A qLPV Nonlinear Model Predictive Control with Moving Horizon Estimation

1.1. Contributions and rganization

oAs detailed in the prequel, the topic of NMPC through qLPV embedding has been studied by a handful of papers and deserves further attention. It seems that the development of these strategies can surely be established as a competitive category for nonlinear MPC design, regarding time-critical applications.

Pursuing this matter and motivated by the previous discussion, this paper proposes an alternative formulation to the recent algorithm by Cisneros and Werner^[22]. In their work, the nonlinear proxy has to be evaluated online w.r.t. to the future state evolution prediction originated through the QP. The alternative procedure proposed herein relies on approximating the nonlinear proxy by a time-varying auto-regressive function, whose parameters are found through another QP, based on a Moving-Horizon Estimation (MHE) method. This alternative is able to slightly boost the numerical performances of the whole algorithm, which only needs to evaluate three QPs to find the control law.

Accordingly, the contributions presented are the following:

• An alternative formulation for NMPC is proposed: using qLPV embeddings, the MPC operates together with an MHE layer, which estimates the future behaviour of the scheduling parameters.

• The convergence of the algorithm is demonstrated.

• A benchmark example is used to demonstrate the effectiveness of the proposed scheme, in terms of performance and numerical burden.

Regarding organization, this paper is structured as follows. Idn the next Section, the preliminaries and formalities are presented, especially regarding how nonlinear processes can be embedded into a qLPV representation through LDI. Moreover, the problem setup regarding MPC applied to such qLPV model is presented. Furthermore, the proposed MHE-MPC formulation and the discussion about stability and an offline LMI-solvable remedy for the computation of the terminal ingredients is addressed. Lastly, Section simulation results and general conclusions are drawn.

1.2. Basic efinitions

Definition 1. Nonlinear Programming Problem

Consider an arbitrary real-valued nonlinear function f_c(x_c). A Nonlinear Programming Problem determines the vector x_c that minimizes f_c(x_c) subject to g_i(x_c) ≤ 0, h_j(x_c) = 0 and $$ x_c\in\mathcal{X}_c $$, where g_i and h_j are also nonlinear.

Definition 2. Quadratic Programming Problem

A Quadratic Programming Problem (or simply Quadratic Problem) is a linearly constrained mathematical optimization problem of a quadratic function. A QP is a particular type of NP. The quadratic function may be defined with respect to several variables, all of which may be subject to linear constraints. Considering a vector$$ c\in\mathbb{R}^{n_c} $$, a symmetric matrix $$ Q_c\in\mathbb{R}^{n_c\times n_c} $$, a real matrix $$ A_{ineq}\in\mathbb{R}^{m_c\times n_c} $$, a real matrix $$ A_{eq}\in\mathbb{R}^{m_c\times n_c} $$, a vector $$ b_{ineq}\in\mathbb{R}^{m_c} $$ and another vector $$ b_{eq}\in\mathbb{R}^{m_c} $$, the goal of a QP is to determine the vector $$ x_{c}\in\mathbb{R}^{n_c} $$ that minimizes a regular quadratic function of form$$ \frac{1}{2}(x_{c}^{T}Q_xx_c+c^Tx_x) $$subject to constraints A_ineqx_c ≤ b_ineq and A_eq^x_c = b_eq. The solution x_c to this kind of problem is found by many solvers seen in the literature, based on Interior Point algorithms, quadratic search, etc.

1.3. Notation

In this work, the set of non-negative real number is denoted by $$ \mathbb{R}_+ $$, whist the set of non-negative integers including zero is denoted by $$ \mathbb{N} $$. The index set $$ \mathbb{N}_{[a,b]} $$ represents $$ \{i\in\mathbb{N}\mid a\le i\le b\} $$, with 0 ≤ a ≤ b. The identity matrix of size j is denoted as $$ \mathbb{I}_j $$; col {a, b, c} denotes the vectorization (collection) of the entries and diag{v} denotes the diagonal matrix generated with the line vector v.

The value of a given variable v(k) at time instant k + i, computed based on the information available at instant k, is denoted as v(k + i\k).

$$ \mathcal{K} $$ refers to the class of positive and strictly increasing scalar functions that pass through the origin. A given function $$ f:\mathbb{R}\to \mathbb{R} $$ is of class $$ \mathcal{K} $$ if f(0) = 0 and lim_ξ→+∞f(ξ) → +∞. A real-valued scalar function $$ \phi :\mathbb{R}_+\to \mathbb{R}_+ $$ belongs to class $$ \mathcal{K}_\infty $$ if it belongs to class $$ \beta :\mathbb{R}_+\times\mathbb{R}_+\to \mathbb{R}_+ $$ and it is radially unbounded (this is lim_s→∞ϕ(s) → +∞. A function $$ \beta :\mathbb{R}_+\times \mathbb{R}_+ \to \mathbb{R}_+ $$ belongs to class $$ \mathcal{KL} $$ if, for each fixed $$ m\in\mathbb{R}_+,\beta (\cdot ,m)\in \mathcal{K} $$ and, for each fixed $$ s\in\mathbb{R}_+ $$, β(s,·) is non-increasing and holds for lim_m→+∞β(s,m) = 0.

$$ C^n $$ denotes the set of all compact convex subsets of $$ \mathbb{R}^n $$. A convex and compact set $$ X\in C^n $$ with non-empty interior, which contains the origin, is named a PC-set. A subset of $$ \mathbb{R}^n $$ is denoted a polyhedron if it is an intersection of a finite number of half spaces. A polytope is defined as a compact polyhedron. A polytope can be analogously represented as the convex hull of a finite number of points in $$ \mathbb{R}^n $$. A hyperbox is a convex polytope where all the ruling hyperplanes are parallel with respect to their axes.

Finally, consider two sets $$ A\subset \mathbb{R}^n $$ and $$ B\subset \mathbb{R}^n $$. The Minkowski set addition is defined by A ⊕ B := {a+b|a ∈ A, b ∈ B},while the Pontryag in set difference is defined by A ⊖ B := {a |a ⊕ B ⊆ A}. The cartesian product between two sets is defined as $$ \mathcal{A}\times\mathcal{B} $$.

2.1. The Nonlinear System and its qLPV Embedding

We consider the following generic discrete-time nonlinear system:

where $$ k\in\mathbb{N} $$ represents the sampling instant, $$ x:\mathbb{N}\to\mathcal{X}\subset \mathbb{R}^{n_x} $$ represents the system states, $$ u:\mathbb{N}\to\mathcal{U}\subset \mathbb{R}^{n_u} $$ is the vector of control inputs and $$ y:\mathbb{N}\to\mathcal{Y}\subseteq \mathbb{R}^{n_y} $$ stands for the measured outputs of the process.

We begin by characterizing this process, which should satisfy the following key Assumptions:

Assumption 1.The admissible zone for the states is given by a 2-norm upper bound on each entry x_j, this is:

Assumption 2.The admissible region for the control inputs is given by a 2-norm upper bound on each entry u_j, this is:

Assumption 3.The nonlinear maps $$ f:\mathcal{X}\times\mathcal{U}\to\mathcal{X} $$ and $$ f_y:\mathcal{X}\times \mathcal{U}\to \mathcal{Y} $$ are continuous and continuously differentiable with respect to x, i.e. class C^∞

Assumption 4.This nonlinear system is controllable in terms of x and y through the input trajectory u.

To represent any nonlinear system, as the one in Eq. (1), under a qLPV formalism, this last Assumption must be verified, since it is the LDI property that furnishes the settings for such representation.

The LDI property is as follows: suppose that, for each x, u and y and for every sampling instant k, there exists a matrix $$ \mathcal{H}(x,u,k):\mathcal{X}\times\mathcal{U}\times\mathbb{N}\to\mathcal{H} $$ such that

where $$ \mathcal{H}\subseteq \mathbb{R}^{(n_x\times (n_x+n_u)} $$ is the set within which the LDI property holds. Then, when there exists a matrix H(·) that verifies Eq. (4), the nonlinear model from Eq. (1) can be equivalently expressed as:

which is a qLPV formulation where $$ f\rho:\mathcal{X}\times\mathcal{U}\to\mathcal{P}\subset \mathbb{R}^{n_p} $$ represents the endogenous nonlinear function for the scheduling parameters. Note that ρ(k) is bounded and known online at each instant k, but generally unknown for any future instant k + j, $$ \forall j \in \mathbb{N}_{[1,\infty]} $$.

We consider that the qLPV scheduling parameters have bounded rates of variations, this is: ρ(k + 1) = ρ(k) + ∂ρ(k + 1), being $$ \partial \rho \in \partial \mathcal{P} $$, ∀ k their variation rates. This is very reasonable for any practical application. To assume that ρ varies arbitrary implies in quite conservative control synthesis ^[23].

3.1. Process constraints

The qLPV MPC problem solution proposed in this paper is formulated with respect to the scheduled state evolution equation, as gave Eq. (15), with H(P_k), g(·) and κ(·) being as passed as inputs to the resulting MPC optimization. This means that the MPC optimization does not treat state evolution X_k as optimization variables, but the whole problem is formulated singularly in terms of U_k.

For this reason, the admissibility process constraints $$ u(k+i-1\mid k)\in \mathcal{U} $$ and $$ x(k+i-1\mid k)\in \mathcal{X} $$ are conversely written in the following fashion, w.r.t. $$ \mathcal{A}(\cdot) $$, $$ \mathcal{B}(\cdot) $$, U_k and x(k), instead of u(k + j) and x(k + j):

The above formulations also facilitates the inclusion of slew rates on u, i.e. constraints on the control variation δu(k + i) = u(k + i) – u(k + i – 1), which can be done directly by adapting the vector-wise set $$ \mathcal{\breve{U}} $$.

The terminal constraint x(k + Np|k) ∈ X_f is stated in terms of U_k as:

where

Additional output constraints are also easily formulated. If the process has some outputs y_c (which are not necessarily equal to y, but could be) that must be hard constrained, i.e. $$ y_c\in \mathcal{Y}_c $$. Then, assume that these outputs can be described as:

In what follows, we take

Then, the additional constraint is formulated as follows [Note that the last $$ y_c(k+N_p\mid k) $$ is not constrained due to unavailability of $$ \rho (k+N_p) $$ inside $$ P_k $$. This issue is amendable by taking a longer scheduling prediction guess $$ P_k $$, but out of the scope.]:

thus

3.2. Reference tracking

Finally, before showing the proposed mechanism to guess P_k and solve the qLPV MPC problem, a comment must be made regarding reference tracking. The considered cost function J(x, u, k) from Eq. (6) (or its vector form of Eq. (16)) is set in order to minimize the variations from the desired set-point target p_r= (x_r, u_r).

The majority of processes that require reference tracking, require it regarding the controlled outputs and not the states. This is, to ensure that y(k) tracks some steady-state value y_r. Since the controlled outputs in Eq. (5) are given by

we can find a linear (parameter varying) combination of the states x that, if tracked, ensures that y(k) → yr. We denote the output tracking target as $$ p_r^y=(y_r,u_r^y), $$, which is known.

Then, following the lines of previous reference tracking frameworks ^[24-26], we use an offline reference optimization selector, which is set to find the set-point target p_r that abides to the constraints and ensures an output tracking of y_r. This nonlinear optimization procedure is as follows:

This procedure ensures some steady-state x_r= A(ρ_r)x_r + B(ρ_r)u_r ) that abides to the states constraints and guarantees that the output tracking goal y_r is followed.

Note that this optimization procedure has a steady-state target point p_r= (x_r, u_r) as output, and not the full state and input trajectories towards this target.

The state reference selection problem can be solved online, at each sampling instant, if the output reference goal y_r changes over time. By doing so, an additional computational complexity appears, which can be smoothed if the scheduling parameter guess P_k is used instead of solving the nonlinear optimization itself. A full discussion on periodically changing reference tracking for nonlinear MPC has been recently presented ^[27]. The focus of this paper is constant reference signals, either given in terms of states x_r or outputs y_r

It is important to notice that, in order for the method to hold, the state reference x_r must be contained inside the terminal set of the MPC problem from Eq. (8). This ensures that the stability and recursive feasibility guarantees (as verified in Section Stability and Offline Preparations) hold.

4.1. Convergence property

In order to demonstrate the convergence of the proposed method (as in its implementation form given in Algorithm 1), we will proceed by verifying a well-known results for Newton based SQPs from the literature ^[28-30]. This is the same path followed in a previous paper ^[22], which invokes established results to demonstrate that, under certain conditions, the MHE-MPC mechanism that is solved at each iteration is equivalent to a quadratic sub-problem used in standard Newton SQP. Therefore, a local convergence property is readily found.

Algorithm 1. Proposed qLPV MPC

Proposition 2.A quadratic sub-problem program of SQP algorithms is derived by a second-order approximation of the SQP optimization cost and a linearization of its constraints.

Proof. Found in ^[28]. □

For illustration purposes regarding this matter, we consider the following generic NP (as given in Definition 1):

This problem can be given as a quadratic sub-problem directly, as follows:

where H_fc (x_c) denotes the Hessian of the optimization cost f_c(x_c) and ∇h_j(x_c) and ∇g_i(x_c) denote divergent operators. We note that this sub-problem is evaluated at a given solution estimate x̅_c (at some given iteration), for which $$ \breve{x}_c=x_c-\bar{x}_c $$.

Regarding the proposed MHE-MPC mechanism, we can easily show that if either simple Jacobian linearization or Linear Differential Inclusion are used to find a qLPV model (as in Eq. (5)) for the nonlinear system (as in Eq. (1)), then, the proposed mechanism iterates in equivalence to a Newton SQP sub-problem. Notice how such sub-problem in Eq. (27) is identical to the MHE-MPC optimization given through the consecutive iterations of the MHE (Eq. (25)) and the MPC (Eq. (8)). The terminal constraint in the MPC optimization adds no convergence trade-off.

Thence, it follows that if local convergence of the equivalent Newton SQP can be established, the proposed MHE-MPC also yields convergence. The sufficient conditions for local convergence of a Newton SQP sub-problem at x_c =x̅_c, as given by prior references ^[29-32], are that (i) the problem is set simply with equality constraints (not the MPC case) or that (ii) the subset of active inequality constraints are known before the optimization solution. The second condition is also not true for general MPC paradigms. However, one can iterate the sub-problem until convergence is found at another point $$ x_c=\overset{=}{x}_c $$, as previously discussed ^[22,33,34].

In practice, the proposed MHE-MPC will not be set to freely iterate until the convergence of P_k. This is not desirable because the number of iterations needed for convergence may require more time than the available sampling period. Therefore, a stop criterion is added to the mechanism, so that iterations stop at a given threshold. A warm-start is also included by shifting the result regarding X_k and P_k from one sampling k as the initial guess for the optimization at k + 1, which ensures that the proposed algorithm reaches convergence after a few discrete-time samples. We note that convergence of Newton SQP sub-problems with warm-start have been assessed ^[33,34] and shown to be practicable for real-time schemes.

5. STABILITY AND OFFLINE PREPARATIONS

In this Section, we offer a Theorem to construct the terminal ingredients of the MPC algorithm: (a) the terminal set within which x(k + N_p|k) is bounded to, and (b) the terminal offset cost V(x(k + N_p|k)) minimized by the MPC.

Since the establishment of terminal ingredients toolkit ^[2,35] as the key way to ensure stability and recursive feasibility of state-feedback predictive control loops, MPC grew on both theory and industrial practice.

The usual approach with terminal ingredients resides in some ensuring that conditions are met by (a) the terminal set X_f and (b) the terminal cost V(x(k + N_p|k)) with respect to a nominal state-feedback controller u(k) = kⁿx(k), which is usually the unconstrained solution of the MPC problem. For the tracking case, the nominal feedback is given by u(k) = kⁿ (x(k) – x_r) (and so is the terminal constraint (x(k + N_p|k) – x_r) ∈ X_f and the terminal cost V(x(k + N_p|k) – x_r)). Accordingly we develop a sufficient stability condition for the proposed MHE-MPC mechanism in order to verify these conditions.

Firstly we consider that there exists a parameter-dependent nominal state-feedback gain $$ K^n:\mathbb{R}^{n_\rho}\to\mathbb{R}^{n_x\times n_u} $$. For demonstration simplicity and notation lightness, we will proceed with x_r null [The tracking equivalency is easily done with $$ \frac{dx_r}{dt} =0 $$ and by computing the qLPV model with the states dynamics are given with respect to $$ x_{tracking}(k)=x(k)-x_r $$.].

This nominal controller is purely fictional, used to demonstrate stability and recursive feasibility properties of the proposed MHE-MPC mechanism. Anyhow, it stands for the infinite-horizon LPV Linear Quadratic Regulator (LQR) solution, which verifies $$ K^{\mathrm{n}}(\rho)=\text{arg min} _{K \in \mathbb{R}^{n x \times n u}}\left(\sum_{i=1}^{+\infty}\|x(k+i \mid k)\|_{Q}^{2}+\|K x(k+i-1)\|_{R}^{2}\right) $$ and the admissibility of x(k + i) and u(k + i – 1).

Of course, there is a complexity barrier to solve this problem, because the states have parametric nonlinearities that impact their trajectories (the qLPV scheduling parameters). Therefore, we determine this nominal feedback gain together with the terminal ingredients, which are also taken as parameter-dependent on ρ. We consider, for regularity, an ellipsoidal set as the terminal constraint, which is given by:

This ellipsoid is centered at the origin and has a radius of α_p. Furthermore, this terminal set is a sub-level set of terminal cost V(·), which is taken as a Lyapunov function as follows:

This parameter-dependent nominal feedback gain Kⁿ(ρ)) and the parameter dependent terminal ingredients verbalized through the symmetric parameter dependent Lyapunov matrix P(ρ) are so that the following input-to-state stability Theorem is guaranteed.

Theorem 1.Input-to-State Stable MPC^[2,22,35,36]

Let Assumptions 4 and 7 hold. Assume that a nominal control law u = Kⁿ(ρ)x exists. Consider that the MPC is in the framework of the optimization problem in Eq. (8), with a terminal state set given by X_f(ρ) and a terminal cost V(x,ρ). Then, input-to-state stability is ensured if the following conditions are hold $$ \forall\rho\in \mathcal{P} $$:

• (C1) The origin lies in the interior of X_f(ρ);

• (C2) Any consecutive state to x, given by (A(ρ) + B(ρ)Kⁿ(ρ)) x lies withinX_f(ρ) (i.e. this is an invariant set);

• (C3) The discrete algebraic Ricatti equation is verified within this invariant set, this is, ∀ x ∈ X_f (ρ(k)):

  $$ \begin{align*} V((A(\rho(k))+B(\rho(k)) K^{n}(\rho(k))) x, \rho(k+1))-V(x, \rho(k)) \\ \leq-x^{T} Q x-x^{T}\left(K^{n}(\rho(k))\right)^{T} R K^{n}(\rho(k)) x . \end{align*} $$

• (C4) The image of the nominal feedback always lies within the admissible control input domain:.

• (C5) The terminal set X_f (ρ) is a subset of the admissible state domain $$ \mathcal{X} $$.

Assuming that the initial solution of the MPC problem $$ U^\star $$, computed with respect to the initial state x(0), is feasible, the MPC algorithm is indeed recursively feasible, asymptotically stabilizing the state origin.

Proof. Provided in Appendix Proof of Theorem 1. □

In order to find some nominal state-feedback gain Kⁿ(ρ), some terminal set X_f and some terminal offset cost V(·), an offline LMI problem is proposed in the sequel. This LMI problem is such that a P(ρ) positive definite parameter-dependent matrix is found to ensure that the conditions of Theorem 1 are satisfied. Due to condition (C3), the LMI is solved over a sufficiently dense grid over ρ, consider its admissibility domain $$ \mathcal{P} $$. We note that (C3) is a time-variant condition, which depends explicitly on ρ(k) and ρ(k + 1) due to the nature of the parameter dependent V(·).

This LMI problem is provided through the following Theorem, which aims to find the largest terminal set X_f that is invariant under the nominal control policy u(k) = Kⁿ(ρ(k))x(k) for all k, while remaining admissible, i.e. $$ \mathcal{P} $$, $$ \forall x \in \mathcal{X} $$ and $$ \rho\in \mathcal{P} $$. Note that the largest ellipsoidal set as in the form of Eq. (28) is posed through the maximization of α_p.

Theorem 2.Terminal Ingredients^[22,37]

The conditions (C1)-(C5) of Theorem 1 are satisfied if there exist a symmetric parameter-dependent positive definite matrix $$ P\rho:\mathbb{R}^{n_p}\to \mathbb{R}^{n_x\times n_x} $$ a parameter-dependent rectangular matrix $$ W\rho:\mathbb{R}^{n_p}\to \mathbb{R}^{n_u\times n_x} $$ and a scalar $$ 0<\hat{\alpha}_P \in \mathbb{R} $$ such that Y(ρ) = (P(ρ))^–1> 0, W(ρ) = Kⁿ (ρ)Y (ρ) and that the following LMIs hold for all$$ \rho\in \mathcal{P} $$and$$ \partial \rho\in \partial\mathcal{P} $$, while minimizingα̂_P:

where I_j denotes the j-throw of the identity matrix$$ \mathbb{I} $$. In LMI (31), it is given w.r.t. to an identity$$ \mathbb{I}_{n_u} $$, while in LMI (32), it is given w.r.t. to an identity$$ \mathbb{I}_{n_x} $$.

Proof. Refer to^[37]. □

We must note that the above proof demonstrates that the solution of the LMIs presented in Theorem 2 ensure a positive definite parameter dependent matrix P(ρ) which can be used to compute the MPC terminal ingredients V(·) and X_f such that input-to-state stability of the closed-loop in guaranteed, verifying the conditions of Theorem 1. Furthermore, when the MPC is designed with these terminal ingredients, for whichever initial condition x(0) ∈ X_f it starts with, it remains recursively feasible for all consecutive discrete time instants k > 0.

Anyhow, Theorem 2 provides infinite-dimensional LMIs, since they should hold for all $$ \rho\in\mathcal{P} $$ and for all $$ \partial \rho\in \partial \mathcal{P} $$. To address this issue, one can handle the LMIs considering an sufficiently dense grid ^[38] of $$ \mathbb{R}^{n_p\times n_p} $$ points in $$ \mathcal{P} \times \partial\mathcal{P} $$, for which the LMIs must be enforced. This solves the infinite dimension of the problem, which is converted into an n_g-dimensional LMI problem, being n_g the number of grid points. For this solution to be practically implementable, continuity on matrices A(ρ) and B(ρ) should be verified. We must also note that parameter-dependency on ρ may be dropped if the system is quadratically stabilizable, which is a conservative assumption.

6. BENCHMARK EXAMPLE

In this Section, we pursue the application of the proposed MHE-MPC mechanism with terminal ingredients found through Theorem 2. For such, we consider the application of our control method upon a benchmark system, detailed in the sequel.

6.4. Simulation results

Considering a realistic nonlinear CSTR model, we obtain simulation results to demonstrate the effectiveness of the proposed control scheme. The following results were obtained in a 2.4 GHz, 8 GB RAM Macintosh computer, using Matlab, yalmip and Gurobi solver.

First, we show how the MHE operation is able to accurately predict the behaviour of the scheduling trajectory, as depicts Figure 2. At each instant k, the MHE provides estimation for P_k, which is composed of the following N_p entries of the scheduling variables. In this Figure, we observe the real scheduling variables ρ₁ and ρ₂ (dot-dash black line) and the estimates P_k provided at different samples (coloured x-marked lines). Within some samples, we can see that the predicted trajectory converges to the real one, which confirms the effectiveness of the MHE operation.

Figure 2. MHE: scheduling trajectory estimates.

Based on the scheduling behaviours predicted by the MHE loop, the predictive controller determines the control input (Figure 4) in order to drive the system states from x₀ to the reference goal x_r. The corresponding state trajectories are depicted in Figure 3, which also shows the state admissibility set and the terminal set $$ \mathcal{X} $$ constraint X_f (a parameter-dependent ellipsoid generated via Theorem 2). As one can see, the behaviour of the process variables is a smooth trajectory towards x_r.

Figure 3. State trajectories, state admissibility set, terminal set.

Figure 4. MPC: control input.

Finally, we demonstrate the dissipating properties of the proposed control scheme. In Figure 5, we show the evolution of MPC stage cost J over time. As expected, J decays and converges to the origin, which verifies the dissipation properties required by Theorem 1.

Figure 5. MPC: stage cost.

7. CONCLUSIONS

In this paper, a new method for the fast, real-time implementation of Nonlinear Model Predictive Control is proposed. The method provides a near-optimal, approximated solution, which is found through the online operation of sequential Quadratic Programming Problems. The main necessary argument to develop the method is that the nonlinear process should be described by quasi-Linear Parameter Varying model, for which the embedding is ensured through a scheduling proxy. Then, the online operations resides in the consecutive operation of the MPC program together with a Moving-Horizon Estimation scheme, which is used to match the future values of the scheduling proxy along the prediction horizon, which are unknown. Input-to-state stability and recursive feasibility properties of the algorithm are ensured by parameter-dependent terminal ingredients, which are computed offline. Using a benchmark example, the method is tested. We highlight that it proves itself more effective for stronger nonlinearities in the qLPV scheduling proxy, for which the MHE scheme operates faster than the application of the scheduling proxy upon each entry of the future state variables, as in many other techniques. For future works, the Authors plan on assessing the issue of periodically-changing (possibly unreachable) output reference signals.