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Published in IET Electric Power ApplicationsReceived on 14th November 2007Revised on 22nd February 2008doi: 10.1049/iet-epa:20070455

ISSN 1751-8660

Input filter state feed-forward stabilisingcontroller for constant power load systemsX. Liu A.J. ForsythSchool of Electrical and Electronic Engineering, The University of Manchester, PO Box 88, Sackville StreetManchester M60 1QD, UKE-mail: andrew.forsyth@manchester.ac.uk

Abstract: An input filter state feed-forward stabilising controller is presented to stabilise a constant power loadand is validated using a brushless DC motor drive system. The strategy is to feed-forward a stabilising signal whichis a function of the DC-link filter variables, capacitor voltage and the inductor current, into the current controlloop of the motor drive to modify the magnitude and phase of the system input admittance around the inputfilter natural frequency and thereby damp the input filter. The controller design and parameter selection aredescribed. The impact of the stabilising controller is examined on the motor controller performance andfinally the effectiveness of the controller is verified by simulation and experimentally.

NomenclatureCsrc capacitance of the input filter

D duty-ratio of the inverter

Gc(s) current controller transfer function

GCL(s) closed loop current control transfer function

H motor inertia constant ¼ 1=2v20 J=(v0

Tload0)

Ia averaged motor armature current

IDC averaged DC-link current after the inputfilter

IL input filter inductor current

Iref current reference from speed controller

Kfb current feedback constant

Lsrc inductance of the input filter

M1(s), M2(s) motor transfer functions

R1, R2 parameters of the stabilising controller

Tload load torque

te, tm motor electrical and mechanical timeconstants

�Va averaged motor armature input voltage

VDC DC-link voltage after the input filter

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vp corner frequency of the stabilising controller

v motor speed

YA(s), YD(s) low-frequency transfer functions of systeminput admittance

SA(s), SD(s) sensitivity transfer functions

Yin(s) input admittance of the system

z1 zero of YA(s)

z2, z3, z4 zeros of YD(s)

0 denotes the steady-state value of a variable

1 IntroductionPower electronics-based (PEB) systems are increasingly usedin aerospace, industrial and military areas because of thebenefits they may bring, such as higher efficiencies, reducedsize and lower maintenance. However, in a PEB system,tightly regulated power electronic devices like inverters,converters and motor drives may behave as constant powerloads to the power supply and have negative inputadmittance characteristics, which may lead to instability inthe input filter [1]. Commonly used techniques to preventthis instability include designing an appropriate input filteror using passive/active damping techniques to make the

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system admittances satisfy a certain stability criterion [1–5].These techniques usually require an electrolytic capacitorin the input filter or the damping network to achievegood damping. However, large electrolytic capacitors areunwelcome in some environments where high reliability isrequired and/or space is limited, for example in aerospaceapplications.

Little work has been reported on stabilising constantpower PEB systems, the main exceptions being thefeedback linearization technique presented in [6, 7], thenonlinear system stabilising controller (NSSC) in [8], anegative input resistance compensator (NIRC) in [9, 10]and a high-pass filter method in [11]. However, some ofthe methods are complex to implement and lack practicalvalidation [6, 7], or have reduced effectiveness under someoperating conditions, for example when the motor drivecurrent reference approaches zero [8]. The linearcompensator presented in [9, 10] is simple, but hasthe weakness of a high sensitivity to DC-link voltagetransients.

In this paper, an input filter state feed-forward stabilisingcontroller (SFSC) that offers a high immunity to DC-linkdisturbances while providing effective damping ispresented. The stabilising controller injects a stabilisingsignal to the control loop when disturbances occur andcompensates the system input admittance to a positivevalue around the natural frequency of the input filter. Thestabilising controller can effectively stabilise the systemwithin a wide range of operation providing that theparameters and dominant operating point of the system areknown. The stabilising controller is verified using a three-phase brushless DC (BLDC) drive. The controller wasintroduced in [9], but its characteristics were not fullyanalysed and there was no experimental validation.

2 Input filter state feed-forwardstabilising methodThe input filter state feed-forward stabilising controlalgorithm is derived from the following differentialequations describing the generic constant power loadsystem in Fig. 1, where the infinite bandwidth constantpower load, represented by a voltage dependent current

Figure 1 Generic constant power load and input filter

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source, is connected to a typical L–C filter

_I L ¼ �1

Lsrc

VDC þ1

Lsrc

Vs

_V DC ¼1

Csrc

IL �1

Csrc

P þ q

VDC

8>><>>: (1)

Lsrc and Csrc are the equivalent inductance and capacitance ofthe input filter. IL is the inductor current, VDC is the inputvoltage of the constant power load and Vs is the sourcevoltage. P is the demanded load power and q is theadditional power that is drawn by the load due to theaction of the stabilising controller.

The state variables IL and VDC can each be written as thesum of two components, the ideal or desired value denotedas I �L and V �DC, respectively, plus an error term,eI L and eV DC,which must approach zero during transients. Applying thissubstitution for IL and VDC in (1) and adding two dampinginjection terms, R1

eI L and eV DC=R2, respectively, to eachside of the first and second equations in (1) results in

Lsrc_eI L þ

eV DC þR1eI L ¼ Vs � Lsrc

_I�

L � V �DC þR1eI L

Csrc_eV DC �

eI L þeV DC

R2

¼ I �L �P þ qeV DC þ V �DC

�Csrc_V�

DC þeV DC

R2

8>>>>>>><>>>>>>>:(2)

where R1 and R2 are real positive numbers. R1 and R2 arealso known as the damping injection constants and will bethe main parameters for the stabilising controller.Equation (2) can be written in matrix form as follows in

terms of the error variableex ¼ eI L, eV DC

n o`

DDDDD _exþRRRRRex ¼ G (3)

where DDDDD ¼Lsrc 00 Csrc

� �, RRRRR ¼

R1 1�1 1=R2

� �and

G ¼

�Lsrc_I�

L � V �DC þR1eI L þ Vs

�Csrc_V�

DC þ I �L �P þ qeV DC þ V �DC

þeV DC

R2

8><>:9>=>; (4)

When G ¼ 0, the error variable vectorex will asymptoticallyapproach zero, which ensures the system state variables, IL

and VDC, the inductor current and the DC-link voltage,asymptotically approach their desired values I �L and V �DC.Therefore the equation G ¼ 0 may be used to form thecontroller for the system with I �L and V �DC effectively beingthe state variables of the controller. Equating the expressionfor G in (4) to zero and rearranging results in

Lsrc_I�

L ¼ �V �DC þ R1eI L þ Vs

Csrc_V�

DC ¼ I �L �P þ qeV DC þ V �DC

þeV DC

R2

8><>: (5)

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which describes the relation between the compensating signalq and the state variables of the controller I �L and V �DC. Withtwo state variables, the controller design has two degrees offreedom, but since there is only one output, then forsimplicity one of the state variables, I �L or V �DC, can bearbitrarily set to follow the trajectory of the desired steady-state value. The controller then has a single variable and asingle output. In this paper, I �L is set to be a constantvalue of I �L ¼ IL0

¼ P=VDC0, where the 0 subscript denotes

the steady-state value of the variable. Therefore thederivative of I �L becomes zero and, after rearrangement, (5)becomes

V �DC ¼ VDC0þ R1 IL �

P

VDC0

!

q ¼ VDC

P

VDC0

� CsrcR1_I L

!� P þ

(VDC � V �DC)VDC

R2

8>>>><>>>>:(6)

The control law in (6) requires a knowledge of the systempower P, which may not be readily available in practice. Tosimplify the implementation it is assumed that the DC-linkvoltage only varies by a small amount around the steady-state value, that is VDC/VDC0

’ 1 when there is adisturbance in the system, and (6) becomes

V �DC ¼ VDC0þ R1(IL � IL0

)

q ¼ �VDCCsrcR1_I L þ

(VDC � V �DC)VDC

R2

8<: (7)

Eliminating V �DC between the two equations in (7) results inthe simplified control algorithm

q ¼ �R1

R2

VDC(CsrcR2_I L þ IL � IL0

) (8)

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3 SFSC for a BLDC motordrive system3.1 System structure with the SFSC

In this paper, the simplified control law (8) is used to stabilisea BLDC motor drive system. The block diagram of thesystem with the stabilising controller is shown in Fig. 2.The system comprises a low-voltage, BLDC motor rated at125 W, which is supplied by a three-phase MOSFETinverter and loaded by a brushed DC generator. Hall effectsensors in the motor provide commutation signals to theinverter every sixty electrical degrees, and within each sixtyelectrical degree interval only two of the inverter legs areactive and only two motor phase windings are inconduction. The position decoding, current and speedcontrol loops are implemented using an industry-standardUC3625 integrated circuit. The parameters of the systemcan be found in Table 1.

The SFSC block takes two inputs, the DC-link voltageVDC and the input filter inductor current IL and injects asignal Imod into the current reference Iref of the motorcontroller

I �ref ¼ Iref þ Imod (9)

where Imod ¼ Kpq and Kp is the power to machine currenttransform constant. For simplicity, Kp is fixed according tothe dominant operating point of the system in this paper,although it should ideally vary with the system operatingpoint. Since the machine speed and therefore terminalvoltage vary relatively slowly, then the change in powerdrawn by the machine will, transiently, be proportional tothe change in machine current.

Figure 2 Block diagram of the BLDC system with the SFSC

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A block diagram for the SFSC may be constructed from(8), Fig. 3, where the calculation of IL 2 IL0

is realisedthrough a high-pass filter, that is

IL � IL0¼

s

s þ vp

IL (10)

where vp is the corner frequency of the filter.

3.2 Modelling the motor drive

The averaged-value modelling technique is used to model theDC–AC inverter. The input voltage of the motor Va isdetermined by the PWM signal and the system DC-linkvoltage VDC. Neglecting circuit parasitics, the local averageof Va, denoted �V a can be written as

�V a ¼ DVDC (11)

where 21 � D � 1 is the inverter duty-ratio.

Similarly, the local average of the inverter input currentIDC maybe expressed as

�I DC ¼ D �I a (12)

where �I a is the local average of the active phase current.

The motor is modelled by the standard differentialequations describing the electrical and mechanical parts ofthe system. Converting the equations to the Laplace

Table 1 Test parameters

source voltage 35 V

maximum motor torque 0.3 Nm

maximum motor continuous speed 418 rad/s

input filter capacitor Csrc 47.6 mF

input filter inductor Lsrc 500 mH

system inertia J 1.56E-4 kgm2

motor armature inductance La 0.799 mH

motor armature resistance Ra 0.78 V

motor torque constant Kt ¼ Ke 0.084 Nm/A

Figure 3 Detailed structure of the SFSC

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domain and rearranging results in the following transferfunction relation

�I a ¼Js

Js(Las þ Ra)þ KeKt

�V a þKe

Js(Las þ Ra)þ KeKt

Tload

¼M1(s) �V a þM2(s)Tload (13)

where La and Ra are the motor inductance and resistance,respectively. Ke and Kt are the motor back emf constant,and torque constant, respectively, J is the total mechanicalinertia and Tload is the load torque.

The averaged-value expressions for the inverter, (11) and(12) may be linearised by considering small changes in thevariables then removing the steady-state components andneglecting the products of small changes

d �V a ¼ VDC0dD þD0dVDC (14)

d�I DC ¼�I a0

dD þD0d�I a (15)

where the d prefix denotes the small-signal change in avariable.

3.3 System small-signal model withthe SFSC

First, working from the block diagram of the SFSC in Fig. 3,a transfer function may be written down which relates dImod

to the small changes in the DC-link voltage and current,dVDC and d�I DC

dImod ¼ sCsrcB(s)dVDC þ B(s)d�I DC (16)

where

B(s) ¼ �KpR1CsrcVDC0

s(s þ z4)

s þ vp

(17)

z4 ¼ vp þ1

R2Csrc

(18)

The derivation of (16) also uses (19), obtained by summingthe local average values of the component currents in theDC-link capacitor

IL ¼�I DC þ sCsrcVDC (19)

Then, by combining the machine transfer function in (13)with the equations for the small-signal model of theinverter (14),(15) and using the model of the SFSC, (16),the overall small-signal model of the system may be drawnas shown in Fig. 4.

3.4 Modified current control loop

It can be seen from Fig. 4 that a signal path from d�I a to dIref

has been introduced through B(s), which will modify theclosed-loop current control transfer function. The newclosed-loop current control transfer function can be derived

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Figure 4 Small-signal model of the system with the SFSC

from Fig. 4 as

d�I a

dIref

¼GCL(s)

1þ Kfb[1� B(s)YC(s)]GCL(s)(20)

where

GCL(s) ¼ VDC0M1(s)Gc(s) (21)

YC(s) ¼�I DC0

KfbVDC0D0M1(s)

þD0

Kfb

¼D0

Kfb

(1þH=tm)

(1=2þH=tm)

(s=z2 þ 1)(s þ z3)

s(22)

z2 ¼2

te

1þH

tm

� �(23)

z3 ¼1=2

tm þH(24)

Gc(s) ¼Kc(s þ zc)

s(25)

here te ¼ La/Ra, tm ¼ JRa/(KeKt) are the electrical andmechanical time constants of the motor andH ¼ 1=2v2

0J=(v0Tload0) is the inertia constant of the

motor. Kfb is the original current feedback constant.

It can be seen from (20) that effectively the feedback termin the current control loop has been changed from Kfb toKfb[1 2 B(s)YC(s)]. To understand better this effect, (20) isexpanded using (17), (21) and (22), which results in

d�I a

dIref

¼1

R1K1Kfb

(s þ vp)

s3þ ðz2 þ z4 þ

1

R1K1zG

�s2

þ z2z4 þzG þ vp

R1K1zG

� �s þ

vp

R1K1

(26)

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where

K1 ¼KpCsrcD0VDC0

te

Kfb(1þ 2H=tm)(27)

zG ¼KfbVDC0

Kc

La

(28)

By assuming that

vp �1

R1K1(z2 þ z4)(29)

which may be satisfied by appropriate choice of vp, (26) maybe factorised as

d�I a

dIref

¼1

R1K1Kfb

� �(s þ vp)

(s þ zp)(s2 þ jvBs þ v2B)

(30)

where

zp ’ vp

1

1þ R1K1z2z4

(31)

vB ’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

R1K1

þ z2z4

s(32)

It can be seen from (30) that the new closed loop current controltransfer function has a pole at zp and a zero at vp. In order tominimise the impact on the current control loop bandwidth,the pole zp should coincide with vp, that is from (31)

vp

1

1þ R1K1z2z4

’ vp (33)

which may be satisfied by letting

R1 ¼0:1

K1z2z4

(34)

Furthermore, since the new current control loop bandwidth isdetermined by the pair of complex poles in (30), the complex

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pole frequency vB must ideally be set at approximately theoriginal current loop bandwidth.

3.5 System input admittance withthe SFSC

To investigate the effect of the SFSC on the system inputadmittance, the small-signal expression for the system inputadmittance Yin(s) is derived using Fig. 4 and the small-signal expressions for �I DC, (15) and motor current �I a, (13)

Yin(s) ¼d�I DC

dVDC

¼KfbGCL(s)

1þ Kfb[1� B(s)YC(s)]GCL(s)

� [YA(s)þ YD(s)] (35)

where

YA(s) ¼ ��I DC0

VDC0

(s=z1 þ 1)

(s=zc þ 1)(36)

YD(s) ¼ sCsrcB(s)YC(s) (37)

z1 ¼Kfb

�I DC0Kczc

Kfb�I DC0

Kc �D20

(38)

The expression for Yin in (35) is a product of the closed-looptransfer function of the current control loop, Kfb, and the sumof the terms YA(s) and YD(s). The product of Kfb and theclosed-loop current control transfer function will be unitywithin the loop bandwidth. YA(s) is independent of thestabilising controller parameters and represents the inherentinput admittance, whereas YD(s) represents the contributionmade by the stabilising controller. At low frequency YA(s)takes the value ��I DC0

=VDC0, which is the classical negative

input admittance of an ideal constant power load. Theoverall input admittance, Yin(s), deviates from the idealvalue of ��I DC0

=VDC0at higher frequency due to the finite

bandwidth of the current controller. To damp the inputfilter, Yin(s) must have a positive real value around the inputfilter natural frequency, which may be achieved byarranging for YD(s) to have a positive real value at thisfrequency that is greater than the magnitude of the negativereal part of YA(s).

The expression for YD(s), (37) can be expanded using (17)and (22)

YD(s) ¼ �R1CsrcK1

s(s þ z2)(s þ z3)(s þ z4)

s þ vp

(39)

The magnitude asymptotes of YD(s) and YA(s) are sketched inFig. 5 where the compensating admittance term YD(s) has agreater magnitude than YA(s) at the input filter naturalfrequency vsrc and a phase angle that is close to zero. Thezero frequency z3 is related to the mechanical time constantof the machine and is therefore the lowest corner frequencyin the YD(s) transfer function. The zero frequency z2 isrelated to the electrical time constant of the machine and

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will occur at a higher frequency. The pole frequency vp is aparameter of the stabilising controller, Fig. 3, which ischosen in the design process according to the inequality(29) and is assumed here to occur between z2 and z3. Thefourth zero z4 depends on vp and the controller parameter,R2. To ensure that YD(s) has a small phase angle in theregion of vsrc, z4 is placed below vsrc. Here vsrc is assumedto be less than z2, however, if vsrc were higher than z2, thepositions of z2 and z4 must be exchanged to provide astabilising effect at the input filter natural frequency.

3.6 Impact on system performance

To examine the impact that the stabilising controller has onthe drive system performance the small-signal sensitivitytransfer function, relating d�I a, which is directlyproportional to motor torque, and dVDC is derived usingthe system small-signal model in Fig. 4 and (13) and (15)

d�I a

dVDC

¼KfbGCL(s)

1þKfb[1�B(s)YC(s)]GCL(s)[SA(s)þ SD(s)] (40)

where

SA(s)¼D0s

VDC0KfbKc(sþ zc)

(41)

SD(s)¼�KpR1C2

srcVDC0

Kfb

s(sþ z4) s��I DC0

VDC0Csrc

!sþvp

(42)

Similar to the expression in (35) for the input admittance, thesensitivity transfer function in (40) is a product of the closed-loop current control transfer function, Kfb, and the sum of theinherent sensitivity transfer function, SA(s) and the impactcaused by the SFSC, SD(s).

The magnitudes of SA(s) and SD(s) are sketched in Fig. 6.Ideally SA(s) should be zero, indicating that the motor torquehas complete immunity to supply voltage transients, howeverjSA(s)j rises with frequency due to the finite bandwidth of thecurrent controller. The transfer function of SD(s) has a pole atvp and three zeros: at the origin, �I DC0

=VDC0Csrc in the right

half plane and z4 in the left half plane. The magnitude ratio

Figure 5 Sketched magnitude plots of YA(s) and YD(s)

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jSD(s)/SA(s)j at low frequency may be expressed asR1KpKczcCsrcz4VDC0

�I a0=vp. To reduce the impact of the

stabilising controller to the torque control performance ofthe drive, the stabilising parameters should ideally bechosen to make jSD(s)/SA(s)j as small as possible.

Beyond the frequency of z4 the magnitude of SD(s) starts toincrease quickly. However, beyond the bandwidth of thecurrent control loop the first term in (40) will fall withincreasing frequency, which will offset the increasing valueof SD(s).

4 Design and performance of theSFSC4.1 Controller design

To demonstrate the performance of the stabilising controllerand validate the small-signal analysis, the controller wasimplemented on the BLDC motor drive system shown inFig. 2 using an AN734 analogue multiplier and severalstandard op-amps. The system parameters are shown inTable 1. The natural frequency of the input filter vsrc was6.3 krad/s, well within the bandwidth of the currentcontrol loop, which was 22 krad/s in the uncompensatedsystem, and the system was unstable at the nominaloperating power of 125 W. The switching frequency was25 kHz.

The stabilising controller was designed by selectingz4 ¼ 2.5 krad/s and vp ¼ 500 rad/s, resulting in an overallinput admittance of jYinj ¼ 0.2S, /Yin ¼ 328 at the filtercorner frequency and a damping factor of 0.9 for the inputfilter poles. The values of the associated dampingcoefficients R1 and R2 were calculated to be 50 and 10 V,respectively. This resulted in a small reduction in thecurrent control loop bandwidth to 15.3 krad/s.

The choice of the parameters z4 and vp was a compromisebetween achieving a satisfactory damping of the inputfilter, while fulfilling the design expression identified inthe analysis, (29), and maintaining the current loopbandwidth, (32), close to its original figure of 22 krad/s,

Figure 6 Sketched magnitude plots of SA(s) and SD(s)

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without seriously degrading the rejection of supply voltagedisturbances (40).

4.2 Frequency domain responses

The HP3577B network analyser was used to measure theinput admittance of the system with the SFSC controller.The practical system included an outer speed feedback loopwith a bandwidth of 844 rad/s, however the presence of thespeed control loop had virtually no noticeable effect on theadmittance frequency response plots, thereby confirmingthat it was valid to neglect the speed control loop in theanalysis of Section 3. The input admittance measurementwas made by injecting a small, variable frequencydisturbance signal between the DC-link capacitor and theDC input terminals of the inverter. The disturbance signalwas generated by the network analyser and coupled into theDC-link using a wide-band power amplifier. The resultingsmall changes in inverter input voltage and current weremeasured with the network analyser and used to determinethe input admittance.

The measured admittances (in solid lines) are presented inFig. 7 for a range of values of the parameter R1 withR2 ¼ 10 V along with the predicted input admittance (indash-dotted lines) calculated by (35). The design values ofR1 ¼ 50 V, R2 ¼ 10 V are also indicated in Fig. 7.Overall, the results show close agreement between thepredictions and measurements, confirming the accuracy ofthe small-signal analysis. Without the compensatingcontroller the admittance has a phase angle of 1808 acrossthe bandwidth of the current control loop, up to around22 krad/s.

Figure 7 Measured and predicted input admittance at 50%full speed, v ¼ 209 rad/s, Tload ¼ 0.3 Nm, power ¼ 67 W,R2 ¼ 10 V and vp ¼ 500 rad/s, with different R1 ¼ 20,50, 100 V (equivalent to z4 ¼ 6.5, 2.5, 1.2 krad/s) currentcontrol loop bandwidth ¼15.3 krad/s, vsrc ¼ 6.3 krad/s(measured data: solid lines; predicted data: dash-dot lines)

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It can be seen that the SFSC doesn’t change the systemadmittance at frequencies below 1 krad/s. Beyond afrequency of 3.6 krad/s, the SFSC compensates the phaseangle of the input admittance to be approximately zeroaround the natural frequency of the input filter vsrc ¼

6.3 krad/s. The magnitude of the input admittance at theinput filter natural frequency is higher with greater R1 orsmaller z4 and the phase angle is further away from zerowith higher R1, which all agree with the previous analysis.The variation of R2 (vp) has very little effect on themagnitude of the input admittance and is therefore notshown.

To validate the sensitivity transfer function derived in (40),the network analyser was used to measure the relationshipbetween the disturbance injected into the DC-link and theresultant disturbance observed in the armature current. Thespeed feedback was disabled in the measurement toeliminate the effect of the speed control loop.

The measured and predicted results are shown in Fig. 8,again for several values of R1 and R2. The steady-stateoperating point was maintained at the same level as inFig. 7. The measured and predicted results show closeagreement but with some divergence above a frequency of50 krad/s. It can be seen that the SFSC increases thesensitivity magnitude by less than 10 dB at frequenciesbelow 1 krad/s with the maximum R1, whereas theincrease is greater at higher frequencies, above 20 dB forfrequencies above 10 krad/s. Compared with the NIRCcontroller, [10], the SFSC produces a much smallerincrease in the sensitivity transfer function magnitude,typically 15 dB less, which results in a better immunity to

Figure 8 Measured and predicted d Ia/d VDC sensitivitytransfer function at 50% full speed, v ¼ 209 rad/s,Tload ¼ 0.3 Nm, power ¼ 67 W, R2 ¼ 10 V and vp ¼

500 rad/s, with different R1 ¼ 50, 100 V (equivalent toz4 ¼ 2.5, 1.2 krad/s) and R1 ¼ 50 V and z4 ¼ 2.5 krad/s,R2 ¼ 100 V (equivalent to vp ¼ 2.1 krad/s) currentcontrol loop bandwidth ¼ 15.3 krad/s, vsrc ¼ 6.3 krad/s(measured data: solid lines; predicted data: dash-dot lines)

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the disturbances on the DC-link voltage. Smaller R1 (largerz4) and larger R2 (larger vp) result in a smaller increase inthe magnitude, which confirms the conclusion drawnearlier in the paper.

4.3 Time domain responses

The averaged-value Simulink model for the BLDC motordrive system with the SFSC is used to simulate thebehaviour of the system. Fig. 9 shows the response of themotor speed and the associated transient in the DC-linkvoltage to a 30 rad/s step increase in speed reference withfour different values of R1. The motor speed is shown onthe left and the DC-link voltage on the right. Simulationresults are shown in dashed lines and measured results areshown in solid lines. There is close agreement between thesimulated and measured results which further confirms theoperation of the stabilising controller and validates the theory.

The results show that the speed control response isunaffected by the presence of the stabilising controller,however, with a very low value of R1 corresponding to largez4, wild oscillations are seen in the DC-link voltage,indicating that the system is on the margin of instability.With larger values of R1 the DC-link transient becomessmall and well damped. The measured transient in theDC-link voltage is more heavily damped than the

Figure 9 Measured and simulated system response to smallstep increase in speed reference for a range of values of R1,initial v ¼ 209 rad/s, Tload ¼ 0.3 Nm, R2 ¼ 10 V, vp ¼

500 rad/s (measured data: solid lines; predicted data:dash-dot lines)

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prediction for small values of R1 due to the presence of circuitlosses, which were not included in the model. However, withhigher values of R1 the damping of VDC was less effectivethan predicted, and this was attributed to non-linear effectssuch as the current limit in the motor controller. With afixed R1 ¼ 50 V, z4 ¼ 2.5 krad/s, the system response isrelatively insensitive to R2 and therefore is not included inthis paper.

Fig. 10 shows the response of the motor speed and DC-link voltage to a 3 V step change in the DC supply voltage,again with four different values of R1. The speed signal ison the left and the DC-link voltage on the right. It can beseen that the effect of R1 remains the same as in the speedstep transients, namely increasing R1 (reducing z4)improves the damping. The results also illustrate thatincreasing the value of R1 to damp the input filter polesproduces a small transient in the motor speed, however, themotor speed deviation is ,0.1% of the steady-state speedwith the largest value of R1.

The system responses to large step changes have also beensimulated and measured and the results show that the largesignal response of the system remains well behaved and isin accordance with the small-signal analysis.

Figure 10 Measured and simulated system response to a3 V step increase in the DC supply voltage for a range ofvalues of R1, v ¼ 209 rad/s, Tload ¼ 0.3 Nm, R2 ¼ 10 V,vp ¼ 500 rad/s (measured data: solid lines; predicteddata: dash-dot lines)

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5 SummaryAn input filter state feed-forward stabilising controller for aconstant power load system is presented and verifiedthrough simulations and experiments. The stabilisingcontroller monitors the changes in both the DC-linkvoltage and the input filter inductor current and injects astabilising signal into the current reference.

A small-signal model was derived for a BLDC motor drivesystem with the SFSC and, based on this model, thecharacteristics of the controller were explained and a designprocedure for the system parameters was formulated. Theanalysis and experimental results indicate that thestabilising controller has very little effect on the overallsystem characteristics well below the natural frequency ofthe DC-link filter, and only compensates the system inputadmittance to a positive value within the frequency regionwhere the system is prone to oscillating. Through thistechnique, the system shows very good immunity todisturbances in the input voltage and the speed controlresponse is unaffected.

However, since the stabilising controller only modifies thesystem input admittance within a specific range offrequencies, it is essential for input filter parameters to beknown accurately. Another disadvantage of this stabilisingmethod is that under some design conditions it may benecessary to reduce the current control loop bandwidth inorder to achieve sufficient damping.

6 References

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[2] BELKHAYAT M., COOLEY R., WITULSKI A.: ‘Large signal stabilitycriteria for distributed systems with constant powerloads’. Proc. IEEE Power Electronics Specialists Conf.,PESC, 1995, pp. 1333–1338

[3] FENG X., LIU J., LEE F.C.: ‘Impedance specifications forstable DC distributed power systems’, IEEE Trans. PowerElectron, 2002, 17, (2), pp. 157–162

[4] HANKANIEMI M., KARPPANEN M., SUNTIO T.: ‘Load-imposedinstability and performance degradation in a regulatedconverter’, IEE Proc. Electr Power Appl., 2006, 153, (6),pp. 781–786

[5] SUNTIO T., HANKANIEMI M., KARPPANEN M.: ‘Analysing thedynamics of regulated converters’, IEE Proc. Electr. PowerAppl., 2006, 153, (6), pp. 905–910

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[6] EMADI A., EHSANI M.: ‘Negative impedance stabilisingcontrols for PWM DC/DC converters using feedbacklinearization techniques’. IEEE Intersociety EnergyConversion Engineering Conf., 2000, vol. 1, pp. 613–620

[7] MOSSKULL H.: ‘DC-link stabilization of an inductionmachine drive’. Proc. European Conf. Power Electronicsand Applications, EPE, 2003

[8] GLOVER S.F., SUDHOFF S.D.: ‘Experimentally validatednonlinear stabilizing control for power electronics-basedpower systems’, SAE Trans., 1998, 107, (1), pp. 68–77

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[9] LIU X., FORSYTH A.J.: ‘Comparative study of stabilizingcontrollers for brushless DC motor drive systems’. IEEEInt. Electric Machines and Drives Conf., IEMDC’05, 2005

[10] LIU X., FORSYTH A.J., CROSS A.M.: ‘Negative input-resistancecompensator for a constant power load’, IEEE Trans. Ind.Electron, 2007, 54, (6), pp. 3188–3196

[11] RIVETTA C.H., EMADI A., WILLIAMSON G.A., JAYABALAN R., FAHIMI B.:‘Analysis and control of a buck DC–DC converter operatingwith constant power load in sea and undersea vehicles’,IEEE Trans. Ind. Appl., 2006, 42, (2), pp. 559–572

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