## 2 Problem Statement  
In this work, we consider the class of nonlinear systems described by the following state-space model:  
\[\left\{ \begin{array}{rcl}\dot{x}(t)&=&f(x(t),u(t))\\ y(t)&=&g(x(t))\end{array} \right. \tag{1}\]  
where \(x(t)\in\mathbb{R}^{n}\) is the state vector, \(u(t)\in\mathbb{R}^{m}\) is the manipulate input vector and \(y(t)\in\mathbb{R}^{p}\) is the output. In order to solve the problem of control in the same way as the linear quadratic regulator, first the continuous nonlinear model of the process \((1)\) is discretized using the Euler integration method and re-arranged into the state-dependent coefficient form [12] as:\[\left\{ \begin{array}{lcl}x(k+1)&=&A(x(k))x(k)+B(x(k))u(k)\\ y(k)&=&C(x(k))x(k)\end{array} \right. \tag{2}\]  
where \(x(k)\in\mathbb{R}^{n}\), \(y(k)\in\mathbb{R}^{p}\) and \(u(k)\in\mathbb{R}^{m}\) are the state, the output and the input vectors respectively at the \(k\)th sampling instant. The general formulation of the problem (Equations (3)-(8)) consists of the optimization of a cost function that represents the control and economic objectives, subject to a set of constraints. The objective function includes the penalization of control error, the penalization of control efforts and a term \((f_{exo})\) that accounts for the economic objectives:  
\[ \begin{split} J=&\sum_{i=1}^{n_{y}}||w_{1}(r(k+i|k))-y(k+i|k) ||_{2}^{2}\\ &+\sum_{i=0}^{n_{x}-1}||w_{2}\Delta u(k+i|k)||_{2}^{2}\\ &+\sum_{i=1}^{n_{y}}w_{3}f_{exo}(u(k+i|k),y(k+i|k))\end{split}  \tag{3}\]  
and the minimization of \(J\) at each sampling time is subject to the following constraints:  
\[u_{min} \leqslant u(k+i|k)\leqslant u_{max},\quad i=0,\ldots,n_{u}-1 \tag{4}\] \[y_{min} \leqslant y(k+i|k)\leqslant y_{max},\quad i=1,\ldots,n_{y}\] (5) \[\Delta u_{min} \leqslant\Delta u(k+i|k)\leqslant\Delta u_{max},\quad i=0,\ldots, n_{u}-1\] (6) \[x(k+i|k) =A(x(k+i|k))x(k+i|k)+B(x(k+i|k))u(k+i|k),\quad k\geqslant 1\] (7) \[y(k+i|k) =C(x(k+i|k))x(k+i|k),\quad k\geqslant 1 \tag{8}\]  
where \(n_{y}\) and \(n_{u}\) are the output and input horizon, respectively; \(u(k+i|k)\) is the control input computed at time \(k\) to be predicted at time step \(k+i\); \(y(k+i|k)\) is the output prediction at time step \(k+i\); \(r\) is the desired value of the output; \(\Delta u(k+i|k)=u(k+i|k)-u(k+i-1|k)\); \(w_{1}\), \(w_{2}\) and \(w_{3}\) are positive definite matrices. Note that the different terms of the cost function must be weighted such that the economic criterion and the dynamic compensation of the output error have a similar influence on the values of the overall cost. It is assumed that the state variables are measurable. The control strategy proposed in this paper is described schematically in Figure 1, This controller is achieved by using a new closed loop nonlinear predictive control paradigm that combines an unconstrained economic nonlinear Generalized Predictive Control law \(F(x(k))\) with the parameterization \(c(k)\), associated with the closed loop paradigm that allows taking into account the process constraints and improving the performance of the controller. Some specific characteristics of this control strategy are:  
* The optimizer shown in the control scheme (Figure 1) constitutes the economic nonlinear closed loop paradigm. The predicted control moves are centered around an unconstrained stabilizing control law, \(u(k)=F(x(k))x(k)\), over the whole prediction horizon, and some additive degrees of freedom, \(c(k)\), are added over a finite horizon to handle constraints. The resulting control \(u(k)=F(x(k))x(k)+c(k)\) is applied to the plant. * In the objective function the nonlinear economic term is replaced by its gradient making this function a quadratic one. * The prediction model, a nonlinear phenomenological model of the plant, is written as a state dependent coefficient model, also called extended linearization, which consists of factorizing the nonlinear system in a linear structure with state dependent matrices.