**Algorithm 1**

**Offline** Step 1—Initialize **Θ**(0) and **Qˆ** (0) **Online** Step 2—Obtain *ρ*1(*k*), *y*(*k*) and *u*(*k*) Step 3—Construct *γT*(*k*) vector Step 4—Calculate scalar *c* Step 5—Obtain vector *σ*(*k*) Step 6—Obtain **Θ**(*k*) Step 7—Obtain **Qˆ** (*k*) Step 8—Obtain *ρ*1(*k*) Step 9—Set *k* = *k* + 1, If *k* < *Np* go to Step 10, else, go back to step 3 Step 10—Construct **Pˆ**(*k*) = - *ρ*1(*k*), *ρ*1(*k* + 1), ..., *ρ*1(*k* + *Np*) 

## **6. Quadratic Stability in MPC-LPV Approach**

To ensure Quadratic Stability in the MPC-LPV approach, system (7) can be considered to be a parametric uncertain system. In parametric uncertain systems, the scheduling variable is limited to vary in a range Δ*ρ*1*min* ≤ Δ*ρ*1*<sup>k</sup>* ≤ Δ*ρ*1*max*. To ensure stability in parametric uncertain systems, the following condition needs to be met as presented in [31].

$$\left(\mathbf{A}(\rho\_1) + \mathbf{B}\mathbf{K}\right)^T \mathbf{P}\left(\mathbf{A}(\rho\_1) + \mathbf{B}\mathbf{K}\right) - \mathbf{P} < 0\tag{28}$$

which is the Riccati Equation for parametric uncertain systems where **P** > 0 is a positive definite matrix of appropriate dimensions and **K** a static feedback gain matrix. Then, (28) can be pre- and post-multiplied by a matrix **Q** = **P**−<sup>1</sup> and **KQ** = **R** to obtain:

$$\left(\mathbf{Q}\mathbf{A}^T(\rho\_1) + \mathbf{R}^T\mathbf{B}^T\right)\mathbf{Q}^{-1}\left(\mathbf{A}(\rho\_1)\mathbf{Q} + \mathbf{B}\mathbf{R}\right) - \mathbf{Q} < 0\tag{29}$$

To cope with the MPC paradigm, the Schur complement is applied to (29) to obtain the following LMI:

$$
\begin{bmatrix}
\mathbf{Q} & \mathbf{Q}\mathbf{A}^T(\rho\_1) + \mathbf{R}\mathbf{B} \\
\mathbf{A}(\rho\_1)\mathbf{Q} + \mathbf{B}\mathbf{R} & \mathbf{Q}
\end{bmatrix} > 0
\tag{30}
$$

for every possible value of *ρ*<sup>1</sup> at time instant *k* which leads to an infinite number of LMI. However, as system (7) is considered to be a parametric uncertain system, (30) can be evaluated on the vertex of matrix **A** to consider the worst-case scenarios. Therefore, (30) can be written as:

$$
\begin{bmatrix}
\mathbf{Q} & \mathbf{Q}\mathbf{A}\_{i,j}^T(\rho\_1) + \mathbf{R}\mathbf{B} \\
\mathbf{A}\_{i,j}(\rho\_1)\mathbf{Q} + \mathbf{B}\mathbf{R} & \mathbf{Q}
\end{bmatrix} > 0\tag{31}
$$

The previous condition must be met <sup>∀</sup>*<sup>j</sup>* <sup>∈</sup> [*k*, *<sup>k</sup>* <sup>+</sup> *Np*] and <sup>∀</sup>*<sup>i</sup>* <sup>∈</sup> [1, 2*<sup>l</sup>* ], where *l* is the number of scheduling variables *ρ*1, **Q** > 0 is a positive definite stability matrix to be determined, and **KQ** = **R**, where **K** is the static feedback gain matrix. With these adjustments, the number of LMI to be solved is now finite and equal to 2*<sup>l</sup> Np*. Since there is the consideration of a static feedback gain, the control law is determined as *u*(*k*) = **Kx**(*k*), but to comply with the MPC paradigm, the previous expression can be considered to be an inequality as *u*(*k*) < **Kx**(*k*). This leads to a conservative MPC performance due to the limitations of the input variable. However, this problem will be addressed in Section 7 with the inclusion of terminal sets. Therefore, using (17)–(19) and (31) the optimization problem needs to find the optimal control sequence at each time step *k* is the following:

$$\min\_{\mathbf{U}} J \text{ s.t.(18), (19) & (31)}\tag{32}$$
