## 4 Speed Controller for PMSM Drive  
In order to verify the proposed control concept for a PMSM-based drive, a series of simulation tests were conducted for its simplified model. The object structure was assumed to be in the form (10), where the output is the rotating speed \(\omega(s)\) and the input is the \(q\)-axis reference current \(i_{qref}(s)\) for a reference system \(dq\) rotating synchronously with the rotor:  
\[G_{ob}(s)=\frac{\omega(s)}{i_{qref}(s)}=e^{-s\tau}\cdot\frac{k_{ob}}{T_{cu}s+ 1}\cdot\frac{1}{s}, \tag{10}\]  
where \(k_{ob}\) denotes the total gain of the control object, which includes the torque-current factor \(k_{m}\) and the total moment of inertia \(J\) (Figure 6); \(T_{cu}\) is the time constant of the closed \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(J_{1}\) & **Overshoot** & **Minimum Value** & **Settling Time** & **Recovery Time** \\ \hline  & & **1** & **2** & **3** & **4** & **5** \\ \hline Ref. PID & 1 & 5.30 & 49.0 \% & 0.57 & 10.3 s & 21.2 s \\ Ref. opt. PID & 2 & 2.31 & 56.0 \% & 0.81 & 10.9 s & 6.5 s \\ MPID \(\gamma_{1}=2\) & 3 & 2.49 & 21.0 \% & 0.78 & 12.0 s & 10.8 s \\ MPID \(\gamma_{1}=0\) & 4 & 1.16 & 42.0 \% & 0.89 & 8.3 s & 5.4 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quality indices for the I2 PD object type. loop current control; and \(\tau\) corresponds to the time delay in the control loop. Such an object can be called a FOLIPD (_First Order Lag Integral Plus time Delay_). The C controller acts as the speed controller under this test. In testing, it has a PID structure for the reference system, and in modified form, it has an MPI or MPID form for the active derivative component. The simulation model is supplemented with measured signal quantization blocks and a measurement noise source, and the controller has an output value limit and an anti-windup. Next, the parameters of the model used were identified based on the recorded waveforms of speed (\(\omega\)) and reference current in the \(q\)-axis (\(i_{q\_ref}\)) using a laboratory stand. The settings and weights were determined in the initial tests on the simulation model using the optimization method for the quality function presented earlier. As part of the proof of concept, the behavior of the speed control system was tested for different setpoint shapes--for a step change and for a ramp. This was to demonstrate that the proposed controller works well for more than just one setpoint shape. These tests were realized for the same speed controller settings and the same FIR coefficients--for both simulation and laboratory bench experiment, but for another form of the quality function (16), as discussed in Section 5. The concordance of simulation and measurement waveforms shown in the following illustrations confirms the correctness of the model and concept. Figures 7-10 present the selected waveforms for the test, which involves a sequential change in the set speed (either stepped or ramped). Additionally, at the constant upper speed, there is a load disturbance acted by a stepping value of 0.2 p.u. and released. The speed is changed within a range of 0.6 \(\cdot\) 98 % \(\rightarrow\) 0.6 p.u., which corresponds to a change of about 199.7 \(\rightarrow\) 203.8 rad/s. The range of speed changes was chosen small enough so that, in tests with the real drive system, the current would stay in the zone of the limitation value as short as possible because the testing of control systems for the operating point in the limitation does not give much information about the system. On the other hand, the range of variation in the set speed should be large enough to maintain the correct ratio of measured signal value to measurement noise. The chosen range of variation is a compromise between these conflicting requirements. The current limit value is set to 0.4 p.u. This refers to the value of the maximum current measured by the measurement system on the laboratory bench, not to the value of the rated current. This is a feature of the laboratory system used for which this model was created.