# Advanced Construction of the Dynamic Matrix in Numerically Efficient Fuzzy MPC Algorithms
## 1 Introduction
### MPC Algorithms Based on Nonlinear Models  
Assume that we have a nonlinear process model:  
\[\mathbf{\widehat{y}}_{k}=\mathbf{f}(\mathbf{y}_{k-1},\mathbf{y}_{k-2},\ldots,\mathbf{y}_{k-n_{x} },\mathbf{u}_{k-1},\mathbf{u}_{k-2},\ldots,\mathbf{u}_{k-n_{y}})\enspace, \tag{17}\]  
where \(\mathbf{y}_{k-i}=\left[y_{k-i}^{1},\ldots,y_{k-i}^{n_{y}}\right]^{T}\) is a vector that contains output values measured at the \((k-i)\)th sampling instant and \(\mathbf{u}_{k-i}=\left[u_{k-i}^{1},\ldots,u_{k-i}^{n_{y}}\right]^{T}\) is a vector that contains control values applied at the \((k-i)\)th sampling instant; denote the output values generated by the model in the \((k+i)\)th sampling instant as \(\mathbf{\widehat{y}}_{k+i}=\left[\mathbf{\widehat{y}}_{k+i}^{1},\ldots,\mathbf{\widehat{y }}_{k+i}^{n_{y}}\right]^{T};n_{a},n_{b}\) determine how many past output and control values the model needs. If one wants to use the model (17) directly in the optimization problem (1)-(4), then the prediction takes the form of the following formulas, passed to the optimization problem as a set of equality constraints:  
\[\mathbf{y}_{k+i|k}= \mathbf{f} (\mathbf{\widehat{y}}_{k+i-1},\ldots,\mathbf{\widehat{y}}_{k+1},\mathbf{y}_{ k},\ldots,\mathbf{y}_{k-n_{x}+i-1},\] \[\mathbf{u}_{k+i-1|k},\ldots,\mathbf{u}_{k|k},\mathbf{u}_{k-1},\ldots,\mathbf{u}_{k -n_{b}+i-1})+\mathbf{d}_{k}\enspace,\]  
where \(\mathbf{u}_{k+i|k}=\left[u_{k+i|k}^{1},\ldots,u_{k+i|k}^{n_{x}}\right]^{T}\) is a vector containing future control values, depending on the decision variables from the vector \(\Delta\mathbf{u}\) and:  
\[\mathbf{d}_{k}=\mathbf{y}_{k}-\mathbf{\widehat{y}}_{k}\enspace, \tag{19}\]  
where \(\mathbf{y}_{k}=\left[y_{k}^{1},\ldots,y_{k}^{n_{y}}\right]^{T}\) is a vector of recently measured output values; it is assumed that \(\mathbf{d}_{k}\) is the same for all instants in the prediction horizon--an approach proposed in the Dynamic Matrix Control (DMC) algorithm and therefore called the DMC-type disturbance model; see, e.g., [9]. Unfortunately, the optimization problem (1)-(4) with the prediction (18), based on a nonlinear model, is, in general, a non-convex nonlinear optimization problem; see, e.g., [14; 15; 16]. Thus, it is a hard-to-solve, time-consuming computational task, and there is no guarantee of finding a global solution, while the time needed to obtain the solution cannot be foreseen in advance. One of the methods to overcome this problem is to use so-called fast NMPC algorithms in which a suboptimal solution is generated faster than in the standard approach; see, e.g., [17; 18; 19]. To this group of methods belongs also the explicit approach in which most of the calculations are done off-line; see, e.g., [20; 21; 22] (it is interesting that this approach is optimal if the linear model is used [23; 24; 25]). Unfortunately, in this approach, the complexity of the controller grows significantly with the number of constraints taken into consideration. If the process behavior is described by means of a fuzzy model, then the standard NMPC approach can be used, but also the structure of the model can be exploited to formulate algorithms that are easier to solve. One group of such algorithms is based on Linear Matrix Inequalities (LMIs); see, e.g., [26; 27; 28; 29; 30]. The other group is based on the classical fuzzy Takagi-Sugeno approach in which a few algorithms in the form of control laws, based on linear models, are used to obtain the fuzzy controller; see, e.g., [31; 32]. In the other method, the linearization of the process model for the MPC algorithm is obtained in each time step, and the linear prediction relative to control changes is formulated; see, e.g., [9; 33; 34; 35; 36; 37; 38; 39]. As a result, the optimization problem solved by the MPC algorithm in each time step is formulated as the quadratic one (like in LMPC algorithms). In the linearization-based algorithms, the prediction can be based on a classical linearization or a method of prediction generation can exploit the structure of the nonlinear model on which it is based.