Applying EKF for Nonlinear System

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Applying the unscented Kalman filter for nonlinear state estimation

Rambabu Kandepu a, Bjarne Foss a,*, Lars Imsland b

a Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim 7491, Norwayb SINTEF ICT, Trondheim 7465, Norway

Received 9 July 2007; accepted 4 November 2007

Abstract

Based on presentation of the principles of the EKF and UKF for state estimation, we discuss the differences of the two approaches.Four rather different simulation cases are considered to compare the performance. A simple procedure to include state constraints in theUKF is proposed and tested. The overall impression is that the performance of the UKF is better than the EKF in terms of robustnessand speed of convergence. The computational load in applying the UKF is comparable to the EKF. 2007 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear state estimation; Kalman filter; Constraint handling

1. Introduction

In the process industries one of the main goals is tomake the end product at the lowest possible cost while sat-isfying product quality constraints. State estimation oftenplay an important role in accomplishing this goal in pro-cess control and performance monitoring applications.There are many uncertainties to deal with in process con-trol; model uncertainties, measurement uncertainties anduncertainties in terms of different noise sources acting onthe system. In this kind of environment, representing themodel state by an (approximated) probability distributionfunction (pdf) has distinct advantages. State estimation isa means to propagate the pdf of the system states over time

in some optimal way. It is most common to use the Gauss-ian pdf to represent the model state, process and measure-ment noises. The Gaussian pdf can be characterized by itsmean and covariance. The Kalman filter (KF) propagatesthe mean and covariance of the pdf of the model state inan optimal (minimum mean square error) way in case oflinear dynamic systems [2,11].

All practical systems posses some degree of nonlinearity.Depending on the type of process and the operating region

of the process, some processes can be approximated with alinear model and the KF can be used for state estimation.In some cases the linear approximation may not be accu-rate enough, and state estimator designs using nonlinearprocess models are necessary. The most common way ofapplying the KF to a nonlinear system is in the form ofthe extended Kalman filter (EKF). In the EKF, the pdf ispropagated through a linear approximation of the systemaround the operating point at each time instant. In doingso, the EKF needs the Jacobian matrices which may bedifficult to obtain for higher order systems, especially inthe case of time-critical applications. Further, the linear

approximation of the system at a given time instant mayintroduce errors in the state which may lead the state todiverge over time. In other words, the linear approximationmay not be appropriate for some systems. In order to over-come the drawbacks of the EKF, other nonlinear state esti-mators have been developed such as the unscented Kalmanfilter (UKF) [1], the ensemble Kalman filter (EnKF)[10]and high order EKFs. The EnKF is especially designedfor large scale systems, for instance, oceanographic modelsand reservoir models [10]. The UKF seems to be a promis-ing alternative for process control applications [15,16,20].

0959-1524/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2007.11.004

* Corresponding author. Tel.: +47 73594476; fax: +47 73594599.E-mail address: [email protected] (B. Foss).

www.elsevier.com/locate/jprocont

Available online at www.sciencedirect.com

Journal of Process Control xxx (2008) xxxxxx

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Please cite this article in press as: R. Kandepu et al., Applying the unscented Kalman filter for nonlinear state estimation, J. ProcessContr. (2008), doi:10.1016/j.jprocont.2007.11.004
mailto:[email protected]:[email protected]


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The UKF propagates the pdf in a simple and effective wayand it is accurate up to second order in estimating meanand covariance [1]. The present paper focuses on usingthe UKF for nonlinear state estimation in process systemsand the performance is evaluated in comparison with theEKF. The paper proposes a simple method to incorporate

state constraints in the UKF.Section 2 describes the principles and algorithms ofEKF and UKF. Section 3 introduces a method to incorpo-rate the state constraints in the UKF state estimation andcompares it with the EKF. Four examples are studied inSection 4 to compare the performances of the UKF andEKF. Discussion on the differences between the UKFand EKF is presented in Section 5 and conclusions aredrawn in Section 6.

2. The EKF and UKF algorithms for nonlinear state

estimation

We present the principles and algorithms of the EKFand UKF. At the end of the section, different characteris-tics of the EKF and UKF are compared.

2.1. EKF principle and algorithm

To illustrate the principle behind the EKF, consider thefollowing example. Let x 2 Rn be a random vector andy gx 1be a nonlinear function, g : Rn ! Rm. The question is howto compute the pdf ofy given the pdf ofx? For example, in

the case of being Gaussian, how to calculate the mean yand covariance (Ry) of y? If g is a linear function and thepdf of x is a Gaussian distribution, then Kalman filter(KF) is optimal in propagating the pdf. Even if the pdf isnot Gaussian, the KF is optimal up to the first two mo-ments in the class of linear estimators [9]. The KF is ex-tended to the class of nonlinear systems termed EKF, byusing linearization. In the case of a nonlinear function(g(x)), the nonlinear function is linearized around the cur-rent value of x, and the KF theory is applied to get themean and covariance of y. In other words, the meanyEKF and covariance PEKFy of y, given the mean xand covariance (Px) of the pdf of x are calculated as

follows:

yEKF gx; 2PEKFy rgPxrgT; 3where ($g) is the Jacobian of g(x) at x.

Algorithm. Let a general nonlinear system be representedby the following standard discrete time equations:

xk fxk1; vk1; uk1; 4yk hxk; nk; uk; 5where x

2R

nx is the system state, v

2R

nv the process

noise, n 2 Rnn the observation noise, u the input and

y the noisy observation of the system. The nonlinearfunctions f and h are need not necessarily be continu-ous. The EKF algorithm for this system is presentedbelow:

Initialization at k= 0:bx0 Ex0;Px0 Ex0 bx0x0 bx0T;Pv Ev vv vT;Pn En nn nT:

For k= 1, 2, . . ., 1:(1) Prediction step.

(a) Compute the process model Jacobians:

Fxk rxfx; v; uk1jxbxk1 ;Gv rvf

bxk1; v; ukjvv:

(b) Compute predicted state mean and covariance(time update)

bxk fbxk1; v; uk;Pxk FxkPxkFTxk GvPvGTv :

(2) Correction step.(a) Compute observation model Jacobians:

Hxk rxhx; n; ukjxbxk

;

Dn rnhbxk ; n; ukjnn:(b) Update estimates with latest observation (mea-surement update)

Kk PxkHTxkHxkPxkHTxk DnPnDTn 1

;

bxk bxk Kkyk hbxk ; n;Pxk I KkHxkPxk:

2.2. UKF principle and algorithm

Consider now the same example as in the pre-vious section. The question is how the UKF com-pute pdf of y given the pdf of x, in other words, howto calculate the mean yUKF and covariance PUKFy of y, in the case of being Gaussian? Consider a set ofpoints

xi; i 2 f1; . . . ;pg; p 2n 1;(similar to the random samples of a specific distributionfunction in Monte Carlo simulations) with each pointbeing associated with a weight w(i). These sample pointsare termed as sigma points. Then the following steps are in-volved in approximating the mean and covariance: Propa-gate each sigma point through the nonlinear function,

yi

gxi

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the mean is approximated by the weighted average ofthe transformed points,

yUKF Xpi0

wiyi; Rwi 1

and the covariance is computed by the weighted outer

product of the transformed points,

PUKFy Xpi0

wiyi yyi yT:

Both the sigma points and the weights are computeddeterministically through a set of conditions given in [1].

Algorithm. The UKF algorithm is presented below; forbackground theory, refer to [1,8,9]. Let the system berepresented by (4) and (5). An augmented state at timeinstant k,

xak,xk

vk

nk

264 375 6is defined. The augmented state dimension is,

N nx nv nn 7Similarly, the augmented state covariance matrix is built

from the covariance matrices of x, v and n,

Pa,

Px 0 0

0 Pv 0

0 0 Pn

264

375

8

where Pv and Pn are the process and observation noisecovariance matrices.

Initialization at k= 0:bx0 Ex0; Px0 Ex0 bx0x0 bx0Tbxa0 Exa Ebx0 0 0 TPa0 Exa0 bxa0xa0 bxa0T

Px 0 0

0 Pv 0

0 0 Pn

264

375

For k= 1, 2, . . .,

1:

(1) Calculate 2N+ 1 sigma-points based on the presentstate covariance:

Xai;k1

,bxak1; i 0;,bxak1 cSi; i 1; . . . ;N;,bxak1 cSi; i N 1; . . . ; 2N;

8>: 9

where Si is the ith column of the matrix,

S ffiffiffiffiffiffiffiffiffiPak1p :In (9) c is a scaling parameter [8],

c ffiffiffiffiffiffiffiffiffiffiffiffiN kp ; k a2N j N;

where a and j are tuning parameters. We mustchoose jP 0, to guarantee the semi-positive definite-ness of the covariance matrix, a good default choiceis j = 0. The parameter a, 0 6 a 6 1, controls the sizeof the sigma-point distribution and it should ideallybe a small number [8].The ith sigma point (aug-

mented) is the ith column of the sigma point matrix,

Xai;k1 Xxi;k1Xvi;k1Xni;k1

264

375;

where the superscripts x, v and n refer to a partitionconformal to the dimensions of the state, processnoise and measurement noise, respectively.

(2) Time-update equations:Transform the sigma points through the state-updatefunction,

Xxi;k=k

1

f

Xxi;k

1; X

vi;k

1; uk

1

; i

0; 1; . . . ; 2N:

10Calculate the apriori state estimate and aprioricovariance,

bxk X2Ni0

wim Xxi;k=k1; 11

Pxk X2Ni0

wic Xxi;k=k1 bxk Xxi;k=k1 bxk T: 12The weights wim and w

ic are defined as,

w0

m k

N k ; i 0;w0c

k

N k 1 a2 b; i 0;

wim wic 1

2N k ; i 1; . . . ; 2N;

where b is a non-negative weighting parameter intro-duced to affect the weighting of the zeroth sigma-point for the calculation of the covariance. Thisparameter (b) can be used to incorporate knowledgeof the higher order moments of the distribution. Fora Gaussian prior the optimal choice is b = 2 [8].

(3) Measurement-update equations:

Transform the sigma points through the measure-ment-update function,

Yi;k=k1 hXxi;k=k1; Xnk1; uk; i 0; 1; . . . ; 2N13

and the mean and covariance of the measurementvector is calculated,

byk X2Ni0

wim Yi;k=k1;

Pyk

X2N

i0wic

Yi;k=k1

byk

Yi;k=k1

byk

T:

R. Kandepu et al. / Journal of Process Control xxx (2008) xxxxxx 3

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The cross covariance is calculated according to

Pxkyk X2Ni0

wic Xxi;k=k1 bxk Yi;k=k1 byk T:The Kalman gain is given by,

Kk PxkykP1

yk ;

and the UKF estimate and its covariance are com-puted from the standard Kalman update equations,

bxk bxk Kkyk byk ;Pxk Pxk KkPykKTk: 14

2.3. Discussion

The difference in the principles of state estimation usingUKF and EKF is illustrated based the Fig. 1 by consider-ing the following example,

y gx x2; x 2 R; x Xmean 6; Rx 16:The figure illustrates how the mean and variance of x are

propagated to obtain mean and variance ofy for EKF andUKF. The true mean and variance are calculated by usingthe following equations,

Ytruemean Egx Z1

1x2:

1ffiffiffiffiffiffi2p

pr

e12

xxr 2 dx;

Ptruey Egx Egx2:

where r ffiffiffiffiffiRxp is the standard deviation. The true meanand variance are 51.43 and 2686, respectively. The EKFmean is obtained by using (2) giving

YEKFmean 36:00;and the variance is obtained by performing the lineariza-tion as shown in the figure and using (3),

PEKFy 2304:

For the UKF, three sigma points are propagatedthrough the function, and the mean and variance are calcu-lated accordingly,

YUKFmean 52:00;PUKFy 2816:

The UKF approximates the propagation of the pdfthrough the nonlinearity more accurately when comparedto the EKF as illustrated by the numbers in the exampleabove.

In the EKF algorithm, during the time-update (predic-

tion) step, the mean is propagated through the nonlinearfunction, in other words, this introduces an error since ingeneral y6 gx. In case of the UKF, during the time-update step, all the sigma points are propagated throughthe nonlinear function which makes the UKF a betterand more effective nonlinear approximator. The UKFprinciple is simple and easy to implement as it does notrequire the calculation of Jacobians at each time step.The UKF is accurate up to second order moments in thepdf propagation where as the EKF is accurate up to firstorder moment [8].

Later, we will see that it is possible to implement stateconstraints by proper conditioning of the sigma points.

An example (state estimation of a reversible reaction) willbe considered to illustrate the constraint handling capabil-ity of the UKF.

0 5 10

0

10

20

30

40

50

60

70

80

90

100

110

x

y=g(x)=x

2

EKF

Xmean

Ymean

EKF

Ymean

true

linearization

0 5 10

0

10

20

30

40

50

60

70

80

90

100

110

x

y=g(x)=x

2

UKF

Xmean

Ymean

ukf

Ymean

true

transformed sigma points

Px=16

Py

true=2686

Py

EKF=2304

Px=16

Py

true=2686

Pa

UKF=2816

Fig. 1. Illustration of principle of EKF and UKF.


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3. State estimation with constraints

Constraints on states to be estimated are importantmodel information that is often not used in state estima-tion. Typically, such constraints are due to physical limita-tions on the states; for instance, estimated concentrations

should remain positive. In Kalman filter theory, there isno general way of incorporating these constraints intothe estimation problem. However, the constraints can beincorporated in the KF by projecting the unconstrainedKF estimates onto the boundary of the feasible region ateach time step [18,19]. An other way of nonlinear state esti-mation with constraints is moving horizon estimation(MHE), in which the constraints can be included in the esti-mation problem in a natural way [17]. In MHE, the statetrajectory is computed taking state constraints into account

at the expense of solving a nonlinear programing problemat each time step. The numerical optimization at each timestep may be a challenge in time-critical applications. In thissection, a new and simple method is introduced to handlestate constraints in the UKF and it is compared to the stan-dard way of constraint handling in the EKF, known as

clipping [14].Assume that the state constraints are represented by boxconstraints,

xL 6 x 6 xH:

We will illustrate the method for x 2 R2. In case of a sec-ond order system, the feasible region by the box constraintscan be represented by a rectangle as in Fig. 2. The figureshows the illustration of the steps of constraint handlingin case of the EKF and UKF from one time step to the

xk-1

Initial set up, t=k-1

EKF, t=k

xkEKF

UKF, t=k

covariance

covariance

covariance

x-kUKF

xkEKF, C

xkEKF, C

xkEKF

covariance

Transformed sigma points

Fig. 2. Illustration of estimation with state constraints.


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next. At t = k 1, the true state (xk), its estimate bxk andstate covariance are selected as shown in the figure. Att = k, the unconstrained EKF estimate bxEKFk is outsidethe feasible region and is projected to the boundary ofthe feasible region to get the constrained EKF estimatexEKF;Ck as shown in the figure. While projecting the EKFestimate, the covariance of the EKF estimate is notchanged and thus the constraints have no effect on thecovariance. Hence, the covariance does not include anyconstraint information. This way of the handlingconstraints in the EKF is termed as clipping in literature[14].

The constraints information can be incorporated in theUKF algorithm in a simple way during the time-updatestep. After the propagation of the sigma points from(10), the (unconstrained) transformed sigma points whichare outside the feasible region can be projected onto theboundary of the feasible region and continue the furthersteps. In Fig. 2, at t = k three sigma points which are out-

side the feasible region are projected onto the boundary(lower right plot in the figure). The mean and covariancewith the constrained sigma points now represents the apri-ori UKF estimate xUKFk and covariance, and they are fur-ther updated in the measurement-update step. Theadvantage here is that the new apriori covariance includesinformation on the constraints, which should make theUKF estimate more efficient (accurate) compared to theEKF estimate. An example (reversible reaction) is consid-ered in the next section to illustrate the state estimationwith constraints with the proposed method.

Extension of the proposed method to higher dimension

is straightforward. Alternative linear constraints, e.g.,

Cx 6 d

are easily included by projecting the sigma point violatingthe inequality normally onto the boundary of feasible re-gion. It is observed that the new (constrained) covarianceobtained at a time step is lower in size compared to theunconstrained covariance. If, in case, the estimate afterthe measurement-update (refer (14)) is outside the feasibleregion, the same projection technique can be extended.Further, the technique can be extended to the sigma pointsfrom (9).

The proposed algorithm is outlined below:Algorithm (outline). For k= 1, 2, . . . 1:

(1) Calculate 2N+ 1 sigma points based on the presentstate covariance according to (9) and project thesigma points which are outside the feasible regionto the boundary to obtain the constrained sigmapoints,

Xx;Ci;k1 PXxi;k1 i 0; 1; . . . ; 2N;

where P refers to the projections.(2) Time-update equations: Transform the sigma points

through the state-update function,

Xxi;k=k1 fXx;Ci;k1; Xvi;k1; uk1; i 0; 1; . . . ; 2N:Again apply the constrains on the transformed sigmapoints to obtain the constrained transformed sigmapoints,

Xx;Ci;k=k

1

P

Xxi;k=k1

i

0; 1; . . . ; 2N:

Calculate the apriori state estimate and aprioricovariance as given in (11) and (12) using the con-strained transformed sigma points Xx;C

i;k=k1.(3) Measurement-update equations: Transform the con-

strained sigma points through the measurement-update function as in (13) and obtain the UKF esti-mate by following the same steps given in Section 2.If UKF estimate violates the constraints, the sameprojection technique can be used.

4. Simulation studies

Four rather different examples are considered to com-pare different characteristics of the UKF and the EKF.First, the Van der Pol oscillator is considered to studythe behavior of the UKF and also to compare the robust-ness of the estimator due to model errors with that of theEKF. Second, an estimation problem in an inductionmachine is chosen to evaluate the performances for a non-linear system. Third, state estimation of a reversible reac-tion is studied to illustrate the constraint handlingcapability of the UKF. Finally, a solid oxide fuel cell(SOFC) combined gas turbine (GT) hybrid system is con-sidered to evaluate the performances in the case of higher

order, nonlinear system.In all the examples, the following assumptions are made:

The measurement update frequency of the KF coincideswith the system discretization sampling frequency.

The system model and the state estimator model are thesame unless otherwise specified.

Process noise and measurement noise are applied to thesystem. The noise is Gaussian with zero mean value.

The tuning parameters (the initial covariance and pro-cess and measurement noise covariances) are chosen tobe the same for both the EKF and UKF.

For UKF algorithm, a,b and j in Section 2.2 are set tothe following values,

a 1; b 2 and j 0:

4.1. Van der Pol oscillator

The Van der Pol oscillator is considered as it is a widelyused example in the literature. It is highly nonlinear and,depending on the direction of time, it can exhibit both sta-ble and unstable limit cycles [12]. In case of an unstablelimit cycle, if the initial state is outside the limit cycle, it

diverges and if the initial state is inside the limit cycle, it


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converges to zero as time progresses. In case of stable limitcycle, any non-zero initial state converges to a stable limitcycle. First, the unstable limit cycle is considered and theinitial state is chosen to be just inside the limit cycle andthe initial state estimate is considered outside the limit cycleto check the convergence of the UKF and EKF estimates.

4.1.1. Unstable limit cycle

The unstable limit cycle (Van der Pol oscillator inreverse time) is represented by the following state differen-tial equations [12]

x

1 x2;x

2 l1 x21x2 x1;l 0:2:The output vector is defined as

y x1 x2 T:The system is discretized with a sampling interval of 0.1.

Process noise with a covariance of 103

I2 and measurementnoise with a covariance of 103I2 is added to the systemstates and measurements, respectively. These noise charac-teristics are used throughout Section. 4.1.

The initial state is chosen as

x0 1:4 0 T;which is just inside the limit cycle. The initial state estimatebx0 is chosen to be outside the limit cycle, and the initialstate covariance matrix Px0, Rv and Rn are chosen as

bx0 0 5 T; 15

Px0 kxI2; Rv kpI2 and Rn knI2; withkx 5; kp 103 and kn 103: 16

The true and estimated states using the UKF and EKFare shown in Fig. 3. The UKF estimate converges to thetrue state and stays with it, whereas the EKF estimatecould not converge to the true state as the time progresses.Fig. 4 shows the phase portraits of the UKF and EKF esti-mates for the first 5 s. It also includes the corresponding

covariances of UKF and EKF estimates at each second,drawn according to

fxjx bxkTP1xk x bxk 1g:The covariances may also give an idea of the distribu-

tion of the sigma points around the mean at a given timeinstant in the case of the UKF. The covariances of theUKF and the EKF decrease a lot between from t = 0and t = 1 s. Further it may be observed that the latercovariances for the EKF are smaller than for the UKF.The choice ofPx0 is reasonable here as the initial state esti-mate is far from the true initial state. The UKF is robust to

the choices of the Px0 , Rv and Rn compared to the EKF as isillustrated with the following choices,

kx 102; kp 103 and kn 1: 17Figs. 5 and 6 show the plots and phase portraits corre-

sponding to the choices of a small Px0 and a large Rn. Eventhough the small Px0 is a bad choice for the considered bx0,the UKF estimates are still able to converge to the truestates because of the measurement correction and alsodue to the increase of the size of the covariance until theestimates converge to the true states. On the other hand,the EKF estimate is not converging to the true state asshown in Fig. 5. The difference can be attributed to the bet-

ter nonlinear approximation by the UKF at each time step.Compared to Figs. 5 and 6, both the UKF and EKF

0 5 10 15 20 25 30 35 401.5

1

0.5

0

0.5

1

1.5

2

time (sec)

x1

true

UKF

EKF

0 5 10 15 20 25 30 35 402

1

0

1

2

3

4

5

time (sec)

x2

true

UKF

EKF

Fig. 3. Estimated states for the Van der Pol oscillator in reverse time using UKF and EKF with an initial state estimate far from the limit cycle: large

initial state covariance and small measurement noise.


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estimates converge very fast in Figs. 3 and 4. The reason isthat the assumed measurement noise is less in Figs. 3 and 4,and hence the feedback in the measurement-update makesthe estimates converge quite fast. In Figs. 5 and 6, theassumed measurement noise is higher meaning that thefeedback from the measurement-update is less effective.

As the UKF gives a better approximation in time-update step, the UKF estimate is able to converge quitefast, even though the measurement noise is higher andthe initial covariance is a bad choice. In addition, the Kal-man gain is quite different in the two estimators due to thedifference in the covariances as can be observed in Fig. 6.

Hence, it is reasonable to assume that the correction step

in the UKF also improves convergence as compared tothe EKF.

To verify these results, two Monte Carlo simulationswith 100 runs are performed varying the initial state, andthe process and measurement noise realizations. In addi-tion the initial state estimate is also varied. Both states ofx0 andbx0 were chosen from a normal distribution with zeromean and standard deviation 0.4. The Monte Carlo simu-lations differ in the choices of the tuning parameters (Px0 ,Rv and Rn) as given in (16) and (17). The correspondingmean square errors (MSE)s of the state estimate errorsfor the EKF and UKF are 0.18 and 0.02, and 0.23 and

0.09, respectively. The MSEs corresponding to the UKF

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5

1

0

1

2

3

4

5

6

7

8

x1

x2

UKF

EKF

4 3 2 1 0 1 2 34

3

2

1

0

1

2

3

4

5

6

UKF

EKF

x2

x1

Fig. 4. Phase portraits of UKF and EKF estimates with an initial state estimate far from limit cycle, large initial state covariance matrix and smallmeasurment noise and with covariances plotted at each second, until 5 s.


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are considerably lower compared to that of EKF, andhence it can be concluded that the UKF gives improvedperformance compared to the EKF.

4.1.2. Stable limit cycle with model error

In [13], it is observed that when there is a significantmodel error the EKF is not able to converge to the true

states in case of the Van der Pol equation with a stable limit

cycle. This cycle is considered to evaluate the robustness ofthe UKF to model errors and it is compared to that ofEKF. The state and measurement equations are

x

1 x2; 18x

2 l1 x21x2 x1; l 0:2; 19

y x1 x2T

: 20

0 5 10 15 20 25 30 35 40

4

2

0

2

4

time (sec)

true

UKF

EKF

0 5 10 15 20 25 30 35 40

4

2

0

2

4

time (sec)

true

UKF

EKF

x1

x2

Fig. 5. Estimated states for Van der Pol equation using UKF with an initial state estimate far from limit cycle, small initial state covariance matrix andlarge measurment noise covariance.

4 3 2 1 0 1 2 34

3

2

1

0

1

2

3

4

5

6UKF

EKF

x2

x1

Fig. 6. Phase portraits of UKF and EKF estimates with an initial state estimate far from limit cycle, small initial state covariance matrix and largemeasurment noise and with covariances plotted at each second, until 5 s.


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In the state estimator model the value ofl is chosen as0.5. The system is discretized with sampling interval of 0.1.An initial state is chosen as x0 0:5 0 T. A initial stateestimate bx0, and initial state covariance matrix Px0, Rvand Rn are chosen as before with the following kx, kp andkn values,

bx0 5 1 T; kx 1; kp 103 and kn 103:21

The estimates states using UKF and EKF are comparedin Fig. 7. The errors in the estimated states in polar coor-dinates are shown in Fig. 8. From Figs. 7 and 8, it is clearthat the EKF estimates are sensitive to the model error inl, whereas the UKF shows robust performance to themodel error.

4.2. State estimation in an induction machine

The UKF and EKF are applied to a highly nonlinearflux and angular velocity estimation problem in an induc-tion machine. To develop high-performance inductionmotor drives, it may be necessary to have rotor flux estima-tors and rotor time constant identification schemes [3]. In[3] and [4], new EKF-based algorithms are developed forthe considered estimation problem as the EKF estimatorcannot produce effective results. The state space modelfor a symmetrical three-phase induction machine is [4],

x

1 k1x1 z1x2 k2x3 z2;x

2 z1x1 k1x2 k2x4;

x

3 k3x1 k4x3 z1 x5x4;

x

4 k3x2 z1 x5x3 k4x4;x

5 k5x1x4 x2x3 k6z3;

22

where x1, x2 and x3,x4 are the components of the stator andthe rotor flux, respectively, and x5 is the angular velocity.All the state variables are normalized. The inputs, the fre-quency and the amplitude of the stator voltage are denotedby z1 and z2 respectively, and the load torque is denoted byz3. k1, . . . ,k6 are parameters depending on the considered

drive. The outputs are the normalized stator currents y1and y2. The output equations are given by,

y1 k7x1 k8x3;y2 k7x2 k8x4;with parameters k7 and k8. For simulation, the modelparameters and inputs are set to the values given in Table1. Process noise with a covariance of 104I5 and measure-ment noise with a covariance of 102I2 is added to the sys-tem states and measurements, respectively. The model isdiscretized with a sampling interval of 0.1.

The UKF and EKF are used to estimate the states with

the actual initial conditionx0 0:2 0:6 0:4 0:1 0:3 T:

The initial estimated state is assumed to be

bx0 0:5 0:1 0:3 0:2 4 T:The tuning parameters are selected as,

Px0 kxI5; Rv kpI5 and Rn knI2;with kx 1; kp 104 and kn 102:The state estimation results are shown in Fig. 9 where

the UKF and EKF estimates of states x1, x3 and x5 are

compared with the true states. With the initial predictedstate far from the actual initial state, the simulation resultsshow that the UKF performs better than the EKF. Thereason for the better performance of the UKF is attributed

0 5 10 15 20 25 30 35 40 45 50

2

0

2

4

time (sec)

true

UKF

EKF

0 5 10 15 20 25 30 35 40 45 503

2

1

0

1

2

3

time (sec)

true

UKF

EKF

x2

x1

Fig. 7. Comparison of state estimates using UKF and EKF with a model error.


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to the better nonlinear approximation at each step. It canbe mentioned that the behavior of the EKF is similar tobehavior observed in [3].

A Monte Carlo simulation with 100 runs was performedand the MSEs of the EKF and UKF estimates are 0.13 and0.05, respectively. The MSE corresponding to the UKF issubstantially lower compared to those of the EKF. Againthe UKF performs significantly better, this time in the caseof a highly nonlinear semi-realistic example.

4.3. State estimation with constraints: a reversible reaction

example

We will here consider an example to illustrate the con-straint handling capability of the UKF compared to thatof the EKF. Consider the gas-phase, reversible reaction,

2A!k B; k 0:16;with stoichiometric matrix

v 2 1 and reaction rate

r kC2A:The state and measurement vectors are defined as

x

CA

CB ; y 1 1 x;

where Cj denotes the concentration of species j. It is as-sumed that the ideal gas law holds and that the reaction oc-curs in a well-mixed isothermal batch reactor. Then, fromfirst principles, the model for this system is

x fx vTr:

The system is discretized with sampling interval of 0.1.The UKF and EKF are used for state estimation, withthe following setup as used in [14]:

x0 3 1 T; bx0 0:1 4:5 T;Px0

36 0

0 36

; Rv 106

1 0

0 1

;

Rn 1021 0

0 1

:

The estimation result for the unconstrained case isshown in Fig. 10. The result shows that the dynamic per-formance of the UKF estimates is better compared to thatof the EKF. The EKF performance is very similar to thereported results in [14].

However, during the dynamic response, both the UKFand EKF estimates become negative, (meaning negativeconcentrations) which is not possible physically. State con-straints are incorporated according to the proposedmethod in Section 3 for the UKF and standard clippingis used for the EKF. The results are shown in Fig. 11. FromFig. 11 the UKF estimates converge to the true states with-

out violating the constraints. Because of the clipping in the

0 5 10 15 20 25 30 35 40 45 50

2

1

0

1

time (sec)

errorinamplitude UKF

EKF

0 5 10 15 20 25 30 35 40 45 50

1.5

1

0.5

0

0.5

1

1.5

time (sec)errorinphase(radians)

UKF

EKF

Fig. 8. Errors in amplitude and phase of the UKF and EKF state estimates.

Table 1Model parameters and inputs

k1 k2 k3 k4 k5 k6 k7 k8 z1 z2 z3

0.186 0.178 0.225 0.234 0.081 4.643 4.448 1 1 1 0


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EKF, CA estimate of EKF did not converge to the truestate and the estimate ofCB takes much longer time to con-verge to the true state.

Fig. 12 shows the phase portraits of the unconstrainedand constrained UKF estimates for the first 4 s. The figurealso includes the corresponding covariances plotted att = 0, 1 and 3 s. From Fig. 12, it is clear that it takes longertime for the unconstrained estimate to converge as the cor-responding covariances do no include the constraint infor-mation. The constrained UKF estimate converges fasteras the covariances decrease faster, which include the con-

straint information. The results from this example con-

firm that the proposed constraint handling method ispromising.

4.4. SOFC/GT hybrid system

The solid oxide fuel cell (SOFC) stack integrated in a gasturbine (GT) cycle, known as SOFC/GT hybrid system, isan interesting power generation system due to its high effi-ciency in the range of 6575%. A schematic diagram of theSOFC/GT hybrid system integrated in an autonomouspower system is shown in Fig. 13. The fuel, natural gas

(CH4), is partially steam reformed in a pre-reformer before

0 5 10 15 20 25 30 35 40 45 50

0.5

0

0.5

time (sec)

true

UKF

EKF

0 5 10 15 20 25 30 35 40 45 501

0.5

0

0.5

time (sec)

true

UKF

EKF

0 5 10 15 20 25 30 35 40 45 50

0

2

4

6

time (sec)

true

UKF

EKF

x3

x5

x1

Fig. 9. Comparison of estimated states: components of stator flux.

0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

time (sec)

CA

trueUKFEKF

0 1 2 3 4 5 6 7 8 9 1025

20

15

10

5

0

5

time (sec)

CB

true

UKF

EKF

Fig. 10. Comparison of estimated states: no constraint handling.


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it enters the SOFC anode. A part of anode flue gas is recy-cled to supply the necessary steam required for the steamreformation in the pre-reformer. The remaining part ofthe anode flue gas is supplied to a combustor where theunused fuel is burnt completely in presence of oxygen com-ing from the cathode flue gas. Air is compressed and pre-heated in a heat exchanger before entering the SOFCcathode. The hot stream from the combustor is expandedusing a high pressure turbine (HPT) which drives the com-pressor. The HPT flue gas is expanded to atmospheric pres-

sure using low pressure turbine (LPT) which drives an

alternator. The DC power from SOFC stack is invertedto AC using an inverter. The inverter and the alternatorare connected to the electric load through a bus bar. Typ-ically 6070% of the total power is supplied by the SOFCstack.

All the models of the system are developed in the mod-ular modeling environment gPROMS [7] as reported in [5].In [5], a low complexity, control relevant SOFC model isevaluated against a detailed model developed in [6]. Thecomparisons indicate that the low complexity model is suf-

ficient to approximate the important dynamics of the

0 1 2 3 4 5 6 7 8 9 101

0

1

2

3

time (sec)

CA

trueUKF

EKF

0 1 2 3 4 5 6 7 8 9 101

2

3

4

time (sec)

CB

trueUKF

EKF

Fig. 11. Comparison of state estimates: state constraint handling.

6 4 2 0 2 4 6 82

0

2

4

6

8

10

12UKF unconstrained

UKF constrained

x1

x2

Fig. 12. Phase portraits of the UKF unconstrained and constrained estimates with covariances.


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SOFC and can hence be used for operability and controlstudies. A regulatory control layer is designed and it isincluded in the model.

The purpose of a state estimator is to design anadvanced controller and as well for monitoring and faultdiagnosis purposes. In this section, EKF and UKF areused to design the state estimator, and their performancesand computational load are compared.

The SOFC/GT hybrid system model has 3 inputs, 18

states, and 14 measured outputs which are listed in Table 2.A state estimator of the hybrid system is designed usingboth the UKF and EKF, and the simulation results arepresented in Figs. 14 and 15. The figures show selectedstates for a typical run. Both the state estimators are tunedto obtain good performance. In the EKF, the Jacobians arecalculated numerically at each time step. The tuningparameters for both the UKF and EKF are the same toensure a fair comparison.

The initial state estimate is different from the actual ini-tial state and from the simulation results it can be con-cluded that UKF estimate converges to the true statesvery fast compared to the EKF estimate. The initial stateestimate cannot be chosen too far from the actual stateto avoid numerical problems in gPROMS. From thisexample, it is clear that the UKF can be used to designstate estimators for higher order systems and the UKF per-formance is again favorable. The computational load of theUKF is almost the same as that of the EKF, in addition toproviding better performance.

5. Discussion

The UKF as a tool for state estimation has been com-pared to the standard method for nonlinear state estima-

tion; the EKF. The state estimation methods have been

compared using the same tuning parameters to make thecomparative study as credible as possible. Four differentexamples have been tested. The UKF shows consistentlyimproved performance as compared to the EKF. In severalcases the improvement is substantial, both in the conver-gence rate as well as in the long term state estimation error.Monte Carlo simulations have also been run with differentinitial states. These runs reinforced the hypothesis that theUKF is indeed an interesting alternative to the EKF.

A method for constraint handling in UKF is proposedin Section 3. The results albeit for one example are prom-ising. An important feature of the proposed constraint han-dling method is the fact that it adjusts the covariance inaddition to the state estimates themselves. This is quite dif-ferent from the EKF where clipping usually is performedwithout adjusting the state estimate covariance. A rationalefor using the MHE is often attributed to its constraint han-dling ability. Our results indicate that the UKF with the

proposed constraint handling may be an interesting alter-

Anode

Cathode

Fuel

Exhaust

Air

Mixer

Ejector

Reformer

SOFC stack

Splitter

Combustor

Compressor

Heat exchanger

shaft

HPT

LPT Generator

DC

AC

Inverter

Bus bar

g gI

0ac

V

0g

V

0acI dcI

t tV

t tI

R RI RL RLI RC RCI m mI PK PK I IK IKI

LX CX

RL

RC

RPK

PK

P

Q

d

d

P

Qm

P

TX

Air blow-off

Electrolyte

Fig. 13. SOFC-GT hybrid system integrated in an autonomous power system.

Table 2Available measurements

No. Measurement

1 Pre-reformer temperature (K)2 Shaft speed (rad/s)3 Heat exchanger hot stream temperature (K)4 Heat exchanger cold stream temperature (K)5 SOFC outlet temperature (K)6 Combustor outlet temperature (K)7 Fuel mass flow rate (kg/s)

8 Anode recycle flow rate (kg/s)9 Flow to the combustion chamber (kg/s)10 Air blow-off flow rate (kg/s)11 Air mass flow rate (kg/s)12 SOFC current (A)13 SOFC voltage (V)14 Generator power (kW)


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native to MHE. Furthermore, it is very simple to imple-ment the proposed method to incorporate the constraintsin the UKF compared to the MHE.

The improved performance of the UKF compared tothe EKF is due to two factors, the increased time-updateaccuracy and the improved covariance accuracy. Thecovariance estimation can be quite different for the two fil-ters as shown in Fig. 6. This again makes a difference interms of different Kalman gains in the measurement-update equation and hence the efficiency of the measure-ment-update step.

There is usually a price to pay for improved perfor-

mance. In this case the computational load increases when

moving from the EKF to the UKF if the Jacobians are com-puted analytically. In many cases, particularly for higherorder systems, the Jacobians for the EKF are computedusing finite differences. In this case the computational loadfor the UKF is comparable to the EKF. The latter impliesthat the computational load using the UKF is significantlylower than what can be expected for the MHE.

6. Conclusions

This paper shows that the UKF is an interesting option

to the EKF because of improved performance. Further,

0 10 20 30 40 509.94

9.96

9.98

10

10.02

10.04x 10

3

time (sec)

Con

c.ofO2

incathode

true

UKF

EKF

0 10 20 30 40 500.0166

0.0168

0.017

0.0172

time (sec)

Co

nc.ofH2

inanode

true

UKF

EKF

0 10 20 30 40 50

0.101

0.1015

0.102

time (sec)

Conc.ofH2

Oi

nanode

true

UKF

EKF

0 10 20 30 40 50

0.043

0.0432

0.0434

0.0436

time (sec)

Conc.ofCO

2

inanode

true

UKF

EKF

Fig. 14. Comparison of state estimation using UKF and EKF: concentration of O2 in cathode and concentrations of H2, H2O and CO2 in anode of SOFC.

0 10 20 30 40 507450

7452

7454

7456

7458

7460

7462

time (sec)

shaftspeed(rad/s)

true

UKF

EKF

0 10 20 30 40 501350.02

1350.04

1350.06

1350.08

1350.1

1350.12

1350.14

time (sec)

TSOFC

true

UKFEKF

0 10 20 30 40 50910

911

912

913

time (sec)

Tref

true

UKF

EKF

0 10 20 30 40 501190

1195

1200

1205

1210

time (sec)

THX,cold true

UKF

EKF

Fig. 15. Comparison of state estimation using UKF and EKF: Compressor shaft speed and SOFC, prereformer and heat exchanger cold streamtemperatures.


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due to the proposed constraint handling method it mayalso be an interesting alternative to the MHE. The pro-posed method is much simpler to implement compared tothe MHE. The computational load for the UKF is compa-rable to the EKF for the typical case, where the Jacobiansare computed numerically.

Acknowledgements

The authors are grateful to Professor Biao Huang fordiscussions on the UKF. The financial support from TheGas Technology Center, NTNU-SINTEF and NorwegianResearch Council is acknowledged. Further the financialsupport from the NFR BIGCO2 project is alsoacknowledged.

References

[1] S. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estima-

tion, Proceedings of the IEEE 92 (2004) 401422.[2] C.T. Chen, Linear System Theory and Design, Oxford University

Press, New York, 1999.[3] L. Salvatore, S. Stasi, L. Tarchioni, A new EKF-based algorithm for

flux estimation in induction machines, IEEE Transactions onIndustrial Electronics 40 (5) (1993) 496504.

[4] K. Reif, F. Sonnemann, R. Unbehauen, An EKF-based nonlinearobserver with a prescribed degree of stability, Automatica 34 (9)(1998) 11191123.

[5] R. Kandepu, L. Imsland, B. Foss, C. Stiller, B. Thorud, O. Bolland,Modeling and control of a SOFC-GT based autonomous powersystem, Journal of Energy 2006 36 (2007) 406417.

[6] C. Stiller, B. Thorud, S. Seljbeo, H. Karoliussen, O. Bolland, Finite-volume modeling and hybrid-cycle performance of planar and tubularsolid oxide fuel cells, Journal of Power Sources 141 (2005) 227240.

[7] gPROMS, gPROMS introductory user guide, Process SystemsEnterprise Ltd, 2004.

[8] R. Van der Merwe, Sigma-Point Kalman Filters for probabilityinference in dynamic state-space models, PhD Thesis, Oregon Healthand Science University, 2004.

[9] B. Huang, Q. Wang, Overview of emerging Bayesian approach tononlinear system identification, in:RoundTableson Non-linearModelIdentification, International Workshop on Solving Industrial Controland Optimization Problems, Cramado, Brazil, April 67, 2006.

[10] G. Evensen, Data Assimilation, The Ensemble Kalman Filter,Springer, Berlin, 2007.

[11] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Mathe-matics in Science and Engineering, vol. 64, Academic Press, NewYork and London, 1970.

[12] H.K. Khalil, Nonlinear Systems Third Edition, Prentice Hall, 2000.[13] M.S. Ahmed, S.H. Riyaz, Design of dynamic numerical observers,

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[14] E.L.Haseltine, J.B. Rawlings, A criticalevaluationof extendedkalmanfiltering and moving horizon estimation, Technical Report, Texas-Wisconsin Modeling and Control Consortium (TWMCC), 20022003.

[15] A. Romanenko, J.A.A.M. Castro, The unscented filter as analternative to the EKF for nonlinear state estimation: a simulationcase study, Computer and Chemical Engineering 28 (2004) 347355.

[16] B. Akin, U. Orguner, Aydin Ersak, State estimation of inductionmotor using unscented Kalman filter, IEEE Transactions on ControlApplications 2 (2003) 915919.

[17] C.V. Rao, J.B. Rawlings, D.Q. Mayne, Constrained state estimationfor nonlinear discrete-time systems: stability and moving horizonapproximations, IEEE Transactions on Automatic Control 48 (2003)246258.

[18] D. Simon, T.L. Chia, Kalman filtering with state equality constraints,IEEE Transactions on Aerospace and Electronic Systems 38 (2002)128136.

[19] S. Ungarala, E. Dolence, K. Li, Constrained extended kalman filterfor nonlinear state estimation, in: Proceedings of 8th InternationalIFAC Symposium on Dynamics and Control Process Systems,Cancun, Mexico, June 2007, vol. 2, pp. 6368.

[20] W. Li, H. Leung, Simultaneous registration and fusion of multipledissimilar sensors for cooperative driving, IEEE Transactions onIntelligent Transportation Systems 5 (2004) 8498.


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