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Speed Sensorless State Estimation for Induction

Motors: A Moving Horizon Approach

Lei Zhou1, Yebin Wang21 Massachusetts Institute of Technology

2 Mitsubishi Electric Research Laboratories

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Motivations

The tracking bandwidth of speed sensorless induction drive islimited by the state estimation: estimation scheme with fastconvergence is needed;

The tuning of the classic model reference adaptation estimation isdifficult: systemetic estimation scheme is desired.

The moving horizon estimation (MHE) provides a systematicframework and demonstrates fast estimation convergence, but issensitive to model parameter mismatch.

In this work, we study:

The application of MHE for induction motor speed sensorlessestimation;

Dual-stage adaptive MHE that accounts for parametric modeluncertainties.

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Moving Horizon Estimation

Moving horizon estimation and fullinformation problem[1].

[1] Rawlings, James B. Moving horizonestimation. Encyclopedia of Systems andControl (2015): 799-804.

Full information estimation:

minx0,{wk}

T1k=0

(x0) +T1k=0

Lk(wk, vk),

where

(x0) = x0 x02

P10

,

Lk(wk, vk) = wk2Q1k

+ yk h(xk)2R1k

.

Moving horizon estimation:

minz,{wk}

T1k=TN

ZTN(z)+T1

k=TN

Lk(wk, vk).

ZTN(z): arrival cost, summarizing theinformation fromk= 0 toT N.

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Induction Motor Model

The classical induction motor model in the stationary two-phase reference frame canbe written as:

ids = ids+ dr+ qr+ 1

uds,

iqs = iqs Lm

dr+ qr +

1

uqs,

dr =Lmids dr qr,qr =Lmiqs+ dr qr ,

=

J(idsqr + driqs)

TLJ

.

The parameters are

=Ls 1 L2mLsLr

; = Rs

; = RrLr

= LmLr

; = 32

LmLr

,

where (Rs, Ls) and (Rr, Lr) are the resistance and inductance of the stator and rotorrespectively, and Lm is the mutual inductance between stator and rotor.

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Adaptive MHE: Problem Formulation

Consider a general nonlinear, discrete system:xk+1=fk(xk, uk, wk),

yk =hk(xk) +vk.

The MHE formulation for state and parameter estimation at time T is

minp,z,{wk}

T1k=TN

ZTN(z, p) +T1

k=TN

Lk(wk, vk, p)

subject to xk(k; z, {wj}, uk) Xk, k=TN, ..., T

wk Wk, k= (TN), ..., (T1)vk =ykhk(xk) Vk, k=TN, ..., T1

p P

Remark: the conventional (non-adaptive) MHE does not have parameters p in the

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MHE for Induction Motor State Estimation

Stage costs Lk(wk, vk)

Lk(wk, vk) =wTkQ

1wk+ vTkR

1vk.

where Q and R are the covariance matrices for process noise and measurementnoise respectively.

Arrival cost ZTN(z)Define the cost for the initial estimation error as (x0) = (x x0)

T10 (x x0).The approximated arrival cost can be calculated by

ZTN(z) = (z xTN)T1TN(z xTN) +

TN

where TN is updated by the one-step predictor equation as

k+1=GQGT + AkkA

Tk AkkC

T(R + CTkCT)1CkA

Tk .

ConstraintsInequality constraints are removed to simplify the computation. Theoptimization problem only subject to the system dynamics.

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Dual-stage MHE for parameter adaption

In an augmented-state MHE, estimating parameters and states together can leadto highly nonlinear optimization problems and add difficulties to solving;

In dual-stage adaptive MHE, the state estimation and parameter estimation areseparated into two sequential steps, and significantly simplify the solvingdifficulties.

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!"#$

)#*#+$"$* &'(

!"%$

&'()*+,-*&.)*+,- /

*+0 &1)*+,-*

&.)*+,2 /*+0 &1)*+,2

*

3+0

45

State estimation:

minz,{wk}

T1TNx

xT=ZTNx(z)+

T1

k=TNx

Lxk(wk,vk)

Parameter estimation:

minp

pT=

pTNp

+

T1

k=TNp

vTkR1vk

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Dual-stage MHE with RLS parameter estimation

Define parameter vector as = [,,, 1/]T. The current equations can be written into the regression form:

ik+1ds

ikdsdt

ik+1

qs ik

qsdt

y

= ikds

kds

kqsk ukds

ikqs kqs kdsk ukqs T

1/

,

The recursive least square (RLS) estimation can be readily applied as

min RLST =

1

T

Tk=1

yk| yk2. Remark: for induction motor parameter estimation (linear, unconstrained system

with no dynamics), RLS estimation is equivalent to MHE parameter estimationwith infinite estimation horizon.

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MHE Simulation for Induction Motor

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%

&'"#$('

()

(*+, &'-&'

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B "?32#

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Block diagram of induction motor field-oriented control.

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20-steps horizon. Samping rate 10 kHz. P0 =I66 103. Initial values

x= [1, 1, 0, 0, 5, 0]T; x= [0, 0, 0, 0, 0, 0.1]T.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4MotorSp

eed(rad/s)

0

50

100

150

Reference SpeedPlant SpeedEstimated Speed

Time (s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

SpeedEstimation

Error(rad/s)

-2

0

2

4

6

MHE Speed Estimation ErrorEKF Speed Estimation Error

State estimation for induction motor with MHE and EKF.cMERL June 28, 2016 10 / 13


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20-steps horizon. Samping rate 10 kHz. P0 =I66 103. Initial values

x= [1, 1, 0, 0, 10, 0]T; x= [0, 0, 0, 0, 0, 0.1]T.A torque disturbance of1 Nm is added to motor at T = 0.2 s.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Motor

Speed(rad/s)

0

50

100

150

Reference SpeedPlant Speed Q

TL= 1

Plant Speed QTL= 10Plant Speed Q

TL= 100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

SpeedEstimation

Error(rad/s)

-20

-10

0

10

20Speed Error Q

TL= 1

Speed Error QTL

= 10

Speed Error QTL

= 100

Time (s)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Loadtorque(Nm

)

-0.5

0

0.5

1

1.5

True load torqueEstimated, Q

TL= 1

Estimated, QTL

= 10

Estimated, QTL

= 100

Induction motor speed with MHE under step torque disturbance.

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Dual-stage MHE with RLS for parameter estimation

Initial parameters: 0 = 0.7true; 0 = 0.7true; 0 = 0.9true; 0 = 0.9true.

Dynamic horizon length is used for better convergence.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4M

otorSpeed(rad/s)

0

50

100

150

ReferencePlant speedEstimated

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4SpeedEstimation

Error(rad/s)

-20

0

20

40

Time0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Parameter

EstimationError(%)

-50

0

50

Dual-stage adaptive MHE with RLS parameter estimation for induction motor system.

Non-adaptive MHE cannot provide converging estimation.

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Conclusion

Moving horizon estimation scheme for induction motor stateestimation considering the rotor mechanical dynamics is studied;

Dual-stage adaptive MHE can successfully achieve convergingparameter and state estimation despite the initial model parametricerror.

Future work

Analysis and tuning for the dual-stage adaptive MHE;

Computational cost reduction of MHE for real-time implementation.

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