# Multi-Phase Stator Current Tracking with Gradual Penalization of Commutations
## 1 Introduction  
Model predictive control (MPC) is a flexible method for electrical systems using power converters [1; 2]. In particular, Predictive Stator Current Control (abbreviated PSCC) has been explored in the specialized literature. PSCC relies on the sample-wise optimization of a cost function (CF), penalizing deviations from control objectives. In particular, for multi-phase drives, reference tracking in \(\alpha-\beta\) (torque producing) and \(x-y\) (harmonic) subspaces constitutes an important control objective [3]. In addition, many papers expanded the CF by adding a term penalizing commutations in the legs of the power converter. This is achieved to keep the average switching frequency (ASF) below acceptable limits. Limiting the maximum ASF has a relevant impact on the drive and enables the selection of the hardware (e.g., standard IGBTs or wideband switches). Also, ASF affects the energetic efficiency of the drive, as commutation losses are an important portion of all losses [4]. Average switching frequency reduction constitutes a challenge since the switching frequency in PSCC is not constant [5]. Also, the ASF shows a nonlinear behavior with speed and load [6; 7]. Previous papers dealing with ASF in predictive control can be classified into two categories: (a) those that include a term in the CF and (b) those that use physical insight to reduce the available set of voltage vectors (VVs). These two categories apply to different predictive methods, including those using a single-vector per sampling period and those relying on the multi-vector concept. This latter lot includes the virtual VV method (also known as synthetic VV method) [8; 9] and methods derived from the deadbeat (DB) concept [10]. For single-vector PSCC based on the CF, the work of [11] presented a neural solution for online weighting factor tuning. A previous work used scheduled tuning with no adaptation [6]. In [12], VV pre-selection was used mainly to reduce the computational burden. Similar ideas were used in [13]. A memory stage was proposed in [14] to reduce the number of commutations considering the past. Analytical normalization was proposed in [15] to eliminate the weighting factor, thus reducing the complexity of the problem. Quadratic current prediction and boundary constraint design were used in [16] to enable a CF that reduces the ASF with some increase in the computational cost. A further step along these lines consists of the elimination of the average switching frequency term by reducing the set of available voltage vectors on the fly, such as in [17]. Intermediate solutions do use a CF but avoid exploring all possible switching sequences, as in [18]. Regarding the multi-vector approach, several avenues have been investigated. In [19], four candidate voltage vectors were considered during each sampling period. A look-up table was used to derive the optimal VV. The table allowed just one commutation, thus reducing the ASF. In [20], a deadbeat method with a hybrid exploration state was used to reduce the computational burden for a given average switching frequency. A similar approach was used in [21] in conjunction with an asymmetric space vector pulse width modulation method. In [22], a multi-vector solution for a dual three-phase permanent magnet motor was presented. Just two VVs were applied, where switching commands were changed in two phases at most. In [23], the DB concept was used in a multi-variable setup to control all system variables in a single predictive loop and using modulation techniques. Similar concepts were proposed in [24]. Multi-vector approaches consider an optimal (or reference) voltage \(V^{*}\) to be synthesized using basic VVs. This is important for multi-phase systems, where torque-producing currents (i.e., currents in \(\alpha-\beta\) plane) and non-torque-producing currents (i.e., currents in \(x-y\) plane) must be dealt with.