3.1.1. Definition 1

(Multi-loop integral controllability for nonlinear 2ISO processes) Consider the closed loop system depicted in Figure 2.

i. For the nonlinear process P described by Eq. (1), if a multi-loop integral controller C exists, such that the unforced closed loop system (r ¼ 0) is globally asymptotically stable (GAS) for the equilibrium x ¼ 0 and such that the globally asymptotically stability is satisfied if each individual loop can be detuned independently by a factor ki (0 ≤ ki ≤ 1, i ¼ 1, 2), then the nonlinear process P is said to be multi-loop integral controllable (MIC) for the equilibrium x ¼ 0.

Figure 2. Multi-loop integral controllability for 2ISO system.

ii. If the closed loop system is asymptotically stable (AS) near the region of the equilibrium x ¼ 0, then the nonlinear process P is said to be locally multi-loop integral controllable around the equilibrium x ¼ 0 [19, 20].

In Figure 2, we assume the state equation of the general process P~ (which includes original process P and the two scalar non-integral controllers c<sup>1</sup> andc2) is modeled as follows (with the same assumptions for Eq. (1) of process P):

$$\tilde{P}: \begin{cases} \dot{\mathfrak{x}} &=& f(\mathfrak{x}, \tilde{\mathfrak{u}}) \\ \mathfrak{y} &=& \mathfrak{g}(\mathfrak{x}, \tilde{\mathfrak{u}}) \end{cases} \tag{2}$$

w1ð Þþ u~<sup>1</sup> w<sup>2</sup> ð Þ ð Þ u~<sup>2</sup> k<sup>1</sup>

then the nonlinear 2ISO process is MIC for the equilibrium.

perturbation theory [18] and can be found in [19].

speed-HR and gradient-HR subsystems is DIC respectively.

(when <sup>w</sup>1ð Þþ <sup>u</sup>~<sup>1</sup> <sup>w</sup>2ð Þ <sup>u</sup>~<sup>2</sup> 6¼ 0 and <sup>u</sup><sup>~</sup> <sup>∈</sup> <sup>U</sup><sup>~</sup> <sup>⊂</sup>R<sup>2</sup>

g hð Þ u~Þ; u~ > r w1ð Þþ u~<sup>1</sup> w<sup>2</sup> j j ð Þ u~<sup>2</sup> <sup>2</sup>

w1ð Þþ u~<sup>1</sup> w2ð Þ¼ u~<sup>2</sup> 0.

to be 0 ≤ ki ≤ 1, i ¼ 1, 2.

processes based on Theorem 1.

able or not under faulty conditions [22, 23].

4.1. MATLAB®/Simulink® simulation verification

4. Results

∂w1ð Þ u~<sup>1</sup> ∂u~<sup>1</sup>

þ k<sup>2</sup>

Multi-Loop Integral Control-Based Heart Rate Regulation for Fast Tracking and Faulty-Tolerant Control…

Then there exists η > 0, such that the equilibrium is GAS. That is, if the two scalar controllers c<sup>1</sup> and c<sup>2</sup> can be found such that the generalized process P~ can satisfy Conditions (i), (ii) and (iii),

Remark: If the generalized process P~can be guaranteed to satisfy Conditions (i), (ii) and (iii), based on Theorem 1 it is said that the nonlinear 2ISO process can be MIC. Once the control system is assumed as nonlinear 2ISO MIC, the stability of system can be easily guaranteed by independently tuning the factor ki. In practice, the factor ki usually can be manually configured

The Proof of the above theorem (similar as that of Theorem 1 in [11]) is based on singular

Based on Definition 1, we can easily check that a necessary MIC condition for a 2ISO process in each single loop is DIC respectively. For HR regulation system, the necessary condition for the

A sufficient DIC condition for SISO system is the passivity in steady state, that is, the sector condition for passivity. We can easily prove that this condition is also sufficient for 2ISO

For the HR responses during walking or running exercises, it is not hard to see that the incremental increasing of speed or gradient respectively will lead to the incremental increasing in HR for the same exerciser, that is, each single loop is DIC in either walking zone or running zone. However, the HR variation during walking and running transition is not clear. The following parts simulated the transition of walking/running as well as performed several experiments to

We also explore the offset free tracking when one of the actuators is in faulty conditions. We consider the case when one of the motors (speed motor and gradient motor) is broken, whether HR tracking is still possible or not. We also investigate whether offset free tracking is achiev-

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

investigate the HR response during transition between walking and running [21].

∂w2ð Þ u~<sup>2</sup> ∂u~<sup>2</sup>

g h<sup>ð</sup> ð Þ <sup>u</sup>~Þ; <sup>u</sup><sup>~</sup> <sup>&</sup>gt; <sup>0</sup> (4)

(for some scalar r > 0) for in a neighborhood of

∂w1ð Þ u~<sup>1</sup> <sup>∂</sup>u~<sup>1</sup> þ k<sup>2</sup>

http://dx.doi.org/10.5772/intechopen.71855

∂w2ð Þ u~<sup>2</sup> ∂u~<sup>2</sup>

281

) and w1ð Þþ u~<sup>1</sup> w<sup>2</sup> ð Þ ð Þ u~<sup>2</sup> k<sup>1</sup>

The state equation for the linear integral controller is expressed as:

$$\mathbf{C}\_{l}: \begin{cases} \dot{\xi} = \begin{bmatrix} \dot{\xi}\_{1} \\ \dot{\xi}\_{2} \end{bmatrix} \ = \ \eta \begin{bmatrix} k\_{1} \\ k\_{2} \end{bmatrix} e = -\eta \begin{bmatrix} k\_{1} \\ k\_{2} \end{bmatrix} y \\ \tilde{\boldsymbol{\mu}} = \boldsymbol{\xi} \end{cases} \tag{3}$$

The following theorem presented a sufficient condition for MIC:

#### 3.1.2. Theorem 1

(Steady-state MIC conditions for nonlinear 2ISO processes).

Consider the closed loop system in Figure 1, and assume that the general process P~ and the linear part of the controller Cl are described by Eqs. (2) and (3), respectively. If the following assumptions are satisfied:


Multi-Loop Integral Control-Based Heart Rate Regulation for Fast Tracking and Faulty-Tolerant Control… http://dx.doi.org/10.5772/intechopen.71855 281

$$\left(\varphi\_1(\check{u}\_1) + \varphi\_2(\check{u}\_2)\right) \left(k\_1 \frac{\partial \varphi\_1(\check{u}\_1)}{\partial \check{u}\_1} + k\_2 \frac{\partial \varphi\_2(\check{u}\_2)}{\partial \check{u}\_2}\right) \mathbf{g}(h(\check{u}), \check{u}) > 0 \tag{4}$$

(when <sup>w</sup>1ð Þþ <sup>u</sup>~<sup>1</sup> <sup>w</sup>2ð Þ <sup>u</sup>~<sup>2</sup> 6¼ 0 and <sup>u</sup><sup>~</sup> <sup>∈</sup> <sup>U</sup><sup>~</sup> <sup>⊂</sup>R<sup>2</sup> ) and w1ð Þþ u~<sup>1</sup> w<sup>2</sup> ð Þ ð Þ u~<sup>2</sup> k<sup>1</sup> ∂w1ð Þ u~<sup>1</sup> <sup>∂</sup>u~<sup>1</sup> þ k<sup>2</sup> ∂w2ð Þ u~<sup>2</sup> ∂u~<sup>2</sup> g hð Þ u~Þ; u~ > r w1ð Þþ u~<sup>1</sup> w<sup>2</sup> j j ð Þ u~<sup>2</sup> <sup>2</sup> (for some scalar r > 0) for in a neighborhood of w1ð Þþ u~<sup>1</sup> w2ð Þ¼ u~<sup>2</sup> 0.

Then there exists η > 0, such that the equilibrium is GAS. That is, if the two scalar controllers c<sup>1</sup> and c<sup>2</sup> can be found such that the generalized process P~ can satisfy Conditions (i), (ii) and (iii), then the nonlinear 2ISO process is MIC for the equilibrium.

Remark: If the generalized process P~can be guaranteed to satisfy Conditions (i), (ii) and (iii), based on Theorem 1 it is said that the nonlinear 2ISO process can be MIC. Once the control system is assumed as nonlinear 2ISO MIC, the stability of system can be easily guaranteed by independently tuning the factor ki. In practice, the factor ki usually can be manually configured to be 0 ≤ ki ≤ 1, i ¼ 1, 2.

The Proof of the above theorem (similar as that of Theorem 1 in [11]) is based on singular perturbation theory [18] and can be found in [19].

Based on Definition 1, we can easily check that a necessary MIC condition for a 2ISO process in each single loop is DIC respectively. For HR regulation system, the necessary condition for the speed-HR and gradient-HR subsystems is DIC respectively.

A sufficient DIC condition for SISO system is the passivity in steady state, that is, the sector condition for passivity. We can easily prove that this condition is also sufficient for 2ISO processes based on Theorem 1.

For the HR responses during walking or running exercises, it is not hard to see that the incremental increasing of speed or gradient respectively will lead to the incremental increasing in HR for the same exerciser, that is, each single loop is DIC in either walking zone or running zone. However, the HR variation during walking and running transition is not clear. The following parts simulated the transition of walking/running as well as performed several experiments to investigate the HR response during transition between walking and running [21].

We also explore the offset free tracking when one of the actuators is in faulty conditions. We consider the case when one of the motors (speed motor and gradient motor) is broken, whether HR tracking is still possible or not. We also investigate whether offset free tracking is achievable or not under faulty conditions [22, 23].
