11.2 Cost Function Classification
Although the cost function's main objective is to keep track of a particular variable and control the system, it is not limited to this, as explained in the previous chapter. In fact, one of the main advantages of MPC is that the cost function admits any necessary term that could represent a prediction for another system variable, system constraint, or system requirement. Since these terms most likely can be of a different physical nature (current, voltage, reactive power, switching losses, torque, flux, etc.) their units and magnitudes can also be very different. This issue has been commonly dealt with in MPC by including weighting coefficients or weighting factors λ, for each term of the cost function
11.1 
Depending on the nature of the different terms involved in the formulation of the cost function, they can be classified in different groups. This classification is necessary in order to facilitate the definition of a weighting factor adjustment procedure that could be applied to similar types of cost functions or terms.
11.2.1 Cost Functions without Weighting Factors
In this kind of cost function, only one, or the components of one variable, are controlled. This is the simplest case, and since only one type of variable is controlled, no weighting factors are necessary. Some representative examples of this type of cost function are obtained for: predictive current control of a voltage source inverter [1]; predictive power control of an active front-end (AFE) rectifier [2]; predictive voltage control of an uninterruptible power supply (UPS) system [3]; predictive current control with imposed switching frequency [4]; and predictive current control in multi-phase inverters [5–7], among others. The corresponding cost functions are summarized in Table 11.1.
Table 11.1 Cost functions without weighting factors
| Application | Cost function |
| Current control of a VSI |  |
| Power control of an AFE rectifier | |Qp| + |P* − Pp| |
| Voltage control of a UPS |  |
| Imposed switching frequency in a VSI |  |
| Current control of a multi-phase VSI |  |
Note that all the terms in a cost function are composed of variables of the same nature (same unit and order of magnitude). Moreover, some of them are a decomposition of a single vector into two or more components. Therefore, no weighting factors or their corresponding tuning processes are necessary.
11.2.2 Cost Functions with Secondary Terms
Some systems have a primary goal or a more important control objective that must be achieved in order to provide a proper system behavior, and additional secondary constraints or requirements that should also be accomplished to improve system performance, efficiency, or power quality. In this case the cost function contains primary and secondary terms, where the importance of the secondary terms can vary over a wide range, depending on the application and its specific needs. Some examples are: predictive current control with reduction of the switching frequency to improve efficiency [8]; predictive current control with reduction of common-mode voltage to prevent motor damage [9]; and predictive current control with reactive power reduction to improve power quality [10, 11]. The corresponding cost functions are listed in Table 11.2.
Table 11.2 Cost functions with secondary terms
| Application | Cost function |
| Switching frequency reduction |  |
| Common-mode voltage reduction |  |
| Reactive power reduction |  |
The importance of the second term (i.e., how much the switching frequency, the common-mode voltage, or the reactive power is reduced), will depend on the specific needs of the application and will impose a trade-off with the primary control objective, in this case current control. Note that in each cost function a weighting factor λ is included with the corresponding secondary term. Hence, solving the trade-off can be seen as a weighting factor adjustment to the cost function.
11.2.3 Cost Functions with Equally Important Terms
Unlike the previous case, there are systems in which several variables need to be controlled simultaneously with equal importance in order to control the system. Here the cost function can include several terms with equal importance, and it is the job of the weighting factors to compensate the difference in nature of the variables. Such is the case for the torque and flux control of an induction machine, where both variables need to be controlled accurately to achieve proper system performance [12, 13]. Another example is the current control of a neutral-point clamped (NPC) inverter, in which the DC link capacitor voltage balance is essential in order to reduce voltage distortion and avoid system damage (exceed the permitted voltage level of the capacitors, otherwise overrated capacitors should be used) [8]. Both cost functions are included in Table 11.3. Note that there are two additional terms in each cost function used to normalize the quantities in relation to their nominal values (denoted by subscript n); the reason for this will be explained later.
Table 11.3 Cost functions with equally important terms
| Application | Cost function |
| Torque and flux control |  |
| Capacitor voltage balance |  |