3.1. Multi-loop integral controllability analysis for HR responses to treadmill exercise

In [8], the multi-loop PI controller has been designed for the regulation of HR for treadmill exercise. Now, we consider the case when one of the actuators is in faulty condition. First, we introduce a definition of MIC, which is a direct extension of DIC for a non-square 2ISO process [19].

As shown in Figure 1, assume the HR response can be described by the following equations with an input vector u∈ R<sup>2</sup> and an output vector y∈R<sup>1</sup> :

$$\begin{cases} \dot{\mathbf{x}} &=& f(\mathbf{x}, \boldsymbol{\mu}) \quad \mathbf{x} \in \mathbf{X} \subset \mathbb{R}^n, \boldsymbol{\mu} \in \mathbf{U} \subset \mathbb{R}^2\\ \mathbf{y} &=& g(\mathbf{x}, \boldsymbol{\mu}) \quad \mathbf{y} \in \mathbf{Y} \subset \mathbb{R}^1 \end{cases} \tag{1}$$

where the state x tð Þ is determined by its initial value xð Þ0 and the input function u tð Þ. Considering the system (1) has equilibrium at origin, that is, fð Þ¼ 0; 0 0 and gð Þ¼ 0; 0 0, if the equilibrium xe is not at origin, a translation is then needed by redefining the state x as x � xe [19, 20].
