Controlling Human Heart Rate Response During Treadmill Exercise

Transcription

1 Controlling Human Heart Rate Response During Treadmill Exercise Frédéric Mazenc (INRIA-DISCO), Michael Malisoff (LSU), and Marcio de Queiroz (LSU) Special Session: Advances in Biomedical Mathematics 2011 AMS Spring Southeastern Section Meeting Statesboro, GA, March 12-13, 2011

2 Motivation

3 Motivation Human heart rate (HR) is a widely used indicator of exercise intensity, because it is easy to monitor, and can play a role in detecting and preventing overtraining.

4 Motivation Human heart rate (HR) is a widely used indicator of exercise intensity, because it is easy to monitor, and can play a role in detecting and preventing overtraining. Hence, controlling HR in real time during exercise can help develop training or rehabilitation protocols, e.g., for athletes and patients with cardiovascular or obesity problems.

5 Motivation Human heart rate (HR) is a widely used indicator of exercise intensity, because it is easy to monitor, and can play a role in detecting and preventing overtraining. Hence, controlling HR in real time during exercise can help develop training or rehabilitation protocols, e.g., for athletes and patients with cardiovascular or obesity problems. The HR increase is not proportional to the walking speed, and the transient HR behavior depends on both the walking speed and the exercise intensity.

6 Motivation Human heart rate (HR) is a widely used indicator of exercise intensity, because it is easy to monitor, and can play a role in detecting and preventing overtraining. Hence, controlling HR in real time during exercise can help develop training or rehabilitation protocols, e.g., for athletes and patients with cardiovascular or obesity problems. The HR increase is not proportional to the walking speed, and the transient HR behavior depends on both the walking speed and the exercise intensity. Hence, the existing linear models e.g. (Brodan, Hajek, & Kuhn, 1971) and (Cooper, Fletcher-Shaw, & Robertson, 1998) based on PID control may not be accurate.

7 Model (Cheng, Savkin, et al., IEEE-TBE)

8 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b)

9 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate.

10 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR

11 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects,

12 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature,

13 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature, and loss of body fluids).

14 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature, and loss of body fluids). u = treadmill speed.

15 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature, and loss of body fluids). u = treadmill speed. a i = constant parameter.

16 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature, and loss of body fluids). u = treadmill speed. a i = constant parameter. Motivation:

17 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature, and loss of body fluids). u = treadmill speed. a i = constant parameter. Motivation: Metabolic cost from walking on level ground is approximately proportional to the square of the walking speed.

18 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature, and loss of body fluids). u = treadmill speed. a i = constant parameter. Motivation: Metabolic cost from walking on level ground is approximately proportional to the square of the walking speed. Model has been validated with human subjects.

19 Model (Cheng, Savkin, et al., IEEE-TBE) ẋ 1 = a 1 x 1 + a 2 x 2 + a 2 u 2 (1a) ẋ 2 = a 3 x 2 + a 4 x 1 1+e (x 1 a 5 ), (1b) x 1 = deviation of the HR from the at rest heart rate. x 2 = slower, local peripheral effects on the HR (e.g., hormonal effects, increase in body temperature, and loss of body fluids). u = treadmill speed. a i = constant parameter. Motivation: Metabolic cost from walking on level ground is approximately proportional to the square of the walking speed. Model has been validated with human subjects. Unlike conventional linear models, it captures peripheral effects and is suitable for long duration exercise.

20 Control Objective

21 Control Objective Given any bounded C 0 x 1r, x 2r, u r : [0, ) [0, ) such that ẋ 1r = a 1 x 1r + a 2 x 2r + a 2 u 2 r (2a) ẋ 2r = a 3 x 2r + a 4 x 1r 1+e (x 1r a 5 ), (2b)

22 Control Objective Given any bounded C 0 x 1r, x 2r, u r : [0, ) [0, ) such that design the controller u ẋ 1r = a 1 x 1r + a 2 x 2r + a 2 u 2 r (2a) ẋ 2r = a 3 x 2r + a 4 x 1r 1+e (x 1r a 5 ), (2b)

23 Control Objective Given any bounded C 0 x 1r, x 2r, u r : [0, ) [0, ) such that ẋ 1r = a 1 x 1r + a 2 x 2r + a 2 u 2 r (2a) ẋ 2r = a 3 x 2r + a 4 x 1r 1+e (x 1r a 5 ), (2b) design the controller u so that the tracking error variable x = ( x 1, x 2 ) = (x 1 x 1r, x 2 x 2r ) dynamics x 1 = a 1 x 1 + a 2 x 2 + a 2 [u 2 u r (t) 2 ] (3a) [ ] x x 2 = a 3 x 2 + a e (x 1 a 5 ) x 1r (t) 1+e (x (3b) 1r(t) a 5 ) is globally exponentially stable to zero

24 Control Objective Given any bounded C 0 x 1r, x 2r, u r : [0, ) [0, ) such that ẋ 1r = a 1 x 1r + a 2 x 2r + a 2 u 2 r (2a) ẋ 2r = a 3 x 2r + a 4 x 1r 1+e (x 1r a 5 ), (2b) design the controller u so that the tracking error variable x = ( x 1, x 2 ) = (x 1 x 1r, x 2 x 2r ) dynamics x 1 = a 1 x 1 + a 2 x 2 + a 2 [u 2 u r (t) 2 ] (3a) [ ] x x 2 = a 3 x 2 + a e (x 1 a 5 ) x 1r (t) 1+e (x (3b) 1r(t) a 5 ) is globally exponentially stable to zero, i.e., there are constants c i > 0 so that x(t) c 1 e c 2t x(0) for all trajectories of (1).

25 Admissible Trajectories

26 Admissible Trajectories Knowing x 1r allows us to solve for x 2r from (2b). x 1r (τ) x 2r (t) = x 2r (0)e a3t t +a 4 0 ea 3(τ t) dτ. (4) 1+be x 1r (τ)

27 Admissible Trajectories Knowing x 1r allows us to solve for x 2r from (2b). x 1r (τ) x 2r (t) = x 2r (0)e a3t t +a 4 0 ea 3(τ t) dτ. (4) 1+be x 1r (τ) Then, we use (2a) to calculate u r (t) = J x1r (t), where J x1r (t) = 1 a 2 (ẋ1r (t)+a 1 x 1r (t) ) x 2r (0)e a 3t a 4 t 0 ea 3(τ t) x 1r (τ) dτ. 1+be x 1r (τ) (5)

28 Admissible Trajectories Knowing x 1r allows us to solve for x 2r from (2b). x 1r (τ) x 2r (t) = x 2r (0)e a3t t +a 4 0 ea 3(τ t) dτ. (4) 1+be x 1r (τ) Then, we use (2a) to calculate u r (t) = J x1r (t), where J x1r (t) = 1 a 2 (ẋ1r (t)+a 1 x 1r (t) ) x 2r (0)e a 3t a 4 t 0 ea 3(τ t) x 1r (τ) dτ. 1+be x 1r (τ) That means nonnegativity of J x1r (t) for all t 0 is necessary and sufficient for x 1r to be the first component of a trajectory. (5)

29 Admissible Trajectories Knowing x 1r allows us to solve for x 2r from (2b). x 1r (τ) x 2r (t) = x 2r (0)e a3t t +a 4 0 ea 3(τ t) dτ. (4) 1+be x 1r (τ) Then, we use (2a) to calculate u r (t) = J x1r (t), where J x1r (t) = 1 a 2 (ẋ1r (t)+a 1 x 1r (t) ) x 2r (0)e a 3t a 4 t 0 ea 3(τ t) x 1r (τ) dτ. 1+be x 1r (τ) That means nonnegativity of J x1r (t) for all t 0 is necessary and sufficient for x 1r to be the first component of a trajectory. This condition holds for x 1r (t) = c + (t) with positive constant c s, provided the functions and are sufficiently small. (5)

30 Admissible Trajectories Knowing x 1r allows us to solve for x 2r from (2b). x 1r (τ) x 2r (t) = x 2r (0)e a3t t +a 4 0 ea 3(τ t) dτ. (4) 1+be x 1r (τ) Then, we use (2a) to calculate u r (t) = J x1r (t), where J x1r (t) = 1 a 2 (ẋ1r (t)+a 1 x 1r (t) ) x 2r (0)e a 3t a 4 t 0 ea 3(τ t) x 1r (τ) dτ. 1+be x 1r (τ) That means nonnegativity of J x1r (t) for all t 0 is necessary and sufficient for x 1r to be the first component of a trajectory. This condition holds for x 1r (t) = c + (t) with positive constant c s, provided the functions and are sufficiently small. However, we need another condition to ensure trackability of x r. (5)

31 Standing Assumption

32 Standing Assumption We always assume that there is a constant ε (0, 1] such that } def > P ε = max{ 1+ε ε, sup t 0 (SA) a 1 a 3 a 2 a 4 where b = e a 5. 1+b(1+{1+ε}x 1r (t))e x 1r (t) [1+be {1+ε}x 1r (t) ][1+be x 1r (t) ]

33 Standing Assumption We always assume that there is a constant ε (0, 1] such that } def > P ε = max{ 1+ε ε, sup t 0 (SA) a 1 a 3 a 2 a 4 where b = e a 5. 1+b(1+{1+ε}x 1r (t))e x 1r (t) [1+be {1+ε}x 1r (t) ][1+be x 1r (t) ] This condition is robust with respect to perturbations of the a i s.

34 Standing Assumption We always assume that there is a constant ε (0, 1] such that } def > P ε = max{ 1+ε ε, sup t 0 (SA) a 1 a 3 a 2 a 4 where b = e a 5. 1+b(1+{1+ε}x 1r (t))e x 1r (t) [1+be {1+ε}x 1r (t) ][1+be x 1r (t) ] This condition is robust with respect to perturbations of the a i s. This robustness is important because the a i s are uncertain.

35 Standing Assumption We always assume that there is a constant ε (0, 1] such that } def > P ε = max{ 1+ε ε, sup t 0 (SA) a 1 a 3 a 2 a 4 where b = e a 5. 1+b(1+{1+ε}x 1r (t))e x 1r (t) [1+be {1+ε}x 1r (t) ][1+be x 1r (t) ] This condition is robust with respect to perturbations of the a i s. This robustness is important because the a i s are uncertain. Cheng et al. use the Levenberg-Marquardt method to estimate the a i s.

36 Theorem 1

37 Theorem 1 The nonlinear controller { ( ) u c (x, t)= max 0, u r (t) 2 1+ R( x 1,t) P ε x 2 }, [ 1+be x 1r(t) 1+x 1r (t) ] 0 1 e x 1m dm where R( x 1, t) = (1+be ( x 1+x 1r (t)) )(1+be x1r(t), ) solves the aforementioned control problem. (6)

38 Theorem 1 The nonlinear controller { ( ) u c (x, t)= max 0, u r (t) 2 1+ R( x 1,t) P ε x 2 }, [ 1+be x 1r(t) 1+x 1r (t) ] 0 1 e x 1m dm where R( x 1, t) = (1+be ( x 1+x 1r (t)) )(1+be x1r(t), ) solves the aforementioned control problem. (6) Proof:

39 Theorem 1 The nonlinear controller { ( ) u c (x, t)= max 0, u r (t) 2 1+ R( x 1,t) P ε x 2 }, [ 1+be x 1r(t) 1+x 1r (t) ] 0 1 e x 1m dm where R( x 1, t) = (1+be ( x 1+x 1r (t)) )(1+be x1r(t), ) solves the aforementioned control problem. (6) Proof: Take V( x 1, x 2 ) = 1 2 x k 2 x 2 2

40 Theorem 1 The nonlinear controller { ( ) u c (x, t)= max 0, u r (t) 2 1+ R( x 1,t) P ε x 2 }, [ 1+be x 1r(t) 1+x 1r (t) ] 0 1 e x 1m dm where R( x 1, t) = (1+be ( x 1+x 1r (t)) )(1+be x1r(t), ) solves the aforementioned control problem. (6) Proof: Take V( x 1, x 2 ) = 1 2 x k 2 x 2 2, where k = a 2 a 4 P ε.

41 Theorem 1 The nonlinear controller { ( ) u c (x, t)= max 0, u r (t) 2 1+ R( x 1,t) P ε x 2 }, [ 1+be x 1r(t) 1+x 1r (t) ] 0 1 e x 1m dm where R( x 1, t) = (1+be ( x 1+x 1r (t)) )(1+be x1r(t), ) solves the aforementioned control problem. (6) Proof: Take V( x 1, x 2 ) = 1 2 x k 2 x 2 2, where k = a 2 a 4 P ε. Then V σv along all trajectories of the closed loop system, where σ = 2c 0 / max{k, 1}, c 0 = min{ λ max, a 1, ka 3 }, and λ max < 0 is the maximal eigenvalue of [ ] a1 a M = 2. a 2 ka 3

42 Observer Design

43 Observer Design Assume that x 2 (0) is unknown.

44 Observer Design Assume that x 2 (0) is unknown. Use { ( u c (x 1,ˆx 2, t) = max 0, u r (t) 2 1+ R( x } 1, t) )ˆx 2. (7) P ε

45 Observer Design Assume that x 2 (0) is unknown. Use { ( u c (x 1,ˆx 2, t) = max 0, u r (t) 2 1+ R( x } 1, t) )ˆx 2. (7) P ε The estimate ˆx 2 of x 2 is from the observer ˆx 1 = a 1ˆx 1 + a 2ˆx 2 + k 1 x 1 + a 2 [u 2 c (x 1,ˆx 2, t) u r (t) 2 ] ˆx 2 = a 3ˆx 2 + a 4 R( x 1, t) x 1 + k 2 x 1. (8)

46 Observer Design Assume that x 2 (0) is unknown. Use { ( u c (x 1,ˆx 2, t) = max 0, u r (t) 2 1+ R( x } 1, t) )ˆx 2. (7) P ε The estimate ˆx 2 of x 2 is from the observer ˆx 1 = a 1ˆx 1 + a 2ˆx 2 + k 1 x 1 + a 2 [u 2 c (x 1,ˆx 2, t) u r (t) 2 ] ˆx 2 = a 3ˆx 2 + a 4 R( x 1, t) x 1 + k 2 x 1. Here k 1 > 0 and k 2 > 0 are tuning constants, and x 1 = x 1 ˆx 1. (8)

47 Observer Design Assume that x 2 (0) is unknown. Use { ( u c (x 1,ˆx 2, t) = max 0, u r (t) 2 1+ R( x } 1, t) )ˆx 2. (7) P ε The estimate ˆx 2 of x 2 is from the observer ˆx 1 = a 1ˆx 1 + a 2ˆx 2 + k 1 x 1 + a 2 [u 2 c (x 1,ˆx 2, t) u r (t) 2 ] ˆx 2 = a 3ˆx 2 + a 4 R( x 1, t) x 1 + k 2 x 1. Here k 1 > 0 and k 2 > 0 are tuning constants, and x 1 = x 1 ˆx 1. Proposition. The ( x, x) dynamics in closed loop with (7) is globally exponentially stable to the origin. (8)

48 Observer Design Assume that x 2 (0) is unknown. Use { ( u c (x 1,ˆx 2, t) = max 0, u r (t) 2 1+ R( x } 1, t) )ˆx 2. (7) P ε The estimate ˆx 2 of x 2 is from the observer ˆx 1 = a 1ˆx 1 + a 2ˆx 2 + k 1 x 1 + a 2 [u 2 c (x 1,ˆx 2, t) u r (t) 2 ] ˆx 2 = a 3ˆx 2 + a 4 R( x 1, t) x 1 + k 2 x 1. Here k 1 > 0 and k 2 > 0 are tuning constants, and x 1 = x 1 ˆx 1. Proposition. The ( x, x) dynamics in closed loop with (7) is globally exponentially stable to the origin. Proof: (8)

49 Observer Design Assume that x 2 (0) is unknown. Use { ( u c (x 1,ˆx 2, t) = max 0, u r (t) 2 1+ R( x } 1, t) )ˆx 2. (7) P ε The estimate ˆx 2 of x 2 is from the observer ˆx 1 = a 1ˆx 1 + a 2ˆx 2 + k 1 x 1 + a 2 [u 2 c (x 1,ˆx 2, t) u r (t) 2 ] ˆx 2 = a 3ˆx 2 + a 4 R( x 1, t) x 1 + k 2 x 1. Here k 1 > 0 and k 2 > 0 are tuning constants, and x 1 = x 1 ˆx 1. Proposition. The ( x, x) dynamics in closed loop with (7) is globally exponentially stable to the origin. Proof: Take V ( x, x) = V( x)+ L x 2 for a big enough L > 0. (8)

50 Simulations

51 Simulations We took a 1 = 2.2, a 2 = 19.96, a 3 = , a 4 = , a 5 = 8.32 (Cheng et al., IEEE-TBE).

52 Simulations We took a 1 = 2.2, a 2 = 19.96, a 3 = , a 4 = , a 5 = 8.32 (Cheng et al., IEEE-TBE). We generated the reference trajectory x r by designing u r and then solving the reference dynamics with x r (0) = 0.

53 Simulations We took a 1 = 2.2, a 2 = 19.96, a 3 = , a 4 = , a 5 = 8.32 (Cheng et al., IEEE-TBE). We generated the reference trajectory x r by designing u r and then solving the reference dynamics with x r (0) = 0. Reference treadmill speed km hr Time s

54 Simulations We took a 1 = 2.2, a 2 = 19.96, a 3 = , a 4 = , a 5 = 8.32 (Cheng et al., IEEE-TBE). We generated the reference trajectory x r by designing u r and then solving the reference dynamics with x r (0) = 0. Reference treadmill speed km hr Time s The resulting x 1r satisfies (SA) with ε = 0.5 so our results apply.

55 Tracking using State Feedback Control u c (x, t)

56 Tracking using State Feedback Control u c (x, t) HR above rest bpm Time s

57 Tracking using State Feedback Control u c (x, t) HR above rest bpm Time s x 1r (blue and dashed) and state x 1 (red and solid).

58 Tracking using State Feedback Control u c (x, t) HR above rest bpm Time s x 1r (blue and dashed) and state x 1 (red and solid). Initial state: x(0) = (2, 0).

59 Tracking using Output Control u c (x 1,ˆx 2, t)

60 Tracking using Output Control u c (x 1,ˆx 2, t) HR aove rest bpm Time s

61 Tracking using Output Control u c (x 1,ˆx 2, t) HR aove rest bpm Time s x 1r (blue and dashed) and state x 1 (red and solid).

62 Tracking using Output Control u c (x 1,ˆx 2, t) HR aove rest bpm Time s x 1r (blue and dashed) and state x 1 (red and solid). Initial states: x(0) = (0.01, 0.05), ˆx(0) = (2, 0.3).

63 Certifying Good Performance under Uncertainty Since our global Lyapunov functions are strict, we can prove input-to-state stability of the augmented tracking error dynamics with respect to additive uncertainty on the output controller.

64 Certifying Good Performance under Uncertainty Since our global Lyapunov functions are strict, we can prove input-to-state stability of the augmented tracking error dynamics with respect to additive uncertainty on the output controller. x 1 = a 1 x 1 + a 2 x 2 + a 2 [ (uc (x 1, x 2 x 2, t)+d) 2 u r (t) 2] x 2 = a 3 x 2 + a 4 R( x 1, t) x 1 ẋ 1 = a 1 x 1 + a 2 x 2 k 1 x 1 ẋ 2 = a 3 x 2 k 2 x 1 (9)

65 Certifying Good Performance under Uncertainty Since our global Lyapunov functions are strict, we can prove input-to-state stability of the augmented tracking error dynamics with respect to additive uncertainty on the output controller. x 1 = a 1 x 1 + a 2 x 2 + a 2 [ (uc (x 1, x 2 x 2, t)+d) 2 u r (t) 2] x 2 = a 3 x 2 + a 4 R( x 1, t) x 1 ẋ 1 = a 1 x 1 + a 2 x 2 k 1 x 1 ẋ 2 = a 3 x 2 k 2 x 1 Theorem: For each constant δ > 0, we can find constants c i > 0 depending on δ so that along all trajectories of (9) for all measurable functions d : [0, ) [ δ, δ], we have ( x(t), x(t)) c 1 ( x(0), x(0)) e c 2t + c 3 d [0,t] for all t 0. (9)

66 Certifying Good Performance under Uncertainty Since our global Lyapunov functions are strict, we can prove input-to-state stability of the augmented tracking error dynamics with respect to additive uncertainty on the output controller. [ x 1 = a 1 x 1 + a 2 x 2 + a 2 (uc (x 1, x 2 x 2, t)+d) 2 u r (t) 2] x 2 = a 3 x 2 + a 4 R( x 1, t) x 1 ẋ 1 = a 1 x 1 + a 2 x 2 k 1 x 1 ẋ 2 = a 3 x 2 k 2 x 1 Theorem: For each constant δ > 0, we can find constants c i > 0 depending on δ so that along all trajectories of (9) for all measurable functions d : [0, ) [ δ, δ], we have ( x(t), x(t)) c 1 ( x(0), x(0)) e c 2t + c 3 d [0,t] for all t 0. This is input-to-state stability with exponential transient decay. (9)

67 Ideas for Input-to-State Stability Proof Step 1: Pick a positive definite quadratic function W(x) such that Ẇ 4a2 2 (1+8 δ 2 ) c 0 x 2 (10) along all trajectories of the x subdynamics in (9).

68 Ideas for Input-to-State Stability Proof Step 1: Pick a positive definite quadratic function W(x) such that Ẇ 4a2 2 (1+8 δ 2 ) c 0 x 2 (10) along all trajectories of the x subdynamics in (9). This can be done because the x dynamics are exponentially stable to zero.

69 Ideas for Input-to-State Stability Proof Step 1: Pick a positive definite quadratic function W(x) such that Ẇ 4a2 2 (1+8 δ 2 ) c 0 x 2 (10) along all trajectories of the x subdynamics in (9). This can be done because the x dynamics are exponentially stable to zero. Step 2: Show that along all trajectories of (9), the function V ( x, x) = V( x)+w(x) satisfies V 1 8 c 0 x 2 2a2 2 c 0 x 2 + a2 2 c 0 ( d +2 u r + 4(1+ x r )) 2 d 2.

70 Ideas for Input-to-State Stability Proof Step 1: Pick a positive definite quadratic function W(x) such that Ẇ 4a2 2 (1+8 δ 2 ) c 0 x 2 (10) along all trajectories of the x subdynamics in (9). This can be done because the x dynamics are exponentially stable to zero. Step 2: Show that along all trajectories of (9), the function V ( x, x) = V( x)+w(x) satisfies V 1 8 c 0 x 2 2a2 2 c 0 x 2 + a2 2 c 0 ( d +2 u r + 4(1+ x r )) 2 d 2. This means that V is an ISS Lyapunov function for (9) with disturbances d bounded by δ in the sup norm.

71 Robust Tracking using Output Control u c (x 1,ˆx 2, t)

72 Robust Tracking using Output Control u c (x 1,ˆx 2, t) HR above rest bpm Time s

73 Robust Tracking using Output Control u c (x 1,ˆx 2, t) HR above rest bpm Time s x 1r (blue and dashed) and state x 1 (red and solid).

74 Robust Tracking using Output Control u c (x 1,ˆx 2, t) HR above rest bpm Time s x 1r (blue and dashed) and state x 1 (red and solid). x(0) = (0.01, 0.05), ˆx(0) = (2, 0.3).

75 Robust Tracking using Output Control u c (x 1,ˆx 2, t) HR above rest bpm Time s x 1r (blue and dashed) and state x 1 (red and solid). x(0) = (0.01, 0.05), ˆx(0) = (2, 0.3). d(t) = 0.15e t.

76 Conclusions

77 Conclusions The control of human heart rate in real time during exercise is an important problem in biomedical engineering.

78 Conclusions The control of human heart rate in real time during exercise is an important problem in biomedical engineering. We designed a bounded exponentially stabilizing controller for a nonlinear human heart rate dynamics.

79 Conclusions The control of human heart rate in real time during exercise is an important problem in biomedical engineering. We designed a bounded exponentially stabilizing controller for a nonlinear human heart rate dynamics. The reference trajectory gives a desired heart rate profile, and the control input is the treadmill speed.

80 Conclusions The control of human heart rate in real time during exercise is an important problem in biomedical engineering. We designed a bounded exponentially stabilizing controller for a nonlinear human heart rate dynamics. The reference trajectory gives a desired heart rate profile, and the control input is the treadmill speed. Using an observer, the tracking is guaranteed for all possible initial values and gives ISS to actuator errors.

81 Conclusions The control of human heart rate in real time during exercise is an important problem in biomedical engineering. We designed a bounded exponentially stabilizing controller for a nonlinear human heart rate dynamics. The reference trajectory gives a desired heart rate profile, and the control input is the treadmill speed. Using an observer, the tracking is guaranteed for all possible initial values and gives ISS to actuator errors. For complete proofs, see [FM, MM, and MdQ, Tracking control and robustness analysis for a nonlinear model of human heart rate during exercise," Automatica, accepted.]