Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 2515107 |

Abdiddaim Katkout, Tamou Nasser, Ahmed Essadki, "Novel Predictive Control for the IPMSM Fed by the 3L-SNPC Inverter for EVAs: Modified Lyapunov Function, Computational Efficiency, and Delay Compensation", Mathematical Problems in Engineering, vol. 2020, Article ID 2515107, 17 pages, 2020.

Novel Predictive Control for the IPMSM Fed by the 3L-SNPC Inverter for EVAs: Modified Lyapunov Function, Computational Efficiency, and Delay Compensation

Academic Editor: Laurent Dewasme
Received13 May 2020
Revised22 Jul 2020
Accepted05 Aug 2020
Published27 Aug 2020


This paper proposes a novel predictive strategy based on a model predictive control (MPC) for the interior permanent magnet synchronous motors (IPMSMs) driven by a three-level simplified neutral-point clamped inverter (3L-SNPC) for electric vehicle applications (EVAs). Based on the prediction of the future behavior of the controlled variables, a predefined multiobjective cost function incorporates the control objectives which are evaluated for every sampling period to generate the optimal switching state applied directly to the inverter without the modulation stage. The control objectives in this paper are tracking current capacity, neutral-point voltage balancing, common-mode voltage control, and switching frequency reduction. The principal concepts of the novel scheme are summarized as follows: first, the delay compensation based on the long horizon of prediction is adopted by a multilevel power converter structure. Second, based on the modified Lyapunov candidate function, both stability and recursive feasibility are ensured of the proposed predictive scheme. Third, the practicability of the real-time implementation is improved by the proposed “static voltage vector” (SVV) and “single state variation” (SSV) principles. Finally, the proposed concepts are implemented in the novel predictive control formulation as additional constraints without compromising the complexity and the good performances of the predictive controller. Therefore, only the switching states that guarantee the stability and the reduction of calculation burden criteria are considered in the evaluation of cost function. The proposed predictive scheme based on the “SVV” principle has demonstrated superior performance in simulation compared with the proposed scheme with the “SSV” principle. The computational burden and switching frequency rates are reduced by 35% and 56.22%, respectively.

1. Introduction

Currently, permanent magnet synchronous machines (PMSMs) are widespread in many electric vehicle applications (EVAs) [1, 2]. Depending on the magnet placement, two categories of the PMSM are defined: surface-mounted PMSM and interior PMSM [3]. Particularly, interior PMSM (IPMSM) demonstrates excellent properties, such as compact structure with small size and weight, higher power density, high torque to inertia ratio, wide speed range operations, low noise, and robustness [4]. Many control strategies for the IPMSM are proposed: linear control with a modulator and nonlinear controls without a modulator [5, 6]. In different aspects, other classifications include robust controllers (RC), adaptive controllers (AC), and intelligent controllers (IC) [710].

There are many research studies accorded to ensure a satisfactory control performance of IPMSMs. In [11], the authors propose an online maximum torque per ampere (MTPA) control based on the numerical optimization technique to avoid the large memory usage and the accurate interpolation algorithm required using the look-up table (LUT) methods. A deadbeat-direct torque and flux control (DB-DTFC) of IPMSM drive for machine parameter variation is presented in [12]. The flux-weakening control strategy for the IPMSM with maximum torque per volt in VAs is presented in [13]. The desired dq-optimal current references are calculated analytically to achieve the operating point regarding the electrical limits. A comparative study on DTC of IPMSM for EVs between DTC combined with space vector modulation (DTC-SVM) and deadbeat DTC (DB-DTC) is presented in [14]. Nevertheless, these existing control strategies are limited by several challenges [15]:(i)Low dynamic-state performances and need of the internal loop(ii)Complex modulation stage and sensibility of the system variables(iii)Controlled variable ripples

To overcome these aforementioned issues, numerous predictive controls (PCs) are proposed to improve the control performances for the power converter applications [16]. Based on the dynamic system model, the PC predicts the future behavior of the controlled system. A predefined cost function incorporates the controlled variables which are minimized every sampling period to generate the optimal switching state [17]. In comparison with the existing control strategies, PC offers many advantages: easy application in a variety of processes and multiple-input-multiple-output (MIMO) systems, system constraints and nonlinearities which can be incorporated directly into the control law, simple controller concept and implementation, and good dynamic and steady-state performances [18]. Furthermore, the computational delay and dead time can be compensated, and several control loops can be incorporated into one control law. The predictive controllers can be divided into two principal categories: Continuous control set MPC and finite control set MPC (FMPC). Particularly, FMPC is the most attractive predictive strategy for power converters [19]. Taking into account the discrete nature of the power converter, FMPC uses the available switching states to formulate the predictive problem algorithm without requiring any table of switches, external modulator stage, or regulator. Little has been published regarding FMPC of IPMSM for EVAs. Linear inequality matrices (LMIs) represent an effective tool to transform the problem of robust PC design into the form of a convex optimization problem [20]. LMIs are a powerful tool to deal with the many optimization problems in control theory, system identification, and signal processing. A robust predictive control with nonsymmetric constraints (NSC) using the LMI framework is presented in [20]. The method is based on three principles: time-varying vector of constraints, switching indicator function to determine the current boundaries, and support controller that certifies the robust stability. Ojaghi et al. [21] proposed an LMI approach to robust predictive control of nonlinear systems (NLS) with state-dependent uncertainties. An LMI approach to mixed-integer predictive control of uncertain hybrid systems with binary and real-valued control inputs is proposed in [22].

Generally, different topologies of power converters are available in the market. The aforementioned control strategies have been mainly limited to a two-level power converter in a variety of IPMSM applications [2325]. However, the limited switching state combinations available in the two-level power converters (eight possible combinations) lead to high controlled variable ripples as well as voltage and current waveforms with a higher total harmonic distortion. Additionally, this results in higher switching losses and high switching device stress due to the high derivative [26].

To address the aforementioned issue and to fulfill the high-performance IPMSM application requirements (e.g., EVAs), numerous multilevel power converter structures are recognized: neutral-point clamped (NPC), flying capacitor (FC), cascade H-bridge (CHB), and modular multilevel (MM) [2729]. Several researchers have investigated the multilevel power converter, thanks to its benefits: low voltage applied to each component, less and low THD of the output voltage and current waveforms, and low switching loss using the redundancy of the switching states [30, 31]. In recent years, the NPC and T-type are the most used multilevel power converter topologies in the industry [3234]. The system complexity and calculation cost are increasing significantly with the number of switching devices. Maintaining the multilevel output, the 3L-SNPC requires a few switching devices. A comparison in terms of semiconductor device number between 3L-NPC, T-type, and 3L-SNPC is tabulated in Table 1 [35]. Needless to say, the main source of neutral-point voltage fluctuations in NPC and T-type topologies does not exist in the 3L-SNPC topology.

Semiconductor device3L-NPCT-type3L-SNPC

DC-link capacitor222
Active switch121210
Passive switch181210
Switch block 1264
Switch block 066

In [33], the authors proposed a predictive control for a grid-tie three-level neutral-point-clamped inverter with dead-time compensation. This predictive control is based on the orientation of grid voltage to predict the performance of control variables required for the system. Using the stability condition through a control Lyapunov function, the execution time is decreased by 26%. In [36], an asymptotically stable mono-objective predictive control of a grid-connected converter based on discrete space vector modulation is proposed. The presented approach combines the concept of discrete space vector modulation (DSVM) into the FMPC structure to improve the input current quality of the power converter. However, the additional virtual voltage vectors can cause an unacceptable computational burden to the controller. To alleviate this issue, the number of possible voltage vectors for the optimization process is reduced by a preselection technique. In this work, we propose to reduce computation burden by the proposed “static voltage vector” (SVV) and “single state variation” (SSV) principles. The computational burden is reduced by 35%. The aspect of reduction of switching frequency is introduced and decreased by 56.22%.

Literally, there is very little research, if any, performed on the predictive control of the multilevel 3L-SNPC inverter. Furthermore, a little has been published regarding the feasibility and stability, delay compensation, and the computational burden reduction of IPMSM driven by a 3L-SNPC inverter.

Taking into account the EVA requirements, the principal contributions of this paper are summarized as follows:(i)This paper proposes a novel and promising configuration of EVAs: FMPC predictive control strategy for IPMSM fed by the 3L-SNPC inverter.(ii)A modified Lyapunov candidate function which guarantees the feasibility and stability by taking into consideration the discrete nature of the power converter is proposed. Both recursive feasibility and stability are embedded in MPC problem formulation as additional constraints.(iii)Multiobjective predictive problem formulation that includes four control objectives is considered: tracking current capacity, neutral-point voltage balancing, common-mode voltage control, and switching frequency reduction.(iv)By using a two-step horizon of the prediction approach, the computational delay due to the calculation and communication time is compensated.(v)In order to guarantee the practicability of the real-time implementation and computational efficiency of the proposed predictive control, “static voltage vector” (SVV) and “single state variation” (SSV) principles are proposed and embedded in MPC problem formulation as additional constraints.

In the novel predictive control in this paper, the aforementioned contributions are explicitly included in the predictive problem formulation without affecting the controller performances and simplicity.

Section 2 presents the preliminaries of principal concepts of this paper. Section 3 describes the system dynamics. The FMPC and proposed predictive control formulation are illustrated in Sections 4 and 5, respectively. In Section 6, detailed simulation verification studies are carried to show the efficiency of the proposed predictive strategies. Finally, Section 7 draws the conclusion. The nomenclature of this paper is summarized in Table 2.

Reference value
Predicted value
Measured value
Delay compensation

Switching states
Neutral point
Common mode
Switching frequency

MPCModel predictive control
FMPCFinite control set MPC
CMI-FMPCComputationally efficient FI-FMPC
SVVStatic voltage vector
Cost function
Weighting coefficient
Control of the 3L-SNPC inverter
Lyapunov candidate function (LCF)
dq-positive gains of (LCF)
SSVSingle state variation

2. Preliminaries

The FMPC is a very attractive solution for controlling power converters and electrical drives. Nonetheless, stability, computational delay, and computational burden reduction are still open issues to be investigated. In this section, a statement about the three aspects is presented.

2.1. Stability

In conventional predictive problem formulation, the stability in the closed loop is not taken into account. It is worth mentioning that the stator reference current cannot be obtained by applying the voltage vectors resulting from the application of available 21 switching states in the 3L-SNPC. Consequently, the equilibrium point is not achieved by some reference values. Accordingly, the optimization loop of the predictive control aims to generate the current value bounded around the reference value, the reason why we focus on practical stability. In this work, the 3L-SNPC multilevel inverter structure is designed to produce the desired output voltage by multiple available switching states instead of limited switching state combinations in two-level inverters [37].

In spite of good dynamic and steady-state performances of the conventional FMPC predictive strategy, the stability, still an open issue, is to be studied. A notable tool to overcome this issue is Lyapunov stability theory [38]. To guarantee stability in the predictive problem formulation, the cost function is considered as Lyapunov candidate function (LCF) between two sampling periods. The stability analysis focuses on showing that such cost function satisfies the conditions of practical LCF [39].

2.2. Impact of Computational Delay

The FMPC algorithm consists of the following steps [40]:(i)Measurement of controlled variables(ii)Prediction of future values of controlled variables for the combination of each switching state available in the 3L-SNPC inverter(iii)Evaluation of predefined cost function for every sampling period(iv)Selection and application of the optimal switching state that minimizes the cost function

The operating cycle of the FMPC is illustrated in Figure 1. The predicted, reference, and measured values of stator current are presented by dashed black lines, dashed blue lines, and solid black lines, respectively. In the perfect case, due to the negligibility of the calculation time , FMPC algorithm steps are accomplished spontaneously at instance k to generate the optimal switching state , Figure 1(a). Consequently, the stator current reaches its reference value at the k + 1 instance [41]. In the real case, a significant computational delay is required for the operating cycle and communication time, Figure 1(b). The delay between the measurements and the generation of can cause some problems if not considered. Consequently, the previous optimal switching state will continue to be applied during and stator current which will oscillate around its reference , increasing the current undulation. The computational delay affects the system performances and increases the difficulty of its controllability.

To compensate the computational delay, we propose using a long horizon of prediction approach as shown in Figure 1(c). Using the optimal switching state and controlled variable values at the k instance, the system state values at the k + 1 instance are predicted; denotes the switching states available in the inverter. For each predicted value at the k + 1 instance, future values are predicted at k + 2. Accordingly, the optimal switching state that minimizes the cost function at the k + 2 instance is applied directly to the inverter without the modulation stage at the k + 1 instance [40].

2.3. Computational Efficiency

To compensate the computational delay for the 3L-SNPC inverter, using a two-step horizon of prediction, a discrete set of possible trajectories of the switching states has to be enumerated for the evaluation of the cost function, Figure 2(a). This proposition led to a large number of computational burden, which makes the real challenge to the actual digital system implementation [42]. To reduce the number of real-time evaluations, this paper proposes a modified two-step horizon prediction using a proposed “static voltage vector” (SVV), Figure 2(b), and “single state variation” (SSV), Figure 2(c), principles.

3. System Dynamics

3.1. IPMSM

A typical dq-coordinate rotating reference frame-based mathematical model in the time domain of IPMSM drive systems is given by [24]

The parameters of the IPMSM dynamic model are defined in Table 3.

dq-stator voltages (V)
dq-stator currents (A)
Stator resistance ()
dq-stator inductances (H)
Electric rotatory frequency (rad/s)
Flux linkage (Wb)
Electromagnetic torque (Nm)
External torque (Nm)
Number of pole pairs [1]
Moment of inertia (kg·m2)
Viscous coefficient (Nm·s)
Sampling period (s)

The electromagnetic torque of the IPMSM can be calculated as [43]

Remark 1. Due to the nonuniform air-gap flux in the IPMSM, the stator q-inductance is smaller than the stator d-inductance . Opposed to the IPMSM, surface-mounted PMSM is specified by uniform air-gap, and the stator dq-inductances are identical to each other: [3].
In Figure 3(a), with the assumption of constant DC-bus voltage , equal DC-link capacitors , and [44],where .

3.2. 3L-SNPC Inverter

The 3L-SNPC inverter consists of a dual Buck converter integrated with a conventional two-level inverter [45]. The 3L-SNPC inverter is supplied by two-series-connected capacities and . The dual Buck converter consists of four transistors with neutral point Z to connect different voltage levels, Table 4, to the input of the two-level inverter. The combinations of the switching states of 3L-SNPC can be calculated by combinations, where and are the switching states of the two-level inverter and dual Buck converter, respectively. The combinations of switching states are devised to 18 active and 14 zero voltage vectors leading to 21 inverter output voltages obtained in abc-coordinate as described in Table 5 [46]. Three groups of voltage vectors are defined: zero voltage vectors , small voltage vectors , and large voltage vectors .

Switching stateTerminal voltage


OrderSwitching statesOutput voltage




The topology of the IPMSM fed by 3L-SNPC inverter configuration is depicted in Figure 3(a). In the abc-coordinate, the common-mode voltage measured from the stator winding neutral to neutral point Z is expressed as [47]where is the phase voltage from an output terminal to neutral point Z.

4. FMPC Formulation

Generally, FMPC generates the optimal switching state by optimizing a predefined cost function, for each possible switching state available in the inverter, that describes the desired system behavior until a predefined control horizon. The multiobjective cost function considers the tracking error between the predicted and reference values of controlled variables allowing simultaneous control of all of them. FMPC is a receding horizon approach, namely, its optimization problem is formulated and resolved every sampling period updating the actual system state data.

In general, the cost function consists of two parts: the predicted control errors over the prediction horizon N and the penalty term. Typically, the following quadratic cost function form is used [48]:where is a vector of error between the predicted and reference values of the controlled variable at instant . is the vector of increments of the future values of the manipulated variable (control inputs) at instant . and represent the control and prediction horizons, respectively. Matrices and denote weighting coefficients. Then, the optimal switching state is

4.1. Cost Function Selection

Cost function designing is one of the most important stages in the definition of any MPC strategy. In this work, the purposes of the proposed predictive control are as follows:(i)Reducing the mechanical component stress of the IPMSM by fast and accurate reference current tracking(ii)Limiting the output 3L-SNPC inverter voltage sensibility to the ripples in DC-link voltage by capacitor voltage balancing(iii)Increasing the safety and reliability of the IPMSM by limiting the peak of common-mode voltage(iv)Reducing the switching losses by minimizing the switching frequency

When matrices and in equation (7) are diagonal, the aforementioned control objectives are included in a single cost function of the FMPC given in the following:

The parameters of FMPC cost function are defined in Table 6.

Minimization terms
Current tracking control
Common-mode voltage control
Neutral-point voltage control
Reduction of switching frequency

Predicted values
Extrapolated reference current
Neutral-point voltage
Predicted current
Common-mode voltage

Weighting coefficients
Current tracking control
Common-mode voltage control
Neutral-point voltage control
Reduction of switching frequency

4.2. Weighting Coefficient Design

Cost function (9) synthesizes multiple controlled variables with different natures (e.g., current and voltage), order and magnitude, and effect. Accordingly, the tradeoff between the four controlled variables is defined by the weighting coefficients , , , and. .

The classification of the cost functions is depended on the controlled variables incorporated in its formulation. Three kinds of cost functions are defined [48]:(i)Cost function without weighting coefficients: one type of variable is controlled without using the weighting coefficients.(ii)Cost function with secondary terms: primary goal and secondary constraints are defined to accomplish a proper system behavior and control performances. In this case, the weighting coefficient of the second term is forced to zero (). The procedure of tuning the weighting coefficients is converted to the cost function without weighting coefficient procedure. Based on a steady-state error and total harmonic distortion (THD) criteria, the system behavior without the weighting coefficients is evaluated to fulfill the control requirements. Finally, by evaluating the system behavior, the simulation started with and increases progressively until it defined a good tradeoff between the primary goal and secondary constraints. A common approach to select the optimal weighting coefficient is branch and bound technique.(iii)Cost function with equally important terms: in this case, several variables have equal importance to control the system which are incorporated in the cost function. Firstly, the cost function is normalized. Accordingly, all terms will be similarly important (). The second and third steps are the same as cost function with secondary terms; is considered instead of zero value.

4.2.1. Optimization Algorithm

The discrete-state space model of the IPMSM can be obtained by forward Euler’s method using equations (1) and (2) as follows [43]:where and are the stator current and output voltage vectors, respectively, , , and denote system, input, and feed-through matrices, and is the sampling period.

The future values of reference current are extrapolated using the Lagrange method [49]:

By using equation (5) and actual information of at the k instance, the predicted value of the neutral-point voltage at the k + 1 instance is obtained as

To compensate the computational delay, according to the long horizon of the prediction approach, we propose to use a two-step horizon of prediction, Section 2.2 and Figure 1. The improved cost function with computational delay compensation of improved FMPC (I-FMPC) is evaluated at time to generate the optimal switching state applied at time :

By using the second-order Lagrange extrapolation, the reference current at the k + 2 instance is [49]

Based on equations (10) and (12), the predicted system state and neutral-point voltage data at the k + 2 instance are expressed aswhere is the stator current in -coordinates.

4.3. Stability: Modified Lyapunov Function Constraint

In order to avoid overcurrent/voltage in the power converter and IPMSM, the stator current reference is limited by maximum permissible current and safety factor (to reduce the voltage and provide robustness against model uncertainties) [50]:

By taking into account constraints (16) and (17) and the improved cost function with delay compensation (13), optimization problem (8) is converted to

and denote the state and control input constraint sets, respectively. The improved cost function (18a) synthetizes the control objectives for the two-step horizon of prediction. This cost function is evaluated by taking the system constraints into account. The dynamic of the system is fulfilled by (18b) using the measured system state at the k instance. The state and control input constraint sets are implemented by (18c) and (18d), respectively.

The conventional predictive control does not guarantee stability. Accordingly, the modified Lyapunov candidate function (LCF) is proposed to ensure that the neighborhood of the reference signal set is reached in finite time.

Definition 1. For all , , there exists a preliminary LCF , , , and :where is a tuning parameter used to compromise the controlled variables.

Proof 1. For every , there exists an admissible input such that , i.e., : is the convex control set and is defined as the convex hull of [38]: is the sum of the feedback controller .

Definition 2. Considering parameter, is a sublevel set of the LCF:Due to the discrete nature of the 3L-SNPC inverter, the controlled variable cannot always achieve its reference value, namely, the capacity of the controlled system state towards the origin by feasible inputs is limited.

Proposition 1. To describe how fast the LCF decreases beyond a certain value, we define

has a direct effect on the LCF performances; if is fixed as a large value, LCF decreases faster, and the algorithm robustness against the model uncertainties is improved. Accordingly, equation (18c) is implemented in the predictive problem formulation as a stabilizing constraint to ensure the recursive feasibility in the problem optimization by the finite-time convergence of the system states to the set .

In equation (23), the dynamic of the current error is defined as the difference between the reference and actual stator current:

The proposed modified LCF in this work is defined as

Proposition 2. The time-varying function is proposed as

is a time instant when the modified LCF constraint suits the standard one. Conditioned by the constraint in equation (18c) is relaxed, the recursive feasibility of the proposed control is guaranteed regardless of the cost function.

By considering (19), the value of the LCF can be decreased robustly for all . Consequently, any state is driven towards the set. In addition, for all there exists when . Consequently, there exists a sequence such that

The FMPC system is said to be global and robust asymptotically set stabilizable.

Proposition 3. Using the specific structure of the and sets, the largest set is

Therefore, the modified LCF iswithwhere and are the positive gains.

Considering the modified LCF (29) and (30) in improved cost function (18b), the following modified I-FMPC (MI-FMPC) is proposed:

It can be shown that, in predictive problem formulation (31a), the modified LCF stability criteria constraint is imposed with (31e). With the aim to ensure the stability of the closed loop, the switching states which violate the constraints are discarded, while the remaining states are compared according to the cost function. Two cases are considered: the first one corresponds to and the second one . If , only switches obtaining are considered, and if , only switches assuring are considered. Hence, can be used as a stabilizing constraint that guarantees recursive feasibility and stability for the optimization problem.

A simplified algorithm of the proposed MI-FMPC is summarized in Algorithm 1.

Require: , , , ;
 Extrapolation of references and ;
 % Initialize the values of and ;
 Set ; ;
for i = 1 : 21 do
  Compute predictions: and ;
  Calculate the current error ;
  Evaluate the Lyapunov candidate function: , , ;
   Prediction of and ;
   for j = 1 : 21 do
    Predict , , and ;
    Estimate ;
    Evaluate and store the cost function value;
   end for
  end if
end for
 Select and apply the optimal switching state ;

5. Proposed Computationally Efficient MI-FMPC Algorithms

As mentioned in Section 2.3, an amount of iterations is required to evaluate the improved cost function (31a) which makes a challenge in MI-FMPC algorithm implementation. To overcome this issue, in this work, the MI-FMPC in (31a) is developed with two proposed computation-efficient principles: “static voltage vector” (SVV), Figure 2(b), and “single state variation” (SSV), Figure 2(c). It is worth to state that the “SVV” and “SSV” principles are based on the approach of limiting the evaluation of cost function (31a) in case of a long horizon of a prediction, and they are applied separately to the proposed computationally efficient MI-FMPC (CMI-FMPC) strategy.

5.1. Computationally Efficient MI-FMPC-SVV

In order to reduce the computational burden, the proposed static voltage vector (SVV) principle is based on the evaluation of (32a) using fixed vector voltage during the two-step horizon of prediction instead of different voltage vectors, Figure 2(b). In computationally efficient MI-FMPC-SVV (CMI-FMPC-SVV) formulation in (32a), the “SVV” principle is implemented using constraint (32f):

The simplified algorithm of the proposed CMI-FMPC-SVV is presented in Algorithm 2. The two loops depicted in the predictive control formulation presented in Algorithm 1 are embarked in a single loop in Algorithm 2.

Require: , , , ;
 Extrapolation of references and ;
 % Initialize the values of and ;
 Set ; ;
for i = 1 : 21 do
  Compute predictions: and ;
  Calculate the current error ;
  Evaluate the Lyapunov candidate function: , , ;
   Prediction of and ;
   Predict , , and ;
   Estimate ;
   Evaluate and store the cost function value;
  end if
end for
 Select and apply the optimal switching state ;
5.2. Computationally Efficient MI-FMPC-SSV

To avoid the high computational burden in two-step horizon of prediction in (31a) formulation of the proposed MI-FMPC strategy, the single state variation (SSV) principle is proposed, namely, in the first stage of prediction, the 21 system states are predicted. In the second stage, the transition of the switching state is limited to the combinations of inputs that have only one state variation in k + 2, Figure 2(c). For example, if the vector voltage