LMI-based Lipschitz Observer Design with Application in Fault Diagnosis

Transcription

1 Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006 LMI-based Lipschitz Observer Design with Application in Fault Diagnosis A. M. Pertew, H. J. Marquez and Q. Zhao s: pertew, marquez, Addresses: Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada, T6G 2V4 Author to whom correspondence should be addressed Abstract The problem of fault detection and diagnosis in the class of nonlinear Lipschitz systems is considered. An observerbased approach offering extra degrees of freedom over classical Lipschitz observers is introduced. This freedom is used for the sensor faults diagnosis problem with the objective to make the residual converge to the faults vector achieving detection and estimation at the same time. The use of appropriate weightings to solve this problem in a standard convex optimization framework is also demonstrated. A LMI design procedure solvable using commercially available software is presented. I. INTRODUCTION The fault diagnosis problem is gaining increasing consideration worldwide in both theory and application. This is due to the growing demand for higher reliability in control systems, and hence the importance of having a monitoring system to detect the existing faults and specify their location and significance. The observer-based approach, in which an observer plays the role of the monitoring system, is one of the most popular techniques used for fault diagnosis. Many standard observer-based techniques exist in the literature providing different solutions to the problem for the linear time-invariant (LTI) case (see [1], [2] for good surveys). The basic idea behind this approach is to estimate the outputs of the system from the measurements by using either Luenberger observers in a deterministic framework [3], [4] or Kalman filters in a stochastic framework [5], [6]. The weighted output estimation error is then used as the residual (monitoring signal) in this case. Unlike the LTI case, however, the nonlinear counterpart of this problem still lacks a universal approach and is currently an active area of research (see [7]-[13] for different attempts to extend some of the LTI results to the nonlinear case). The main obstacle in the solution of the observer-based nonlinear fault detection problem is the lack of a universal approach for nonlinear observer synthesis. In this paper, we focus on the class of Lipschitz systems of the form: ẋ(t) =Ax(t)+Γ(u, t)+φ(x, u, t) (1) y(t) =Cx(t), A R n n, C R p n (2) with (A, C) detectable and where Φ(x, u, t) satisfies a uniform Lipschitz condition globally in x as follows: Φ(x 1,u,t) Φ(x 2,u,t) α x 1 x 2 (3) for all u R m and t R and for all x 1 and x 2 R n. Here α R is referred to as the Lipschitz constant and is independent of x, u and t. Lipschitz systems constitute a very important class. Any nonlinear system ẋ = f(x, u) can be expressed in the form of (1) as long as f(x, u) is continuously differentiable with respect to x. Many nonlinearities satisfy (3) at least locally. Examples include trigonometric nonlinearities occurring in robotic systems, the nonlinearities which are square or cubic in nature, etc. The function Φ(x, u, t) can also be considered as a perturbation affecting the system as in [14]. Due to its importance, the observer design problem for Lipschitz systems has seen much attention in the literature (see [15]-[19] for different design techniques). All of these techniques apply the classical constant gain Luenberger-like structure. Besides, the fault diagnosis application was not considered in all of these works. In this paper, however, we consider a more general framework, making use of the dynamic observer structure introduced in [20], [21] where an observer gain is seen as a filter designed so that the error dynamics has some desirable frequency domain characteristics. We apply this dynamic structure for the sensor fault estimation problem where the objective of estimating the fault magnitude is considered (in addition to detection and isolation). We show that, unlike the classical constant gain Luenberger structure, this objective is achievable by minimizing the faults effect in a narrow frequency band on the observer s state estimation error. The sensor fault diagnosis problem is then modeled as a convex optimization problem and an LMI design procedure is presented. The use of weightings to consider different frequency patterns for the sensor faults is also illustrated. II. PRELIMINARIES AND NOTATION Much research has been done on the observer design problem for Lipschitz systems of the form (1)-(3) (see [15]- [19] for different design techniques). All of these works, however, use the same observer structure which falls in the class of Luenberger-like observers, where the only available freedom is the static observer gain L: ˆx = Aˆx +Γ(u, t)+φ(ˆx, u, t)+l(y ŷ), L R n p (4) ŷ = C ˆx (5) In [20] a different approach was adopted, by making use of dynamical observers of the form: ˆx(t) =Aˆx(t)+Γ(u, t)+φ(ˆx, u, t)+η(t) (6) ŷ(t) =C ˆx(t) (7) where η(t) is obtained by applying a dynamical compensator K of order k ( k being arbitrary) on the output estimation /06/$ IEEE. 514

2 error. In other words η(t) is given from ξ = A L ξ + B L (y ŷ), A L R k k, B L R k p (8) η = C L ξ + D L (y ŷ), C L R n k, D L R n p (9) We will also write [ ] AL B K = L C L D L (10) to represent the compensator in (8)-(9), and it was shown in [20] that K must solve a regular H problem in order to achieve asymptotic convergence. The observer gain K can be represented by a set of controllers in this case, bringing additional degrees of freedom in the design. In this paper, we make use of this freedom in the sensor fault diagnosis problem showing that the classical structure in (4)-(5) can not solve the problem in this case. In our formulation, we consider the system s output to be affected by sensor faults, i.e we assume equation (2) to be replaced by: y(t) =Cx(t)+f s (t) (11) where f s R p is a vector representing the sensor faults that affect the system. We consider different frequency patterns for f s, proposing a numerical approach based on LMIs to compute the dynamic observer gain that can diagnose the faults for these different cases. To this end, we borrow the following fault diagnosis definitions from [22]. These definitions are widely accepted by the Fault Detection and Diagnosis (FDD) community and are related to the different tasks of a residual generator (note that in the following definitions f(t) R l is a general fault vector, r(t) R q is the residual and the transient period of the residuals is not considered): Definition 1: Fault detection : The residual generator achieves fault detection (strong fault detection) if the following condition is satisfied: r i (t) =0; for i =1,,q ; t if (if and only if) f i (t) =0; for i =1,,l ; t Definition 2: Fault isolation : The residual generator achieves fault isolation if the residual has the same dimension as f(t) (i.e, q = l) and if the following condition is satisfied: (r i (t) = 0; t f i (t) =0; t) ; for i =1,,l Definition 3: Fault identification : The residual generator achieves fault identification if the residual has the same dimension as f(t) and if the following condition is satisfied: (r i (t) = f i (t) ; t) ; for i =1,,l According to these definitions, in fault detection a binary decision could be made either that a fault occurred or not, while in fault isolation the location of the fault is determined and in fault identification the size of the fault is estimated. In this paper, we consider the fault diagnosis problem for Lipschitz systems of the form (1)-(3) (with the faulty output in (11)), by using the residual generator as the observer (6)- (10) along with the residual: r(t) = y(t) ŷ(t) (12) We develop conditions on the observer gain K that guarantee fault detection and fault identification for different frequency patterns of the sensor faults. We tackle the case when the sensor faults f s are in a narrow frequency band by showing that the sensor fault identification problem is equivalent to an output zeroing problem which is solvable only with a dynamic observer. The problem in this case (as well as the low frequency and the high frequency cases) is modeled as a convex optimization problem solvable using LMI numerical techniques. The following definitions and notation will be used throughout the paper: Definition 4: (L 2 space) The space L 2 consists of all Lebesque measurable functions u : R + R q,havinga finite L 2 norm u L2, where u L2 = u(t) 0 2 dt, with u(t) as the Euclidean norm of the vector u(t). For a system H : L 2 L 2, we will represent by γ(h) Hu the L 2 gain of H defined by γ(h) = sup L2 u u L2. It is well known that, for a linear system H : L 2 L 2 (with a transfer matrix Ĥ(s)), γ(h) is equivalent to the H- infinity norm of Ĥ(s) defined as: γ(h) Ĥ(s) = sup σmax(ĥ(jω)), where σ ω R max represents the maximum singular value of Ĥ(jω). The matrices I n, 0 n and 0 nm represent the identity matrix of order n, the zero square matrix of order n and the zero n by m matrix respectively. Diag r (a) represents the diagonal square matrix of order r with [ a a a ] as its diagonal vector, while 1 r diag(a 1,a 2,,a r ) represents the diagonal square matrix of order r with [ ] a 1 a 2 a r as its diagonal vector. The symbol ˆT yu represents the transfer matrix from input u to output y. The symbol RH denotes the space of all proper and real rational [ stable] transfer matrices. The partitioned A B matrix H = (when used as an operator from u C D to y, i.e, y = Hu) represents the state space representation ( ξ = Aξ + Bu, y = Cξ + Du), and in that case the transfer matrix is Ĥ(s) =C(sI A) 1 B + D. We will also make use of the following property on the rank of Ĥ(s) [23]: [ ] ) A si B rank = n + rank (Ĥ(s) (13) C D if s is not an eigenvalue of A and where n is the dimension of the matrix A. The standard setup in Fig. 1 will also be used throughout the paper along with the state space representation for the plant G in (14). Ĝ(s) = A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 (14) We will also make use of the following result on the H norm of ˆT ζτ in this case: Theorem 1: [24] For the generalized plant in (14), assume stabilizability and detectability of (A, B 2,C 2 ) and that D 22 = 0, and let N 12 and N 21 denote orthonormal bases of the null spaces of ( ) B2 T,D12 T and (C 2,D 21 ) respectively. There exists a controller K such that ˆT ζτ <γif and only if there exist two symmetric 515

3 τ ν G K ζ ϕ φ ˆTe φ e 1 Fig. 1. Standard setup. e + e 2 ˆTefs f s + matrices R, S R[ n n satisfying ] the following [ system ] of N12 0 N21 0 LMIs (where X = and Y = ): 0 I 0 I AR + RA T RC T X T 1 B 1 C 1 R γi D 11 X < 0 (15) B1 T D11 T γi A T S + SA SB 1 C T Y T 1 B1 T S γi D11 T Y < 0 (16) C 1 D 11 γi [ ] R I 0 (17) I S III. FAULT DETECTION OBSERVER In this section, we focus on the sensor fault detection problem (stated in Definition 1) for the nonlinear Lipschitz system defined in (1)-(3) and subject to the faulty output in equation (11). General sensor faults are assumed in this section (i.e, no assumptions are made on the time-domain or frequency-domain properties of the faults). The dynamic observer in (6)-(10),(12) is used as the residual generator, and the objective is to develop conditions on the observer gain K that guarantee fault detection according to Definition 1 in this case. To this end, using the observer (6)-(10) as a residual generator along with the residual in (12), it can be seen that the residual dynamics are given by: ė(t) = Ae(t) η(t)+ φ(t) (18) r(t) = Ce(t)+f s (t) (19) where e = x ˆx is the observer estimation error, and φ = Φ(x, u, t) Φ(ˆx, u, t). Defining the variables: [ ] [ ] τ1 φ τ =, ν = η = K (y ŷ), τ 2 f s ζ = e = x ˆx, ϕ = y ŷ = Ce + f s (20) Then, the error dynamics in (18) can be represented by: ż = [ A ] z + [[ ] ] [ ] τ I n 0 np I n (21) ν [ [ ] [[ ] ] ζ In 0n 0 = z + [0pn np ] 0 n τ (22) ϕ] C I p 0 pn][ ν which can also be represented by the standard form in Fig. 1 where the plant G has the state space representation in (14) with the matrices defined in (21)-(22) and where the controller K is the observer gain in (10). The following theorem is an extension of the observer asymptotic convergence result (obtained in [20], [21]) to the fault detection problem. Fig. 2. Feedback interconnection. Theorem 2: Given the nonlinear system in (1)-(2), the residual generator (6)-(10),(12) achieves fault detection for the sensor faults in (11) for all Φ satisfying (3) with a Lipschitz constant α if the observer gain K is chosen such that: sup ω R σ max [ ˆT ζτ1 (jω)] < 1 (23) α Proof : The proof is similar to Theorem 4 in [20] and is hence omitted. However, it is important to note that the observer s estimation error in (18) can be represented by the feedback interconnection of ˆT e φ, ˆT efs and as shown in Fig. 2, where is the static nonlinear time-varying operator defined as follows: (t) :e φ =Φ(x, u, t) Φ(ˆx, u, t) =Φ(e +ˆx(t),u(t),t) Φ(ˆx(t),u(t),t) and satisfying γ( ) α due to the Lipschitz condition in (3). The feedback interconnection is locally asymptotically stable when f s =0. Therefore, condition (23) is sufficient for achieving fault detection according to Definition 1. IV. NARROW FREQUENCY BAND SENSOR FAULT DIAGNOSIS Throughout the paper, we will assume the fault detection objective (i.e, the condition stated in Theorem 2) to be satisfied for the residual generator (6)-(10),(12). We will further consider the fault identification problem (according to Definition 3) with the objective to make the residual converge to the fault vector achieving detection and estimation at the same time. In this section, we focus on the case when the sensor faults are in a narrow frequency band around a nominal frequency ω o. Existence conditions for solving this problem are provided, and a numerical LMI design is presented using the dynamic observer structure in section II. A. Fault identification: problem definition Since the residual r is given from equation (19), it is then clear that the observer estimation error e constitutes a part of the residual response, and that by minimizing e the residual converges to f s which guarantees fault identification in this case. However, it was seen in section III that the estimation error e can be represented by the feedback interconnection in Fig. 2 where f s is the sensor fault vector that affect the system. Hence, minimizing e is equivalent to minimizing the effect of f s on the feedback interconnection of Fig. 2. In this section, we consider the solution of this 516

4 minimization problem in an L 2 sense (when f s is in a narrow frequency band around a nominal frequency ω o ) by using the dynamic observer introduced in section II, showing that the problem is not tractable for the classical structure in (4)-(5). Towards that goal, we will first assume that the fourier transform of the sensor fault F s (jω) have a frequency pattern restricted to the narrow band ω o ± ω as described by equation (24). { A ; ω ωo < ω F s (jω) (24) δ ; otherwise where δ is a small neglected number for the frequency magnitudes outside the region of interest. We will then define an observer gain K as optimal if e L2 can be made arbitrarily small for all possible sensor faults satisfying (24). But by applying the small gain theorem to Fig. 2 when fault detection is satisfied (i.e, when K satisfies ˆT e φ = µ< 1 α )wehave: e L µα e 2 L2. And since (as ω 0), ˆTefs ( (jω) ) ˆT efs (jω o ) then we have e 2 L2 σ max ˆTefs (jω o ) f s L2, and therefore, it is easy to see that an optimal gain K is one that satisfies ˆT efs (jω o )=0. By assuming that the fault detection objective is satisfied (as stated in Theorem 2), it follows that fault identification according to Definition 3 is satisfied if the following two conditions are satisfied: (i) ˆT e φ < 1 α, (ii) ˆT efs (jω o )=0, where the first one is a sufficient condition in order to achieve fault detection. Based on the previous discussion, we will define an optimal residual generator as follows: Definition 5: (Optimal residual for narrow freq. band) An observer of the form (6)-(10),(12) is said to be an optimal residual generator for the sensor faults identification problem (with faults in a narrow frequency band around ω o )ifthe observer gain K satisfies ˆT e φ < 1 α and ˆT efs (jω o )=0, for the standard setup in Fig. 1 where the plant G has the state space representation in (14) with the matrices defined in (21)-(22). Now we present the main result of this section in the form of a theorem showing that the classical observer structure can never be an optimal residual, which shows the importance of having a dynamic observer gain in this case: Theorem 3: An observer of the form (4)-(5) can never be an optimal residual generator according to Definition 5. Proof : First, using (21)-(22) and the gain K in (10) it can be shown that ˆT efs is given from: A D L C C L D L ˆT efs (s) = B L C A L B L (25) I n 0 nk 0 np The proof follows by noting that (when the observer gain K is replaced by the static gain L) the transfer [ matrix from] f s to e in (25) is given by ˆT A LC L efs (s) =. I n 0 np And since the gain L is chosen to stabilize (A LC), then ( ω o ) jω o is not an eigenvalue of (A LC). Therefore, by using (13), we have ( ) [ ] A LC jωo I rank ˆTefs (jω o ) = rank n L n. [ ] I n [ 0 np ] A LC jωo I But rank n L L 0n = rank = I n ( 0 np ) 0 np I n n+rank(l). Therefore, rank ˆTefs (jω o ) 0unless L =0. This implies that no gain L can satisfy ˆT efs (jω o ) = 0, and therefore the static observer strcuture can never be an optimal residual generator according to Definition 5. The previous theorem shows that the classical observer structure can not solve the identification problem for any Lipschitz constant α. In the following section (section IV- B), we provide a numerical approach based on LMIs by modeling the problem as a convex optimization problem using the dynamic observer structure in (6)-(10). B. A LMI design procedure After defining the fault identification problem as the two objective problem in Definition 5, we now show that the second objective, i.e ˆT efs (jω o )=0, can also be modeled as a weighted H problem. To this end, we first note that for a gain K that satisfies the fault detection condition (as stated in Theorem 2), the following two statements are equivalent: (i) ˆTefs (jω o )=0. (ii) W (s) ˆT efs (s) RH. where W (s)=diag p ( 1 s ) if ω 1 o=0 and W (s)=diag p ( s 2 +ω ) if o 2 ω o 0. The equivalence of the previous two statements can be seen by first noting that the condition in Theorem 2 implies that ˆT e φ < 1 α and hence that ˆTe φ RH. It then follows that ˆTefs (s) RH since ˆT efs in (25) and ˆT e φ both have the same state transition matrix. Finally, since ˆT efs (jω o )=0is equivalent to cancelling the poles of W (s) on the imaginary axis, it is easy to see that ˆTefs (jω o )=0 is equivalent to having W (s) ˆT efs (s) RH. According to the previous discussion, it follows that the objective ˆT efs (jω o )=0can be restated as follows: { ɛ >0 such that ɛ W (s) ˆT efs (s) < 1 } α where the scalar ɛ is used for compatibility with the first objective (i.e, ˆT e φ < 1 α ). It then follows that the two objectives can be combined in the unified framework in Fig. 3, where the plant G has the state space representation in (14) with the matrices defined in (21)-(22), and with the weighting W defined above. [ τ I 0 0 ɛw Ḡ Fig. 3. ] τ G K Weighted standard setup. ζ 517

5 It can be seen that Ḡ in Fig. 3 is given by: Ā B1 B2 ˆḠ(s) = C 1 D11 D12 C 2 D21 D22 [ ] [ ] [ ] Aθ 0 ln 0ln B θ 0ln = [ 0 nl A I n 0 np I n ] [ ] [ ] [ 0nl I n 0 n 0 np 0n ɛcθ C ] [ ] 0 pn 0 p 0 pn (26) where the matrices for the two cases ω o =0and ω o 0are as follows: ω o =0: A θ =0 p, B θ = I p, C θ = I p and l = p ([ ]) ([ ]) A ω o 0: θ =diag p ωo 2,B 0 θ =diag p 1 ([ ]) C θ =diag p 1 0 and l =2p (27) Based on the previous results, the following theorem redefines the identification problem in Definition 5 as follows: Theorem 4: Given the nonlinear system of equations (1)- (2), there exists an optimal residual according to Definition 5, for all Φ(,, ) satisfying (3) with a Lipschitz constant α if and only if ɛ > 0 and a controller K satisfying ˆT ζ τ < 1 α for the setup in Fig. 3 where Ḡ has the state space representation in (26). Proof : a direct result of Definition 5 and the discussion in the beginning of this section. However, standard H tools can not be directly applied for the H problem defined in Theorem 4. For instance the Riccati approach in [23] can not be implemented since the augmented plant Ḡ in (26) does not satisfy the needed regularity assumptions. Also, the LMIs in equation (15)-(17) are not feasible due to the poles that Ḡ has on the imaginary axis, making the use of the LMI approach in [24] impossible. In the following, we propose a numerical approach to solve this problem by replacing the weightings W (s) by the modified weightings W (s) where W (s) = diag p ( 1 s+λ ) if ω o =0and W 1 (s) =diag p ( s 2 +2λω os+ω ) if ω o 2 o 0, with λ R +. With this change, the augmented plant Ḡ in Fig. 3 is still given by equation (26), but where A θ is now given by: A θ = ([ diag p ( λ) ; ω o =0 0 1 diag p ωo 2 2λω o ]) ; ω o 0 (28) which has no poles on the imaginary axis. Using this modified augmented plant and the result in Theorem 1, we then propose the following convex optimization problem to solve the problem defined in Theorem 4: min R,S λ subject to the 3 LMIs in (15)-(17) with γ = 1 α where the matrices in (15)-(17) are replaced by the corresponding matrices in (26)- (27) with A θ in (28). The set of all admissible observer gains K for a given λ can then be parameterized using R, S by using the result in [24]. It can also be seen, that these LMIs are feasible for all λ>0, and that minimizing ( ) λ in this case is equivalent to minimizing σ max ˆTefs (jω o ). This guarantees that the proposed optimization problem converges to the existing solution as λ 0. Comments - According to the previous definition, an optimal residual generator guarantees sensor faults estimation and at the same time state estimation (with minimum energy for the estimation errors). An advantage of having state estimation in presence of faults is the possibility to use the observer in fault tolerant output feedback control. - From the special cases of interest is the case of sensor bias, where the previous approach can be used to get an exact estimation of all sensor biases at the same time. - In case of multiple frequency bands, a bank of observers can be used where each one estimates the faults vector in a specific range. These multiple estimates can then be used to restore the original fault vector. V. THE LOW AND HIGH FREQUENCY RANGES In this section we consider two different cases: the low frequency range and the high frequency range. For the first case, we assume the system to be affected by sensor faults of low frequencies determined by a cutoff frequency ω l, i.e the frequency pattern for f s (t) is confined to the region [0,ω l ]. On the other hand, in the high frequency case, we assume these faults to have very high frequencies above a minimum frequency ω h, i.e the frequencies are confined to the region [ω h, ). For these two cases (similar to section IV), we define and solve the fault identification problem using the dynamic observer strcuture in (6)-(10),(12) as the residual generator. Numerical design procedures that can be solved using available software packages are also provided. A. The low frequency range case We now consider sensor faults of low frequencies determined by a cutoff frequency ω l. The SISO weighting ŵ l (s) = as+b s, [23], emphasizes this low frequency range with b selected as ω l and a as an arbitrary small number for the magnitude of ŵ l (jω) as ω. Therefore, with a diagonal transfer matrix Ŵ (s) that consists of these SISO weightings (and similar to what was done in section IV- B), the identification objective can be combined with the detection objective in the unified framework represented by the weighted setup of Fig. 3. It can be seen that the augmented plant Ḡ (consisting of the weighting W cascaded with G in (21)-(22) is given by: [ ] [ ] [ ] Aθ 0 pn 0pn B θ 0pn ˆḠ(s) = [ 0 np A I n 0 np I n ] [ ] [ ] [ 0np I n 0 n 0 np 0n ɛcθ C ] [ ] 0 pn ɛd θ 0 pn (29) where A θ =0 p, B θ =I p, C θ =diag p (b) and D θ =diag p (a). 518

6 However, this standard form violates the assumptions of Theorem 1 (note that (Ā, B 2 ) is not stabilizable). Therefore, we introduce the modified weighting ŵ lmod (s) = as+b s+λ ; with arbitrary small positive λ. It is easy to see that, with this modification, the augmented plant Ḡ is the same as (29) except for A θ which is now given by the stable matrix diag p ( λ) and C θ given by diag p (b aλ). Similar to the narrow frequency band case, the assumptions of Theorem 1 are now satisfied and the LMI approach in [24] can then be used to solve the associated H problem. To this end, we define the continuous H problem associated with the low frequency range as follows: Definition 6: (Low frequency H ) Given λ > 0 and ɛ > 0, find S, the set of admissible controllers K satisfying ˆT ζ τ <γfor the setup in Fig. 3 where Ḡ has the state space representation (29) with A θ = diag p ( λ), B θ = I p, C θ = diag p (b aλ) and D θ = diag p (a). Based on the previous results, we now present the main result of this section in the form of the following definition for an optimal residual generator in L 2 sense: Definition 7: (Optimal residual for low frequencies) An observer of the form (6)-(10) along with r = y ŷ is an optimal residual generator for the sensor fault identification problem (with low frequency faults below the cutoff frequency ω l ) if the dynamic gain K S (the set of controllers solving the H problem in Definition 6 with γ =1/α and with the minimum possible λ). Note that the constants a and λ should be selected as arbitrary small positive numbers, while b must approximately be equal to ω l (the cutoff frequency). Different weightings could also be used for the different sensor channels. In this case A θ = diag( λ 1,, λ p ), C θ = diag(b 1 a 1 λ 1,, b p a p λ p ) and D θ = diag(a 1,, a p ). B. The high frequency range case Similar to the low frequency range, a SISO weighting ŵ h (s) = s+(a b) b could be selected to emphasize the high frequency range [w h, ) with b selected as w h and a as an arbitrary small number for ŵ h (jω) as ω 0. Since this weighting is not proper, a modified weighting can be used, [23], as ŵ hmod (s) = s+(a b) λs+b with an arbitrary small λ>0. With the help of ŵ hmod (s), a suitable weighting W that emphasizes the high frequency range for the observer problem in Fig. 3 can be designed. The augmented Ḡ is also given from (29) (same as the low frequency case), but with A θ, B θ, C θ and D θ given as diag p ( b λ ), I p, diag p ( a b λ b λ ) 2 and diag p ( 1 λ ) respectively. It is straightforward that this augmented plant Ḡ satisfies all of the assumptions of Theorem 1 and therefore, similar to the low frequency range, a continuous H problem related to the high frequency range can be defined. Also, an optimal residual generator can be defined in a similar way to Definition 7 (details are omitted due to similarity). VI. CONCLUSION A new LMI observer design for Lipschitz nonlinear systems is proposed and is applied in the fault diagnosis problem. This design offers extra degrees of freedom over the classical Luenberger structure and we show how this freedom can be used for the sensor fault and state estimations problems. For the narrow frequency band case, the problem is shown to be equivalent to an output zeroing problem for which a dynamic gain is necessary. The use of appropriate weightings, for different frequency patterns, to transform these problems into standard H optimal control problems is also demonstrated. A systematic design procedure that can be carried out using commercially available software products is presented. REFERENCES [1] A. S. Willsky, A survey of design methods for failure detection in dynamic systems, Automatica, vol. 12, pp , [2] P. M. Frank, S. 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