Flux-Vector Control#

Flux vector control directly controls the stator flux magnitude and the electromagnetic torque, which simplifies reference generation especially in the field-weakening region [1]. We use the control law that results from feedback linearization of the flux and torque dynamics [2]. Here, the inclusion of integral action is briefly discussed, as an extension to [3]. Furthermore, these notes aim to provide a link between the column-vector and complex-vector formulations of the control law. Here, a synchronous machine is considered as an example (see the Synchronous Machine document for the model and notation), but the approach is analogous for induction machines.

Flux and Torque Dynamics#

From the machine model, the flux and torque dynamics can be written as

(1)#\[\begin{split}\frac{\mathrm{d} \psi_\mathrm{s}}{\mathrm{d} t} &= \frac{1}{|\boldsymbol{\psi}_\mathrm{s}|} \mathrm{Re} \{\left(\boldsymbol{u}_\mathrm{s} - R_\mathrm{s} \boldsymbol{i}_\mathrm{s} - \mathrm{j}\omega_\mathrm{m}\boldsymbol{\psi}_\mathrm{s}\right) \boldsymbol{\psi}_\mathrm{s}^* \} \\ \frac{\mathrm{d}\tau_\mathrm{M}}{\mathrm{d} t} &= \frac{3 n_\mathrm{p}}{2} \mathrm{Im} \{ \left(\boldsymbol{u}_\mathrm{s} - R_\mathrm{s} \boldsymbol{i}_\mathrm{s} - \mathrm{j}\omega_\mathrm{m}\boldsymbol{\psi}_\mathrm{s} \right) \boldsymbol{i}_\mathrm{a}^*\}\end{split}\]

These expressions are valid also for saturated machines, when the auxiliary current \(\boldsymbol{i}_\mathrm{a}\) is defined according to (11) in the Synchronous Machine document.

Proportional Control#

Consider proportional control of the form, resulting from the feedback linearization, [3]

(2)#\[\boldsymbol{u}_\mathrm{s,ref} = \hat R_\mathrm{s} \boldsymbol{i}_\mathrm{s} + \mathrm{j}\omega_\mathrm{m} \hat{\boldsymbol{\psi}}_\mathrm{s} + e_\psi \boldsymbol{t}_{\psi} + e_\tau\boldsymbol{t}_{\tau}\]

where \(\hat R_\mathrm{s}\) is the stator resistance estimate, \(\hat{\boldsymbol{\psi}}_\mathrm{s}\) is the estimated flux vector, and \(e_\psi = \alpha_\psi(\psi_\mathrm{s,ref} - |\hat{\boldsymbol{\psi}}_\mathrm{s}|)\) and \(e_\tau = \alpha_\tau(\tau_\mathrm{M,ref} - \hat{\tau}_\mathrm{M})\) are the control errors in the flux and torque, respectively, scaled by the respective bandwidths. The complex quantities \(\boldsymbol{t}_{\psi}\) and \(\boldsymbol{t}_{\tau}\) are the direction vectors.

(3)#\[\boldsymbol{t}_{\psi} = \frac{ |\hat{\boldsymbol{\psi}}_\mathrm{s}|}{ \mathrm{Re}\{\hat{\boldsymbol{\psi}}_\mathrm{s}\hat{i}_\mathrm{a}^*\}}\hat{i}_\mathrm{a} \qquad \boldsymbol{t}_{\tau} = \frac{2}{3 n_\mathrm{p} \mathrm{Re}\{\hat{\boldsymbol{\psi}}_\mathrm{s}\hat{i}_\mathrm{a}^*\}}\mathrm{j}\hat{\boldsymbol{\psi}}_\mathrm{s}\]

Notice that \(\mathrm{Re}\{\hat{\boldsymbol{\psi}}_\mathrm{s}\hat{i}_\mathrm{a}^*\} > 0\) in the whole feasible operating region.

Assuming accurate estimates, linearization of the closed-loop system consisting of (1)–(3) results in

(4)#\[\frac{\mathrm{d}\Delta \psi_\mathrm{s}}{\mathrm{d} t} = \alpha_\psi (\Delta \psi_\mathrm{s,ref} - \Delta \psi_\mathrm{s} ) \qquad \frac{\mathrm{d}\Delta\tau_\mathrm{M}}{\mathrm{d} t} = \alpha_\tau(\Delta\tau_\mathrm{M,ref} - \Delta \tau_\mathrm{M})\]

These closed-loop small-signal dynamics are valid also for saturated machines, if the auxiliary current is estimated based on (11).

Inclusion of Integral Action#

Consider a 2DOF PI control law consisting of (2) and (3) and the following integral states and control error terms,

(5)#\[\begin{split}\frac{\mathrm{d} x_\psi}{\mathrm{d} t} &= \alpha_\psi\alpha_\mathrm{i}(\psi_\mathrm{s,ref} - |\hat{\boldsymbol{\psi}}_\mathrm{s}|) \\ \frac{\mathrm{d} x_\tau}{\mathrm{d} t} &= \alpha_\tau\alpha_\mathrm{i}(\tau_\mathrm{M,ref} - \hat{\tau}_\mathrm{M}) \\ e_\psi &= \alpha_\psi(\psi_\mathrm{s,ref} - |\hat{\boldsymbol{\psi}}_\mathrm{s}|) + x_\psi - \alpha_\mathrm{i} |\hat{\boldsymbol{\psi}}_\mathrm{s}| \\ e_\tau &= \alpha_\tau(\tau_\mathrm{M,ref} - \hat{\tau}_\mathrm{M}) + x_\tau - \alpha_\mathrm{i} \hat{\tau}_\mathrm{M}\end{split}\]

where \(\alpha_\psi\) and \(\alpha_\tau\) are the reference-tracking bandwidths, \(\alpha_\psi\) and \(\alpha_\tau\) are the integral states, and \(\alpha_\mathrm{i}\) is the integral action bandwidth. It can be shown that the resulting linearized closed-loop reference-following dynamics remain the same as (4). However, in this case, both the flux and torque dynamics have additional pole at \(s = - \alpha_\mathrm{i}\) resulting from the integral action. The integral action pole is canceled from reference-tracking dynamics (see the 2DOF PI Controller document).

This control law is a variant of [2], having channel-specific reference-tracking bandwidths. It has been implemented in the motulator.drive.control.sm.FluxVectorController and motulator.drive.control.im.FluxVectorController classes for synchronous and induction machines, respectively. The disturbance observer structure is used in the implementation to avoid the need for anti-windup mechanism (see the Disturbance-Observer Structure document).

References