving an estimate of the form 4 for 3 is important because the overshoot term γ d 0,t quantifies the effect of the uncertainty d. To state our robustness results, we use the constants ū = u r, R = 1 + c, and b = u + x r 1/ and this lemma: Lemma u c x, t b for all t, x 0, 0,. Proof. Notice that Lemma 1 gives R x 1, t R when x 1 = x 1. If u c > 0, then u c = u r t 1 + R x1,t x x r t. Since x is never negative and R is everywhere nonnegative, we deduce that 0 u c u r t R x1,t x r t. Since R x 1, t R everywhere, we get 0 u c ur t + x r t u + x r, because r + s r + s for all r 0 and s 0. Corollary 1 The system 3, in closed loop with the state feedback u c x, t, is ISS with respect to d D. Proof. Let V be as in 8 and the constant > 0 be as in 13. Then 13 holds along all trajectories of the unperturbed tracking dynamics 3, so V x 1 + x + a x 1 d + u c d holds along all trajectories of 3. Hence, Lemma gives V x 1 + x + a x 1 d + b d. By the triangle inequality rs 0.r + 0.s applied with r and s chosen to be the terms in braces in a x 1 d + b d { c0 x } {a d + b d / c0 }, V 0. x 1 + x + a d + b d, which is the ISS Lyapunov function decay condition along the perturbed error dynamics 3. Since V is also proper and positive definite, we conclude that it is an ISS Lyapunov function for 3. By standard arguments, the ISS Lyapunov function implies ISS; see Remark 4. 3 Here K is the class of all continuous functions γ : 0, 0, that are strictly increasing and unbounded and satisfy γ0 = 0. Also, KL is the class of all continuous functions β : 0, 0, 0, such that a for each fixed s 0, the function β, s belongs to class K, and b for each fixed r 0, βr, is non-increasing and βr, s 0 as s.

6 The preceding proof relies on the global boundedness of the state feedback u c x, t, which follows from the nonnegativity of x. If we instead use the output feedback u c x 1, ˆx, t from Section 7, and assume that d enters the observer as well i.e., 19 holds with u c x 1, ˆx, t replaced by u c x 1, ˆx, t + d, then we obtain the perturbed augmented error dynamics x 1 = a 1 x 1 + a x +a uc x 1, x x, t + d u r t x = a 3 x + a 4 R x 1, t x 1 6 ẋ 1 = a 1 x 1 + a x k 1 x 1 ẋ = a 3 x k x 1 for the combined variable x, x = x x r, x ˆx, where the tuning constants k i > 0 are chosen so that the x subdynamics in 6 is exponentially stable. Assuming as before, we can then prove: Corollary For any constant δ > 0, the system 6 is ISS with respect to disturbances d : 0, δ, δ. See Appendix A.3 for the proof of Corollary. When we use the output feedback, the functions β and γ in the ISS estimate depend on δ. The corollary follows from the relation ˆx = x x r x and the bound u c x 1, ˆx, t u r t + x r + x ū x r + x, 7 which is shown by a slight variant of the proof of Lemma. To illustrate the robustness of our feedback, we simulated the perturbed tracking error dynamics 3 using the same parameter values a i, the same controller u c x 1, ˆx, t, and the same reference trajectory x r as in the previous simulation, but we take dt = 0.1e t, which produces an exponentially decaying disturbance. We took x0 = 0.01, 0.0 and ˆx0 =, 0.3. We report the simulation in Fig. 4. HR above rest bpm Fig. 4. Output feedback controller with dt = 0.1e t : blue and dashed and state x 1 red and solid. Remark 4 Using a variant of a standard argument, we can use our Lyapunov functions to get formulas for the functions β KL and γ K in the ISS estimate 4. Hence, we can precisely quantify the effects of the disturbance d using the transient term β x 0, t and the overflow term γ d 0,t. Here is a sketch of the argument when the state feedback u c x, t is used. Choose the K functions α 1 s = 0. min{k, 1}s and α s = 0. max{k, 1}s. Then, α 1 x V x α x everywhere. Set α 3 s = s / and γ 1 d = a d + b d /{ }. 8 Then gives V α 3 x +γ 1 d along all trajectories of 3. Setting αs = α 3 α 1 s = s/ max{k, 1}, 9 we get V αv + γ1 d along all trajectories of 3. Hence, along any trajectory of 3, V α 1 γ 1 d implies V 0.αV. 30 Fix any d D and any initial condition x0 = x 0 for 3. Let xt denote the corresponding trajectory of 3. If V 0.αV on any interval of the form I = 0, t, then xt β x0, t on I, where βs, t = max{k,1} min{k,1} e lt/ s 31 and l = 0. / max{k, 1}. A standard invariance argument shows that if V xt α 1 γ 1 d at a given time t = t, then this inequality remains true for all t t. Hence, we can take 31 and γr = α 1 1 α 1 γ 1 r. 11 Conclusion We designed a bounded tracking controller for a nonlinear human heart rate dynamics that rendered the tracking error dynamics globally exponentially stable. The reference trajectory represents a desired heart rate profile during exercise, and the control input is the treadmill speed. Using an observer, the tracking is guaranteed for all possible initial values. Our strict Lyapunov function approach made it possible to quantify the effects of actuator errors, using input-to-state stability. This quantified the effects of variations of the treadmill speed from the controller values. Appendices A.1 Proof of Lemma 1 If x 1 0, then 0 1 e c x1m dm 1 since c > 0, so R x 1, t 1+be c t 1+ ct 1+be c x 1 + t 1+be c t 1+be c t 1+ ct 1 + c x 1+be c t 1r. When x 1 < 0, we have R x 1, t 1 + cbe c c x 1 1 e c x 1 1+be c x be c c x 1 1 e c x 1 1 e c + x 1 b 1 + cbe c 1+be c A.1 = 1 + ce c e c x 1 1e c c x c x 1+be c 1r because e c x1 1/{ c x 1 } 1 for all x 1 < 0. This proves the lemma. A. Proof of Proposition 1 Consider the function δx, Y = 1 X Y X Y 1+be cx 1+be cy on the domain {X, Y R : X > Y 0}. Fix any constant ε > 0. We consider two cases:

7 1 X {1 + ε}y. Then X Y εx/{1 + ε}. Since X Y 0 when X > Y, we have 1+be cx 1+be cy δx, Y 1+ε X Y εx 1+ε 1+be cx 1+be cy ε. A. Y < X < {1 + ε}y. We have δx, Y = γx, Y 1+be cx+y Xe cx Y e cy X Y A.3 where γx, Y = {1 + be cx 1 + be cy } 1. Since φm = me cm and φ m are increasing over 0,, we deduce from the Mean Value Theorem that Xe cx Y e cy /X Y cx + 1e cx. Therefore, since X < {1 + ε}y, condition A.3 gives δx, Y γx, Y 1 + be cy cx be cy c{1+ε}y +1 1+be c{1+ε}y 1+be cy. A.4 Next note that R x 1, t = δx 1, t when x 1 > 0. In this case, A. and A.4 give 1 with P = P ε as defined in 16, which proves the proposition. A.3 Proof of Proposition and Corollary Our proof of Corollary will include the proof of Proposition as the special case where d = 0. Notice that u c x 1, ˆx, t u r t ka4 a R x 1, t x A. x + x everywhere, by separately considering the possibilities u c x 1, ˆx, t = 0 and u c x 1, ˆx, t 0. If u c x 1, ˆx, t 0, then A. holds because 1 + R x1,t ˆx R x1,t x x, by Lemma 1. Otherwise, u r 1 + ka 4R x 1, t/a ˆx, so x u r t ka 4 R x 1, t/a x x. Pick a positive definite quadratic function W x so that 4a Ẇ + L x A.6 along all trajectories of the x subdynamics in 6, where the constant > 0 is from 13 and L = 3a δ /. This is possible since we chose the k i s to make the x dynamics exponentially stable. Let V be the Lyapunov function 8 in the proof of Theorem 1, and set V x, x = V x + W x. Along the trajectories of 6, the argument that led to 9 with u c x, t replaced by u cx 1, ˆx, t gives V = a 1 x 1 ka 3 x + x 1 a u c x 1, ˆx, t u r t +1 + ka4 4a a R x 1, t x + L x + x 1 a uc x 1, ˆx, td + d x 4a + L x + { x 1 } { a x } + x 1 a uc x 1, ˆx, td + d 0. x a + L x + x 1 a {ū x r + 4 x + d } d 1 8 x a x + a d + ū x r d. The first inequality used A. and our definition 13 of. The second inequality followed from the triangle inequality pq 0. p + 0.q / with the choices p = x 1 and q = a x, and the bound on u c x 1, ˆx, t from 7. The last inequality followed from the relations a x 1 ū+41+ x r + d d 0. x + a ū x r + d d / and 4 x 1 a x d x /8 + 3a x d /, and our choice of L. Since V is a positive definite quadratic function, this gives the exponential stability of 6 when δ = 0, and the ISS estimate for any δ > 0, by a slight variant of the argument from Remark 4. References Achten, J., Jeukendrup, A.E Heart rate monitoring: Applications and limitations. Sports Medicine, 337, Brodan, V., Hájek, M., & Kuhn, E An analog model of pulse rate during physical load and recovery. Physiologia Bohemoslovaca, 0, Cheng, T., Savkin, A., Celler, B., & Su, S Nonlinear modeling and control of human heart rate response during exercise with various work load intensities. IEEE Transactions on Biomedical Engineering, 11, Cheng, T., Savkin, A., Su, S., Celler, B., & Wang, L A robust control design for heart rate tracking during exercise. In Proceedings of the Annual International IEEE EMBS Conference, Vancouver, Canada pp Cooper, R.A., Fletcher-Shaw, T.L., & Robertson, R.N Model reference adaptive control of heart rate during wheelchair ergometry. IEEE Transactions on Control Systems Technology, 64, Coyle, E.F., Alonso, G Cardiovascular drift during prolonged exercise: New perspectives. Exercise and Sport Sciences Reviews, 9, Hájek, M., Potucek, J., & Brodan, V Mathematical model of heart rate regulation during exercise. Automatica, 16, Johnson, J.M., Rowell, L.B Forearm and skin vascular responses to prolonged exercise in man. Journal of Applied Physiology, 396, Kawada, T., Sunagawa, G., Takaki, H., Shishido, T., Miyano, H., Miyashita, H., Sato, T., Sugimachi, M., & Sunagawa, K Development of a servo-controller of heart rate using a treadmill. Japanese Circulation Journal, 631, Rowell, L.B Human Cardiovascular Control. New York: Oxford University Press. Sontag, E.D Stabilizability, i/o stability and coprime factorizations. In Proceedings of the IEEE Conference on Decision and Control, Austin, USA pp Sontag, E.D Input-to-state stability: Basic concepts and results. In Nonlinear and Optimal Control Theory. Berlin, Germany: Springer pp Su, S.W., Wang, L., Celler, B.G., Savkin, A.V., & Guo, Y Identification and control for heart rate regulation during treadmill exercise. IEEE Transactions on Biomedical Engineering, 47,