*3.1. Bilinear Representation*

To develop a controller based on the PI-PBC approach, let us make the following definition.

**Definition 1.** *An admissible equilibrium point (x*-*) exists for non-linear dynamical Model (4) if its variables are represented in Park's reference frame (direct- and quadrature-axis), such that*

$$0 = [\mathcal{J}\left(\mu^\star\right) - \mathcal{R}]x^\star + \mathcal{J},\tag{8}$$

*for some constant control input u*-*.*

Considering the definition of the equilibrium point, now we use some auxiliary variables to develop a PI-PBC controller as follows: *u*˜ = *u* − *u*- and *x*˜ = *x* − *x*-, where *x*˜ and *u*˜ represent the error of the state variables and control inputs.

Note that if we subtract Equation (8) from (4), the following result is obtained

$$\mathcal{D}\dot{\mathbf{x}} = \left[ \mathcal{J}\left(\mathbf{u}\right)\mathbf{x} - \mathcal{J}\left(\mathbf{u}^\*\right)\mathbf{x}^\* \right] - \mathcal{R}\ddot{\mathbf{x}}.\tag{9}$$

To simplify Equation (9), let us define the property related to bilinear systems as follows:

**Definition 2.** *The matrix product* J (*u*) *x has a bilinear structure if it can be separated as a sum as follows*

$$\mathcal{J}\left(\boldsymbol{u}\right)\mathbf{x} = \mathcal{J}\_0 + \sum\_{i=1}^{m} \mathcal{J}\_i \mathbf{x} u\_{i\prime} \tag{10}$$

*where* J0 *and* J*i are constant matrices with skew-symmetric structure.*

Now, if we consider the Definition 2 in Equation (9) and make some algebraic manipulations, then the below result is obtained:

$$\mathcal{D}\dot{\mathbf{x}} = [\mathcal{J}\_0 - \mathcal{R}]\dot{\mathbf{x}} + \sum\_{i=1}^{m} \mathcal{J}\_i \mathbf{\tilde{x}} u\_i + \sum\_{i=1}^{m} \mathcal{J}\_i \mathbf{x}^\* \,\mathbf{\tilde{u}}\_i. \tag{11}$$

**Remark 3.** *Expression (11) is the essential structure to design PI-PBC controllers for bilinear systems, as demonstrated in [14].*

### *3.2. Lyapunov's Requirements for Stability Analysis*

To guarantee that the dynamical System (11) is stable in the sense of Lyapunov for the equilibrium point *x*˜ = 0, i.e., *x* = *x*-, let us define a candidate Lyapunov function *V* (*x*˜) with hyperboloid structure as presented below

$$\mathcal{V}(\mathfrak{x}) = \frac{1}{2} \mathfrak{x}^T \mathcal{D} \mathfrak{x}.\tag{12}$$

Observe that *V* (*x*˜) meets the first two conditions of the Lyaponov's stability theorem, i.e., *V* (0) = 0, and *V* (*x*˜) > 0, ∀*x*˜ = 0. In addition, if we take the temporal derivative of Equation (12) and substitute Equation (11), the following result is obtained:

$$\dot{\mathcal{W}}(\tilde{\mathbf{x}}) = \tilde{\mathbf{x}}^T \mathcal{D} \dot{\tilde{\mathbf{x}}} = -\tilde{\mathbf{x}}^T \mathcal{R} \tilde{\mathbf{x}} + \sum\_{i=1}^m \tilde{\mathbf{x}}^T \mathcal{J}\_i \mathbf{x}^\star \tilde{u}\_{i\prime} \tag{13}$$

which implies guarantee in stability, if the second term on the right hand side of Equation (13) is negative definite or at least negative semidefinite. To simplify this expression, let us use the input–output relation *u*˜ → *y*˜ being *y*˜ the passive output as follows

$$\dot{\mathcal{V}}(\vec{x}) \le \vec{y}^T \vec{u},\tag{14}$$

where *y*˜*i* = *<sup>x</sup>*˜*<sup>T</sup>*J*ix*-

.

Note that Expression (14) can help us to design a stable controller if and only if the set of control inputs *u*˜ is selected such that this expression is always negative semidefinite. These characteristics are presented in the next section using a PI controller.
