## 2 Model Predictive Control Algorithms  
Future control signal trajectories are generated in the MPC algorithms in such a way that predicted, many sampling instants ahead, future behavior of the control plant fulfills assumed criteria. It is usually formulated as an optimization problem in which, it is demanded that a performance function is minimized subject to the constraints put on control and output signals. The performance function takes into consideration future, predicted control errors and the shape of control signals.  
where \(\mathbf{y}_{k-i}=\left[y^{1}_{k-i},\ldots,y^{n_{y}}_{k-i}\right]^{T}\) is a vector which contains output values measured at the \((k-i)\)th sampling instant, \(\mathbf{u}_{k-i}=\left[u^{1}_{k-i},\ldots,u^{n_{u}}_{k-i}\right]^{T}\) is a vector which contains control values applied at the \((k-i)\)th sampling instant; denote output values generated by the model in the \((k+i)\)th sampling instant as \(\widehat{\mathbf{y}}_{k+i}=\left[\widehat{y}^{1}_{k+i},\ldots,\widehat{y}^{n_{y }}_{k+i}\right]\), \(n_{a},n_{b}\) determine how many past output and control values the model needs. If one wants to use the model (12) 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:  
\[\mathbf{y}_{k+i|k}=\mathbf{f}\left(\widehat{\mathbf{y}}_{k+i-1},\ldots,\widehat{\mathbf{y}}_{k +1},\mathbf{y}_{k},\ldots,\mathbf{y}_{k-n_{a}+i-1},\right. \tag{13}\]  
where \(\mathbf{u}_{k+i|k}=\left[u^{1}_{k+i|k},\ldots,u^{n_{u}}_{k+i|k}\right]^{T}\) is a vector containing future control values, depending on decision variables from the vector \(\Delta\mathbf{u}\) and  
\[\mathbf{d}_{k}=\mathbf{y}_{k}-\widehat{\mathbf{y}}_{k}\, \tag{14}\]  
where \(\mathbf{y}_{k}=\left[y^{1}_{k},\ldots,y^{n_{y}}_{k}\right]^{T}\) 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 for example, Reference [7].  
Unfortunately, the optimization task with the prediction (13), based on a nonlinear model, is not a quadratic optimization task. The optimization problem ((1)-(4)) becomes, in general, non-convex nonlinear optimization problem. Solving such a problem is difficult and time-consuming task, there is no guarantee to find a global solution and time needed to obtain solution cannot be estimated in advance. Different kinds of nonlinear process models can be used in the algorithm, but they are exploited in a similar way, see for example, References [13; 40; 41; 42]; such algorithms will be referred to as Nonlinear MPC (NMPC). Due to problems caused by formulation of NMPC algorithms, in practice MPC algorithms based on linear approximations of process models, obtained at each iteration of the algorithm are considered. Such linear approximation is used to formulate the, easy to solve, quadratic optimization problem [35]. Such an optimization problem is also used in the MPC algorithms based on linear models, therefore now their formulation will be reminded. In the next section, the proposed algorithms, based on an advanced approximation of the nonlinear model, will be proposed.