On the Design of Unknown Input Observers and Fault Detection Filters*

Kemin Zhou Zhang Ren and Wei Wang Dept. of Electrical & Computer Engineering College of Automation

Louisiana State University Beijing University of Aero. & Astro. Baton Rouge, LA 70820, USA Beijing,, China

kemin@ece.lsu.edu renzhang@buaa.edu.cn,wwinney@126.com

* This work was supported in part by NASA (NCC5-573), LEQSF (NASA/LEQSF(2001-2004)-01), the NNSFC Young Investigator Award for OverseasCollaborative Research (60328304) and a NNSFC grant (10377004).

Abstract - This paper examines the design of unknown inputobservers from a frequency domain point view and shows that a UIO design can be formulated as a perfect

2H or H disturbance

rejection problem that many tools in robust control field can beapplied. In particular, it is shown that a more general non-perfect disturbance rejection or approximate UIO problem may be considered and potentially interesting open problems are outlined. The paper also establishes the connection betweenresidual error based on the UIO fault detection filter andresidual error based on a standard observer. It is shown that aUIO based filter may be useless for fault detection. Hence it is critical to find the suitable UIO filters from the family of all possible UIO filters to fit the specific fault detection needs.

Index Terms - Unknown Input Observer, Fault Detection, State Estimation.

I. INTRODUCTION

State estimation under the influence of external disturbancesor model uncertainties is critical in many practicalapplications. Ideally, it is desirable to design state estimators(observers) so that the estimation errors are least sensitive toor completely decoupled with disturbances and modeluncertainties. A well-known technique is called unknown input observer (UIO) design. See for example, [1,2,8] and references therein. The necessary and sufficient conditions for the existence of a UIO are given in [2] and furthermore a full order UIO can be constructed using the procedure given inthat paper. When a UIO exists, it is in general not unique. Inthe case of full order UIOs, the freedom lies in the choice of a gain matrix. There are also techniques of designing reduced order UIOs. However, we will illustrate later that it is sometimes important to find all orders of UIOs, i.e. aparameterization of UIOs. In additional to being useful asdisturbance decoupling state estimators, UIOs are also widely used in fault diagnosis to detect actuators, sensors, or components faults in a dynamical system under the influenceof external disturbances or modeling errors [1,2,3,4,5].Unfortunately, it has not been emphasized enough that a UIOgiving good estimation may not be actually useful for faultdetection and vice verse. It is critical to find a UIO that will fit the particular needs. In this paper, we shall try to demonstrateall those problems through some simple examples. We show that we can formulate and potentially solve a much moregeneral class of approximate UIO design problems using somestandard robust control techniques [6,7,9]. We also establish a

connection between the UIO approach to fault detection and the standard observer approach to fault detection.

The paper is organized as follows. In Section 2, we shall lookat the UIO design from a frequency domain point of view and show that we can formulate a general UIO problem as somestandard optimization problem that can potentially be solvedusing existing robust control techniques and point out somepotential obstacles and further research challenges. In Section 3, we shall establish a connection between the residual error based on the UIO fault detection filter and the residual errorbased on a standard observer. We then demonstrate through a simple example that a UIO based filter may be useless forfault detection. On the other hand, a good UIO based filter for fault detection may not be a good state estimator (or observer)with system faults. Hence it is critical to find the suitable UIO filters from the family of all possible UIO filters to fit thespecific fault detection needs.

II. A FREQUENCY DOMAIN APPROACH TO UIO

In this section, we first review some existing results in theliterature and then we give a frequency domain interpretation.As is well known, we can assume without loss of generalitythat a dynamic system with an additive unknown disturbance (or inputs) under consideration is given by

( ) ( ) ( ) ( )

( ) ( )

x t Ax t Bu t Ed t

y t Cx t(1)

where ( ) nx t R is the state vector, is the output,is the known input vector and is the

unknown disturbance (or input) vector. It is also assumedwithout loss of generality that has full column rank.

( ) my t R( ) ru t R ( ) qd t R

E

An observer for the system (1) is called an unknown inputobserver (UIO) if its state estimation error approaches zeroasymptotically regardless of the presence of the unknown input (disturbance) In other words, the state estimationerror is completely decoupled from the disturbance. Necessaryand sufficient conditions for the existence of a UIO have been obtained in [2] and are stated in the following lemma.

.d

Lemma 1. There exists a UIO for system described in (1) ifand only if the following conditions hold

1. ( ) ( )rank CE rank E

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Proceedings of the 6th World Congress on Intelligent Controland Automation, June 21 - 23, 2006, Dalian, China

2. is detectable where1( , )C A 1A A HCA and1[( ) ' ] ( ) 'H E CE CE CE or equivalently the

dynamic system0

A E

C is strictly minimum phase.

Furthermore, a full order UIO can be constructed as follows

1( ) ( )ˆ( )

0 ( )

F I HC B K FH u tx t

I H y t (2)

where F is given by 1 1 1F A HCA K C A K C

and 1K is such that F is stable.

Note that the only freedom we have is the choice of 1K . Ofcourse, it is also possible to design reduced order UIOs.

To compare this state space solution with other solutions to bepresented later, we shall consider a simple example below.

Example 1: Consider an example from Yang and Wilde [8]1 1 0 1

1 0 01 0 0 , , 0

0 0 10 1 1 0

A C E

Then .1

0 0 0

1 0 0

0 1 1

A

It is clear that rank(CE)=rank (E) 1 and 1( , is observable. Hence the existence of a UIO is guaranteed.

Let . Then

)C A

1

1 0

1 1

0 0

K

1 1 1

1 0 0 0 0

0 0 1 , 1 1

0 1 1 0 0

F A K C K FH

We shall now look at UIO from a transfer function point of view. For technical reasons, we shall assume that the systemunder consideration is stable, i.e., A is stable. Note that

( )( )

0 0 ( )

A B E u tx t

I d tAn observer is a stable dynamic system

1 2

( )ˆ( )

( )

u tx t G G

y tsuch that approaches zero asymptotically.Since

ˆ( ) ( ) ( )e t x t x t

1 2( ) ( )0 0

A B A Be t G G u t

I C

2 ( )0 0

A E A EG d

I Ct

We need

20 0

A E AG

I C

E (3)

and

1 20 0

A B AG G

I C

B (4)

From (3), it is clear that since0

A E

Ihas no transmission

zero, any zero of 0

A E

Cmust be cancelled by a pole of

Since is a stable transfer matrix,2 .G 2G0

A E

C must be

strictly minimum phase. Moreover, expanding (3) in terms of 1/s and compare the 1/s term, we have

(5) 1

Hence it is clear that the conditions in Lemma 1 are necessary for the existence of any UIO. Of course, those conditions are also sufficient since Lemma 1 shows that there is a full order UIO if all those conditions are satisfied. In fact, one set of

solutions of and is given by

( ) .E G CE

1G 2G 1

( )

0

F I HC BG

I

1

2

F K FHG

I HFurthermore, the solutions are generally not unique.

In general, we would like to find a so that2G

20 0

A E A EG

I Cis minimized in some sense even when the conditions inLemma 1 are not satisfied.

Example 2: Let1 1 1 1

, , ,0 1 0 1

A B C E

Then

2 2

21 ( ),

0 01( 1) ( 1)

A E A Es sI Css s

(2 )

and the conditions in Lemma 1 are satisfied if (2 ) /( ) 0 and in this case 2 has a unique solution

G

2

21.

1( ) (2 )

sG

ss

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Furthermore,

1 .( ) (2

Gs )

Now suppose 2, and 3 . Then the conditions in Lemma 1 are not satisfied since

2

10 ( 1)

A E sC s

is not minimum phase. Nevertheless, it can be shown that for either 2 norm or norm, we have

2 22 2

22 2

2

21 10 0 1( 1) ( 1)

21 1 1

11 ( 1) ( 1)

3 / 2 1/ 21 1.

11 1s sIt can be shown that 0

A E A E s sG G

I C ss s

ss sG

ss s s

G

2

22

2

3 / 21 2min

0 0 11 4G RH

A E A EG

I C s

6

and the optimal 2 norm solution is .2

1 / 2

0G

On the other hand, the optimal norm solution is

2

2 11 4sHowever, the optimal norm is not achievable but can be approximated to arbitrary accuracy by a stable

2

2ˆ

min0 0

3 / 21 13ˆmin

G RH

G RH

A E A EG

I C

G

2

1 / 2 ( 1) ˆ0

k sG G

s k 2

as .k

Example 3: Consider the same system as in Example 1 andnote that

2

2

( 1)1

1 ,0 ( 1)( 1)

1

( 1)10 1( 1)( 1)

s sA E

sI s s s

A E s sC s s s

and the conditions in Lemma 1 are satisfied. In this casehas many solutions and one of the solutions is

2G

2

1 0

11 .

10 1

Gs

This is interesting since it says that we only need to estimatethe second state which is obvious because the first and thethird states are already available from measurement.

We can actually parameterize all solutions as

2

1 0

1 11 ( )

1 10 1

G Q ss s

s

where is any 3( )Q s 1 strictly proper and stable transfer matrix.

It is easy to verify that

1

2

2 2

2

1 0 0 01

0 0 ( 1) ( 1)( 1)( 1)

0 0 1 1

F K FHG

I H

s ss s s

s sas given in Example 1 if

2

01

( )1

1

Q s ss s

s

.

Some further thoughts reveal that for practical purpose it is not necessary to make state estimation error

20 0

A E AG

I C

E

perfectly zero or small for all frequencies since thedisturbance is usually limited to certain frequency range,particularly low frequency range. Hence it is more appropriateto consider the following 2 norm or norm minimizationproblem

( )d t

2

2min0 0G RH

A E A EG

I CW (6)

where is a stable transfer function that reflects the frequency contents of the disturbance even when the conditions in Lemma 1 are not satisfied.

( )W s( )d t

Nevertheless, Problem (6) is by no means easy to solve. It isnoted that the standard state space 2 norm or normoptimization techniques cannot be applied directly since

0

A E

Cis strictly proper. Furthermore, this transfer matrix

may not be minimum phase. It is also possible that the optimalsolutions do not exist and we have to find some suboptimal

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solutions. In the norm optimization case, it is possible to use Nevanlinna-Pick interpolation method to compute the optimal performance level [6,7]. The parameterization of allsolutions can be complicated even when the conditions in Lemma 1 are satisfied. These problems are of interest for further research.

III. UIOS AS FAULT DETECTION FILTERS

It is well known that one of the important applications of UIO is in fault detection. Consider for example a dynamic systemwith possible vector fault ( )f t

1

2

( )( ) ( ) ( ) ( )

( ) ( ) ( )

x t Ax t t Ed t R f t

y t Cx t R t

Bu

f(7)

where 1 and 2 are known fault distribution matrices. Nowlet an observer be given by

R R

1 2

( )ˆ( )

( )

u tx t G G

y tand let . Suppose the conditions in Lemma 1are satisfied. Then using a UIO described by (2), we get

ˆ( ) ( ) ( )e t x t x t

1 1

22

1 1 1 2 2

2

( ) ( )0

( )

A R A Re t G f t

I C R

F R HCR K R FHRf t

I HRand a disturbance decoupled residual vector can be generated as

( )r t

(8) ˆ( ) ( ) ( )UIOr t y t Cx tIndeed, it is easy to verify that

1 1 1 2 2

2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )UIO

e t Fe t R HCR K R f t HR f t

r t Ce t R f tHence is completely decoupled from the disturbance( )UIOr t

( ).d tIt is also known that a residual vector can be generated using a standard observer as

OBr t

ˆ ˆ ˆ( ) ( ) ( ) ( ( ) ( ))

ˆ( ) ( ) ( )ob ob ob

OB ob

x t Ax t Bu t L y t Cx t

r t y t Cx t (9)

Let . Then we have ˆ( ) ( ) ( )ob obe t x t x t

1 2

2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )ob ob

ob ob

e t A LC e t Ed t R LR f t

r t Ce t R f tIt is clear that this residual is not decoupled from the disturbance ( ).d t

An immediate question is if there is any relationship between the standard observer based residual and the UIO based residual if the conditions for the existence of UIOs are satisfied. The following theorem answers this question.

OBr t( )UIOr t

Theorem 1. Assume that the conditions for the existence of UIOs are satisfied. Then

( ) ( ) ( )UIO OBr t S s r twhere is a stable filter given by( )S s

( )( )

F K I HC LS s

C I CH

and 1 .K K FH

We should point out that there are some potentialshortcomings with fault detection using a UIO technique whenthe measurement is noisy. For example, suppose the measurement in (7) is noisy with a measurement noise ( )v t

2

Then the residual equations are given by( ) ( ) ( ) ( )y t Cx t R f t v t

1 1 1 2 2 1

2

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )UIO

e t Fe t R HCR K R f t HR f t K v t Hv t

r t Ce t R f t v tIt is now clear that the forcing function ( )Hv t can be verylarge even though itself can be very small. Hence the residual response due to this noise term can be very significant and fault detection may be impossible. It is therefore in manycases a standard observer approach may be preferred.

( )v t

In the case when the conditions in Lemma 1 are satisfied and let and solving (3) and (4). Then we get1G 2G

1 1

22

( ) ( )0

A R A Re t G f t

I C Rand

1 1

2 22 2

( ) ( ) ( ) ( )UIO

A R A Rr t Ce t R f t CG f t

C R C RFor the purpose of fault detection, it is desirable toparameterize all solutions of 2 in terms of a free stable transfer matrix Q so that the freedom Q can be used to maximize the residual error due to the faulty signal

G

( ).f t Morespecifically, it is desirable to maximize the signal

UIOr in

some sense especially in the frequency range where the faultyis significant. A suitable mathematical problem can be

formulated as

( )t

( )f t

1 1

22 2

max inf ( ) ( ) ( )Q

A R A Rj CG j j

C R C Rwhere is the set of frequency range that correlates to the significance of the faulty signal frequency contents.

Example 4: Continue from Example 3 with the same A, C, Eand

1 2, 01 1 0 T

R R

Then

01

( ) 1 ( )1

0

e t Q f ts

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and 1

2

3

1( ) ( ) ( ) ( )

1UIO

qr t Ce t R f t f t

qswhere

is any strictly proper and stable transfer matrix or zero.

1 2 3( )T

Q s q q q 3 1

It is clear that even though gives a UIO but it is not useful for fault detection. This also points out that a reducedorder UIO may not be desirable for fault detection. It isdesirable to choose 1 3 to be fast dynamics so that thefaulty signal can be detected quickly. For example,

and

0Q

and q q( )f t

2 0q 1 3

kq q

s k for large 0 and 0.k Then

2

1( 1)( )

11

1

1( 1)( )

k ks

s s k s k

Gs

k ks

s s k s k

.

In contrast to the fault detection problem, the state estimationproblem under external disturbance and failure would like thestate estimation error insensitive to the fault and disturbance.Hence suppose the conditions in Lemma 1 are satisfied and let

and solving (3) and (4). Then we get1G 2G

1 1

22

( ) ( )0

A R A Re t G f t

I C RIn this case, it is desirable to use the freedom in theparameterization of all solutions of to minimize the estimation error due to the faulty signal This is incontrast with the fault detection problem where an error signalis to be maximized. Thus it is clear that a good stateestimation filter may not be a good fault detection filter.

2G( ).f t

Example 5: Continue from Examples 3 and 4 with the same1 2, , , , and .A C E R R To make the state estimation error

insensitive to the system fault, it is desirable to make theestimation error with respect to small. Hence it is

desirable to find a Q so that

( )f t

01

( ) 1 ( )1

0

e t Q f ts

is

small in some sense. The optimal Q would be

, but this is not possible since Q is strictly

proper. A suitable Q can be chosen as

0 1 0 TQ

0T

Qk

s k0

for a large k>0. Then

2

1 01 0

11 0 1

10 1

0 1

s ksG s

s s k s k

as . However, with this Q , we have k ( ) 0UIOr t which is useless for fault detection.

IV. CONCLUSIONS

We have shown that the design of a UIO can be approached from a transfer function point of view and it is in fact a perfectdisturbance rejection problem. A more general weighted disturbance rejection or approximate UIO problem can be formulated in the frequency domain and the rich techniques in interpolation theory and 2 andH H optimization techniquesmay be applied [6,7,9]. It is also possible and desirable to parameterization all optimal or suboptimal solutions to these problems and the available freedom can be used to either making the state estimation insensitive to faults or making the residual signal for fault detection more sensitive to faults.

The examples in the paper show clearly that the objective for fault detection is not necessarily consistent with the optimalstate estimation. A good state estimator may be a bad fault detection filter and vise versa. It is necessary to developmethods that could take the fault detection and disturbance rejection into consideration at the same time in the fault detection filter design. In view of the above conclusion, there are many interesting problems to be considered. For example,parameterization of all UIOs (i.e., perfect disturbance rejection observers) or all approximate UIOs (i.e., non-perfectly disturbance rejection observers) so that the freedomcan be used to maximize the residual signal due to the faults.

REFERENCES

[1] J. Chen, R. J. Patton, and H. Y. Zhang, “Design of unknown input observers and robust fault detection filters,” International Journal of Control, 1996, Vol. 63, No. 1, pp. 85-105.

[2] J. Chen and R. J. Patton, Robust Model-Based Fault Diagnosis ForDynamic Systems, Kluwer Academic Publishers, 1999. J. Chen, R. J.Patton, and H. Y. Zhang, “Design of unknown input observers and robustfault detection filters,” International Journal of Control, 1996, Vol. 63,No. 1, pp. 85-105.

[3] P. M. Frank and X. Ding, “Survey of robust residual generation and evaluation methods in observer-based fault detection systems,” Journal of Process Control, Vol. 7, No. 6, pp. 403-424, 1997.

[4] P. M. Frank and X. Ding, “Frequency domain approach to optimallyrobust residual generation and evaluation for model-based fault diagnosis,” Automatica, Vol. 30, No. 5, pp. 789-804, 1994.

[5] R. J. Patton, “Fault tolerant control: the 1997 situation,” Proceedings ofIFAC Safeprocess’97, Hull, UK, pp. 1033-1055, August 1997.

[6] D. Sarason, “Generalized interpolation in H ,” Trans. AMS, vol. 127,

pp. 179-203, 1967.[7] M. Vidyasagar, Control System Synthesis: A Factorization Approach,

MIT Press, Cambridge, Mass., 1985.[8] F. Yang and R. W. Wilde, “Design of observers for linear systems with

unknown inputs,” IEEE Transactions on Automatic Control, Vol. 33, pp.677-681, 1988.

[9] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control,Prentice Hall, 1996.

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