Fault Detection Observer Design with Unknown Inputs

Feng Tao1 and Qing Zhao1∗

Abstract— This paper deals with Fault Detection observerdesign for linear time-invariant systems with unknown inputs.The design objective is set to minimize a combined performanceindex H∞/H−, which means to achieve a good compromisebetween robustness to the disturbance and sensitivity to thefault. Hence the fault detection observer is designed to detectthe fault of smallest energy possible. In this paper, the designfor both static observer and more general dynamic observeris addressed. For the latter, change of variable method is usedto make matrix inequality linear. In addition, a sequential lin-earization iterative LMI method is used to solve the nonconvexoptimization, which guarantees the convergence.

I. INTRODUCTION

For those safety-critical and advanced automated pro-cesses, component faults, may cause economic, environmen-tal, and social damages. For this reason, fault detectionhas attracted extensive interests and attention from bothindustry and academia during the past three decades. Manymonographs are available, here just name a few, see [1], [2]for different types of fault detection scheme design includingeigenstructure assignment, Unknown Input Observer designand some methods base on statistical signal processing.

Different types of optimization techniques have been ap-plied to deal with the effects of uncertainties and disturbance,e.g. H∞ optimization technique. However, this optimizationapproach can not always achieve good performance. Thereason lies in the fact that for most circumstances, therobustness and sensitivity should be balanced. Otherwise, therobustness of the system may be achieved at the cost of lowerfault sensitivity.

For this reason, the H∞/H− objective, where H− denotesthe minimal singular value of the system, has been used infault detection observer design [3]. Later, such an objectivehas been found to have a clear physical meaning. For thesystem

r(s) = Grd(s)d(s)+Gr f (s) f (s)

If we choose the design objective as low false alarm, anddetermine the threshold, then the value

J = 2inf‖Grd(s)‖∞

‖Gr f ‖− (1)

gives the smallest fault signals that is guaranteed to bedetected [4][5]. Due to this significant property, it motivesus to minimize such a performance index, which results inthe H∞/H− design problem.

1 Feng Tao and Qing Zhao are with Department of Electricaland Computer Engineering, University of Alberta, Canada, T6G 2V4,fengtao,qingzhao@ece.ualberta.ca

∗ Corresponding author: Tel: (780)492-5792; Fax: (780)492-1811

However, obstacles still lie in this design. Since H− isnot a system norm, it prevents us from directly using robustcontrol related theory to carry out the design. For this reason,many efforts are not very successful. Some trial and error isneeded as for [3], and a heuristic iterative LMI algorithmis proposed in [6] without guarantee of convergence. As aresult, a more conservative H∞/H∞ design is used.

In order to merge this gap, this paper treats the H∞/H−design problem via convex optimization, where both staticand dynamic observer design are discussed. And an iterativeLMI algorithm (SLMPP) is used to solve the nonconvexoptimization, which can at least guarantee the local conver-gence. The contribution of this paper is following: First, forstatic FD observer, we use the concept of “inverse system”to convert the H− performance into a H∞ problem, hence iseasier to deal with and no conservatism is introduced. Whenthe common Lyapunov function type of approach is used, thedesign problem can fully be cast into a convex optimizationproblem to find the global optimum; secondly, an iterativeLMI algorithm is presented for the synthesis of static FDobserver, where the common Lyapunov function constraintis removed. Last, such an iterative algorithm, combined with“change of variable” method, is extended to the dynamic FDobserver design, which is a relatively new approach in thecurrent literature.

This paper is organized as follows: in section II, theproblem is formulated with some useful preliminary resultsgiven. The main design is proposed in section III and IV,where both static and dynamic observers are treated. Theiterative LMI algorithm is proposed in section V to solvethe nonconvex optimization problem shows up in section IIIand IV. An example is given in section VI to demonstratethe improved performance of the proposed design.

II. PROBLEM FORMULATION AND PRELIMINARY

RESULTS

Consider the following linear time-invariant system

x(t) = Ax(t)+Bd(t)+F f (t)y(t) = Cx(t)+Ddd(t)+D f f (t) (2)

where x(t) ∈ Rn is the state vector, y(t) is the measurementvector, f (t) is the fault input vector, and d(t) is an unknowninput vector, which contains modelling error, uncertain dis-turbance, process and measurement noise. Both d(t) andf (t) are L2 norm bounded. All the matrices represented bycapital letters are assumed as known constant matrices withappropriate dimensions. Furthermore, we assume that A isstable and (C, A) is observable.

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WA4.1

0-7803-9354-6/05/$20.00 ©2005 IEEE 1275

Two types of Fault Detection Observers are consideredhere, namely, static observer and the more general dynamicobserver. The static observer is given by

Os :

˙x(t) = (A−KC)x(t)+Ky(t)y(t) = Cx(t) (3)

and the more general dynamic observer is given by

Od :

˙x(t) = Akx(t)+Bky(t)y(t) = Ckx(t) (4)

For both cases, we define the state estimation error andthe residual as e(t) = x(t) − x(t) and r(t) = y(t) − y(t),respectively. Therefore, the closed-loop matrices for the twocases can be expressed respectively as

• Static Observer:

Gs =[

A−KC B−KDd F −KD f

C Dd D f

](5)

• Dynamic Observer:

Gd =

⎡⎣

A 0 B FBkC Ak BkDd BkD f

C −Ck Dd D f

⎤⎦ (6)

Actually, the case with static Observer is a special formof the dynamic observer case, e.g.

⎧⎨⎩

Ak = A−KCBk = KCk = C

(7)

Implementing a dynamic FDI observer can relax some un-necessary constraints to achieve better performance.

Now we are ready to put forward our design problem.For effective fault detection, the effect of sensitivity todisturbance in the residual signal should be small while thatdue to fault signal should be large. For this purpose, wepropose our design objective as follows:

H∞/H− Fault Detection Observer Design Problem: Forsystem (2) with FD observer (3) or (4), determine theobserver parameters satisfying the following conditions:

1) the closed-loop system (5) or (6) is asymptoticallystable

2) robustness to disturbance ‖Grd(s)‖∞ < γ1

3) sensitivity to faults ‖Gr f (s)‖− > γ2

4) the robustness/sensitivity ratio (noise-signal ratio)γ1/γ2 is minimized.

The main obstacle when using H− as design objective foroptimization, due to the fact that it is not a system norm.However, we can use the concept of “inverse system” totackle this problem by assuming that the “inverse system”exists.

Next some lemmas are introduced, which are used exten-sively in the derivation of the main results.

Lemma 2.1: (bounded real lemma)[7]: For the systemG(s) with a state-space realization A, B, C, D, thefollowing statements are equivalent:

1) ‖G(s)‖∞ < γ1

2) the LMI⎡⎣

PA+AT P PB CT

BT P −γ1I DC D −γ1I

⎤⎦ < 0 (8)

is feasible with respect to some P > 0.

Lemma 2.2: [7] Suppose LTI system G has a state spacerepresentation as

G =[

A BC D

],

a real rational matrix G−1(s) is called an inverse of a transfermatrix G(s) if G(s)G−1(s) = G−1(s)G(s) = I. Then

G−1 =[

A−BD†C −BD†

D†C D†

]

where D† is the pseudo inverse of D.

Lemma 2.3: For a given system in transfer function asG(s), if its inverse system exists denoted by G−1(s), thenwe have ‖G(s)‖− > γ2 if and only if ‖G−1(s)‖∞ < 1

γ2Proof: The proof is straightforward, using the definition ofthe inverse system and the definition of H− directly.

Lemma 2.4: ([6]) For a system G(s) with a state-spacerealization (A f , B f , Cf , D f ), ‖G(s)‖− > γ2 holds if apositive matrix P > 0 can be found and satisfies the followinglinear matrix inequality:

⎡⎣

PA f +ATf P−CT

f Cf PB f −CTf D f 0

BTf P−DT

f Cf −DTf D f γ2I

0 γ2I −I

⎤⎦ < 0. (9)

Proof: When ‖G(s)‖− > γ2, for system input/output pair( f (t), r(t)), we have

‖r‖2 > γ2‖ f‖2

Define a Lyapunov function as V (e) = eT Pe and assumezero initial conditions, then

γ22

∫ T0 f T f dt − ∫ T

0 rT rdt=

∫ T0 γ2

2 f T f − rT r + dVdt (e)dt −V (e(T ))+V (e(0))

≤ ∫ T0

[rf

]T [PA f +AT

f P−CTf Cf PB f −CT

f D f

BTf P−DT

f Cf γ22 I −DT

f D f

][rf

]dt

< 0(10)

It follows when t → ∞[

PA f +ATf P−CT

f Cf PB f −CTf D f

BTf P−DT

f Cf γ22 I −DT

f D f

]< 0. (11)

Using Shcur complement to expand (2,2) term, one canobtain the result.

III. DESIGN OF STATIC FAULT DETECTION OBSERVER

A. Design Using Inverse Systems Concept

For design objective ‖Grd‖∞ < γ1, bounded real lemmacan be used to obtain the equivalent condition as

1276

⎡⎣

(A−KC)T P1 +P1(A−KC) P1(B−KDd) CT

(B−KDd)T P1 −γ1I DTd

C Dd −γ1I

⎤⎦ < 0.

(12)As to ‖Gr f (s)‖− > γ2, since the minimal singular value is

not a norm, we use lemma 2.2 to convert the above conditionto an H∞ control problem.

Notice that

G−1r f =

[(A−FD†

f C)+K(D f D†f − I)C KD f D†

f −FD†f

D†f C D†

f

]

Define A = A−FD†fC and D = D f D†

f − I, using boundedreal lemma, we can obtain

⎡⎢⎣

AT P2 +P2A+CT DT KT P2 +P2KDC P2K(I + D)−P2FD†f

(I + D)T KT P2 −D†Tf FT P2 −γ−1

2 I

D†f C D†

f

CT D†Tf

D†Tf

−γ−12 I

⎤⎥⎦ < 0.

(13)Theorem 3.1: For some P > 0, the H∞/H− problem has

the following solution:

min γ

s.t.

⎡⎣

AT P+PA− PC−CT PT PB− PDd CT γ2

BT P−DTd PT −γI DT

d γ2

γ2C γ2Dd −γI

⎤⎦ < 0

⎡⎣

AT P+PA+CT DT PT + PDC P(I + D)−PFD†f CT D†T

f

(I + D)T PT −D†Tf FT P −γ2I D†T

f

D†f C D†

f −γ2I

⎤⎦ < 0

(14)where γ2 = γ−1

2 , γ = γ1γ−12 and FD observer gain K is given

by K = P−1P.

Proof: Pre- and post-multiply (12) by diagγ−12

2 ,γ−12

2 ,γ−12

2 ,and then set γ−1

2 P1 = P2 = P, define γ2 = γ−12 , and P = PK,

substitute these relations into (12), (13) then we can get theexpression above.

Remark 1: In Proposition 4 of [6], the same design prob-lem was handled by lemma 2.4, which results in a nonconvexoptimization problem. A heuristic iteration LMI algorithmis proposed. However, proof of the convergence of thealgorithm is missing in [6]. In Theorem 3.1, by using the“inverse system”, we simplify this problem. Furthermore,lemma 2.4 only provides a sufficient condition, whereaswhen using “inverse system”, the exact equivalent conditionscan be obtained.

B. Static Observer Design: The General Case

For the design result shows in section III.A, similarly asin [6], a common Lyapunov function approach is used (theunnecessarily constraints P1 = P2 = P are used in [6], andγ−1

2 P1 = P2 = P are used in Theorem 3.1). Such an approach,generally speaking, will result in a more conservative design.

In this section, these constraints will be removed for betterperformance.

In the derivation of this section, the following lemma isuseful.

Lemma 3.1: (Reciprocal Projection Lemma, [8]): Let P beany given positive-definite matrix, the following statementsare equivalent:

1) Ψ+S +ST < 02) the LMI

[Ψ+P− (W +W T ) ST +W T

S +W −P

]< 0 (15)

is feasible with respect to W .For inequality (12), use reciprocal projection lemma to

obtain⎡⎣

P1 −W1 −W T1 P1(A−KC)+W1 P1(B−KDd) CT

(A−KC)T P1 +W T1 −P1 0 0

(B−KDd)T P1 0 −γ1I DTd

C 0 Dd −γ1I

⎤⎦ < 0

(16)where P1 > 0 and W1 are variables introduced when usingthe lemma.

According to Reciprocal Projection Lemma, P1 can beany positive matrix. Without loss of generality, set P1 = λ1I,where λ1 is a positive scalar.

For the matrix inequality above, define X1 = P−11 and W1 =

X1W1, apply congruence transform with diagX1, I, I, I,we get

⎡⎣

λ1X1X1 −W1X1 −X1W T1 A−KC +W1 B−KDd X1CT

(A−KC)T +W T1 −λ1I 0 0

B−KDd 0 −γ1I DTd

CX1 0 Dd −γ1I

⎤⎦ < 0.

(17)After some manipulations, one can write it as

⎡⎢⎢⎢⎢⎣

−λ−11 W1W T

1 A−KC +W1 B−KDd(A−KC)T +W T

1 −λ1I 0(B−KDd)T 0 −γ1I

CX1 0 Ddλ1X1 −W T

1 0 0X1CT λ1X1 −W1

0 0DT

d 0−γ1I 0

0 −λ1I

⎤⎥⎥⎥⎦ < 0

(18)

Similarly, for the design objective ‖Gr f ‖− > γ2, we get⎡⎢⎢⎢⎢⎢⎣

−λ−12 W2W T

2 A+KDC +W2 KD f D†f −FD†

f(A+KDC)T +W T

2 −λ2I 0(KD f D†

f −FD†f )

T 0 −γ−12 I

D†f C

T X2 0 D†f

λ2X2 −W T2 0 0

X2CT (D†f )

T λ2X2 −W2

0 0(D†

f )T 0

−γ−12 I 00 −λ2I

⎤⎥⎥⎥⎥⎦

< 0

(19)

1277

An alternative approach is using lemma 2.4 by the follow-ing⎡⎢⎢⎢⎢⎣

−λ−12 W2W T

2 A−KC +W2 F −KD f −X2CT D f(A−KC)T +W T

1 −λ2I −CTC 0FT −DT

f KT −DTf CX2 0 −DT

f D f

0 0 γ2Iλ2X2 −W T

2 0 00 λ2X2 −W20 0γ2 0−I 00 −λ2I

⎤⎥⎥⎥⎦ < 0

(20)Notice that in Eqs.(18)-(20), when λ1 and λ2 are given, the

nonlinear terms WiW Ti , i = 1,2, and J = γ1γ−1

2 still exist inthe constraints and the objective function, respectively. Henceit is a nonconvex optimization and is difficult to solve.

However, if we define Qi = WiW Ti , i = 1,2, then both

constraints are linear with respect to the newly definedvariables. At the cost, we have two additional equalityconstraints. As for the objective function, some techniquesare available to convert it into a bilinear objective function.After the conversion, the problem is still nonconvex. In thiscase, an iterative LMI algorithm can be used to solve sucha problem, where the solution is bounded in a convex set,and the objective value is strictly non-increasing, thus theconvergence is guaranteed. The detailed iterative algorithmis described in section V, which can treat both the staticobserver design and the more general dynamic observerdesign in the same manner.

IV. DYNAMIC OBSERVER DESIGN

In this section, the synthesis of a more general dynamicFault Detection observer is considered.

We denote that

Gd =[

A Bd B f

C Dd D f

]

=

⎡⎣

A 0 B FBkC Ak BkDd BkD f

C −Ck Dd D f

⎤⎦

(21)

For the system Gd , using bounded real lemma for H∞ normconstraint and lemma 2.4 for H− constraint, then we get twomatrix inequalities respectively as

⎡⎣

PA+ AT P PBd CT

BTd P −γ1I DT

dC Dd −γ1

⎤⎦ < 0 (22)

⎡⎣

PA+ AT P−CTC PB f −CT D f 0BT P−DT

f C −DTf D f γ2I

0 γ2I −I

⎤⎦ < 0 (23)

Define

P =[

Y NNT U

],P−1 =

[X M

MT V

]

Π1 =[

X IMT 0

],Π2 =

[I Y0 NT

] (24)

For Eq.(22) and Eq.(23), apply congruence transform withdiagΠT

1 , I, I

ΠT1 PAΠ1 =

[AX A

YAX +NBkCX +NAkMT YA+NBkC

]

ΠT1 PBd =

[B

Y B+NBkDd

], ΠT

1 PCT =[

XCT −MCTk

CT

]

ΠT1 PCTCΠ1 =

[(XCT −MCT

k )(CX −CkMT ) (XCT −MCTk )C

CT (CX −CkMT ) CTC

]

ΠT1 PB f =

[F

Y F +NBkD f

], ΠT

1 PCT =[

XCT −MCTk

CT

]D f

Observe that many of the terms above contain nonlinearexpressions in terms of decision variables, which posesdifficulty in using convex optimization to solve the problem.Here we adopt the “change of variable” method [9] tolinearize those original nonlinear terms with respect to thenew variables.Define: ⎧⎨

⎩Ak = YAX +NBkCX +NAkMT

Bk = NBk

Ck = CX −CkMT(25)

After congruence transform in term of newly defined vari-ables, Eq.(22) and Eq.(23) have the following expressions,respectively, as

⎡⎣

AX +XAT A+ ATk B CT

kAT + Ak YA+ATY + BkC +CT BT

k Y B+ BkDd CT

BT BTY +DTd BT

k −γ1I DTd

Ck C Dd −γ1I

⎤⎦ < 0

(26)

⎡⎢⎢⎣

AX +XAT −CTk Ck A+ AT

k −CTk C

AT + Ak −CTCk YA+ATY + BkC +CT BTk −CTC

FT −D f Ck FTY +DTf BT

k −DTf C

0 0F −CT

k D f 0Y F + BkD f −CT D f 0

−DTf D f γ2I

γ2I −I

⎤⎥⎥⎦ < 0

(27)

For above two constraints, similarly to static FD ob-server synthesis case, we can define Q = CT

k Ck, then allthe constrains are linear in terms of decision variablesX , Y, Ak, Bk, Ck, Q, with a nonlinear objective functiondefined as J = γ1/γ2. If we can find solution for thisproblem, then we can construct the corresponding dynamicFD observer.

First, according to our partition of P, P−1, it infers

MNT = I −XY (28)

and from P∗P−1 = I, it is equivalent to[

X II Y

]> 0 (29)

Since I −XY > 0, therefore if only we get X , Y , we canalways construct nonsingular matrices M, N via singularvalue decomposition. And we have:

1278

⎧⎨⎩

Ck = (CX −Ck)M−T

Bk = N−1Bk

Ak = N−1(Ak −YAX −NBkCX)M−T(30)

Remark 2: Till this stage, the FD observer synthesis, forboth static and dynamic observer cases, has already beenexpressed in form of nonconvex optimization, i.e., optimizinga nonlinear objective function subject to nonlinear matrixinequalities. In the next section, we adopt the iterative LMIalgorithm to solve this problem.

V. ITERATION LMIS FOR SOLVING NONCONVEX

OPTIMIZATION

Notice that in Eq.(18), Eq.(19), Eq.(20) and Eq.(27), non-linear term −WiW T

i , i = 1,2 or −CTk Ck exists, which hampers

our efforts to convert the problem into LMI formulation, aconvex problem with global convergence. In addition, thedesign objective min γ1

γ2is not linear in decision variables

either. Current algorithms can not deal with the constrainedminimization problem of this type. In this section, we willuse an iterative LMI algorithm to solve the optimization forboth static and dynamic FD observer synthesis problem.

For static FD observer synthesis problem, whose con-ditions are given in the Section III, we can define, Qi =WiW T

i , to make Eq.(18), Eq.(19), Eq.(20) linear in terms ofWi, Qi,Xi,K,γ1. Such a technique has been used in staticoutput feedback (SOF) design and the related reduced orderoutput feedback (ROF) design with constraint XY = I, whereX , Y are two positive definite matrices.

Many algorithms have been proposed for dealing withconstraint of type XY = I, for example, alternate projection,XY-centering, Min-Max problem and the cone complemen-tarity linearization algorithm (CCP). [10] compared severalalgorithm on convergence. [11] improved the CCP algorithmand proposed a sequential linear programming matrix method(SLPMM), which provided better convergence performance.

In the following, we want solve the optimization problem:

min γ1γ2

s.t. Eq.(18),Eq.(20)(31)

We choose the constraint Eq.(20) instead of Eq.(19) forthe latter objective function J is already in bilinear form.

Define

Ω1 :

[Qi Wi

W Ti I

]> 0

Ω2 :

[γ3 II γ2

]> 0

Ω3 : Eq.(18)∩Eq.(20)

(32)

we then formulate our optimization objective as

J =2

∑i=1

trace(Q0i +Qi −W 0

i W Ti −WiW

0Ti )+ γ0

1 γ3 + γ1γ03

(33)Obviously, if solution is in the set Ω1 ∩ Ω2 ∩ Ω3, the

objective function value is non-increasing, and J > 0, and

J∗ achieved its global optimum if and only if trace(Qi −WiW T

i ) = 0 (which is equivalent to Qi = WiW Ti ),and γ1γ3 is

the upper bound of ‖Grd‖∞/‖Gr f ‖− achieved.

SLPMM Algorithm:

(1) Initialization: Set k = 0, find (Q0i ,W

0i ,γ0

i ) ∈Ω1 ∩Ω2 ∩Ω3.

(2) Determine the unique solution of the following problem

(Qi,Wi, γi)= min∑2

i=1 trace(Qi −Wi(W ki )T −W k

i (Wi)T )+ γk1γ3 + γ1γk

3s.t.(Qi,Wi, γi, Xi, K) ∈ Ω1 ∩Ω2 ∩Ω3

(34)(3) If

trace(Qki + Qi −W k

i W Ti −Wi(W k

i )T )+ γk1 γ3 + γ1γk

3= 2trace(Qk

i −W ki (W k

i )T )+2γk1γk

3(35)

−→ STOP.(4) Solve the following linear search problem

α∗ = minα∈[0,1] f (α)f (α) = ∑2

i=1 trace((Qki +α(Qi −Qk

i ))− (W ki +α(Wi −W k

i ))(W k

i +α(Wi −W ki ))T )+(γk

1 +α(γ1 − γki ))(γk

3 +α(γ3 − γk3))(36)

(5) Set

Qk+1i = (1−α∗)Qk

i +α∗Qi,

W k+1i = (1−α∗)W k

i +α∗Wi,

γk+1i = (1−α∗)γk

i +α∗γi,

Go to Step (2).

In the above algorithm, each step only involves solvingLMIs. Step (1) is a feasibility problem and the step (2)is an “eigenvalue” problem (minimizing a linear functionwith LMI constraints). Hence for each iteration, the globaloptimum can be found via convex optimization, and this isvital for the convergence of the algorithm. The propertiesof convergence of the iterative algorithm can be similarlyproved using results from [10], [11] and hence is omittedhere.

As for dynamic FD observer synthesis problem, the setsand the optimization objective are changed to:

Ω1 :

[Q CT

kCk I

]> 0

Ω2 :

[γ3 II γ2

]> 0

Ω3 : Eq.(26)∩Eq.(27)∩Eq.(29)

(37)

J = trace(Q0 +Q−C0T

k Ck −CTk C0

k )+ γ01 γ3 + γ1γ0

3 (38)

The same SLPMM algorithm is used to find the solutions.

VI. NUMERICAL EXAMPLE

Here we consider a 4th-order system with following de-scription [6]:

1279

A =

⎡⎢⎢⎣−10 0 5 0

0 −5 0 2.50 0 −2.5 00 5 0 −3.75

⎤⎥⎥⎦ , B =

⎡⎢⎢⎣

0.8 0.04−2.4 0.081.6 0.080.8 0.08

⎤⎥⎥⎦

F =

⎡⎢⎢⎣

448−8

⎤⎥⎥⎦ , C =

[1 0 0 11 0 1 1

]

Dd =[

0.2 0.040.4 0.06

], D f =

[2−1

]

The results shown in [6] are: γ1 = 0.8, γ2 = 2.2 and γ1γ2

=0.3636.

In the sequel, three simulations are performed to demon-strate the effectiveness of the algorithms present in this paper.These results are based on the common Lyapunov approachfor static FD observer design, general static FD observerdesign and dynamic FD observer design. For all cases, theperformance indices are listed.

Result 1: Using the algorithm given in the Theorem 3.1, theperformance indices are : γ1 = 0.4529, γ2 = 1.8383, γ1

γ2=

0.2464 and the observer gain is:

K =

⎡⎢⎢⎣

2.4075 0.8392−0.5418 −5.59894.7843 1.6609−2.5476 3.1329

⎤⎥⎥⎦

Result 2: This set of results are based on the static observerdesign but relaxes the constraint on Lyapunov function. Theperformance indexes are γ1 = 0.5435, γ3 = 0.4472, J =0.2431 (where γ3 here is the upper bound of 1/γ2, and hencethe value of J is the upper bound of γ1

γ2, The same notations

are used in Result 3.)

K =

⎡⎢⎢⎣

1.5430 1.59080.5968 −4.35374.3257 1.2159−3.3739 2.1448

⎤⎥⎥⎦

The simulation result shows that when relaxing the com-mon Lyapunov constraints, and using the iterative LMIalgorithm for design, the better performance can be achieved.

Result 3: This set of result is obtained for dynamic FDIobserver synthesis: γ1 = 0.4529, γ3 = 0.5436, J = 0.2462.And the observer is given by:

Ak =

⎡⎢⎢⎣−9.1128 14.0656 13.2293 1.1666−0.5973 −9.5401 −13.5577 −1.5379−0.0528 −3.7876 −7.8062 −1.75430.0384 0.5138 1.2886 −0.2849

⎤⎥⎥⎦

Bk =

⎡⎢⎢⎣

0.0134 −0.0031−0.0024 −0.0030−0.0004 −0.00170.0000 0.0002

⎤⎥⎥⎦

Ck = 1.0e+004∗[−0.0080 −0.0038 −0.0941 1.0015−0.0047 −0.2356 0.0085 −0.7102

]

Remark 3: The simulation results show better perfor-mance than the design result of [6]. It is natural that theResult 2 is better than the result 1, however, it is also betterthan the result 3. It is probably due to two reasons: first, inthe derivation of dynamic FD observer, lemma 2.4 is onlya sufficient condition, and secondly, a common Lyapunovfunction is used, which can result in conservative design.

REFERENCES

[1] R.J. Patton, P.M Frank and R.N. Clark, Fault Diagnosis in DynamicSystems, Theory and Application, Prentice-Hall, Englewood Cliffs, NJ,1989

[2] J. Chen and R.J. Patton, Robust Model-based Fault Diagnosis forDynamic Systems, Kluwer Academic Publishers, 1999

[3] M. Hou and R.J. Patton, “An LMI approach to H−/H∞ fault detectionobservers”, UKACC Int. Conf. on Control,pp 305-310, 1996

[4] M. Rank and H. Niemann, “Norm based design of fault detectors”,Int. J. Control, 72(9): 773-783, 1999

[5] X. Ding and L. Guo, “Observer based optimal fault detector”, Pro-ceedings of 13th IFAC World Congress, San Francisco, CA, USA, pp187-192, 1996

[6] H.B. Wang, J. Lam, S. X. Ding and M.Y. Zhong, “Iterative LinearMatrix Inequality Algorithms for Fault Detection with UnknownInputs”, Journal of Systems and Control Engineering, 219(2): 161-172,2005

[7] K. Zhou, J. Doyle and K. Glover, Robust and optimal control, PrenticeHall, NJ, 1996

[8] P. Apkarian, , H. Tuan and J. Bernussou, “Continuous-time analysis,eigenstructure assignment, and H2 synthesis with enhanced linearmatrix inequalities (LMI) characterizations”, IEEE Trans. AC, 46(12):1941-1946, 2001

[9] C. Scherer, P. Gahinet and M. Chilali, “Multiobjecitve output-feedbackcontrol via LMI optimization”, IEEE Trans. AC, 42(7): 896-911, 1997

[10] L.E. Ghaoui F. Oustry and M.A. Rami, “A cone complementarity lin-earization algorithm for static output-feedback and related problems”,IEEE Trans. AC, 42(8): 1171-1176, 1997

[11] F. Leibfritz, “An LMI-based algorithm for designing suboptimal staticH2/H∞ output feedback controllers”, SIAM J. Control OPtim., 39(6):1711-1735, 2001

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