Control algorithm design

The steps taken to design the motor control algorithm are presented in the following sub-section. The design, tuning and numerical validation of the control strategy have been made in Simulink, in particular, through the use of Motor control blockset toolbox. It provides the development of multirate simulations to detect and correct errors across the complete operating range of the motor before proceeding with hardware testing. Thus, this method permits to reduce the amount of prototype testing and verify the robustness of the control algorithm to fault conditions.

MTPA control reference

The first module designed is the MTPA control reference (fig. 3.3). It computes the d-axis

result in efficient output for the SPM motor in the constant torque region; successively, these references are interfaced with the FWC output.

Figure 3. 3: MTPA control reference

Basically, this block converts the input torque reference in the corresponding q-axis current while setting the d-axis current to zero. Moreover, its execution sample time is equal to 0.005 s. It is important to highlight that all the following control modules that are presented in the next lines are performed at the switching frequency rate to simulate the real microcontroller interrupt service routine i.e., with 𝑇_𝑠 = 5.00𝑒 − 05 s.

Sensorless estimators

Proceeding, the other inputs of the current controller have been processed i.e., the feedback signals. Sensorless control has been chosen for this simulation, thus, either the sliding mode observer or the flux observer are used to estimate the rotor position, as presented in section 2.3.

Figure 3. 4: Sensorless estimators

The SMO is tuned with a current observer gain equal to 0.01 and a unity sliding surface

voltage and current, coming internally from the control, using as reference value for the conversion respectively 𝑣_𝑚𝑥 and 𝑖_𝑚𝑎𝑥. These signals are properly processed by a unit delay and a DC component removal module (fig. 3.5). The latter is inserted to cancel the offset introduced by the observer’s errors. In particular, it is composed by an infinite impulse response filter which modifies the signal with a discrete low pass filter using the following equation:

𝑦(𝑘) = 𝑎 × 𝑥_𝑘+ (1 − 𝑎) × 𝑦_𝑘−1 (3.6)

where 𝑎 is the filter coefficient, 𝑦 is the filtered output value and 𝑥 is the sampled input value. The theoretical cutoff frequency is computed as:

𝑓_𝑐 = 𝑎

(1 − 𝑎) ∙ 2𝜋 ∙ 𝑇_𝑠 (3.7)

Therefore, a lower filter coefficient gives better filtering at the cost of increasing the delay in the response time; in this case, the filter coefficient is set to 0.005.

Figure 3. 5: DC component removal

Clarke and Park transforms

Clarke and Park transforms are needed to translate 𝑖_𝑎 and 𝑖_𝑏 phase current measurements from the motor model in the d and q-axis current feedbacks.

Current controller

Figure 3.7 shows the current controller block scheme.

Figure 3. 7: Current controller block scheme

The inputs are the currents 𝑖_{𝑑𝑞_𝑟𝑒𝑓} and the speed feedback that in this simulation is simply the reference speed imposed to the motor; the outputs are the 𝑣_{𝑑𝑞_𝑟𝑒𝑓} voltages.

Before entering in the PI regulators, the current references are modified by means of the flux-weakening controller contribution and the current saturation. The FWC (fig. 3.8) accepts as input the maximum voltage reference and the magnitude of the 𝑣_{𝑑𝑞_𝑟𝑒𝑓} voltage imposed by the PI controllers. This last variable is first processed through a IIR filter, with coefficient 0.01, and a unit delay.

Figure 3. 8: Flux-weakening controller

• Proportional gain 𝑘_𝑝𝑓𝑤 = 12.5 A/V.

• Integral gain 𝑘_𝑖𝑓𝑤 = 0.06 A/Vs.

• Saturation limit [0, −𝑖_𝑚𝑎𝑥].

Therefore, if increasing the speed of the motor the magnitude of the 𝑣_{𝑑𝑞_𝑟𝑒𝑓} exceeds its maximum value, set to 0.9 ∙ 𝑣_𝑚𝑎𝑥, a negative 𝑖_𝑑 current is imposed to the drive, to maintain this reference. This means that the base speed is reduced with respect to its standard value reached at 𝑣_𝑚𝑥, but the motor can still adequately run with a higher speed properly reducing the flux. In particular, for this machine 𝐼₀ < 𝑖_𝑚𝑎𝑥, thus, the current protection module is designed as in figure 3.9 to complete the work of the FWC.

Figure 3. 9: Current limitation

The overall strategy and the behaviour of the current space vector are the one illustrated in figure 2.14.

Once obtained the final current reference values, these fed the PI current regulator in figure 3.10; only one is needed since the machine is isotropic, so the inductance is the same for d and q axis.

Figure 3. 10: PI current regulator

The 𝑖_{𝑑𝑞_𝑟𝑒𝑓} currents are compared with the feedback currents and depending on the error the voltage references are chosen, with a maximum absolute value equal to 𝑣_𝑚𝑎𝑥. The calibration has been made according to the equations 2.14:

• Current control bandwidth 𝜔_𝑐 = 1000 ∙ 2𝜋 = 6283 rad/s.

• Proportional gain 𝑘_𝑝 = 2.2 V/A.

• Integral gain 𝑘_𝑖 = 427.2 V/As.

Figure 3.11 and 3.12 shows basic frequency control analysis exclusively of the FOC current control system, without FWC, considering the SPM motor parameters in table 3.1.

Figure 3. 11: Bode diagram

Figure 3. 12: Step response

Successively, the feed-forward of the motional terms is added to decouple the d and q-axis control. Finally, the resultant voltages pass through the voltage saturation module below that implements the equations 2.16.

Figure 3. 13: Voltage saturation

Space vector generator

The last module before the physical system model is the space vector generator.

Figure 3. 14: Space vector modulation

Firstly, the d and q-axis voltage references are transformed through the inverse Park transform and converted into per-unit values. Successively, they are sent to the space vector generator block which generates the space vector modulation signals, according to the SVM strategy illustrated in the section 2.2. The outputs are double hump signals between −1 and 1, thus, they are properly rescaled and translated to obtain the final duty cycles to be sent to the inverter (fig. 3.15).

Figure 3. 15: Duty cycles conversion