## 1 Introduction  
The Model Predictive Control (MPC) algorithms during the calculation of the control signals use the model of the control plant directly, predicting the behavior of the process. Thanks to such an approach they can be successfully applied in control systems of control plants with difficult dynamics (delay, inverse response), with constraints and for MIMO (Multiple-Input Multiple-Output) processes, see for example, References [1; 2; 3; 4; 5; 6; 7]. The simplest MPC algorithms are based on linear control plant models; see for example, References [1; 8]. Unfortunately, if a control plant is nonlinear and the MPC algorithm should operate for different operating points, as in control system structures with steady-state setpoint optimization (see e.g., References [7; 9; 10]), then the algorithm based on a linear model may be insufficient in terms of control performance. In such a case operation of the control system can be usually improved using the MPC algorithm based on a nonlinear model. On the other hand, if a nonlinear process model is used in the MPC algorithm directly (such algorithm is called NMPC--Nonlinear MPC), then the optimization problem solved at each algorithm iteration to generate the control signals is a nonlinear optimization problem; see for example, References [11; 12; 13]. Thus, it is usually non-convex, hard to solve, and time-consuming computational task. Moreover, the time needed to find the solution cannot be foreseen in advance. One of the ways to cope with this problem is to apply so called fast NMPC algorithms in which the solution is generated faster than in the standard approach, but unfortunately, at the expense of suboptimality; see for example, References [14; 15; 16]. One of such approaches is the explicit one in which most of calculations are done off-line; see for example, References [17; 18; 19]. It is an interesting approach and it is optimal if the linear model is used [20; 21; 22]. However it has a serious drawback--the complexity of the controller grows significantly with the number of inequality constraints in the problem. When the fuzzy models are used, then of course the nonlinear optimization can be applied in the algorithm. However the form of the model is such that easier to solve algorithms can be formulated. Among the algorithms exploiting the structure of the fuzzy models there is a group which is based on Linear Matrix Inequalities (LMIs); see for example, References [23; 24; 25; 26; 27]. The other group is based on the classical fuzzy Takagi-Sugeno approach in which a few algorithms based on linear models are used to form the fuzzy controller; see for example, References [28; 29]. The other approach consists in online linearization of the process model at each MPC algorithm iteration; see for example, References [7; 30; 31; 32; 33; 34; 35; 36]. Then prediction is formulated in such a way that it is linear relative to decision variables in the optimization problem solved at each iteration of the algorithm. Such form of the prediction leads to the formulation of the optimization problem as the quadratic one. In the linearization-based algorithms the prediction can be based on a classical linearization or a clever method of prediction generation can be used, exploiting structure of the nonlinear model the algorithm is based on. In the algorithms using linearization of the fuzzy Takagi-Sugeno model, described in References [31; 34], both--the free response and the dynamic matrix were obtained using the model obtained after the linearization. In References [32; 36] the model obtained after linearization was used to obtain the dynamic matrix whereas the (classical) free response was calculated using the nonlinear model. The MPC algorithms proposed in the article are an extension of the algorithm presented in Reference [36] and, like their predecessor, are based on two models--a nonlinear model and its, easy to obtain, fuzzy approximation. The nonlinear model can have any form of the model generating outputs on the base of input signals.