**5. Experimental Test**

The experiment aimed to verify the torque performance of this method with surface-mounted permanent magnet synchronous motor (SPMSM) and interior permanent magnet synchronous motor (IPMSM). The experimental platform as shown in Figure 10 was built by using dSPACE1102. The load of the PMSM is provided by the load machine.

**Figure 10.** The controlled 6.5 kW PMSM drive test rig.

#### *5.1. Experiment on SPMSM*

The parameters of the SPMSM are shown in Table 3. The steady-state performance test results of conventional FCS-MPC and proposed FCS-MPC are shown in Figure 11. In the experiment, the reference load torque was 20 N·m, and the results show that the proposed FCS-MPC significantly reduced the torque ripple under the same average switching frequency, while the conventional FCS-MPC had a lot of torque ripples (6 N·m) compared to proposed FCS-MPC (2.8 N·m). This is consistent with the simulation results.

**Figure 11.** Torque steady-state responses (SPMSM).

In order to test the dynamic performance of the proposed FCS-MPC method, a load torque of 25 N·m was suddenly added when the steady speed of the motor was 500 rpm. Figure 12 shows that the response times of proposed FCS-MPC and conventional FCS-MPC are approximately the same; however, the proposed FCS-MPC strategy torque ripple is smaller than that of the conventional FCS-MPC. Meanwhile, the proposed FCS-MPC method enables the motor to quickly recover to the reference speed value compared to conventional FCS-MPC.

**Figure 12.** Speed dynamic and torque dynamic responses (SPMSM).

#### *5.2. Experiment on IPMSM*

Some parameters of the experiment motor are shown in Table 4. The following experiments were carried out in three aspects: torque steady-state responses, speed dynamics, and torque dynamics responses.


**Table 4.** Interior permanent magnet synchronous motor (IPMSM) parameters for simulation.

Figure 13 shows electromagnetic torque waveforms of two methods, with the load torque reference value set to 30 N·m. As can be seen from Figure 13, the method of proposed FCS-MPC can significantly reduce torque ripple by comparing torque waveforms; the average torque ripple of proposed method is only 3 N·m while that of conventional FCS-MPC is 7 N·m.

**Figure 13.** Torque steady-state responses (IPMSM).

Figure 14 shows the waveforms of speed and electromagnetic torque of two methods when the starting moment of the motor and step change of load torque with load torque increasing from 10 to 20 N·m. It can be seen from Figure 14 that at the moment of starting the motor, the speed and electromagnetic torque of the conventional and proposed FCS-MPC methods increase sharply, reaching the given reference value quickly; however, compared with the conventional FCS-MPC, the strategy of proposed FCS-MPC has smaller torque ripple. When the load torque changes abruptly, the proposed FCS-MPC can also track the change of torque quickly, and the corresponding speed is faster than conventional FCS-MPC.

**Figure 14.** Speed dynamic and torque dynamic responses (IPMSM).

Through the analysis of the above experiments, compared with the conventional FCS-MPC control system, the proposed FCS-MPC in this paper effectively reduces the torque ripple and improves the following performance of the motor torque.

#### **6. Summary**

In this paper, a new scheme of direct torque control for PMSM based on a finite control set (FCS) model is proposed. The eight voltage vectors of the two-level converter are utilized as an FCS for the torque prediction of the PMSM. The cost function is used to estimate the duty cycle of each voltage vector. Thus, the optimal voltage vector can be obtained from eight voltage vectors and their duty cycles. Compared with the classical FCS-MPC method, the proposed method has smaller torque ripple and excellent dynamic performance.

**Author Contributions:** Conceptualization and methodology, G.B. and W.Q.; software, T.H.; validation, G.B. and W.Q.; writing—original draft preparation, T.H.; writing—review and editing, G.B. and W.Q.; funding acquisition, G.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by supported by State Key Laboratory of Large Electric Drive System and Equipment Technology (SKLLDJJ032016018) and National Natural Science Foundation of China (51967012) and Scientific Research and Innovation Team Project of Gansu Education Department (2018C-09).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

**Proof of Lemma 1.** Finding partial derivatives of *f*(*id*, *iq*) and *Te* = 1.5*piq*[ψ*<sup>f</sup>* + (*Ld* − *Lq*)*id*] based on current vectors *id* and *iq*, respectively:

$$\frac{\partial f(i\_d, i\_q)}{\partial i\_{dq}} = \left[1 + \frac{2(L\_d - L\_q)}{\psi\_f} i\_{d\prime} - \frac{2(L\_d - L\_q)}{\psi\_f} i\_q\right] \tag{A1}$$

$$\frac{\partial T\_{\varepsilon}}{\partial i\_{dq}} = \left[1.5p(L\_d - L\_q)i\_{q\prime}1.5p(\psi\_f + (L\_d - L\_q)i\_q)\right] \tag{A2}$$

Considering the torque *Te* as constant, the trajectory of the operating point moves along the direction of the α which fulfills (*Te*/*idq*) · α = 0. The vector α can be described as:

$$a = \varepsilon \left[ -1.5p(\psi\_f + (L\_d - L\_q)i\_q), 1.5p(L\_d - L\_q)i\_q \right]^T,\tag{A3}$$

where ε is a positive real number.

$$\frac{\partial f(\mathbf{i}\_d, \mathbf{i}\_q)}{\partial \mathbf{i}\_{dq}} \cdot \alpha = 1.5p\varepsilon \Big[\psi\_F + 3(L\_d - L\_q)\mathbf{i}\_d\Big] - \frac{3p\varepsilon (L\_d - L\_q)^2}{\psi\_f} (\mathbf{i}\_d^2 + \mathbf{i}\_q^2) < 0. \tag{A4}$$

Therefore, the function *f*(*id*, *iq*) is a strict monotonic function along the constant torque curve. -

**Proof of Lemma 2**. The Lyapunov function is derived as:

$$\frac{dV(k+1)}{dt} = \frac{\partial V(k+1)}{\partial I\_{dq}^{k+1}} \cdot \frac{dI\_{dq}^{k+1}}{dt}.\tag{A5}$$

Therefore, the trajectory of the current makes the value of Lyapunov function decrease and there exists a *i k*+1 *dq* which fulfills:

$$\frac{\partial V(k+1)}{\partial \dot{t}\_{d\eta}^{k+1}} \Delta \dot{t}\_{d\eta}^{k+1} \le 0. \tag{A6}$$

*Energies* **2020**, *13*, 234

Therefore, there exists a current derivative:

$$\frac{d\dot{l}\_{d\eta}^{k+1}}{dt} = \mu \cdot \Delta \dot{l}\_{d\eta}^{k+1} \,\tag{A7}$$

so that:

$$\frac{dV(k+1)}{dt} = \frac{\partial V(k+1)}{\partial \dot{\mathbf{r}}\_{dq}^{k+1}} \cdot \mu \cdot \Delta \dot{\mathbf{r}}\_{dq}^{k+1} \le 0. \tag{A8}$$

Here, μ can be a very small positive constant. -

**Proof of Lemma 3**. Any reference vector *u*∗ *dq*(*x*) within the region of feasibility [having a magnitude of less than (2/3)*Udc*] is contained within one of the six nonzero switching sectors of width (π/3) with vertices (*v*<sup>0</sup> *d* , *v*<sup>0</sup> *<sup>q</sup>* ), (*vi d* , *vi <sup>q</sup>*), and (*v<sup>j</sup> d* , *vj <sup>q</sup>*), where *i*, *j* ∈ {1, ... , 6} are the nonzero switching states to the left and right of the reference vector and (*v*<sup>0</sup> *d* , *v*<sup>0</sup> *<sup>q</sup>* ) is one of the two zero vectors. Containment within a switching sector ensures the existence of coefficients γ and η satisfying γ, η ≥ 0 and γ + η ≤ 1 such that the reference vector is expressible as a convex combination of the realizable inputs, given by:

$$
\mu\_{dq}^\*(\mathbf{x}) = \gamma v\_{dq}^i + \eta v\_{dq}^j + (1 - \gamma - \eta) v\_{dq}^0. \tag{A9}
$$

Plugging Equation (A10) into Equation (21) and noting that the system is control affine, we see that [23]:

$$\begin{array}{ll} \frac{dV(k+1)}{dt} &= \frac{\partial V(k+1)}{\partial \dot{t}\_{dq}^{k+1}} (A\_{d\eta}^{k+1} + Bu\_{d\eta}^{\*} + E) = \frac{\partial V(k+1)}{\partial \dot{t}\_{dq}^{k+1}} (A\_{d\eta}^{k+1} + B(\gamma v\_{d\eta}^{j} + \eta v\_{d\eta}^{j} + (1-\gamma-\eta)v\_{d\eta}^{0}) + E) \\ &= \gamma \frac{\partial V(k+1)}{\partial \dot{t}\_{dq}^{k+1}} (A\_{d\eta}^{k+1} + Bv\_{d\eta}^{j}) + \eta \frac{\partial V(k+1)}{\partial \dot{t}\_{dq}^{k+1}} (A\_{d\eta}^{k+1} + Bv\_{d\eta}^{j}) \\ &+ (1-\gamma-\eta) \frac{\partial V(k+1)}{\partial \dot{t}\_{dq}^{k+1}} (A\_{d\eta}^{k+1} + Bv\_{d\eta}^{0}) + \frac{\partial V(k+1)}{\partial \dot{t}\_{dq}^{k+1}} E \end{array} \tag{A10}$$

Because γ, η, and (1 − γ − η) are all nonnegative, the following inequalities hold [12]:

$$\begin{cases} \frac{\partial V(k+1)}{\partial \dot{t}^{k+1}} (A \dot{t}^{k+1}\_{dq} + B \upsilon^{i}\_{dq}) \le 0\\ \frac{\partial V(k+1)}{\partial \dot{t}^{k+1}} (A \dot{t}^{k+1}\_{dq} + B \upsilon^{j}\_{dq}) \le 0\\ \frac{\partial V(k+1)}{\partial \dot{t}^{k+1}\_{dq}} (A \dot{t}^{k+1}\_{dq} + B \upsilon^{0}\_{dq}) \le 0 \end{cases},\tag{A11}$$

which completes the proof in Lemma 3. The theorem also guarantees the stability of the proposed control scheme. -

#### **References**


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