*3.3. PI-PBC Design*

To obtain a general control law to guarantee closed-loop stability in the sense of Lyapunov, let us employ the following PI control structure

$$
\vec{u} = -\mathcal{K}\_p \vec{y} + \mathcal{K}\_i \mathbf{z},\tag{15a}
$$

$$
\dot{z} = -\ddot{y},
\tag{15b}
$$

where <sup>K</sup>*p* 0 and K*i* 0 are the proportional and integral gain matrices and *z* is an auxiliary vector of variables related to the integral action.

To prove stability with the control law defined by Equation (15), let us modify the candidate Lyapunov function in Equation (12) as follows

$$\mathcal{W}(\vec{x}, z) = \mathcal{V}(\vec{x}) + \frac{1}{2} \left( z - z\_o \right)^T \mathcal{K}\_i \left( z - z\_o \right), \tag{16}$$

with *zo* = K−<sup>1</sup> *i u*- and its derivative is

$$\dot{\mathcal{W}}(\vec{x}, \vec{y}) = -\vec{\mathfrak{x}}^T \mathcal{R}\vec{x} - \vec{\mathfrak{y}}^T \mathcal{K}\_P \vec{y} \le 0,\tag{17}$$

which proves that the control input in Equation (15) guarantees stability in the sense of Lyapunov for closed-loop operation. Observe that in Equation (16), we consider that K*i* = K*Ti*.

**Remark 4.** *The control input Equation (15) can guarantee asymptotic stability in the sense of Lyapunov as proved in [15] by referring to Barbalat's lemma [14].*

### **4. Control Structure and Physical Constraint**

This section presents the mathematical structure of the control laws for active and reactive power support with SCES systems, as well as the physical constraint that imposes the interconnection of a supercapacitor for energy storage applications.
