## 1 Introduction
proposed an improved MPVC with an OSS for LC-filtered VSIs, defining an improved vector-switching sequence to reduce output-voltage ripple with a constant switching frequency and tackling the difficulty of extending the OSS to higher-order systems using a \"one-step estimation\" solution [42]. For multilevel inverters, such as the three-level T-type inverter, FCS-MPC with a constant switching frequency can be achieved by extending period control techniques, which measure the periods between the rising and falling edges of gate signals [24]. The Period Control Approach (PCA) for FCS-MPC, introduced by [PERSON] et al. [43], specifically aims to regulate the switching frequency by controlling the switching period directly within the cost function. This method has proven effective in achieving a regular commutation pattern similar to PWM, with a well-defined spectrum, while significantly reducing the computational burden and maintaining the decoupling of control objectives [43]. Further extensions of PCA-FCS-MPC have addressed more complex scenarios, such as achieving precise switching-phase regulation for interleaved DC-DC power converters, thereby enabling the benefits of phase shifting (e.g., ripple cancellation) while retaining the advantages of FCS-MPC [44]. [PERSON] et al. proposed an OSS-MPC for a three-phase, four-level nested neutral point-clamped (4L-NNPC) inverter, where switching sequences are configured to limit peak CMV and the balance of the flying capacitor voltages is realized by involving time in the optimization process [41]. Multi-rate FCS-MPC (MRFCS-MPC) has also been proposed, where the control frequency can be higher than the sampling frequency, allowing for a significantly increased switching frequency without changing the sampling frequency. This is achieved by using a lifting model to predict fast-rate information [25]. A more recent development, multirate MPC (MR-MPC), builds on conventional CCS-MPC, solving for a high-dimensional control sequence within each interrupt interval, offering a more optimized control input and higher computational efficiency than its MRFCS-MPC counterpart [26], effectively decoupling the achievable switching frequency from the sampling/interrupt frequency and thereby relieving the computational burden on digital signal processors. [PERSON] et al. developed a variable-frequency critical-soft-switching (VF-CSS) MPC method, combining MPC with VF-CSS to ensure complete critical soft-switching operation over a varying frequency range, improving efficiency [45]. This Variable-Switching, Constant-Sampling frequency (VSCS) MPC method typically involves a frequency controller to maintain critical soft switching by adjusting the switching frequency based on boundary conditions (while keeping the sampling frequency constant) and an MPC controller to track objectives and ensure soft switching during dynamics [45]. [PERSON] et al. proposed a multimode hybrid-vector MPC strategy, extending the last voltage vector of the previous control period to the present period, including two or three voltage vectors per control period to improve steady-state performance at lower switching frequencies [46]. According to [47], the cost functions in two-level VSIs aim to minimize the switching frequency, reduce the harmonic content, and impose constraints to keep electrical variables within limits. This prevents operational failures from overcurrents during transients, as seen in induction machines [48], and can replace saturators in cascade control. Constraints also affect broader transient behavior [8]. Performance constraints can be integrated into the cost function, relating the control effort to current and voltage variations, and the switching frequency or losses [49; 50]. An additional term, weighted as noted in \(\lambda\), is then added to connect to the main control objective. As shown in [47], increasing the weighting factor (\(\lambda\)) reduces the switch count but increases current distortion, aligning with the primary control objective of the cost function.