# Performance Assessment of Predictive Control--A Survey
## 2 Model Predictive Control
### General MPC Rule
The underlying MPC optimization problem may be extended, for instance taking into consideration stabilizing terminal constraint. Furthermore, additional constraints might be necessary in some specific applications, as for instance, connected with any auxiliary variables or override controls. Quadratic cost function formulation is highly sensitive to any kind of the outliers [34]. Statistics suggests to use other estimates, such as mean absolute error, formulated as a \(\ell_{1}\) norm. An analysis of the \(\ell_{1}\)-MPC may be found in [35]. It also natural that the researchers have considered other norms, e.g., [PERSON] et al. [36] explored \(\ell_{\infty}\)-MPC. Another approach to improve MPC quadratic performance index was proposed by [PERSON] [37], who regularized least squares with \(\ell_{1}\) component. In all MPC algorithms, a dynamic model of the controlled process is used to predict the future values of output variable, \(\hat{y}(k+p|k)\), over the prediction horizon, i.e., for \(p=N_{1},\dots,N_{2}\). The receding horizon predictive control principle formulated above is presented graphically in Figure 2.  
As stated above, there are dozens of versions of the predictive control rule. The difference manifests itself in the selection of:  
* process model;
* performance index formulation;
* utilized optimization algorithm; or
* algorithm numerical representation. Actually, there are no limitations with these selections. The model can be of any form, linear or nonlinear. The performance index is whichever and an optimization algorithm may be in general any nonlinear, non-gradient method unconstrained or constrained in form of the penalty functions [38]. One may use any kind of a local or global optimization approach. Such a generalization only requires repetitive calculation of the entire optimization task during each sampling period. It frequently consumes a lot of calculation resources and may end up in the violation of the sampling interval. Such an optimization is called Nonlinear Optimization MPC (NO-MPC) or repetitive control with a receding horizon. There are various simplifications of this general rule that lead toward simpler and easier to be evaluated and applied algorithms requiring much lower calculation resources. In the general formulation, the original Model Predictive Control optimization problem in Equation (3) leads to constrained nonlinear task, which has to be evaluated repetitively in real time during each sampling period. MPC with Nonlinear Prediction and Linearization Along the Predicted Trajectory (MPC-NPLPT) [39] has been proposed to address this issue. It reduces significantly required calculations with only minor loss in an efficiency. Unlike in simple algorithms with the successive model linearization [40], only the linear approximation of future CVs trajectory prediction over prediction horizon \(N\) is evaluated during each sampling period. Linearization of MVs increments in Equation (1) over an assumed future trajectory is performed. It allows formulating a computationally  
Figure 2: Receding horizon predictive control principle. efficient quadratic MPC-NPLPT problem. Only single trajectory linearization is performed, while the process is close to the required setpoint. However, when the setpoint changes or the process is affected by any disturbance, trajectory linearization is invoked and quadratic optimization is repeated a few times. Research shows that MPC-NPLPT algorithm gives satisfactory performance, similar to the general non-computationally efficient NO-MPC [41]. Despite numerous variants and versions of the MPC control rule, two main algorithmic simplifications seem to be the most popular, i.e., DMC and GPC. They are clearly simpler than NO-MPC. They use linear models which, together with a quadratic performance index, enable to analytically derive control rule. Thereby, no repetitive optimization is needed. The control rule is evaluated only once, enabling simple algorithm coding and applications.