2015--Hindawe- Unknown Disturbance Estimation for a PMSM With a Hybrid

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Research ArticleUnknown Disturbance Estimation for a PMSM with a HybridSliding Mode Observer

Gang Chen,1,2Yong Zhou,3 TingTing Gao,2 and Qicai Zhou1

Tongji University, Shanghai , ChinaZhejiang Textile & Fashion College, Ningbo , ChinaWild SC Intelligent Technology CO., LTD, Ningbo , China

Correspondence should be addressed to Yong Zhou; [email protected]

Received July ; Revised September ; Accepted October

Academic Editor: Xudong Zhao

Copyright Gang Chen et al. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A hybrid sliding mode observer that combines high gain eedback and a high-order sliding mode term is developed to identiy thetime-varyingdisturbance or a permanent-magnet synchronous motor (PMSM). Basedon the measurablecurrent and the position,the unknown disturbance can be identied rom the sliding mode term without digital lter effect. It is then used to enhance therobustness o the speed control dynamics. For ease o implementation in real applications, such as DSP and FPGA, the proposedobserver is properly designed to avoid complex mathematical operators. Simulation results are given to illustrate the perormanceo the proposed observer.

1. Introduction

In real industrial application, the unknown disturbanceon a permanent-magnet synchronous motor (PMSM) isinevitable and it limits the perormance o the controlledprocesses. For such a situation, a robust observer with highestimation accuracy is required to recover the uncertainty inreal time.

Te sliding-mode-based observer has been proven to be

an effective approach or handling uncertain systems, dueto its capability o reconstructing the uncertainties basedon the equivalent injection input concept [], and thereare many effective results presented during past years []. In [], a sliding mode ux observer is developed oran induction motor, and the unknown rotor resistance canbe reconstructed rom a switching unction afer the slidingsurace driven to zero. In[, ], the sliding mode observer isused to identiy the back-EMFs or a PMSM. However, suchresults require a low-pass lter or the uncertainties recon-struction because o the discontinuous switching eedback,and the uncertainty estimation accuracy is greatly dependenton the low-pass lter parameters.

o improve the estimation accuracy by removing theltering effect, some existing works have suggested the usageo a saturation unction to replace the switching term [].However, this results in a trade-off between the robustness oobserver and the estimationaccuracy o uncertainty. Anotherexisting method is based on the higher-order sliding modetechniques, by treating the derivatives o the system input asa new control signal, which results in a continuous integralunction o the switching term; then the chattering/ltering

effect can be avoided [,].Another problem is that the sliding mode gains have to

be chosen large enough to ensure the stability o the observer,which is usually related to the system initial values. o solvethis problem, a hybrid observer that combines high gaineedback and higher-order sliding mode observer has beenproposed in [], in which the high gain eedback worksto constrain the estimation error within an invariant setregardless o initial values; then the sliding mode gains canbe designed to ensure the global stability o the observer. Suchhybrid observershave beenapplied into a series DC motor orthe speed estimation and unknown time-varying parameteridentication or a DC motor [].

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 316360, 7 pageshttp://dx.doi.org/10.1155/2015/316360
http://dx.doi.org/10.1155/2015/316360http://dx.doi.org/10.1155/2015/316360


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Mathematical Problems in Engineering

In this paper, we will consider the unknown disturbanceestimation or a permanent-magnet synchronous motor. Ahybrid sliding mode observer is developed or the distur-bance identication. Te most challenging problem is howto properly design the observer algorithm to simpliy theobserver implementation in real applications, such as in a

digital signal processor (DSP) and FPGA. Te contributiono this work can be summarized as ollows.

() With the proposed observer, the disturbance canbe asymptotically estimated andthen used to enhancethe robustness o the speed control dynamics. Tetheoretical analysis and simulation results are givento demonstrate the effectiveness o the proposedobserver.

() A hybrid observer structure is adopted, in whichthe high gain eedback term works to speed up theconvergence time and guarantees the sliding modegain design is independent o the systeminitial values.

() For disturbance identication, the digital lteringeffect is successully avoided without sacricingrobustness o the observer.

() Te twisting sliding mode algorithm is properly inte-grated in the proposed observer, to make the observerease or implementation in real application.

Te remainder o this paper is organized as ollows.Section introduces the mathematical mode o the PMSMand some background results are presented. InSection , aspeed observer and a disturbance are developed to identiythe unknown external load disturbance. InSection , somesimulations are provided to illuminate the effectiveness o the

two proposed observers.Section concludes this paper.

2. Preliminary

.. Mathematical Model. Te state equation o a surace-mounted magnet brushless AC motor in stationary-reerence rame is given by []:

= +sin +

,=

cos +,

= ,

()

=322sin + cos

, ()where,is the state current in-reerence rame,, isthe state voltage in-reerence rame,is the resistance,is the inductance, is the ux-linkage due to permanentmagnet, is the rotor position, is the electrical angularspeed, is the rotor moment o inertia, is the viscous-riction coefficient, andis the load torque.

Here, the load torque is assumed to be the systemuncertainty that includes the unknown external disturbance,

the parameter variations o rotor inertia, and the viscous-riction coefficient.

Te target o this paper is to develop a robust and highaccuracy observer that can identiy the unknown disturbance, based on the measurable current, voltage, and rotorposition.

.. Background Results. Consider an uncertain nonlinearsystem in the orm o

x= Ax+ (x, )+P (x) () , = Cx, ()

wherex= [1, 2, . . . , ] R andA= 0(1)1 I(1)(1)

011 01(1) ,

C

= 1 0 0 R

()

are constant matrices;() R denotes the lumped systemuncertainty; is the system input; the nonlinear unctions(x, ) and P(x) are smooth Lipschitz vectors with triangularstructures, it has

(x)= 0, . . . ,0 ,1, . . . , , . . . , 1, . . . , ,P (x)= 0, . . . ,0 ,1 ,+11, . . . , , . . . , 1, . . . ,

()

with beingthe relativedegree order between the disturbance()and the measurable output.Ten, a hybrid observer that combines highgain eedback

with higher-order sliding mode can be designed as []

x= Ax+ (x, )+L Cx +P (x) ,= 0, i ,

V, i< ,()

whereLis the high gain linear eedback parameter, given by

L= 12 R, ()and is a positive tuning parameter o nonlinear eedback

, which is designed as an integral unction o a

( + 1)th-

order sliding mode term V; denotes the upper bound o.Ithasbeenprovenin[] that the high gain eedback termworks to constrain the estimation error to within an invariantset and the nonlinear termwill asymptotically track theunknown disturbance() without digital ltering effect i weproperly design the sliding mode eedback term.

Te mechanical dynamics given in () can be consideredas a second-order nonlinear system in (), with the relativedegree order between the disturbance and the measurableoutput being two; that is, = = 2,() = /.Ten, one can design a hybrid observer in the orm o ( ),with V being third-order sliding mode eedback. However,to the best o our understanding, the existing third-order


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or higher-order sliding mode algorithms involve complexmath operations that require more computation resources inreal implementation, such as in digital signal processor andFPGA.

In next section, we will design a hybrid observer that isbased on the second-order twisting sliding mode algorithm

[]; the most advantage o such sliding algorithm is that it onlyrequires the sign o therst derivative operator, but notits realvalue. Tus, the proposed observer takes less computationresources and is easy or implementation in digital signalprocessor.

3. Observer Design

Consider the mechanical dynamic system described in ();the rotor position is measurable, andthe targetis to estimatethe unknown disturbance, as well as the angular speed.

It is well-known that there are many methodologiesto estimate the angular speed based on the measurableposition. One common method is based on the back-orward differentiator; that is,= [ (1)]/, where is the sampling time and subscriptdenotes the variable

value at time = . For such method, the estimationerror is proportional to sampling time. On the other hand,small sampling time may enlarge the numerical error o(1), especially at low rotor speed and low position sensorresolution situation.

As mentioned in the previous section, in order to simpliythe hybrid observer structure and remove the digital lteringeffect, we will consider using the second-order twistingalgorithm or the unknown disturbance estimation.

Te system diagram is shown in Figure . Te rstobserver (speed observer) will be perormed to identiythe rotor speed without involving the operation o inverseo sampling time, that is,1/. Ten the second observer(disturbance observer) ensures the unknown disturbancecan be identied online without digital ltering effect. Finally,one can use the identied disturbance to compensate thecontrol loop to improve the robustness o the control system.

.. Speed Observer. By considering the rotor speed asthe system uncertainty, a hybrid sliding mode observer isdesigned in the orm o

= 11+ 1, ()where1 ,1is a positive high gain parameter, and1is the nonlinear eedback which is given by

1= 0, i1 1,

V1, i1 12

> 1. In other words,


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the second-order sliding surace 1= 1= 0will be reachedin a nite time and remained thereafer. Ten, the rotor speedcan be directly obtained rom (), as = 1.Remark . In real implementation, the derivatives in () canbe replaced with their backward differentiators, respectively;

that is, V

1= 11sign(1) 12sign(1 1(1)). Tesubscriptdenotes the variables obtained at sampling time = . Note that the operation o1/is completely avoided;then one can increase the sampling and control requencywithout sacricing its related numerical accuracy.

.. Disturbance Observer. With the identied rotor speed, ahybrid disturbance observer is designed in the orm o

=3

2

2

sin + cos

22 2

,

2= 0, i2 2,V2, i2< 2,

V2= 21sign 1 22sign 1 ,

()

where 21; 2is a positive high gain parameter;21and22 are two properly chosen positive sliding gains;2 is theupper bound o the nonlinear eedback2, which is designedas an integral unction o a sliding mode term V2.

Teorem .With the proposed speed observer in()and thedisturbance observer in(), the system unknown disturbance can be identied in nite time if properly choosing the

observer parameters; that is, it will have= 2.Proof. According to the result inTeorem , the rotor speed can be identied rom1 afer some time; thereore, thedynamics o the estimation error2= 1= can bededuced rom () and () as

2=

2 22+

2

. ()

Noting that the dynamic orm in () is similar to the onein (), then it can be proven that the second-order slidingsurace 2= 2= 0will be reached and remained thereaferin nite time, i properly choosing the observer parameters;that is,2> ||;21and22are chosen large enough.

By substituting the sliding surace 2= 2= 0into theestimation error dynamics in (), it has

= 2 ()

: Parameters o a PMSM.

Pole pairs Rating speed rad/sResistance .Inductance mHInertia . kgm

Viscous-riction coefficient .Nm/rad/sField ux-related coefficient . Nm/WbA

Real disturbance

Est. disturbance

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

Time (s)

Time (s)

Position

(ra

d)

Real positonEst. position

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

500

1000

Spee

d(ra

d/s)

Real speed

Est. speed

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

10

Time (s)

10

Disturbance(N

m)

F : raditional high gain observer.

which means the unknown disturbance can be directly

obtained rom2without digital ltering effect.Remark . In real implementation, the derivatives in () canbe replaced with their backward differentiators, respectively;that is, V2= 21sign(2) 22sign(2 2(1)). Ten, theoperation o1/is completely avoided.4. Simulation Result

In this section, we consider the simulation on a PMSM withits parameters being given inable .

o demonstrate the proposed observers perormance, aspeed open loop control platorm is used, with the unknown


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Real disturbance

Est. disturbance

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

Time (s)

Time (s)

Position

(ra

d)

Real positon

Est. position

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

500

1000

Spee

d(ra

d/s)

Real speed

Est. speed

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

10

Time (s)

10

Disturbance

(N

m)

F : Pure sliding mode observer.

torque disturbance given as= 4 sin(5) + 3 cos(2)Nm.Ten, the estimation perormance among a traditional highgain observer, a purehigher-order sliding mode observer, andthe proposed hybrid observer is careully compared.

Furthermore, the speed closed loop control perormancewith and without the compensation rom identied distur-bance is also addressed.

.. Traditional High Gain Observer. First, a traditional highgain observer is applied by setting the sliding mode eedbackterms in () and () to zeros; that is,1= 0,2= 0, and

2

. And the high gain parametersarechosen as

1

= 20and2= 20. Te initial rotor position and speed are set to(0) = 50 rad/s and(0) = 200 rad. Ten, the simulationresult is shown inFigure .

Itcan be seen that thehigh gain observer ailedto estimatethe unknown disturbance as it does not have any mechanismto identiy it.

.. Pure Sliding Mode Observer. Ten, a pure sliding modeobserver is used to identiy the unknown disturbance bysetting the high gains in () and () to zeros; that is,1= 0and2= 0. Ten, the sliding mode parameters are chosen as11= 1700,12= 800,21= 51, and22= 20. Te initialconditions are chosen the same as above.

Real disturbance

Est. disturbance

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

Time (s)

Time (s)

Position

(ra

d)

Real positon

Est. position

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

500

1000

Spee

d(ra

d/s)

Real speedEst. speed

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

10

Time (s)

10

Disturbance

(N

m)

F : Proposed hybrid sliding mode observer.

0

500

1000

1500

2000

2500

3000

Position

(ra

d)

0

200

400

600

800

1000

Real speed

Est. speed

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Real positon

Est. position

Spee

d(ra

d/s)

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

F : Closed loop control without disturbance compensation .


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0

5

10

15

0

100

200

300

400

500

Plantinput

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

100

Real disturbance

Est. disturbance

Real input

Compensation

10

15

5

Disturbance

(N

m)

F : Closed loop control without disturbance compensation .

Te simulation result is shown in Figure . Here, wecan see that the unknown disturbance can be successullyidentied afer around . seconds when the sliding modesuraces are reached.

.. Proposed Hybrid Sliding Mode Observer. Last, the pro-posed hybrid sliding mode observers in ()() are applied.Te parameters and initial conditions are chosen the same asabove, and the simulation result is shown inFigure .

It is clear that, with the proposed hybrid sliding modeobserver, the unknown disturbancecan be identied withhigh accuracy and much aster than the pure sliding modeobserver because o the additional high gain eedback terms.

.. Closed Loop Control without Disturbance Compensation.Now, we consider the speed closed loop control system withunknown disturbance and the system diagram is given inFigure .

First, by removing the disturbance compensation romthe speed eedback loop, that is, setting= 0, the simu-lation o speed closed loop control system under a periodicand trapezoidal unknown disturbance is given in Figuresand.

It can be seen that rad/sec maximin speed transientis observed when the disturbance changes rom+4Nm to4Nm at time o seconds... Closed Loop Control with Disturbance Compensation.Ten, by setting the disturbance compensation in the speed

eedback control loop, that is,= 2, all the controller andobserver parameters are chosen the same as previous.

0

500

1000

1500

2000

2500

3000

Position

(ra

d)

0

200

400

600

800

1000

Time (s)

Real positon

Est. position

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Spee

d(ra

d/s)

Real speed

Est. speed

F : Closed loop control with disturbance compensation .

0

5

10

15

0

100

200

300

400

500

Plantinput

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

Real disturbance

Est. disturbance

Real input

Compensation

100

10

15

5

Disturbance

(N

m)

F : Closed loop control with disturbance compensation .

Te simulation results are shown in Figuresandandaround rad/sec speed transient is observed when thedisturbance changes rom+4Nm to4Nm, which meansthe robustness o the speed closed loop controller is improvedwith the disturbance compensation.


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5. Conclusion

Based on the measurable current and rotor position,a hybrid sliding mode observer is developed or the unknowndisturbance estimation on a permanent-magnetic AC motor.Te proposed observer hasa simplestructureand ease orrealimplementation in digital signal processor, and the unknowndisturbance will be identied without digital ltering effect.Moreover, the identied disturbance can be used to compen-sate online the speed closed loop control system to improvethe robustness.

Conflict of Interests

Te authors declare that there is no conict o interestsregarding the publication o this paper.

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