Topic 2

PMSM IMC PI current-controller tuning

Tune a PMSM d-axis or q-axis PI current controller from the first-order RL model, then compare the ideal IMC response with the sampled implementation and computational delay.

Plant\(G(s)=1/(Ls+R)\)
Tuning knob\(\lambda\), desired time constant
Output\(K_p,\ K_i\) and response

1. IMC-inspired tuning idea

After decoupling the speed-dependent terms, one current axis is often approximated by an RL plant. An IMC-inspired PI chooses the controller zero to cancel the plant pole and uses λ to set the desired closed-loop speed.

\[ G(s)=\frac{1}{Ls+R},\qquad C(s)=K_p+\frac{K_i}{s}, \] \[ K_p=\frac{L}{\lambda},\qquad K_i=\frac{R}{\lambda},\qquad \frac{K_i}{K_p}=\frac{R}{L}. \]

Under perfect pole–zero cancellation and without delay, the nominal closed loop is approximately first order:

\[ T(s)\approx\frac{1}{\lambda s+1},\qquad t_{10\text{–}90}\approx\ln(9)\,\lambda\approx2.197\lambda. \]

2. Interactive tuning and sampled response

Ideal first-order targetDigital RL + PIReference
—
\(K_p\)—
\(K_i\)—
Controller zero—
\(T_s\)—
\(\lambda/T_s\)—
Measured 10–90% rise—

3. What the contrast means

When \(\lambda\) contains many sampling periods, the digital response follows the ideal first-order target closely. As \(\lambda\) approaches only a few samples, the discrete PI, integrator update and actuation delay become part of the dominant dynamics. At that point, continuous-time tuning alone is not enough.

Use this page as a baseline, not a production guarantee

The simulation is a decoupled single-axis RL model. It does not include dq cross-coupling, voltage saturation, current measurement noise, parameter mismatch or speed-dependent angle rotation. The next topic closes both controllers against the same exact sampled PMSM model.

Continue to exact discrete design →