Embedded Motor Control

Overview

A proper designed current regulator is the key to control the electric motor. As explained in FOC Overview section, following is an example of the control system:

Current regulator needs to perform following tasks:

• Decoupling of dq frame
• Anti-windup for output voltage commands
• Flux weakening

The items need to be discussed one by one.

Decoupling

With voltage equations on dq frame:

It can be seen that there are cross coupling terms between dq axis. u_d depends on not only i_d, but also i_q. So does u_q. Meanwhile, u_d/u_q outputs of the PI controller are only impacted by errors between i_d/i_q command and feedback. This means the current controller performance will be very poor or even not working at all especially for high speed operation, when cross coupling term is the major factor. One of the ways to compensate this is using the feedforward terms:

The cross coupling terms are now calculated using machine parameters, and are then be added to the outputs of the PI controller. In order to achieve proper behavior, machine parameters including L_q, L_d and φ_pm need to be obtained offline. This will be discussed in Calibration section.

Anti-Windup

Since PI controller is used here, it shall be guaranteed that integration term of the controller will not keep adding up when output becomes saturated. Otherwise, it will take a very long time for output to change the direction later. Furthermore, since final output voltage is defined by sqrt(u_d² + u_q²), controller shall making sure that this value won’t exceed the limitation as well. For maximum output voltage limitation, see PWM Generation section. Following is an example of current regulator code includes the anti-windup feature:

void CRF_T0() {

	Ud_PI.err = IeIDQ_T0_I_Id - IeCRF_T0_I_Id;  //err equals to command (IeIDQ_T0_I_Id) minus feedback (IeCRF_T0_I_Id)
	Uq_PI.err = IeIDQ_T0_I_Iq - IeCRF_T0_I_Iq;  //err equals to command (IeIDQ_T0_I_Iq) minus feedback (IeCRF_T0_I_Iq)

	PI(&Ud_PI, IeIDQ_T0_k_Udff); //voltage output equals to PI value plus feedforward value (IeIDQ_T0_k_Udff)
	PI(&Uq_PI, IeIDQ_T0_k_Uqff); //voltage output equals to PI value plus feedforward value (IeIDQ_T0_k_Uqff)

	IeCRF_T0_MI = Fast_Sqrt(Ud_PI.Out * Ud_PI.Out + Uq_PI.Out * Uq_PI.Out); //output voltage magnitude eqauls to sqrt(ud^2 + uq^2)

	//anti-windup
	if (IeCRF_T0_MI >= KeCRF_T0_k_Umax) { //if output voltage is higher than a pre-defined threshold (KeCRF_T0_k_Umax)

		//ud (Ud_PI.Out) and uq (Uq_PI.Out) are decreased equally
		Ud_PI.Out = KeCRF_T0_k_Umax / IeCRF_T0_MI * Ud_PI.Out; 
		Uq_PI.Out = KeCRF_T0_k_Umax / IeCRF_T0_MI * Uq_PI.Out;

		if (Ud_PI.Out > 0) // if output is bigger than 0, then stop PI from positive integration 
			Ud_PI.Upper_Sat = 1;
		else  // else, stop PI from negative integration
			Ud_PI.Lower_Sat = 1;

		if (Uq_PI.Out > 0) // if output is bigger than 0, then stop PI from positive integration 
			Uq_PI.Upper_Sat = 1;
		else // else, stop PI from negative integration
			Uq_PI.Lower_Sat = 1;

	}

	IeCRF_T0_k_Ud = Ud_PI.Out;
	IeCRF_T0_k_Uq = Uq_PI.Out;

}

Notice that linearly decreasing both u_d and u_q when output get saturated may not be a good idea for ceratin operation, since it will impact both i_d and i_q.

Flux weakening

i_d^* and i_q^* may not be achievable with existing dc link voltage, if they are not selected properly. This can due to miscalibration, motor parameter mismatch, etc. When this happens, current regulator shall be able to modify i_d^* and i_q^*, so the commands become achievable. Usually, it means that more negative id shall be applied to decrease the flux. So it is also called flux weakening. Following shows one of the flux weakening approaches.

Where V is the output voltage magnitude, V^* is the output voltage limitation. V_dc is measured dc link voltage. Notice that ΔV is single direction, and is limited up to 0. So V_dc^* is smaller or equal to V_dc . While the selection of i_d^* and i_q^* will be discussed in Operation Optimization section, smaller V_dc^* will lead to decreasing of the output voltage.

Future Reference

Current regulator is a very complicated topic, or could be the most complicated topic for FOC. For future reference, following two papers can be a good starting point:

F. Briz, “Analysis and design of current regulators using complex vectors”, IEEE Transactions on Industry Applications, vol. 36, pp. 817 – 825, May – Jun. 2000.

F. Briz, “Current and flux regulation in field-weakening operation”, IEEE Transactions on Industry Applications, vol. 37, pp. 42 – 50, Jan. – Feb. 2001.