Jan Erik STELLET,Tobias ROGG

1.Karlsruhe Institute of Technology,76131 Karlsruhe,Germany;

2.Swiss Federal Institute of Technology,8092,Zurich,Switzerland

On linear observers and application to fault detection in synchronous generators

Jan Erik STELLET1†,Tobias ROGG2

1.Karlsruhe Institute of Technology,76131 Karlsruhe,Germany;

2.Swiss Federal Institute of Technology,8092,Zurich,Switzerland

Thisworkintroducesanobserverstructureandhighlightsitsdistinctadvantagesinfaultdetectionandisolation.Itsapplication to the issue of shorted turns detection in synchronous generators is demonstrated.For the theoretical foundation,the convergence and design of Luenberger-type observers for disturbed linear time-invariant(LTI)single-input single-output(SISO)systems are reviewed with a particular focus on input and output disturbances.As an additional result,a simple observer design for stationary output disturbances that avoids a system order extension,as in classical results,is proposed.

Synchronous generators;Field winding;Fault detection;Unknown input observer(UIO);Disturbance observer;Residual generation

DOI10.1007/s11768-014-3036-z

1 Introduction

Initially introduced in 1964 by Luenberger[1],state observers for linear time invariant systems form an integral part of state space control.The following advances termed reduced order observers(ROOs)[2]consider the separation of the state space into a measurable and an immeasurable subspace.Designing observers with minimal or partially reduced order has been studied[3].

Special emphasis is put on fault-tolerant observers.Considerable work has been devoted to the design of unknown input observers(UIOs)[4–9]which converge despite the presence of disturbances in the system equation[10].Bymodellingsensorerrorsinanextended systemstate[11],bothuncertaintiesinsystemandmeasurement equation can be represented as unknown input signals.

More recent approaches include adaptive control techniques for observer design[12]but are limited to constant or slowly time-varying disturbances.High-gain observers[13]can be used to reduce the influence of disturbances to an arbitrarily small level.However,this approachsuffersfromtheamplificationofmeasurement andprocessnoise.Byemployinganextendeddescriptor system,this limitation can be alleviated[14].Moreover,dynamic observers[15]are suitable for fault-tolerantobservation without increasing the dimension of system equations[16].Equivalent-input-disturbance estimators focus on the effect of a disturbance on a control system input rather than on the disturbed states[17].

In this work,several relations between Luenbergertype observers in the presence of disturbances are studiedfromatheoreticalpointofview.Thiscontributionexplicitly details the general results obtained in[4,6]concerning the relationship between unknown input and ROOs in the SISO case.Furthermore,conditions and simplified observer design methods for systems with disturbances in input and measurement are analysed.

Increasing attention is paid to the application of state observers in model based fault detection and isolation(FDI)[18–21].The basic idea is to utilise the guaranteed convergence of an observer in the fault-free case to detect deviations in the system plant[22].Henceforth,the output estimation error(residual)is monitored.Recently,the simple yet comprehensive notion of total measurable fault information residual(ToMFIR)has been studied[23].

A key challenge is that the output residual usually also depends on quantities other than the fault’s magnitude itself.This effect has to be compensated for in threshold-based detection schemes,which poses an additional problem if uncertain or time-varying parameters are involved.In previous results[24]gain-dependent scaling has been investigated for a Luenberger observer design which will be extended in this work to other observer types.

Moreover,a detailed analysis is presented for shorted turns detection in field windings of synchronous generators.Failures of the winding insulation are frequent,difficult to detect[25]and can lead to severe generator damage[26].Theyhaverecentlybeenstudiedin[27]for a machine with constant frequency.Here,special emphasis is put on variable frequency generators as used in wind turbines[28,29]or naval and aircraft systems[30–32].

This work is organised as follows:Section 2 constitutes definitions,background and furthermore derives an explicit formula for a ROO for SISO systems.In Section 3,observer convergence and design in disturbed systems is analysed.Additionally,the close relationship between the ROO and the UIO is highlighted.In Section 4,threshold-based fault detection is studied in general and for the application of shorted-turns detection in synchronous generators.All findings are summarised in Section 5.

2 Background and def i nitions

2.1 Full-state Luenberger state space observer

A linear system is fully characterised by A ∈ Rn×n,b∈Rnand c∈Rnin its state space representation with the state vector x(t)∈Rn,the input u(t)∈R and the output y(t)∈R:

Only observable systems with regular observability matrix QBare considered in this work.The well-known identity observer as proposed by Luenberger[1]is given as

The dynamics of the full-state observer in(2)are determined by the matrix NL:=A−lLc.The observer gain lLcan be obtained using Ackermann’s formula[33]

with f0,...,fn−1being the coefficients of the desired characteristic polynomial.With endenoting the nth canonical unit vector,s1is defined as the last column of the inverted observability matrix:

2.2ROO

Many practical systems possess states that are metrologicallyaccessibleanddonotneedtobeestimated.The idea of a ROO as opposed to the full-state observer is to derive an estimate in the immeasurable state subspace only[2].

For SISO systems(1),consider the special case where the output y(t)would be identical to a particular state xi(t).In this case,the measurable subspace is orthogonal to the immeasurable part r(t)∈ Rn−1.It stands to reason to reorder and split up(1a)to obtain

Considering the last row of(5)as a measurement equation for the system determined by the first n−1 equations,an identity observer for r(t)can be derived according to(2).This yields the ROO formula[2]:

The observer’s dynamic is given by the eigenvalues of(A11−fA21).Therefore,the gain vector f can be chosen bypoleplacementforthisexpression.Bysupplementing ˆr(t)with the measurement y(t),the complete estimate ˆx(t)is obtained.

In general,the measurable subspace is not strictly orthogonal to the immeasurable subspace.However,any observable SISO system(1)can be transformed to its observable canonical form using the transformation

where T is given with s1from(4)as[33]:

In the transformed system,only the nth element zn(t)of the transformed state vector spans the one-dimensional measurable subspace.Therefore,orthogonality to the immeasurable subspace is achieved.

2.3 ROO explicit form

This section details a constructive derivation of an nth order observer formula different to the full-state observer(2).Considering a transformation(7)of system(1)to canonical form,the ROO formula(6)is applied.The main result is an explicit form of the reduced state space observer which will give further insight when compared to other observer types.

Lemma 1(ROO explicit form)A state observer for a system(1)that is derived on the basis of(6)is given by

with

and with the initial value ρ(0)chosen in order to satisfy:

ProofSee Appendix A.

With system(1a)and by differentiating(9b),the observer error can be set forth:

Obviously,the observer error converges to zero if lRis chosen in order to constitute a stable system matrix.The coefficients f0,...,fn−2of its characteristic polynomial are found in(10d).The nth eigenvalue of the system matrix NRequals zero:

Theresultingobserverformula(9)constitutesanequivalencetothegeneral ROO(6).Upon this,newtheoretical insight will be established in the following sections.

3 Observers for systems with disturbances

Next,disturbances on the ideal system(1)are taken into consideration.Commonly experienced causes for such deviations are parameter uncertainties,sensor errors or unmodelled system behaviour.Here,a disturbed system is modelled as

with A ∈Rn×n,b∈Rn,c∈Rn,the state vector x(t)∈Rn,the known input u(t)∈R and the output y(t)∈R.There are two undesired disturbances in the shape of an unknown input v(t)∈R and an additive w(t)∈R output disturbance.

Note that there is a difference between the deterministic but unknown disturbances assumed in this work and disturbances in the form of stochastic processes.In the latter case,the popular Kalman filter[34]yields optimal state estimates under the assumption of white Gaussian noise.

In the following section,necessary conditions for the design of an nth order disturbance observer will be studied.After showing that it is not feasible to achieve resilience to both input and output disturbances of arbitrary nature,observer design for the two cases will be studied.

3.1 Conditions for disturbance observer design

In order to address disturbances v(t)in the system equation(14a),UIOs have been developed.Here,necessary conditions for the design of a UIO will be reviewedinthepresenceofadditionaloutputdisturbances w(t)in measurement equation(14b).

Theorem 1(Disturbance observer distinction) Convergence of a linear Luenberger-type observer of the form

for a system disturbed according to(14)is restricted to the case of either v(t)≠0 or w(t)≠ 0.

Proof Starting from the observer structure(15)whereˆx(t)∈Rnis the state estimate,ρ(t)∈Rn.N∈Rn×n,l∈ Rn,g ∈ Rnand h ∈ Rnare matrices to be determined.The state estimation error for system(14)is

With P:=(In−lc)and the following conditions

(16)is simplified to become

In order for the observer error e(t)to decay it is required that N constitutes a stable system dynamic with eigenvalues in the left half-plane.A necessary condition is thus that N is regular.The proof is completed within the following two lemmata where it is shown that it is not possible to find a regular N which yields independence of(19)against unknown inputs v(t)and arbitrary output disturbances w(t)at the same time.

Lemma 2(UIO)A UIO requires P to be singular.This requires an additive matrix to give a regular system matrix N:

ProofMaking(19)independent of the unknown input v(t)requires:

For d≠0,this holds true if and only if P is singular.

Lemma 3(Output disturbance observer(ODO))Observer convergence in spite of an output disturbance requires P to be regular.

ProofIndependence of w(t)in(19)would require

which in turn determines N according to(18)to

With A assumed to be regular,regularity of N requires that P is regular.

Note that this gives only a necessary condition,as the disturbance’s derivative˙w(t)was not considered.Obviously,there is no simple means for freeing(19)of an arbitrary disturbance at all times.In[7,35]the special case of w(t)being a linear combination of v(t)is considered.However,in the relevant stationary case with ˙w(t)=0,anecessaryandsufficientconditionisprovided by(23).

3.2 UIO design

Considering the disturbed system(14)with v(t)≠0 and w(t)=0 an approach to design a UIO will be studied.Compared with the derivation of the explicit ROO formula(9),a novel simple scheme is identified at the cost of only a minor restriction on the observer’s initial value selection.Given the requirement(21),a way to choose N is presented in[5]and will be briefly reviewed for the SISO case.An alternative design method by a projection operator approach is presented in[36].

It is required that(cd)−1exists.Then,(21)determines

and therefore,

Postmultiplying(18)by d gives hU:

Considering the choice of lUand hU(18)becomes

The general solution for NUin(27)is given by[5]

The approach pursued in(17),(18)and(21)is entirely different to the formulation of the ROO for an undisturbed system in Section 2.3.However,the result in(28)is identical to NRin(10a)except for the additive term βc.

Corollary 1(Equivalence to ROO) Any UIO is a special case of the derived explicit ROO structure where f is not a free parameter but determined by d which characterises the unknown input.

Proof Equation(24)gives

NotethattheUIOonlyconvergesif f constitutesastable polynomial(13).Additionally,β which determines the nth eigenvalue needs to be chosen accordingly to ensure convergence.As the observer eigenvalues can only be partially assigned,the system is not fully observable.This becomes obvious when calculating QBfrom(PA,c)[8].

An interesting consequence of this result is that despite the complex procedure to determine NUfor the UIO in(28),a simple form is obtained if a relatively mild constraint on the initial value ρ(0)is imposed.

Corollary 2(UIO Simplification) If the initial value is restricted in order for

to hold,the system matrix(28)is reduced to NU=PA.

Proof The proof is given by the constructive derivation of the ROO in Section 2.3.Here,condition(11)is imposed in order to achieve the state extension.

3.3 Output disturbance observer design

For completeness,observer design in the presence of additive output disturbances is studied.

Consider system(14)with v(t)=0 and w(t)≠0.As has been pointed out,making the state estimation error(19)independent of w(t)requires additional knowledge on the disturbance.One approach assumes a model of the disturbance’s dynamics and extends the system state[11].The enhanced system state contains w(t)as a combination of k additional states xw(t)which are represented by a linear dynamic:

Observer design for the enhanced system state can be performed using a full-state observer(3)of order n+k.

If however,no information on the disturbance is given except that it exhibits stable dynamics,the extended system can be reduced to include only the stationary state of the disturbance:

Note that another way to represent arbitrary output disturbances is to enhance the above system with an unknown input signal in the(n+1)th component.However,[5]established that in this case,observer design is restricted to systems with stable A and is therefore not practical.

Lemma 4(System matrix and observability)Assuming that the undisturbed system(1)is observable,the extended system(32)is observable if and only if A is regular.

Proof See Appendix B.

Studying observer design for this system of order n+1 using the reduced order formula from Section 2.3 will be presented next.

Theorem 2(Stationary output disturbance observer(SODO))Let system(14)with non-singular A,v(t)=0 and w(t)≠0 be given.Then,the following generalisation of(9)constitutes an nth order state observer for the system with stationary output disturbance:

with

ProofSee Appendix C.

The observer structure(33),hereafter referred to as SODO,is identical to the ROO with the only difference being a generalised gain vector lG.Hence,the error dynamics(12)applyaswell.Notethatthereisnocondition on the initial value ρ(0).

Next,the calculation of the observer gain is explored and related to the pole placement for a full-state Luenberger observer.

Lemma 5(Observer gain) The observer gain lGcan be calculated with s1from(4)and the coefficients of the desired characteristic polynomial f0,...,fn−1as

Here,lLdenotes the gain vector of a full-state Luenberger observer(3)with the same poles.

Proof First,lRfrom(10d)iscalculatedforthe(n+1)-dimensional system(32).Only the first n entries are considered which yields

Here,sGdenotes the first n components of s=where s is defined as in(4)but for the extended system(32).Premultiplying the definition of s with QBgives

The first of these n+1 linear equations determines s2.Comparing the following n equations with the definition of s1for the undisturbed system in(4)gives that sG=.Inserting this relation in(36)gives the result(35).

Furthermore,the eigenvalues of NGachieved byare identical to the eigenvalues of the system matrix NLof a full-state Luenberger observer with gain lL:

Matrix NLis similar to NGwith similarity transformation A.Thus,their eigenvalues are the same.

When compared to a full-state observer design,the reduced order of the observer(33)might give improvements in terms of computational requirements.Another advantage arises in fault diagnosis applications and will be explored in Section 4.

Ontheotherhand,adrawbacksharedbyallobservers in the form of(15)is the immediate dependence on the measurement y(t).In contrast,a Luenberger observer type(2)acts as a low-pass filter on the measurement signal which is beneficial in the presence of measurement noise.

3.4 Summary of main results

In the first part of this section,conditions for observer convergence in the presence of unknown inputs as well as output disturbances are analysed.It is found that independence against both disturbances cannot be achieved with a single observer of order n.

Secondly,design of a UIO is reviewed.It is found that the UIO is a special case of the ROO.From this result,a simplification to the UIO system matrix is proposed which gives an easier path towards finding an observer at the cost of only a minor restriction on how the initial observer state is to be chosen.

Completing the study,the third section details observer design in the presence of output disturbances.As a generalisation of the explicit ROO formula from Section 2.3,the SODO is presented.

4 Observers for fault diagnosis

4.1 Methodology

A state observer incorporates a model of a physical system to estimate state variables.Deviations in the physicalsystem thatarenotreflected inthemodelresult in a residual error in the state estimates.This residual can therefore be used as an indicator of defects and aging[37,38].

Given a system(1a),faults are modelled as

TheerrorcausedbychangesinAandbcanbecombined to form an unknown input∈(t):= δAx(t)+ δbu(t):

The ToMFIR[23]is

which converges to zero in the case of no faults.Of particular interest is the stationary limit(hereinafter called ToMFIR)that is caused by faults with stationary end value:

The ToMFIR of a full-state Luenberger observer(2)depends on the observer gain lL:

Largegainscreateasmallresidual,whilegainsthatplace the eigenvalues of the observer near zero create a high residual value for the same fault.As a remedy,[24]proposes a multiplicative compensation.

The additional correction is avoided for observer designs which generate residuals that are independent of the gain:

Corollary 3(SODO)A SODO(33)can be used to produce ToMFIR values that are gain-independent:

Proof The residual for an observer design(15)is

Giventhattheproposedobserverdesign(33)fulfils(17),(18),(22)and(23),the residual is reduced to(44).

4.2 Application to shorted turns detection

Next,the application of threshold based fault detection in synchronous machines is studied.In[27]a Luenberger observer is employed for diagnosis of shorted turns in the field windings.Here,these results are extended using a SODO(33)for residual generation.Besides the independence on the observer gain that has been discovered in the previous section,emphasis is putontheresidual’sdependenceonsystemparameters,especially the electrical frequency.

A model of a synchronous machine is given by[27]

The system state is given by stator direct and quadrature currents Id(t)and Iq(t).Direct and quadrature voltages Ud(t)and Uq(t)as well as exciter current If(t)form the system input.

Thepresenceofafieldwindingfaultreducestheeffectivenumberofturnsto¯Nfandthuscreatestheunknown input signal:

The synchronous generator is defined by parameters which are explained in Table 1.

Table 1 Parameter description and values used in simulation example.

A full-state Luenberger observer(2)can be employed to calculate residuals under the assumption that both system states Id(t),Iq(t)are measurable.As has been shown in[27],the stationary residual(43)is

with λ2as the second observer eigenvalue and the constant stator-side referred exciter current I′f=2IfNf/3Ns[40].

In order to achieve independence of quantities other than the number of windings in(48),[27]proposes to set:

However,this in turn limits the detection speed of the observer to a fixed value.Moreover,the compensation(49)requiressettingλ2proportionaltoωrwhichmayre-sult in increased noise sensitivity in high frequency machines.Additional difficulties arise in variable frequency applications[28–32].

On the other hand,when using a SODO,the residual(44)is independent of the eigenvalue locations:

Note that A in(46)is always regular,guaranteeing the existence of the observer.

Results of a simulation example visualise the advantage of having additional degrees of freedom in the observer design.Fig.1 shows that varying λ2in relation to the constant value(49)produces improved detection time.

Fig.1 Detection time for a threshold of 80%employing a SODO(33)compared to a conventional Luenberger observer(LBO)(2)for a simulation of one faulted turn with the machine in[39].Having a residual independent of the eigenvalues,the SODO can be designed with an arbitrary λ2.This enables it to detect the fault in only a fraction of the time that the Luenberger observer would have needed.

Another significant advantage is that the first component r∞,1in(50)allows for drastic simplification for typical configurations,leading to the following corollary.

Corollary 4(SODO residual simplification) For typical synchronous generators[41,42]it holds that:

Then,the first component of residual(50)is simplified to

This residual directly gives the per cent of the field windings that have short circuited multiplied by the transformed stator current and Lm/Ld,a factor that is usually close to 1.It is therefore an ideal fault indicator,as minimal dependency on uncertain or varying parameters is achieved.

5 Conclusions

This contribution constructs a unique observer structure for fault detection and isolation.This is achieved by developing a distinct explicit form of a ROO and establishing theoretical relations with other observer types.Second,design of linear observers for systems with stationary output disturbances is considered.In contrast to classical results,an extension of the system order is avoided while maintaining a particularly simple design procedure.

Based on this,recent results in the application of model based fault detection are extended.Compared to a Luenberger identity observer,it is found that gaindependent scaling of the residual is avoided with the novel design.

Exemplifying the general result,further application specific advantages are found for shorted turns detection in synchronous generators.Here,recently obtained results are extended and an improved fault detection scheme is studied.The residual expressions of the proposedobserverdesignstandoutnotonlybytheabsence of undesired scaling,but exhibit further advantages in the form of minimal parameter dependence due to a simplification applicable to most generators in use today.

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Appendix

Appendix A Proof of Lemma 1

First,a ROO for the transformed system is constructed.After artificially expanding the system order to n,it is possible to combineˆr(t)and y(t).Finally,the system is transformed into original coordinates.

Given that system(1)is transformed to canonical coordinates using(7),the system equations are given by

Here,a0,...,an−1are the coefficients of the characteristic polynomial.These equations can be partitioned according to the scheme in(5).With the resulting sub-matrices and a gain vector f∈Rn−1,the ROO estimatingˆz(t)reads:

In order to facilitate the retransformationˆx(t)=Tˆz(t),the observer state in canonical coordinates η(t)will be added by a zero component to increase its order to n:

Next,it is assumed that the value ρ＊(0)is chosen in the form of(a3).The ROO dynamic equation(a2a)is expanded to the order of n maintaining the last row of every matrix and vector to equal zero.Applying some matrix manipulations yields

with

Note that there is a degree of freedom given by the choice of ˜β as the last column of N＊Ris only related to the nth element of ρ＊(t)which equals zero.

The resulting observer(a4)estimates the state vector in observable canonical form.To obtain the desired original state space vector the system has to undergo the transformation ˆx(t)=Tˆz(t).While ρ＊(t)denotes the vector in canonical form,ρ(t)represents the original states:

Next,the initial value condition(a3)and the actual observer(a4)are retransformed.

A.1 Mathematical relationships

First,helpful mathematical relationships are introduced.Premultiplying the definition of s1in(4)with QBgives

Furthermore,the theorem of Cayley-Hamilton states that every A fulfils the characteristic equation:

Combining(a7)with(a8)yields

Moreover,(a8)directly shows that:

It is readily verified with(a7)and(a9)that the product QBT has the form:

Then,relevant entries of the inverse of D are obtained as

A.2 Transformation of

First,(a6b)is considered.When multiplied with the transformation matrix T,the second summand becomes

A.3 Transformation of

Transformation of(a5a)is performed employing(a9),(a11)and(a12):

As˜β can be chosen arbitrarily,it is set to eliminate the second term and simplify(a13)to

A.4 Transformation of

Second,expression(a5b)is manipulated using(a12):

A.5 Transformation of

The third summand in(a6a)is calculated from expression(a5c)by considering(a7),(a10)and(a12):

A.6 Transformation of initial value condition

Equation(a3)sets a constraint on the choice of the initial value ρ＊(0).Because the ROO allows for an arbitrary initial value of the(n − 1)-dimensional η(t),the effective restriction can be expressed as

This yields an equivalent constraint on ρ(0):

Setting ρ(0)=0 or ρ(0)= αlRwith α ∈ R trivially fulfils the requirement.

The underlying reason is that the transformation T−1ρ(t)=ρ＊(t)cannot be fulfilled with regular T−1and an arbitrary ρ(t) ∈ Rn.Choosing ρ(0)in accordance with(a18)and observing that the last row of N＊Requals zero,it holds true for∀t that ρ(t)lies in the measurable subspace of Rnwhere this restriction does not apply.

Appendix B Proof of Lemma 4

The observability matrix of the system in(32)has the following form and its regularity is required for the system to be fully observable:

Since the original system is assumed to be observable,QBis non-singular.Therefore,the regularity of A determines the observability of the extended system.However,if A posses an eigenvalue at zero and is thus singular,the first coefficient of its characteristic polynomial is a0=0.In this case,the theorem of Caley Hamilton(a8)gives that the last row of QBis linearly dependent on rows 2 until n.Therefore,the system is definitely not observable if A is singular.

Appendix C Proof of Theorem 2

To obtain an observer of order n the explicit form of the ROO(9)is used.In the resulting dynamic equation the first n stateestimates,namelyx(t),canbeseparatedfromthe(n+1)th component w(t).

In order to derive the observer,the(n+1)-dimensional observer gain is separated into two components lR=[lGl2]Twith lG∈Rn,l2∈R.The ROO(9)applied to system(32)then reads:

Unlike the ROO,there is no condition on the initial value ρx(0)because for an arbitrary ρx(0)there exists a ρw(0)so that the state in(a19a)lies in the n-dimensional measurable subspace.

7 March 2013;revised 10 September 2014;accepted 11 October 2014

†Corresponding author.

E-mail:jan.stellet@student.kit.edu.

©2014 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Jan Erik STELLET received the B.Sc.and M.Sc.degrees in Electrical Engineering and Information Technology from the Karlsruhe Institute of Technology in 2010 and 2012,respectively.He is currently a Ph.D.student at the Karlsruhe Institute of Technology.E-mail:jan.stellet@student.kit.edu.

Tobias ROGG received the B.Sc.and M.Sc.degrees in Electrical Engineering and Information Technology from the Karlsruhe Institute of Technology in 2011 and 2014,respectively.He is currently a Ph.D.student at the Swiss Federal Institute of Technology in Zurich.E-mail:roggt@hpe.ee.ethz.ch.