Voltage-Constraint-Tracking-Based Field-Weakening Control of IPM Synchronous Motor Drives
V oltage-Constraint-Tracking-Based Field-Weakening Control of IPM Synchronous Motor Drives
Shinn-Ming Sue,Member,IEEE,and Ching-Tsai Pan,Member,IEEE
Abstract—In this paper,?rst,the effect of the interior permanent-magnet synchronous-motor(IPMSM)stator resis-tance to the maximum available motoring or regenerative braking torque is clari?ed.Then,a drive operating point is inter-preted geometrically as the intersecting point of the torque-demand curve with either the maximum torque per ampere (MTPA)trajectory or the voltage-constraint curve inherently imposed by the motor and inverter under different control modes.Based on this principle,a novel voltage-constraint-tracking(VCT)?eld-weakening control scheme for IPMSM drives is proposed.The proposed method can automatically de-termine the desired MTPA or?eld-weakening control modes and provide a smooth transition between these two modes. No machine parameters are required in the?eld-weakening control mode,and no dc-link voltage sensor is used,rendering the proposed scheme rather robust.In addition,the minimum copper-loss operation can be preserved in the VCT-based control to achieve high ef?ciency.The proposed control method has a simple structure so that it can easily be implemented by modifying a con-ventional vector-controlled drive system for practical applications. Finally,a DSP-based prototype drive is constructed to verify the feasibility of the proposed scheme,and some experimental results are provided to demonstrate the satisfactory features.
Index Terms—Field-weakening control,interior permanent-magnet synchronous motor(IPMSM),maximum torque per ampere(MTPA)control,minimum copper-loss operation.
I.I NTRODUCTION
P ERMANENT-magnet synchronous motors(PMSMs)for variable-speed applications are becoming popular because they have some attractive characteristics such as high ef?-ciency,fast dynamic response,high power density,and they are maintenance-free[1].In particular,an interior PMSM (IPMSM)with PMs buried inside the steel rotor core has a me-chanically robust rotor structure which is particularly suitable for high-speed operation[2].Some industrial applications of the IPM machine can be found in the literature works such as air-conditioning compressor[3],electrical scooter[4],electric vehicle[5],and integrated starter generator for hybrid electric vehicles[6],[7].For a high-performance vector-controlled
Manuscript received February7,2007;revised September14,2007.This work was supported in part by the National Science Council of Taiwan under Grants NSC94-2213-E-159-015and NSC95-2221-E-007-266.
S.-M.Sue is with the Department of Electrical Engineering,Minghsin University of Science and Technology,Hsinchu30401,Taiwan,R.O.C.(e-mail: sue7811@https://www.wendangku.net/doc/b04639842.html,.tw).
C.-T.Pan is with the Department of Electrical Engineering,National Tsing Hua University,Hsinchu30013,Taiwan,R.O.C.(e-mail:ctpan@ee. https://www.wendangku.net/doc/b04639842.html,.tw).
Color versions of one or more of the?gures in this paper are available online at https://www.wendangku.net/doc/b04639842.html,.
Digital Object Identi?er10.1109/TIE.2007.909087IPMSM drive,accurate machine parameters are usually needed [8],[9].Unfortunately,the stator resistance and the PM?ux linkage vary with machine temperature[10]–[12].In addition, magnetic saturation may occur under different speed and load conditions resulting in the variation of the d-and q-axis induc-tances[13],[14].It is important to note that the performance of the drive is rather sensitive to machine parameters,particularly in the?eld-weakening region[15].In order to improve the performance of the current regulator and the ef?ciency of the drive,an online parameter-estimation approach was proposed [16],[17]in the constant torque-limit region for an IPM motor drive.However,the online estimated parameters were not used for calculating the corresponding current commands. The authors used a curve-?tting or a lookup-table method.Re-cently,an adaptive parameter estimator applied to both MTPA current-command calculator and current regulator to achieve a robust and ef?cient control performance was presented[18]. Nevertheless,earlier investigations still were not concerned with the?eld-weakening region.Other works used the inverter output voltage and the measured dc-link voltage to decide the onset of?eld weakening and the?ux level to achieve?eld weakening[4],[19].However,they needed the measurement of the inverter dc-link voltage and might cause instability at load detaching during deep?eld-weakening operation[4].An online optimal?eld-weakening control algorithm using the dc-link current,the q-axis current error,and the motor speed as optimization objectives was proposed[20].This paper used a ?xed time-step searching approach incorporated with the feed-forward control to achieve?eld-weakening operation.However, the necessity of the dc-link current,as well as the complex decision process,renders the algorithm less robust.Recently, a fuzzy logic controller for an IPMSM drive was presented in[21].The controller is designed to be robust for high-speed applications.The outer speed loop is a fuzzy logic controller to ful?ll the robust control.However,in the inner current loop,the d-axis current command is calculated by a curve-?tting method which needs motor parameters and information of?xed dc-link voltage.
In this paper,a VCT-based?eld-weakening control scheme for an IPMSM drive is proposed.The scheme can automatically change to a proper control mode,namely,the MTPA or the ?eld-weakening control,and,meanwhile,can maintain the minimum copper-loss operation.Furthermore,as the speed is in the partial?eld-weakening region[22],the control mode can be automatically decided according to the motor speed and the torque demand without complex calculations.The d-and q-axis current commands are generated by using voltage commands and speed error together with some constraints to
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keep the operating point close to the voltage-limit curve in the?eld-weakening region.Particularly,no machine parameters and no measurement of the dc-link voltage are required in the ?eld-weakening control mode.The remaining contents of this paper are organized as follows.In Section II,the mathemat-ical model and operation constraints of an IPMSM will be discussed.Then,the explanation of the proposed scheme is presented in Section III.Also,an implementation example and some experimental results are provided in Section IV.Finally, conclusions are presented in the last section.
II.M ATHEMATICAL M ODEL AND O PERATING
C ONSTRAINTS OF AN IPMSM
The steady-state voltage equations of an IPMSM in the rotor reference frame can be expressed as follows[23]:
νds=R s i ds?ωre L q i qs(1)
νqs=R s i qs+ωre(L d i ds+λPM)(2) whereνds andνqs are the stator d-and q-axis voltages,re-spectively;i ds and i qs are the stator d-and q-axis currents, respectively;R s,ωre,L q,L d,andλPM are the stator resistance, the electrical angular frequency,the q-axis inductance,the d-axis inductance,and the PM?ux linkage,respectively.The motor torque equation can be expressed as
T e=3
2
p
2
[λPM i qs+(L d?L q)i ds i qs](3)
where T e and p are the generated torque and the number of poles,respectively.For a high-performance IPMSM drive, to achieve fast transient response and to have high-ef?ciency operation are the two important concerns.Hence,the maximum torque per ampere(MTPA)control strategy[2],[8]was widely adopted in the constant torque-limit operating range.The cor-responding relationship of i ds and i qs is[8]
i ds=
λPM
2(L q?L d)
?
λ2
PM
4(L q?L d)2
+i2qs.(4)
For a practical motor drive system,the operating states should stay within the system limits,considering both motor and inverter ratings.They can be expressed in terms of the current and the voltage constraints,respectively,as
i2ds+i2qs≤I2sm(5)
ν2ds+ν2qs≤V2sm(6) where I sm and V sm are the maximum line-current and the maximum phase-voltage amplitudes,respectively.The current constraint(5)on i ds?i qs plane can be represented by a circular
region with the following circular boundary:
i2ds+i2qs=I2sm.(7) Substituting(1)and(2)into(6),one can also obtain the voltage constraint which can be represented by an elliptical region with
TABLE I
P ARAMETERS OF THE T ESTED IPM S YNCHRONOUS M
ACHINE Fig.1.Trajectories of the MTPA,the current-limit circle,two constant torque
curves,and?ve voltage-limit ellipses on the i ds?i qs plane for the tested IPMSM.
the following elliptical boundary:
(R s i ds?ωre L q i qs)2+[R s i qs+ωre(L d i ds+λPM)]2=V2sm.
(8) For a convenient explanation of the motor operation,the pa-rameters of the tested IPMSM,as shown in Table I,are used to calculate some characteristic curves.Fig.1shows?ve voltage-limit ellipses on i ds?i qs plane corresponding to?ve motor speeds,the current-limit circle,and two constant torque curves corresponding to±1N·m.In addition,the MTPA trajectory of (4),namely,the AJ0F curve,is also shown in the same?gure. From Table I,one can observe thatλPM>L d I sm,indicating that the tested motor has a?nite maximum speed.To keep the drive operating at a safe and stable state,the available operating point according to the desired speed and desired torque should be inside the overlapped region de?ned by(5)and(6).As the motor speed runs higher,the voltage-limit ellipse shrinks, and the area of the available operating region gets smaller. For instance,the area of the available operating region ofωrD, which is bounded by the DEFID curve,is smaller than that of ωrC,which is bounded by the CDEFG0C curve,as shown in Fig.1.Also,from Fig.1,one can clearly observe the effect of stator resistance R s on the voltage constraint.The long axis of
Fig.2.Torque limit to speed curve forωr>0for the tested IPMSM.
the ellipse is no longer parallel to the i ds-axis.As an illustrative
example,whenωrA=3000r/min,the corresponding voltage-
limit ellipse intersects with the current-limit circle at two points
A and H.One can see that these two points are not symmetrical
with respect to the i ds-axis.Fig.2further shows the torque limit
to speed curve corresponding to Fig.1forωr>0.From Fig.2,
one can observe that the speed range of the positive constant
torque limit is from0to3000r/min and that of the negative
constant torque limit is from0to3750r/min.Hence,one can
see that,asωr is greater than3000r/min,the regenerative
braking power of the motor is larger than the motoring power.
Now,consider the positive torque operating region of Fig.1.
Whenωr∈(0,ωrA),whereωrA is the maximum speed of the positive constant torque limit,the MTPA control of(4),as
denoted by the AJ0curve in Fig.1,can be applied to achieve
high performance.This operating region is in fact the constant
torque region.Whenωr∈(ωrA,ωrC),whereωrC is the speed that the corresponding voltage-limit ellipse passes through the origin,the corresponding operating region is called the partial ?eld-weakening region.In this region,depending on the load condition,either the MTPA or the?eld-weakening control mode can be applied.As an example,consider the case for speedωrB=3300r/min in the partial?eld-weakening region that the corresponding voltage-limit ellipse intersects with the current-limit circle and the MTPA trajectory at points B and J, respectively,as shown in Fig.1.In case the present load is light such that its corresponding torque-demand curve intersects with the MTPA trajectory at a point between J and0,the MTPA control should be applied.On the other hand,when the load is heavy such that its corresponding torque-demand curve intersects with the AJ0curve at a point between A and J, the?eld-weakening control should be applied to achieve high performance.Hence,a proper control scheme should be de-veloped to decide which control mode is suitable to keep the operation stable and high ef?ciency in the partial?eld-weakening region.Whenωr∈(ωrC,ωrE),whereωrE is the maximum available speed of the drive,it is known as the full ?eld-weakening region[22].In the full?eld-weakening region, only the?eld-weakening control is applicable.Similarly for the negative torque operating region where the corresponding characteristic curves are on the negative i qs half-plane in Fig.1, the previous three operation regions can be de?ned.However, due to the asymmetry of the voltage-limit ellipse with respect to the i ds-axis,the corresponding available operating region is also asymmetrical with respect to the i ds-axis.For example, one can observe from Fig.1that,for a positive torque T e= 1N·m and whenωr=ωrC=3600r/min,one should apply the full?eld-weakening control.However,for a negative torque T e=?1N·m and whenωr=ωrC=3600r/min,then the MTPA control can still be applied.In fact,for a negative torque, the drive will reach the full?eld-weakening operation when ωr>ωrD=3750r/min,as shown in Fig.1.One can also check from Fig.2that,atωr=3600r/min,the maximum available positive torque is about+2.12N·m,but that of the regenerative braking is still kept at the maximum absolute constant torque limit of|?2.45|N·m.From the aforementioned observation, one can see that the effect of R s on the performance of the motor drive should not be neglected.
Finally,consider the effect due to the variation of V dc.Ac-cording to(8),it is seen from Fig.1that as V dc is kept constant, then V sm is?xed,and the voltage-limit ellipse shrinks whenωr increases and vice versa.On the other hand,as V dc is varied, then V sm will also change.Hence,ifωr is kept constant,the voltage-limit ellipse also shrinks when V dc is reduced and vice versa.Since the variation of V dc occurs rather often in a battery-powered drive system,such as for a pure or hybrid electric vehicle drive,the control strategy should be properly modi?ed by detecting the magnitude of V dc as quickly as possible to keep the system stable and to operate it in a high-ef?ciency condition again.Conventionally,a voltage sensor is needed for measuring V dc.It needs additional cost and may further introduce system reliability problem.Hence,a control scheme that can automatically and smoothly change the control mode between the MTPA and the?eld-weakening control without the dc-link voltage sensor is essential.
For a high-performance vector-controlled IPMSM drive,the MTPA control strategy is a frequently used method for its high ef?ciency and fast dynamic response in the constant torque-limit region.Fortunately,it had also been proven that the intersection point of the torque-demand and voltage-limit curves is the theoretical minimum copper-loss operating point in the?eld-weakening mode[22].However,for a practical motor drive,parametrical variations are unavoidable,and they may cause some unwanted situations such as instability and performance-degradation problems,particularly in the?eld-weakening region.Thus,a new?eld-weakening control scheme will be presented in the next section to automatically determine the MTPA control or the?eld-weakening control for an IPMSM drive.Particularly,no machine parameters and no measurement of the dc-link voltage are required in the?eld-weakening con-trol mode to achieve minimum copper loss.
III.P ROPOSED VCT-B ASED F IELD-W EAKENING
C ONTROL S CHEME
For reference,a speed-control block diagram of an IPMSM drive is shown in Fig.3.It is seen from Fig.3that the signals
Fig.3.Block diagram of a speed-controlled IPMSM
drive.
Fig.4.Block diagram of the proposed VCT-based controller corresponding
to Fig.3.
of i as and i bs are the motor line currents of phases a and b,respectively.The pulse signals of A,B,and Z are the outputs of the incremental encoder,where A and B are two sequences of pulses with phase shifts of 90?.When the rotor revolves one cycle,a signal pulse will appear in Z for resetting.The
input signal ω?
r
is the speed command.Also,the input signals of the proposed VCT-based controller block are ω?
r
,ωr ,i ds ,i qs ,the control variable u ?,and the rotor position angle θr .Except for the proposed VCT-based controller block,the other blocks are quite typical and will not be explained further.The purpose of the proposed VCT-based controller is to decide the proper inverter gating signals according to the input signals for each possible torque demand over full-speed operation range.For easy explanation,the VCT-based controller block is decomposed into two stages,namely,the d -and q -axis current-commands generator and the current controllers,as shown in Fig.4,respectively.
Considering a stable drive operation,the available current vector according to the desired speed and torque should be inside the overlapped region de?ned by (5)and (6).Hence,as the current vector is located inside the available operating region but not at the intersection point of the torque-demand and voltage-limit curves,the demanded current vector should be moved right to approach the voltage-limit curve to achieve the minimum copper-loss operation.It is implemented by assigning the d -axis current command as
i ?ds (t )=i ?
ds (t 0)+
t
t 0
K a |ω?
r ?ωr |dt
(9)
to obtain no steady-state speed error,where K a is a positive integration constant.On the other hand,as the commended voltage vector is located outside the available operating range,the current controllers tend to lose control because of the limited dc-link voltage.Hence,the demanded current vector should be moved left quickly to avoid lose of control of the current controllers.It is implemented by assigning the d -axis current command as
i ?ds (t )=i ?
ds (t 0)?
t
t 0
K b |ω?
r ?ωr |dt
(10)
to obtain no steady-state speed error,where K b is a positive in-tegration constant.According to the previous integrating tuning
of d -axis current command in the ?eld-weakening region,the operating point can get very close to the intersection point of the torque-demand and voltage-limit curves under steady state.The operational principle of the proposed scheme can be explained by using the block diagram and the ?owcharts,as shown in Figs.4and 5,respectively.As shown in Fig.4,the input signals of d -and q -axis current-commands generator are
u ?,ω?
r
,ωr ,νds ,pu ,and νqs ,pu ,where νds ,pu and νqs ,pu are the per-unit quantities of νds and νqs ,respectively.νds ,pu and νqs ,pu are calculated in the decoupled proportional and integral (PI)d -and q -axis current-controller block.The control variable u ?is the output of the PI speed controller.Now,consider Fig.5.First,the control variable u ?is assigned to be equal
to i ?qs .Then,the ?owchart checks whether the sum of ν2
ds ,pu
and ν2qs ,pu is greater than or equal to V 2sm ,pu ,where V 2
sm ,pu is the square of the per-unit quantity of the maximum phase-voltage amplitude.It is worth mentioning that using per-unit quantities enables the ?eld-weakening control mode to operate successfully without information of the motor parameters and dc-link voltage.If the inverter output voltage is saturated,the value of an index named “mtpaindex”is assigned to be equal to zero to indicate that the ?eld-weakening control should be activated.It should be mentioned that the value of “mtpaindex”is assigned to be one as the program is in the initialized stage.By checking the value of “mtpaindex,”the decision process divides into the MTPA control and the ?eld-weakening control routes.In the succeeding ?owchart of the MTPA control,i ?ds can be calculated by using (4)directly.As to the succeeding ?eld-weakening control process,it is further divided into two ?owcharts named A and B by checking again whether the
sum of ν2ds ,pu and ν2qs ,pu is greater than or equal to V 2
sm ,pu .If the phase voltage is not saturated yet,the program will go to ?owchart A.On the other hand,the ?owchart B will be selected.
In ?owchart A,the per-unit quantity of the inverter output voltage still does not reach V sm ,pu .This condition reveals that the motor drive may either be in the constant torque-limit region or just in the left vicinity of the voltage-limit curve corresponding to the present rotor speed.Therefore,a threshold variable named “i ds ,mtpa ”is calculated by using (4),which will be used to compare with i ?ds to decide whether the MTPA or the ?eld-weakening control mode should be chosen.It is worth mentioning that i ?ds in the ?owchart A is calculated by using
Fig.5.Flowcharts of d-and q-axis current-command generator corresponding to Fig.4.
the previous value of i?
ds plus the absolute value of the rotor-
speed error(ω?r?ωr)times a positive constantα.From the geometrical interpretation about the operation principle of the drive in the previous section,it is seen that,by this scheme, the corresponding current vector will move right to approach the voltage-limit curve.
In?owchart B,the per-unit quantity of the inverter output voltage already reaches V sm,pu.This condition reveals that the operating point of the drive is either on or to the right of the voltage-limit curve corresponding to the present rotor speed. In order to prevent the operating point from going too far to the right-hand side of the voltage-limit ellipse to result in
an unstable operation,i?
ds is calculated by using the previous
value of i?
ds minus the absolute value of the rotor-speed error
(ω?r?ωr)times a positive constantβ.It is seen that the corre-sponding current vector will move left to approach the voltage-limit curve by using this scheme.Furthermore,a minimum d-axis current I ds,min is used to limit the minimum value of
i?ds ,as shown in the?owchart B to prevent from exceeding the
current limit.An empirical method is used to properly select the values ofαandβto guarantee a stable operation.Generally,αcan be selected as large as possible,but it should be less than β.Furthermore,the value ofβfor a low saliency-ratio motor should be larger than that for a high saliency-ratio motor. From the previous discussion,it is seen that,according to
both?owcharts A and B,the proper i?
ds and i?qs can be decided
to keep the operating point as close as possible to the voltage-limit ellipse to achieve minimum copper-loss operation in the ?eld-weakening region without using machine parameters.As the time constant of the inner current loop is far smaller than that of the outer speed loop for a motor drive system,the operating point of the current vector can get very close to the voltage-limit ellipse in the?eld-weakening control mode.Nor-mally,as a heavy load is applied,the desired current amplitude may exceed I sm.Hence,in?owchart C,as the desired current amplitude is greater than I sm,the value of i?qs is modi?ed
according to I sm and the previously decided i?
ds to keep the
commanded current vector on a point of the current-limit circle.In summary,the proposed scheme?rst decides whether MTPA or?eld-weakening control should be executed.In the MTPA control mode,a conventional MTPA control strategy is used.In the?eld-weakening control mode,one can follow?owcharts A,B,and C to generate the desired i?
ds
and i?qs.These two current commands,together with the other input signals i ds, i qs,θr,andωr,are then fed to the decoupled PI d-and q-axis current-controller block to decide the proper inverter gating signals.It should be mentioned that,according to(4), the variations of machine parameters L q,L d,andλPM will affect the calculated commanded current vector in the MTPA control.To overcome such a problem in the MTPA control mode,other online parameter-estimation methods such as that in[18]can be adopted.Similarly,from(3)and(8),one can see that the variations of machine parameters L q,L d,λPM,and R s will also affect the intersecting point of these two equations. Moreover,the intersection point does not have an analytical expression.Therefore,it is not easy to implement the?eld-weakening control of an IPMSM,particularly considering the variations of the system parameters.Fortunately,the proposed ?eld-weakening control,as shown in Fig.5,neither requires solving any equation nor uses any machine parameters to decide the demanded current vector.Hence,the proposed scheme not only can operate at a high-ef?ciency condition over full-speed range but also has a robust speed-control feature.
IV.I MPLEMENTATION AND E XPERIMENTAL R ESULTS
In order to verify the feasibility of the proposed scheme, a vector-controlled IPMSM drive is constructed according to the block diagram of Fig.3by using a?xed-point DSP TMS320F2407A Evaluation Module and a current-regulated pulse-width-modulated inverter.The sampling time period of the current regulation loop is50μs and that of the speed control loop is1ms.The inverter dead time is assigned to be2μs to prevent the three phase legs of the inverter from shooting through.The tested IPMSM has parameters,as shown in Table I,and the motor is coupled on the same shaft with an
Fig.6.Experimental responses of V dc ,ωr ,the square of the voltage ampli-tude,and i qs under variation of V dc
.
Fig.7.Experimental responses of i ds and the line current of phase a corre-sponding to Fig.6.
incremental encoder and a 3-kW PM generator loaded with a three-phase resistor bank.The rotor position for the coordinate transformation and the rotor speed for the speed controller are obtained with the output signals of the incremental encoder.The control algorithm including the proposed scheme is fully implemented with a software,and the program size is about 1.8-k words.For the tested drive,the values of αand βare chosen to be 0.0003and 0.0012,respectively.
In the remaining paragraph,some test results will be pre-sented to illustrate the promising features of the proposed scheme.First,Figs.6and 7show the test results of a step speed
command of ω?
r
=2500r/min starting from rest at t =0s and with a steady-state load torque of 1.3N ·m at 2500r/min.At t =0s,V dc is initially kept at 100V .A sudden change of V dc from 100to 150V occurs at t =1.8s,and it is followed by a sudden change of V dc from 150to 100V at t =3.54s.Fig.6shows the experimental responses of V dc ,ωr ,the square of volt-age amplitude,and i qs under variation of V dc .In addition,Fig.7shows the experimental responses of i ds and the line current of phase a corresponding to Fig.6.As shown in Fig.6,the motor starts accelerating from 0r/min by the MTPA control.As the speed reaches about ωr =1900r/min at t =0.2s,the
?eld-
Fig.8.Experimental responses of ωr ,the square of voltage amplitude,i ds ,and i qs for a sudden change in load torque in the full ?eld-weakening region.
weakening control is activated until ωr =2500r/min before t =1.8s.For 1.8s of the voltage amplitude is less than V 2 sm ,pu ,indicating that the drive is operating in the MTPA control mode.The magnitude of i ds also reduces signi?cantly,as shown in Fig.7.Finally,as t >3.54s,V dc is reduced to 100V ,and the control strategy automatically switches back to the ?eld-weakening control.As t >3.54s in Fig.6,the square of the voltage amplitude is close to V 2 sm ,pu again,which indicates minimum copper-loss operation.The aforementioned experimental results show that the proposed scheme can indeed automatically achieve ?eld-weakening control with minimum copper-loss operation when the dc-link voltage of the inverter is changed. To show the performance under variation of the load torque in the full ?eld-weakening region,a step speed command of ω? r =3900r/min is applied to the drive starting from rest at t =0when V dc =150V and under a no-load condition.Then,at t =1.6s,a load torque of 1.1N ·m at 3900r/min is suddenly added and persists until t =2.6s.From Fig.8,one can see that,at t =1.6s,ωr is suddenly decreased slightly and then quickly recovers to the steady state of 3900r/min.Similarly,at t =2.6s,ωr is increased slightly when the load is suddenly removed,and the speed is then recovered to the steady state of 3900r/min again.From Fig.8,one can see that,as the rotor speed reaches the steady state,the square of the voltage magnitude reaches V 2 sm ,pu to indicate entering to the ?eld-weakening control mode.A similar test in the partial ?eld-weakening region to show the smooth transition between the MTPA and ?eld-weakening control modes is followed,as shown in Fig.9.A step speed command of ω? r =3300r/min is applied to the drive starting from rest at t =0s and with no load.Then,at t =1.46s,a load torque of 1.8N ·m at 3300r/min is suddenly added and persists until t =2.54s.From Fig.9,one can see that the waveform of the speed response is similar to that of Fig.8,and the steady-state speed is kept at 3300r/min.Nevertheless,the square of the Fig.9.Experimental responses ofωr,the square of voltage amplitude,i ds, and i qs for a sudden change in load torque in the partial?eld-weakening region. voltage magnitude is less than V2sm,pu at a no-load condition, which indicates MTPA control.On the other hand,as the load torque is applied,the square of the voltage magnitude increases to V2sm,pu,which indicates?eld-weakening control.One can also observe that the transitions between these two control modes are quite smooth in the partial?eld-weakening region. Finally,the periodic step speed commands between2500and 4500r/min are tested to show the performance under speed-command changes,as shown in Fig.10.In Fig.10,as the motor is?rst accelerating toωr=3000r/min at t=0.29s, the square of the voltage amplitude also reaches the saturated magnitude V2sm,pu.It indicates that the MTPA control mode ends at t=0.29s,and the?eld-weakening control mode starts until the motor speed reaches its steady state of4500r/min.At t=1.39s,a new speed command of2500r/min is assigned, and the motor starts to decelerate fromωr=4500r/min.As ωr is decreased to3800r/min at t=1.46s,the drive ends the transition state and enters to the MTPA control mode.One can observe that the boundary speed for entering the MTPA control is3800r/min during deceleration,which is higher than the boundary speed of3000r/min for leaving the MTPA control mode during acceleration.It shows the performance asymmetry between positive motoring torque and negative regenerative-braking torque,which results from the effect of R s,as discussed in Section II and as shown in Figs.1and2. V.C ONCLUSION A novel VCT-based?eld-weakening control scheme for an IPMSM drive has been presented in this paper.The proposed scheme can achieve high ef?ciency and has a robust speed-control feature for the motor drive over a full-speed range.The effect of the stator resistance to the maximum available torque and the operational principle of the proposed scheme have been discussed and graphically interpreted.As the drive is operated in the?eld-weakening control mode despite the variations of the inverter dc-link voltage and machine parameters,as well as changing load torque and speed command,the corresponding Fig.10.Experimental responses ofωr,the square of voltage amplitude,i ds, and i qs as applying periodic step speed commands. steady-state operating point can track automatically the true intersection point of the torque-demand curve and the voltage-limit ellipse to achieve minimum copper-loss operation.Hence, it is particularly suitable for applications of high-speed and/or battery-powered motor drives.From a practical point of view, the proposed scheme has a rather simple algorithm,and the drive does not need a dc-link voltage sensor.Hence,it can easily be implemented by modifying a conventional vector-controlled drive system for practical applications.Finally,a DSP-controlled prototype has been constructed,and extensive experimental tests are provided to show the promising features of the proposed scheme. 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[19]J.M.Kim and S.K.Sul,“Speed control of interior permanent magnet synchronous motor drive for the?ux weakening operation,”IEEE Trans. Ind.Appl.,vol.33,no.1,pp.43–48,Jan./Feb.1997. [20]Z.Q.Zhu,Y.S.Chen,and D.Howe,“Online optimal?ux-weakening control of permanent-magnet brushless AC drives,”IEEE Trans.Ind. Appl.,vol.36,no.6,pp.1661–1668,Nov./Dec.2000. [21]M.N.Uddin and M.A.Rahman,“High-speed control of IPMSM drives using improved fuzzy logic algorithms,”IEEE Trans.Ind.Electron., vol.54,no.1,pp.190–199,Feb.2007. [22]C.T.Pan and S.M.Sue,“A linear maximum torque per ampere control for IPMSM drives over full-speed range,”IEEE Trans.Energy Convers., vol.20,no.2,pp.359–366,Jun.2005. [23]R.Krishnan,Electric Motor Drives:Modeling,Analysis,and Control. Upper Saddle River,NJ:Prentice-Hall, 2001. Shinn-Ming Sue(M’07)was born in Nantou, Taiwan,R.O.C.,in1965.He received the Ph.D. degree in electrical engineering from National Tsing Hua University,Hsinchu,Taiwan,in2004. He is an Associate Professor with the Depart- ment of Electrical Engineering,Minghsin University of Science and Technology,Hsinchu.His research interests are in the areas of power electronics and permanent-magnet machine drives. Ching-Tsai Pan(M’88)was born in Taipei,Taiwan, R.O.C.,in1948.He received the B.S.degree in electrical engineering from National Cheng Kung University,Tainan,Taiwan,in1970,and the M.S. and Ph.D.degrees in electrical engineering from Texas Tech University,Lubbock,in1974and1976, respectively. Since1977,he has been with the Department of Electrical Engineering,National Tsing Hua Univer- sity,Hsinchu,Taiwan,where he is currently a Tsing Hua Chair Professor and Director of the Center for Advanced Power Technologies.His research interests are in the areas of power electronics,ac motor drives,control and power systems,and numerical analysis. Dr.Pan is a member of the Chinese Institute of Engineers,Chinese Institute of Electrical Engineering,Chinese Institute of Automatic Control Engineering, Chinese Institute of Computer Society,Taiwan Wind Energy Association,Phi Tau Phi,Eta Kappa Nu,and Phi Kappa Phi. 几种特殊行列式的巧算 摘要：在高等代数课程中，n阶行列式的计算问题非常重要，它是行列式理论 的重要组成部分。计算n阶行列式的一般方法有：按行（列）展开,化三角行列式法，降阶法等。对于这些解法，高等代数课本已做了详细介绍，本文重点探索关于三对角，爪型等具有一定特征的行列式的计算，跟几种具有特殊解法的行列式（如范德蒙行列式）计算，突出一个“巧”字，从而提高解题速度。 关键词：“三对角”行列式分离线性因子法“爪型”行列式范德蒙行列式等. 引言： n阶行列式 11121212221 2 n n n n nn a a a a a a a a a 是所有取自不同行、不同列的n 个元素的乘积1212n j j nj a a a 的代数和，其中12 n j j j 是一 个n 阶排列，每个项1212n j j nj a a a 前面带有正负号.当12n j j j 是偶排列时， 项1212n j j nj a a a 前面带有正号，当12 n j j j 是奇排列时，项12 12n j j nj a a a 前面带有负号.即 11 121212221 2 n n n n nn a a a a a a a a a = 121212 () 12() (1) .n n n j j j j j nj j j j a a a τ-∑ 这里 12 () n j j j ∑ 表示对所有的n 阶排行求和. 行列式的计算是高等代数的一个重要内容，同时也是在工程应用中具有很高价值的数学工具，本文针对行列式的几种特殊类型，给出了每一种类型特殊的计算方法，具体如下： 一 三对角行列式的计算 形如 b a b a b a b a b a b a b a b a D n +++++= 0000000000000的行列式称为“三对角”行列式.该 类行列式的计算方法有：猜想法, 递推法, 差分法.下面我们首先用猜想法来解一下这个行 列式. 当b a ≠时 b a b a b a b a b a b a b a b D b a D n n ++++-+=- 000000000000)(1 =21 )(---+n n abD D b a . 即有递推关系式21)(---+=n n n abD D b a D ,为了得到n D 的表达式,可先设b a ≠,采用 浅 一、 特殊行列式法 1．定义法 当行列式中含零元较多时，定义法可行． 例1 计算n 级行列式 α β βαβαβα000000 0000 00 =D . 解：按定义，易见121,2,,,n j j j n === 或 1212,3,,,1n n j j j n j -==== . 得 11(1)n n n D αβ-+=+- 2．三角形行列式法 利用行列式性质，把行列式化成三角形行列式． nn a a a a a a 000n 222n 11211＝nn n n a a a a a a 212212110 0112233nn a a a a = 例2 计算n 级行列式1231 131 211 2 3 1 n n x n D x n x +=++ 解： 将n D 的第(2,3,,)i i n = 行减去第一行化为三角形行列式，则 1230 1000 0200 1 (1)(2)(1) n n x D x x n x x x n -=--+=---+ 3．爪形行列式法 例3 计算行列式 0121 1 220 0000n n n a b b b c a D c a c a = ()0,1,2,,i a i n ≠= 解: 将D 的第i +1列乘以（i i a c - ）都加到第1列()n i ,2,1=，得 10 12 120000000 00n i i n i i n bc a b b b a a D a a - =∑= ＝011()n n i i i i i i b c a a a ==-∑∏ 4． 范德蒙行列式法 1 2 3 2 2221 2 3 11111 2 3 1111n n n n n n n a a a a D a a a a a a a a ----= 1()i j j i n a a ≤<≤= -∏ 例4 计算n 级行列式 2 2221233 333 1 2 3 12 3 11 1 1 n n n n n n n x x x x D x x x x x x x x = 解：利用D 构造一个1n +阶范德蒙行列式 12222 212121111()n n n n n n n x x x x g x x x x x x x x x = 多项式()g x 中x 的系数为3(1)n D +-，而()g x 又是一个范德蒙行列式，即 1 ()() n i i g x x x ==-∏∏≤<≤-n i j j i x x 1)( 1 行列式的定义及性质 1.1 定义[3] n 级行列式 1112121 22 212 n n n n nn a a a a a a a a a 等于所有取自不同行不同列的个n 元素的乘积12 12n j j nj a a a (1)的代数和,这里12 n j j j 是 1,2, ,n 的一个排列,每一项(1)都按下列规则带有符号:当12n j j j 是偶排列时,(1)带正号，当 12n j j j 是奇排列时,(1)带有负号.这一定义可写成 () () 121212 1112121 22 21212 1n n n n j j j n j j nj j j j n n nn a a a a a a a a a a a a τ= -∑ 这里 12 n j j j ∑ 表示对所有n 级排列求和. 1.2 性质[4] 性质1.2.1 行列互换,行列式的值不变. 性质1.2.2 某行(列)的公因子可以提到行列式的符号外. 性质1.2.3 如果某行(列)的所有元素都可以写成两项的和,则该行列式可以写成两行列式的和;这两个行列式的这一行(列)的元素分别为对应的两个加数之一,其余各行(列)与原行列式相同. 性质1.2.4 两行(列)对应元素相同,行列式的值为零. 性质1.2.5 两行(列)对应元素成比例,行列式的值为零. 性质1.2.6 某行(列)的倍数加到另一行(列)对应的元素上,行列式的值不变. 性质1.2.7 交换两行(列)的位置,行列式的值变号. 2 行列式的分类及其计算方法 2.1 箭形（爪形）行列式 这类行列式的特征是除了第1行(列)或第n 行(列)及主(次)对角线上元素外的其他元素均为零，对这类行列式可以直接利用行列式性质将其化为上(下)三角形行列式来计算.即利用对角元素或次对角元素将一条边消为零. 例1 计算n 阶行列式 ()1 2323111100 1 0001 n n n a a D a a a a a =≠. 解 将第一列减去第二列的 21a 倍，第三列的3 1a 倍第n 列的 1 n a 倍，得 1 223 111110 000 000 n n n a a a a D a a ?? -- - ?? ? = 1221n n i i i i a a a ==?? =- ?? ? ∑ ∏. 2.2 两三角型行列式 这类行列式的特征是对角线上方的元素都是c ,对角线下方的元素都是b 的行列式，初看，这一类型似乎并不具普遍性，但很多行列式均是由这类行列式变换而来，对这类行列式，当 b c =时可以化为上面列举的爪形来计算，当b c ≠时则用拆行(列)法[9]来计算. 例2 计算行列式 行列式的几种常见计算技巧和方法 2.1 定义法 适用于任何类型行列式的计算，但当阶数较多、数字较大时，计算量大，有一定的局限性． 例1 计算行列式0 004003002001000. 解析：这是一个四级行列式，在展开式中应该有244=！项，但由于出现很多的零，所以不等于零的项数就大大减少．具体的说，展开式中的项的一般形式是43214321j j j j a a a a ．显然，如果41≠j ，那么011=j a ，从而这个项就等于零．因此只须考虑41=j 的项，同理只须考虑 1,2,3432===j j j 的这些项，这就是说，行列式中不为零的项只有41322314a a a a ，而()64321=τ，所以此项取正号．故 004003002001000=() () 241413223144321=-a a a a τ. 2.2 利用行列式的性质 即把已知行列式通过行列式的性质化为上三角形或下三角形.该方法适用于低阶行列式． 2.2.1 化三角形法 上、下三角形行列式的形式及其值分别如下： nn n n n a a a a a a a a a a a a a K ΛM O M M M K K K 2211nn 333223221131211000000=，nn nn n n n a a a a a a a a a a a a a K Λ M O M M M K K K 22113 2133323122211100 0000=. 例2 计算行列式n n n n b a a a a a b a a a a ++= +K M O M M M K K 21 211211n 1 11 D . 解析：观察行列式的特点，主对角线下方的元素与第一行元素对应相同，故用第一行的()1-倍加到下面各行便可使主对角线下方的元素全部变为零．即：化为上三角形． 解：将该行列式第一行的()1-倍分别加到第2,3…（1n +）行上去，可得 121n 11210000D 000n n n a a a b b b b b += =K K M M M O M K . 2.2.2 连加法 这类行列式的特征是行列式某行（或列）加上其余各行（或列）后，使该行（或列）元素均相等或出现较多零，从而简化行列式的计算．这类计算行列式的方法称为连加法． 毕业论文（设计）作者声明 本人郑重声明：所呈交的毕业论文是本人在导师的指导下独立进行研究所取得的研究成果.除了文中特别加以标注引用的内容外，本论文不包含任何其他个人或集体已经发表或撰写的成果作品. 本人完全了解有关保障、使用毕业论文的规定，同意学校保留并向有关毕业论文管理机构送交论文的复印件和电子版.同意省级优秀毕业论文评选机构将本毕业论文通过影印、缩印、扫描等方式进行保存、摘编或汇编；同意本论文被编入有关数据库进行检索和查阅. 本毕业论文内容不涉及国家机密. 论文题目：几种特殊类型行列式及其计算 作者单位：数学与信息科学系 作者签名： 2012年5月31 日 目录 摘要 (1) 引言 (2) 1行列式的定义及性质 (3) 1.1 定义 (3) 1.2 性质 (3) 2行列式的分类及其计算方法 (4) 2.1 箭形（爪形）行列式 (4) 2.2 两三角型行列式 (4) 2.3 两条线型行列式 (7) 2.4 Hessenberg型行列式 (9) 2.5 三对角型行列式 (10) 2.6 各行(列)元素和相等的行列式 (11) 2.7 相邻两行(列)对应元素相差1的行列式 (12) 2.8 范德蒙德型行列式 (13) 结束语 (14) 参考文献 (15) 致谢 ······································································································································错误！未定义书签。 几种特殊类型行列式及其计算 摘要：行列式的计算是一个普遍的难题.在一些文献中我们已经了解了一些解决它的基本方法，例如:化为上下三角形法，降阶法，加边法，拆项法，递推法，数学归纳法.本文是对几种特殊类型的行列式给以归纳，再根据不同类型给出相应的计算方法.这使得绝大多数行列式能够被归为这其中的某一种，从而能快速简洁的计算出这些行列式. 关键词：行列式;爪形;两三角型;两条线型;范德蒙德型 Several Special Types of Determinants and Its Calculation Abstract: The n-th determinant calculation is a common difficult problem for students. We have already knew some ways in some documents to solve it, for example: the making definition, changing into triangle (upper and low), decreasing the degree, adding the margin, splitting some items, recursive algorithm and induction. This article aims to conclude some special kinds of determinants firstly and then gives the relevant calculation methods.That made most of the determinants can be attributed to one of that kinds,then it can be calculated more quickly and pithily. Key Words: Determinant; Claw; “Two-triangle”type; “Two-wire”type; “Vandermonde”type 1 特殊行列式的计算 摘 要： 运用行列式的定理、性质及推论对一些复杂、特殊行列式进行化简，总结出了一些特殊行列式的计算方法及公式，改变了以往遇到行列式总是通过初等变化按其某行（或某列）展开进行逐次降阶化成阶梯型行列式或依据Laplace 定理进行行列式计算的方法；使行列式的计算更为简洁、灵活，并使得特殊行列式的计算公式化. 关键词： 行列式；行列式的计算；特殊可列阶行列式 1 预备知识 面对一些复杂而又特殊行列式的计算我们往往会不知所措、无从下手，更不知道应该用什么方法去进行化简或计算，就像一只无头的苍蝇只能用各种方法去进行试探.为此我们多么希望一些特殊的可列阶行列式的计算能像一元二次方程一般有其计算公式和特殊的化简方法，从而提高特殊、复杂的行列式的计算效率，简化其计算步骤，改变其算法的冗长性，使之公式化、方法化.现就有关知识做以预习. 定理1.1（Laplace 定理） 设在行列式D 中任意取定了)11(-≤≤n k k 个行，由这k 行元素所组成的一切k 级子式与它们的代数余子式的乘积的和等于行列式D . 性质1.1 行列式与其转置行列式相等. 性质1.2 交换行列式的某两行（或某两列）行列式改变符号. 性质1.3 把行列式某一行（或某一列）的所有元素都乘以一个数k ，等于以k 乘以该行列式. 性质1.4 把行列式的某一行（或某一列）的所有元素乘以同一个数k 后加到另一行或另一列的对应元素上行列式值不变. 性质1.5 如果行列式中有两行（或两列）元素相同,行列式值为0. 性质1.6 行列式中某一行（或某一列）中所有元素的公因子可以提到行列式的外边. 性质1.7 行列式中如果有一行(或一列)的元素全为零,则行列式为0. 性质1.8 如果行列式中有两行（或两列）的元素对应成比例,则行列式等于0. 引理1.1 行列式的任一个子式M 与它的代数余子式A 的乘积中的每一项都是行列式D 的展开式中的项，而且符号也一致. 2 特殊行列式的计算 2.1 二条线型行列式的计算 定义2.1.1 形如1D =n n n a c b a b a b a 11 22 11 -- （或2D = c a a b a b a b n n n 1 12 2 1 1-- ）的行 特殊行列式及行列式计算方法总结 一、 几类特殊行列式 1. 上（下）三角行列式、对角行列式（教材P7例5、例6） 2. 以副对角线为标准的行列式 11112112,1 221222,11,21,1 1,11 2 ,1 (1)2 12,11 000000 0000 0000 (1) n n n n n n n n n n n nn n n n n n nn n n n n n a a a a a a a a a a a a a a a a a a a a a a ---------===-L L L L L L M M M M M M M M M N L L L L 3. 分块行列式（教材P14例10） 一般化结果： 00n n m n n m n m m n m m n m A C A A B B C B ????= =? 0(1)0n m n n m n mn n m m m n m m n A C A A B B C B ????= =-? 4. 范德蒙行列式（教材P18例12） 注：4种特殊行列式的结果需牢记！ 以下几种行列式的特殊解法必须熟练掌握！！！ 二、 低阶行列式计算 二阶、三阶行列式——对角线法则 （教材P2、P3） 三、 高阶行列式的计算 【五种解题方法】 1) 利用行列式定义直接计算特殊行列式； 2) 利用行列式的性质将高阶行列式化成已知结果的特殊行列式； 3) 利用行列式的行（列）扩展定理以及行列式的性质，将行列式降阶进行计算 ——适用于行列式的某一行或某一列中有很多零元素，并且非零元素的代数余子式很容易计算； 4) 递推法或数学归纳法； 5) 升阶法（又称加边法） 行列式的若干计算技巧与方法 内容摘要 1. 行列式的性质 2.行列式计算的几种常见技巧和方法 定义法 利用行列式的性质 降阶法 升阶法（加边法） 数学归纳法 递推法 3. 行列式计算的几种特殊技巧和方法 拆行（列）法 构造法 特征值法 4. 几类特殊行列式的计算技巧和方法 三角形行列式 “爪”字型行列式 “么”字型行列式 “两线”型行列式 “三对角”型行列式 范德蒙德行列式 5. 行列式的计算方法的综合运用 降阶法和递推法 逐行相加减和套用范德蒙德行列式 构造法和套用范德蒙德行列式 行列式的性质 性质1 行列互换，行列式不变．即 nn a a a a a a a a a a a a a a a a a a n 2n 1n2 2212n12111nn n2n12n 2221 1n 1211 . 性质2 一个数乘行列式的一行（或列），等于用这个数乘此行列式．即 nn n2 n1in i2i1n 11211 k k k a a a a a a a a a k nn a a a a a a a a a n2n1in i2i1n 11211. 性质3 如果行列式的某一行（或列）是两组数的和，那么该行列式就等于两个行列式的和，且这两个行列式除去该行（或列）以外的各行（或列）全与原来行列式的对应的行（或列）一样．即 111211112111121112212121 2 1212.n n n n n n n n n nn n n nn n n nn a a a a a a a a a b c b c b c b b b c c c a a a a a a a a a K K K M M M M M M M M M M M M K K K M M M M M M M M M M M M K K K 性质4 如果行列式中有两行（或列）对应元素相同或成比例，那么行列式为零．即 k a a a ka ka ka a a a a a a nn n n in i i in i i n 21 2121112 11nn n n in i i in i i n a a a a a a a a a a a a 212121112 11=0. 性质5 把一行的倍数加到另一行，行列式不变．即 1行列式的定义及性质 1.1定义[3] n级行列式 a 11 a12 (1) a 21 I-a22… a a 2n a a n1 a n2…a nn n元素的乘积的屜…a% （1）的代数和,这里jj…j n是1,2/ ,n的一个排列,每一项（1）都按下列规则带有符号：当jj…j n是偶排列时,⑴带正号，当j l j2…j n 是奇排列时,（1）带有负号.这一定义可写成 an a12 a1n a 21 a22 (2) I-a=无（-1F 山压）?…a nj j1 j2…j n a n1 a n2 a nn 这里V 表示对所有n级排列求和. j1 j2 ■ j n 1.2性质[4] 性质1.2.1行列互换，行列式的值不变. 性质1.2.2某行（列）的公因子可以提到行列式的符号外. 性质1.2.3如果某行（列）的所有元素都可以写成两项的和,则该行列式可以写成两行列式的和；这两个行列式的这一行（列）的元素分别为对应的两个加数之一，其余各行（列）与原行列式相同. 性质1.2.4两行（列）对应元素相同,行列式的值为零. 性质1.2.5两行（列）对应元素成比例，行列式的值为零. 性质1.2.6某行（列）的倍数加到另一行（列）对应的元素上,行列式的值不变. 性质1.2.7交换两行（列）的位置,行列式的值变号. 等于所有取自不同行不同列的个 2行列式的分类及其计算方法 2.1箭形（爪形）行列式 这类行列式的特征是除了第1行（列）或第n 行（列）及主（次）对角线上元素外的其他元素均 为零，对这类行列式可以直接利用行列式性质将其化为上（下）三角形行列式来计算?即利用对 角元素或次对角元素将一条边消为零. 例1计算n 阶行列式 a 1 1 ■ ■ .L 1 1 a 2 0 0 D n = 1 0 a 3… 0 （&2&3…a n 式0） 1 0 … a n 2.2两三角型行列式 这类行列式的特征是对角线上方的元素都是 c,对角线下方的元素都是b 的行列式，初看, 这一类型似乎并不具普遍性，但很多行列式均是由这类行列式变换而来，对这类行列式，当 b 二 c 时可以化为上面列举的爪形来计算，当 b = c 时则用拆行例）法 ［9］ 来计算. 例2计算行列式 将第一列减去第二列的 丄倍，第三列的丄倍…第n 列的 a 2 a 3 倍，得 1 a i - a 2 1 1 a 2 0 a 3 0 0 a n n =''a i i =2 n *1 ' ■- i=2 丄 a i 丿 目录 1 引言 (2) 2 文献综述 (2) 2.1 国内研究现状 (2) 2.2 国内研究现状评价 (3) 2.3 提出问题 (3) 3 预备知识 (3) 3.1 N阶行列式的定义 (3) 3.2 行列式的性质 (4) 3.3 行列式的行(列)展开和拉普拉斯定理 (4) 3.3.1 行列式按一行(列)展开 (4) 3.3.2 拉普拉斯定理 (5) 4 几类特殊N阶行列式的计算 (5) 4.1 三角形行列式的计算 (6) 4.2 两条线型行列式的计算 (7) 4.3 箭形行列式的计算 (8) 4.4 三对角行列式的计算 (8) 4.5 Hessenberg型行列式的计算 (10) 4.6 行（列）和相等的行列式的计算 (11) 4.7 相邻行（列）元素差1的行列式的计算 (12) 4.8 范德蒙型行列式的计算 (13) 5 结论 (15) 5.1 主要发现 (15) 5.2 启示 (15) 5.3 局限性 (15) 5.4 努力方向 (15) 参考文献 (16) 1 引言 行列式是代数学中的一个重要内容,在数学理论上有十分重要的地位.早在17世纪和18世纪初,行列式就在解线性方程组中出现.1772年法国数学家范德蒙(1735-1796)首先把行列式作为专门理论独立于线性方程之外研究.到了19世纪,是行列式理论形成和发展的重要时期,19世纪中叶出现了行列式的大量定理.因此,到19世纪末行列式基本面貌已经勾画清楚. 行列式的计算是高等代数的重要内容之一,也是理工科线性代数的重要内容之一,同时也是学习中的一个难点.在数学和现实中有着广泛的应用,懂得如何计算行列式尤为重要.对于阶数较低的行列式,一般可直接利用行列式的定义和性质计算出结果.对于一般的N阶行列式,特别是当N较大时，直接用定义计算行列式往往是困难和繁琐的，因此研究行列式的计算方法则显得十分必要.通常需灵活运用一些计算技巧和方法，使计算大大简化，从而得出结果.本文归纳了几类特殊N阶行列式的计算方法,从这几类特殊的N阶行列式的计算中,可以总结出归纳出一些行列式的计算方法,只要将这些方法与传统方法结合起来，就可以基本上解决n阶行列式的计算问题. 本文先阐述行列式的定义及其基本性质,然后介绍了几类特殊行列式的计算方法,并结合了相关例题讨论了行列式的求解方法. 2 文献综述 2.1 国内研究现状 现查阅到的文献资料中,大部分只是简单的介绍了行列式的定义、行列式的性质、行列式按行（列）展开、克拉默法则等.其中[1]、[3]介绍了行列式的定义、性质、行列式按行（列）展开，[2]、[4]介绍了利用行列式的性质计算行列式，[4]、[8]直接介绍行列式的计算，主要讲解了行列式的计算在Matlab上的实现，[7]、[9]、[10]介绍了行列式的简单计算和行列式的常用计算方法，[11]、[12]、[13]同样也是介绍了行列式的性质、定义和克拉默法则，[14]在行列式的定义、性质、按行（列）展开克拉默法则等方面介绍得比较完整，[15]-[18]系统介绍了行列式计算中和各种方法,如定义法、降阶法、升降法、拆开法、目标行列式法、乘积法、化三角开法、消去法、加边法、归纳法、递推法、特征值法等行列式的计算方法. 1 行列式的定义及性质 1.1 定义[3] n 级行列式 111212122212n n n n nn a a a a a a a a a L L M M M L 等于所有取自不同行不同列的个n 元素的乘积1212n j j nj a a a L (1)的代数和,这里12n j j j L 是 1,2,,n L 的一个排列,每一项(1)都按下列规则带有符号:当12n j j j L 是偶排列时,(1)带正号，当 12n j j j L 是奇排列时,(1)带有负号.这一定义可写成 () () 121212111212122212121n n n n j j j n j j nj j j j n n nn a a a a a a a a a a a a τ= -∑L L L L L M M M L 这里 12n j j j ∑ L 表示对所有n 级排列求和. 1.2 性质[4] 性质1.2.1 行列互换,行列式的值不变. 性质1.2.2 某行(列)的公因子可以提到行列式的符号外. 性质1.2.3 如果某行(列)的所有元素都可以写成两项的和,则该行列式可以写成两行列式的和;这两个行列式的这一行(列)的元素分别为对应的两个加数之一,其余各行(列)与原行列式相同. 性质1.2.4 两行(列)对应元素相同,行列式的值为零. 性质1.2.5 两行(列)对应元素成比例,行列式的值为零. 性质1.2.6 某行(列)的倍数加到另一行(列)对应的元素上,行列式的值不变. 性质1.2.7 交换两行(列)的位置,行列式的值变号. 2 行列式的分类及其计算方法 2.1 箭形（爪形）行列式 这类行列式的特征是除了第1行(列)或第n 行(列)及主(次)对角线上元素外的其他元素均为零，对这类行列式可以直接利用行列式性质将其化为上(下)三角形行列式来计算.即利用对角元素或次对角元素将一条边消为零. 例1 计算n 阶行列式 ()123 23111100 1000100 n n n a a D a a a a a =≠L L L L L L L L L L . 解 将第一列减去第二列的 21a 倍，第三列的3 1 a 倍L 第n 列的1n a 倍，得 122 3111 11000 00 00 n n n a a a a D a a ?? --- ?? ?= L L L L L L L L L L 1221n n i i i i a a a ==?? =- ?? ? ∑ ∏. 2.2 两三角型行列式 这类行列式的特征是对角线上方的元素都是c ,对角线下方的元素都是b 的行列式，初看， 行列式的计算技巧与方法汇总(修改版) ————————————————————————————————作者：————————————————————————————————日期： 2 行列式的若干计算技巧与方法 内容摘要 1. 行列式的性质 2.行列式计算的几种常见技巧和方法 2.1 定义法 2.2 利用行列式的性质 2.3 降阶法 2.4 升阶法（加边法） 2.5 数学归纳法 2.6 递推法 3. 行列式计算的几种特殊技巧和方法 3.1 拆行（列）法 3.2 构造法 3.3 特征值法 4. 几类特殊行列式的计算技巧和方法 4.1 三角形行列式 4.2 “爪”字型行列式 4.3 “么”字型行列式 4.4 “两线”型行列式 4.5 “三对角”型行列式 4.6 范德蒙德行列式 5. 行列式的计算方法的综合运用 5.1 降阶法和递推法 5.2 逐行相加减和套用范德蒙德行列式 5.3 构造法和套用范德蒙德行列式 3 1 1.2 行列式的性质 性质1 行列互换，行列式不变．即 nn a a a a a a a a a a a a a a a a a a n 2n 1n2 2212n12111nn n2n12n 2221 1n 1211 . 性质2 一个数乘行列式的一行（或列），等于用这个数乘此行列式．即 nn n2 n1in i2i1n 11211 k k k a a a a a a a a a k nn a a a a a a a a a n2n1in i2i1n 11211. 性质3 如果行列式的某一行（或列）是两组数的和，那么该行列式就等于两个行列式的和，且这两个行列式除去该行（或列）以外的各行（或列）全与原来行列式的对应的行（或列）一样．即 111211112111121112212121 2 1212.n n n n n n n n n nn n n nn n n nn a a a a a a a a a b c b c b c b b b c c c a a a a a a a a a K K K M M M M M M M M M M M M K K K M M M M M M M M M M M M K K K 性质4 如果行列式中有两行（或列）对应元素相同或成比例，那么行列式为零．即 k a a a ka ka ka a a a a a a nn n n in i i in i i n 21 2121112 11nn n n in i i in i i n a a a a a a a a a a a a 212121112 11 =0. 性质5 把一行的倍数加到另一行，行列式不变．即 几种特殊类型行列式及其计算 123n n a c c c b a c c D b b a c b b b a =. 解 当 b c =时 12 3n n a b b b b a b b D b b a b b b b a =. 将第2行到第行n 都减去第1行，则n D 化为以上所述的爪形，即 112131 00000 n n a b b b b a a b D b a a b b a a b --=----. 用上述特征1的方法，则有 ()112 12131 100000000 n i i n n a b b a a b b a a b D b a a b b a a b =-----= ----∑ ()() ()()()1111 1 n n i i i n i i a b b a b a b a b a b -+===-+----∑∏. 当b c ≠时，用拆行(列)法[9]，则 112233000n n n x a a a x a a a b x a a b x a a D b b x a b b x a b b b x b b b b x b ++==++- 1 12233000n x a a x a a a b x a b x a a b b x b b x a b b b x b b b b b =+- ()121100 0n n n x a a b a x a a x b D a b a b a x a a b -----=+----. 化简得 ()()()()1211n n n n D b x a x a x a x b D --=---+-. ()1 而若一开始将n x 拆为n a x a +-，则得 ()()()()1211n n n n D a x b x b x b x a D --=---+-. ()2 由()()()()12n n x b x a ?--?-，得 ()()111n n n i j i j D a x b b x a a b ==??=---??-?? ∏∏. 有一些行列式虽然不是两三角型的行列式，但是可以通过适当变换转化成两三角型行列式进行计算. 例 3 计算行列式 ()2n d b b b c x a a D n c a x a c a a x =≥. 解 将第一行a b ?，第一列a c ?，得 22 n a d a a a bc a x a a bc D a a x a a a a a x =.几种特殊行列式的巧算
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