Improving Control Dynamics of PMSM Drive by Estimating Zero-Delay Current Value

1Abstract—Dynamic performance of current control loop still remains crucial for position-, speed-, and torque-controlled drives. In the study, a current loop solution has been designed for field oriented control of permanent magnet synchronous motors (PMSM). It enhances typical PI controller with an estimator of zero-delay current (ZDC) value. The ZDC estimation allows for selecting substantially higher controller gain. It reduces control loop step response rise time to a single control cycle, which is the shortest technically possible value, while avoiding overshoot. The method does not require any hardware changes and it needs only negligible processing overhead. Both simulations made and experimental results obtained in the study have proved the effectiveness of the proposed solution.


I. INTRODUCTION
Dynamic properties of electric drives are crucial to many industrial applications.The most demanding devices are typically fitted with permanent magnet synchronous motors (PMSM), which allow for very fast torque and speed response due to their low stator inductances and low moment of inertia, respectively.
Control structure of electric drive consists of those: an inner current control loop and of outer speed and position loops, optionally.Some recent efforts aim at improving the speed and position control algorithms [1]- [3].However, dynamic response of the drive is ultimately limited by the properties of the inner control loop.Therefore, the dynamic performance of the current control loop remains crucial for position-, speed-, and torque-controlled drives.
A typical current control loop is based on proportionalintegral (PI) controller, highly appreciated in industry due to its simplicity and reliability.Alternative approaches, e.g.predictive controllers, while being superior to PI-based solutions in terms of dynamic properties, are substantially more difficult to implement and require dedicated efforts to reduce their dependency on inaccuracy and variability of pre-determined drive parameters [4].
The work described in this paper addresses the gap between PI and alternative controllers, to find balance between implementation complexity and dynamic performance.The proposed current loop solution enhances the capabilities of typical PI controller by an estimation of motor current value for the beginning of forthcoming control cycle, i.e. zero-delay value.This reduces the control loop step response rise time to a single control cycle, i.e. to the shortest technically possible value, without hardware changes and with negligible processing overhead.

II. APPROACHES TO IMPROVEMENT OF CURRENT CONTROL LOOP PERFORMANCE
Recent works related to current control loop performance can be classified into three groups.The first group aims at improving current measurement accuracy.The second proposes to improve the dynamic properties of current loop by increasing the rate of control algorithm response without increasing the pulse width modulation (PWM) carrier frequency.The last one is focused on predicting the control variables for the forthcoming control cycles.
Impact on the control performance of electric drive from scaling and offset errors in current measurement is analysed by Kim et al. [5].Jarzebowicz analyses the errors resulting from transformation of sampled phase currents into rotating coordinate frame [6].Both papers contain methods for compensating systematic errors which improve general performance of current control.
Bocker and Buchholz prove that updating the PWM generator with a ratio of 8 to 16 may significantly improve control bandwidth [7].Typical controllers allow for single or double update of reference voltage in a PWM cycle [8].Therefore, the proposed solution requires an extraordinary PWM generator.
A predictive approach to field oriented current control algorithm in electric drives is proposed by Cortes et al. [9].Predictive current controller calculates future behaviour of the system based on a model and a set of possible actuations for the horizon of two forthcoming control cycles.Similar method using Smith predictor is applied to an inverter operating in active power filter by Zhou and Liu [4].Both approaches increase computational complexity and share dependency on using exact system parameters, inherent to model-based prediction.In turn, Anuchin and Kozachenko propose to extend PI controller with a current predictor which uses oversampling and digital filtering to calculate the zero-delay value of DC motor armature current [10].
The aim of this work is to indicate an algorithm for estimating zero-delay current (ZDC) values for PMSM drive and to investigate how this estimation will influence the dynamic performance of PI-based current control loop.The structure for field oriented control (FOC) of PMSM is shown in Fig. 1.The motor currents are controlled in d-q rotating reference frame [11].For motors with surface mounted permanent magnets the value of current id is forced to zero.The reference value of current iq is set according to the required torque.Both currents are regulated by PI controllers fitted with decoupling and electromotive force (EMF) compensation block [12].Motor current contains ripple component induced by modulated voltage (Fig. 2).The frequencies of the ripples are much higher than the control bandwidth, thus controller is capable to influence only fundamental current component.This should be extracted by control feedback in order to provide stable and accurate torque control.A convenient solution to this problem was reported by Blasko et al. [13].The authors proved that instantaneous current, when sampled at the mid-points of passive inverter states, corresponds to the fundamental component.This method of sampling currents simultaneously with valleys or peaks of PWM carrier is referred to as synchronous sampling.
PWM voltage sequence for k th control cycle has to be set up before this cycle starts, i.e. before the tV[k] instant in Fig. 2. Therefore the PWM update must be computed using the current value sampled either at tV[k-1] or tP [k-1] instants.This causes a delay of T or T/2, respectively, between the current measurement instant and PWM update instant.The delay reduces the dynamics of current control.The contribution of this work consists of: enhancing PI controller in PMSM drive FOC with linear extrapolation to estimate ZDC value from synchronous sampling; a method for selecting ZDC-enhanced PI controller gain; a timedomain model of digitally controlled drive to verify the effectiveness of the enhancement; implementing the ZDC PI controller; an experimental validation procedure; an assessment of increase in control algorithm computational complexity due to the enhancement.

III. ESTIMATING ZERO-DELAY CURRENT
The fundamental current component for a single PWM cycle can be well approximated by linear dependency [13], [14].Thus the zero-delay current value i(tV[k]) for the k th control cycle can be estimated by 2 .
Estimation requires sampling frequency being only twice the PWM frequency.The last current measurement i(tP[k-1]) used by (1) takes place in the midpoint of (k-1) th PWM cycle, leaving half of cycle for control algorithm execution.As measurements are carried out synchronously with PWM carrier, no signal filtering is required to isolate the fundamental current component.
The impact of the proposed ZDC estimation on the dynamic performance of PMSM drive is investigated below by the analysis of iq current step response.A mixed continuous-discrete model of the drive was implemented in MATLAB-Simulink.The general view of the model consisting of PMSM and digital controller is in Fig. 3.The PMSM model relies on standard continuous-time equations [2], [15].The controller is modelled with triggered subsystems executed at the midpoints or at the endpoints of control cycles.This corresponds to instants of signal sampling and to instants of inverter output update.The implemented control structure is as shown previously in Fig. 1.
To assess fundamental component rise time, one has to get rid of the ripple component of motor currents resulting from changes of inverter states in each PWM cycle.These states depend on such factors as rotor position or instantaneous DC-bus voltage.To make the rise time measurements independent from such uncontrolled factors, the modulated voltage was replaced in the model by voltage mean value calculated for each control cycle.As a result, the motor currents consist of only fundamental component.
The scope of the analysis includes the proposed ZDC approach and two typical current sampling scenarios, i.e. sampling at tV[k-1] or tP[k-1] instants.The model was set up to match laboratory drive parameters listed in Table I.The controller I-term reset time was set to TI = Lq/Rs which follows the common practice of pole-zero-cancelation [8].The controller gain KP was adjusted by experimenting individually for each case to obtain 5 per cent overshoot.Quantitative comparison of control loop dynamic performance is based on measuring rise time T90%, defined as the time required for the response to rise from 0 % to 90 % of its final value.The results of simulation including individual controller gains KP and rise times T90% are given in Fig. 4. The comparison shows the superiority of using ZDC approach over typical synchronous sampling scenarios.ZDC estimation enables for setting substantially higher controller gains to obtain the same overshoot.Therefore the rise time is reduced over 3 or over 2.5 times when compared to sampling at valleys or peaks, respectively.IV.SELECTING CONTROLLER GAIN In typical synchronous sampling scenarios the response time is a trade-off with respect to overshoot.The proposed ZDC approach allows for avoiding overshoot without increasing the rise time considered as number of control cycles.The response may still be achieved in a single control cycle if only the inverter output voltage does not reach the limit resulting from DC bus voltage.
The fastest digital control response would bring the current to its commanded value at the endpoint of the forthcoming cycle, i.e. iq(tv[k+1]) = iq_ref(tv [k]).Simultaneously, such a response would be featured by null overshoot.The following analysis aims to select the controller gain KP to obtain such a response.
Changes of q-axis current in PMSM can be calculated as [16], [17]   The term uq_dec = -Ldpmid -pmf is computed by the decoupling and EMF-compensation block and bypasses the PI controller [12], [15].The voltage drop uR = -Rsiq has a negligible impact on dynamic behavior.Therefore (2) can be simplified in terms of controller activity _ d , d and reformulated into discrete time-domain .
The step response with controller gain set according to (5) was investigated by simulation (Fig. 5).Under step command the controller sets the output voltage uq_PI to a value which cancels the error at the endpoint of the nearest control cycle (Fig. 5(a)).The decoupling and EMFcompensation function produces the uq_dec component which removes the influence of motor speed on controller activity.Therefore, the control performed by the PI controller is not affected by rotor speed as long as maximum inverter output voltage is not exceeded (Fig. 5

Fig. 2 .
Fig. 2. Selected waveforms in PMSM drive supplied by space vector PWM-controlled voltage source inverter.

Fig. 3 .
Fig. 3. Top-level view of a model of digitally controlled PMSM drive implemented in Simulink.

Fig. 4 .
Fig. 4. Comparison of step response rise times for ZDC approach and typical synchronous sampling scenarios.TABLE I. PMSM DRIVE PARAMETERS.Parameter Value PWM carrier frequency f = 1/T 10 kHz Rated phase current In 10 A Rated DC-bus voltage UDCn 216 V Rated speed (mechanical) mn 120 rad/s d-axis component of stator inductance Ld 0.9 mH q-axis component of stator inductance Lq 1.05 mH Flux linkage due to the rotor magnets f 75 mWb Number of pole pairs p 9

4 )
Following the requirements of desired dynamic response the current has to change by iq[k] = iq(tv[k+1]) -iq_ref(tv[k]) during time T of PWM cycle.Thus the gain Kp of the controller should be selected to satisfy _ [ ] [ ] m = 0; b) m = 0.25 p.u.; c) m = 0.5 p.u.For relatively high speeds and high current steps the sum of responses from the controller and EMF-compensation function uq_ref = uq_PI + uq_dec may exceed the maximum inverter output voltage.In such a case, the response takes more than one control cycle (Fig.5(c)).Nevertheless, controller still ensures the shortest technically possible response, considering voltage limitation.