Background and Objectives: To overcome the disadvantages of the traditional two-level inverters, especially in electric drive applications, multi-level inverters (MLIs) are the widely accepted solution. Diode-Clamped Inverters (DCIs) are a well-known structure of multi-level inverters. In DCIs, the voltage balance of the DC-link capacitors and the Common Mode (CM) voltage reduction are two important criteria that should be considered. Methods: This paper concentrates on the current control of 3-phase 4-level DCI with finite control set model predictive control (MPC) strategy. Current tracking performance, DC-link capacitor voltage balance, switching frequency minimization, and CM voltage control have been considered in the objective function of the MPC. Moreover, the multistep prediction method has been applied to improve the performance of the DCI. Results: The effectiveness of the proposed multistep prediction control for the 4-level DCI has been evaluated with different horizon lengths. Moreover, the effect of several values of weighting factors has been studied on the system behavior. Conclusion: Results validate the accuracy of current tracking and voltage balancing in the suggested multistep MPC for the 4-level DCI. In addition, CM voltage control and switching frequency reduction can be included in the predictive control. Decreasing the CM voltage and switching frequency will oppositely affect the dynamic behavior and voltage balancing of the DCI. Therefore, selection of weighting factors depends on the system needs and requirements.

In this paper, we introduce second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.

In this paper, we study the global truncation error of the linear multistep methods (LMM) in terms of local truncation error of the corresponding Runge-Kutta schemes. The key idea is the representation of LMM with a corresponding Runge-Kutta method. For this, we need to consider the multiple step of a linear multistep method as a single step in the corresponding Runge-Kutta method. Therefore, the global error estimation of a LMM through the Runge-Kutta method will be provided. In this estimation, we do not take into account the effects of roundoff errors. The numerical illustrations show the accuracy and efficiency of the given estimation.