# PMSM Current Controller with Pre-Control

Discrete-time permanent magnet synchronous machine current controller with pre-control

• Library:
• Simscape / Electrical / Control / PMSM Control

## Description

The PMSM Current Controller with Pre-Control block implements a discrete-time PI-based permanent magnet synchronous machine (PMSM) current controller in the rotor d-q reference frame with internal feedforward pre-control.

You typically use this block in a series of blocks making up a control structure.

• You can generate a current reference in the d-q frame to be used as an input to this block with a PMSM Current Reference Generator.

• You can obtain a voltage reference in the abc domain by converting the output of this block using an Inverse Park Transform block.

You can see an example of a full control structure, from machine measurements to machine inputs, in the PMSM Field-Oriented Control block.

### Equations

The block is discretized using the backward Euler method due to its first-order simplicity and its stability.

Two PI current controllers implemented in the rotor reference frame produce the reference voltage vector:

`${v}_{d}^{ref}=\left({K}_{p_id}+{K}_{i_id}\frac{{T}_{s}z}{z-1}\right)\left({i}_{d}^{ref}-{i}_{d}\right)+{v}_{d_FF},$`

and

`${v}_{q}^{ref}=\left({K}_{p_iq}+{K}_{i_iq}\frac{{T}_{s}z}{z-1}\right)\left({i}_{q}^{ref}-{i}_{q}\right)+{v}_{q_FF},$`

where:

• ${v}_{d}^{ref}$ and ${v}_{q}^{ref}$ are the d-axis and q-axis reference voltages, respectively.

• ${i}_{d}^{ref}$ and ${i}_{q}^{ref}$ are the d-axis and q-axis reference currents, respectively.

• ${i}_{d}$ and ${i}_{q}$ are the d-axis and q-axis currents, respectively.

• Kp_id and Kp_iq are the proportional gains for the d-axis and q-axis controllers, respectively.

• Ki_id and Ki_iq are the integral gains for the d-axis and q-axis controllers, respectively.

• Ts is the sample time of the discrete controller.

• vd_FF and vq_FF are the feedforward voltages for the d-axis and q-axis, respectively.

The feedforward voltages are obtained from the machine mathematical equations:

`${v}_{d_FF}=-{\omega }_{e}{L}_{q}{i}_{q},$`

and

`${v}_{q_FF}={\omega }_{e}\left({L}_{d}{i}_{d}+{\psi }_{m}\right),$`

where:

• ωe is the rotor electrical velocity.

• Ld and Lq are the d-axis and q-axis inductances, respectively.

• ψm is the permanent magnet flux linkage.

### Zero Cancellation

Using PI control results in a zero in the closed-loop transfer function, which can result in undesired overshoot in the closed-loop response. This zero can be canceled by introducing a zero-cancelation block in the feedforward path. The zero cancellation transfer functions in discrete time are:

`${G}_{ZC_id}\left(z\right)=\frac{\frac{{T}_{s}{K}_{i_id}}{{K}_{p_id}}}{z+\left(\frac{{T}_{s}-\frac{{K}_{p_id}}{{K}_{i_id}}}{\frac{{K}_{p_id}}{{K}_{i_id}}}\right)},$`

and

`${G}_{ZC_iq}\left(z\right)=\frac{\frac{{T}_{s}{K}_{i_iq}}{{K}_{p_iq}}}{z+\left(\frac{{T}_{s}-\frac{{K}_{p_iq}}{{K}_{i_iq}}}{\frac{{K}_{p_iq}}{{K}_{i_iq}}}\right)}.$`

### Voltage Saturation

Saturation must be imposed when the stator voltage vector exceeds the voltage phase limit Vph_max:

`$\sqrt{{v}_{d}^{2}+{v}_{q}^{2}}\le {V}_{ph_max},$`

where vd and vq are the d-axis and q-axis voltages, respectively.

In the case of axis prioritization, the voltages v1 and v2 are introduced, where:

• v1 = vd and v2 = vq for d-axis prioritization.

• v1 = vq and v2 = vd for q-axis prioritization.

The constrained (saturated) voltages ${v}_{1}^{sat}$ and ${v}_{2}^{sat}$ are obtained as follows:

`${v}_{1}^{sat}=\text{min}\left(\mathrm{max}\left({v}_{1}^{unsat},-{V}_{ph_max}\right),{V}_{ph_max}\right)$`

and

`${v}_{2}^{sat}=\text{min}\left(\mathrm{max}\left({v}_{2}^{unsat},-{V}_{2_max}\right),{V}_{2_max}\right),$`

where:

• ${v}_{1}^{unsat}$ and ${v}_{2}^{unsat}$ are the unconstrained (unsaturated) voltages.

• v2_max is the maximum value of v2 that does not exceed the voltage phase limit, given by ${v}_{2_max}=\sqrt{{\left({V}_{ph_max}\right)}^{2}-{\left({v}_{1}^{sat}\right)}^{2}}.$

In the case that the direct and quadrature axes have the same priority (d-q equivalence), the constrained voltages are obtained as follows:

`${v}_{d}^{sat}=\text{min}\left(\mathrm{max}\left({v}_{d}^{unsat},-{V}_{d_max}\right),{V}_{d_max}\right)$`

and

`${v}_{q}^{sat}=\text{min}\left(\mathrm{max}\left({v}_{q}^{unsat},-{V}_{q_max}\right),{V}_{q_max}\right),$`

where:

`${V}_{d_max}=\frac{{V}_{ph_max}|{v}_{d}^{unsat}|}{\sqrt{{\left({v}_{d}^{unsat}\right)}^{2}+{\left({v}_{q}^{unsat}\right)}^{2}}}$`

and

`${V}_{q_max}=\frac{{V}_{ph_max}|{v}_{q}^{unsat}|}{\sqrt{{\left({v}_{d}^{unsat}\right)}^{2}+{\left({v}_{q}^{unsat}\right)}^{2}}}.$`

### Integral Anti-Windup

An anti-windup mechanism is employed to avoid saturation of integrator output. In such a situation, the integrator gains become:

`${K}_{i_id}+{K}_{aw_id}\left({v}_{d}^{sat}-{v}_{d}^{unsat}\right)$`

and

`${K}_{i_iq}+{K}_{aw_iq}\left({v}_{q}^{sat}-{v}_{q}^{unsat}\right),$`

where Kaw_id, Kaw_iq, and Kaw_if are the anti-windup gains for the d-axis, q-axis, and field controllers, respectively.

### Assumptions

• The plant model for direct and quadrature axis can be approximated with a first-order system.

• This control solution is used only for permanent magnet synchronous motors with sinusoidal flux distribution and field windings.

## Ports

### Input

expand all

Desired d- and q-axis currents for control of a PMSM, in A.

Data Types: `single` | `double`

Actual d- and q-axis currents of the controlled PMSM, in A.

Data Types: `single` | `double`

Rotor electrical velocity used for feedforward pre-control, in rad/s.

Data Types: `single` | `double`

Maximum allowable voltage in each phase, in V.

Data Types: `single` | `double`

External reset signal (rising edge) for integrators.

Data Types: `single` | `double`

### Output

expand all

Desired d- and q-axis voltages for control of a PMSM, in V.

Data Types: `single` | `double`

## Parameters

expand all

Control Parameters

Proportional gain of the PI controller used for direct-axis current control.

Integrator gain of the PI controller used for direct-axis current control.

Anti-windup gain of the PI controller used for direct-axis current control.

Proportional gain of the PI controller used for quadrature-axis current control.

Integrator gain of the PI controller used for quadrature-axis current control.

Anti-windup gain of the PI controller used for quadrature-axis current control.

Sample time for the block (-1 for inherited). If you use this block inside a triggered subsystem, set the sample time to -1. If you use this block in a continuous variable-step model, you can specify the sample time explicitly.

Prioritize or maintain the ratio between d- and q-axes when the block limits voltage.

Enable or disable zero-cancellation on the feedforward path.

Enable or disable pre-control voltage.

Pre-Control Parameters

Direct-axis current vector used in the lookup tables for parameters determination. For constant machine parameters, do not change the default.

Quadrature-axis current vector used in the lookup tables used to determine parameters. For constant machine parameters, do not change the default.

Ld matrix used as lookup-table data. For constant machine parameters change only the constant factor, for example, Ld * ```ones(3, 3)```.

Lq matrix used as lookup-table data. For constant machine parameters change only the constant factor, e.g., Lq * ```ones(3, 3)```.

Permanent magnet flux linkage matrix used in the lookup table. For constant machine parameters change only the constant factor, for example psim * `ones(3, 3)`.

## References

[1] Bernardes, T., V. F. Montagner, H. A. Gründling, and H. Pinheiro. "Discrete-time sliding mode observer for sensorless vector control of permanent magnet synchronous machine." IEEE Transactions on Industrial Electronics. Vol. 61, Number 4, 2014, pp. 1679–1691.

[2] Carpiuc, S., and C. Lazar. "Fast real-time constrained predictive current control in permanent magnet synchronous machine-based automotive traction drives." IEEE Transactions on Transportation Electrification. Vol.1, Number 4, 2015, pp. 326–335.