Fault Estimation for Single Output Nonlinear Systems Using an Adaptive Sliding Mode Observer

Transcription

1 Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 28 Fault Estimation for Single Output Nonlinear Systems Using an Adaptive Sliding Mode Observer Xing-Gang Yan Christopher Edwards Department of Engineering, University of Leicester, University Road, Leicester, LE 7RH, UK Abstract: Inthis paper, a class of single outputnonlinear systems with an uncertain parameter is considered. A diffeomorphism is first introduced to simplify the system structure, then, by employing an adaptive approach to identify the unknown parameter, a sliding mode observer is developed to estimate the system state variables. Based on the observer, a fault estimation scheme is proposed based on the minimisation of a weighted L 2 norm of the fault estimation error. A simulation example is given to demonstrate the proposed scheme.. INTRODUCTION Inrecentdecades, some control inspiredapproaches for instance sliding mode techniques Edwards, Spurgeon & Patton 2, modern differential geometric approaches Persis & Isidori 2 and adaptive schemes Zhang, Polycarpou & Parisini 24 have been successfully incorporated with the observer-based FDI approach. In particular, modern geometric approaches have been shown to be effective inthe investigation ofaclass ofnonlinear systems Isidori 995, Marino & Tomei 995. Single output systems have received much attention becuase the structure of such a class of systems is simple see e.g. Marino & Tomei 2, Jo & Seo 22, Zhang et al. 24, Chen & Saif 27. When a system has unknown parameters, adaptive control techniques can be employed to estimate the unknown parameters Xu & Zhang 24, Zhang et al. 24. In Shafai, Pi, Bas & Linder 2, a PI adaptive observer was proposed for the purpose of FDI for SISO linear systems. An adaptive FDI scheme was given in Wang & Daley 996 for linear systems which requires that the observation error dynamics are strictly positive real. More recently, Chen and Saif Chen & Saif 27 considered a FDI problem for a class of SISO linear systems and an adaptive technique was usedtoidentify theunknownparameters. However, in reality, most systems exhibit varieties of nonlinearities and therefore it is necessary to deal with nonlinear systems. Recently, sliding mode techniques have been successfully used in fault detection and isolation Floquet, Barbot, Perruquetti & Djemai 24, Edwards et al. 2, Tan & Edwards 22, Yan & Edwards 27 and it has been proved to be an effective way to estimate/reconstruct system faults. A precise fault reconstruction approach is proposed in Edwards et al. 2 based on an equivalent output errorinjection. Basedon the work in Edwards et al. 2, a sensor fault reconstruction scheme is given in Tan & Edwards 22. It should be noted that in Edwards et al. 2, Tan & Edwards 22, only linear systems are considered and uncertainty is not involved. A fault estimation approach for linear systems with uncertainty was proposed by Tan & Edwards 23. Later, a robust actuator fault reconstruction scheme was presented in Yan & Edwards 27 using the characteristics of the uncertainty structure and the fault distribution. Jiang, Staroswiecki & Cocquempot 24 proposed a fault estimation scheme for a class of systems with uncertainty, and a robust fault detection method for nonlinear systems with disturbances was also considered in Floquet et al. 24. However, in these papers involving uncertainty, the reconstruction/estimation signal is explicitly dependent on the bounds on uncertainties, and they all require that the uncertainties are bounded with known upper bounds. It should be noted that in contrast to multi-output systems, a single output system only has one output which can be employed to estimate a fault and identify a parameter. Therefore in order to make estimation and identification possible simultaneously, both the fault and the uncertain parameter must appear in the output channel of the dynamics. Since the uncertain parameter and the fault both exist in the same channel and can not be separated, precise fault reconstruction as given in Edwards et al. 2, Yan & Edwards 27 is, generally speaking, not possible. However, it is possible to estimate the fault affecting the system and then to detect the fault by establishing an appropriate threshold. In this paper, fault estimation is considered for a class of single output nonlinear systems with an uncertain parameter. Both the fault distribution vector and the distribution vector for the uncertain parameter are allowed to be functions of the system output and input. It is not required that the uncertainty/fault is matched. Undertheassumption that the time derivative of the system output is measurable, a novel adaptive update law is proposed to identify the unknown parameter and then an observer is established using sliding mode techniques. It is shown that the trajectories of the error system dynamics enter a domain around the operating point infinite time and remaininsidethereafter. Based onthe sliding modeobserver, afault estimation scheme is proposedtoestimate thefaultsignal. Boundsforthefault estimation error are also given, which are independent of the uncertain parameter. Moreover, the uncertain parame /8/$2. 28 IFAC / KR-.585

2 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 28 ter is allowed to be arbitrarily large, and a bound on the parameter is not required. This is different in comparison with the existing related work in Jiang et al. 24, Floquet et al. 24, Tan & Edwards 23, Yan & Edwards 27 where the uncertain bounds must be used in the observer design and fault estimation/reconstruction. Notation: The symbol denotes the Euclidean norm or its induced norm, and in particular, 2 is used to represent the L 2 norm. The symbols used in Section 2 are the same as those in Marino & Tomei SYSTEM DESCRIPTION AND ANALYSIS Consider a nonlinear system described by ẋ = Fx + Gx,u + Φx,uθ + Ψx,uft y = hx, x := x 2 where x Ω R n Ω is a neighbourhood of the origin, u U R m U is an admissible control set and y Y R Y is output space are the state variables, the inputs and the output respectively. The vector fields Fx R n, Gx,u R n, Φx,u R n and Ψx,u R n are assumed to be smooth. The scalar θ R represents an unknown constant parameter. The unknown function ft R is a fault affecting the system which satisfies where ρ is assumed to be known. ft ρt 3 In this paper, all the assumptions are only assumed to hold in a domain of the operating point instead of the whole state-space. Without loss of generality, the domain is assumed to be a compact neighbourhood of the origin. Assumption The pair Fx,hx satisfies i rank dh, dl F h,, dl n F h = n; ii ad i F ξ,adj Fξ =, i,j =,,2,...,n ; where ξx is a vector field solution of the equation dh,ξ dl F h,ξ. =... 4 dl n F h,ξ For convenience, the notation O F,h is used throughout the paper and denotes a set of n-dimensional vector fields defined by { O F,h := ζx,u ζ, adj F ξx = } u U, j =,,2,...,n 2 where ξ is determined by 4. Assumption 2. The vector fields Gx,u, Φx,u and Ψx,u satisfy Gx,u,Φx,u,Ψx,u O F,h in the domain x,u Ω U Under Assumptions and 2, it follows from Chapter 5 in Marino & Tomei 995 that there exists a diffeomorphism w = T x such that in the new coordinates w, system 2 can be described by where ẇ = Aw + Gy,u + Φy,uθ + Ψy,uft 5 y = Cw, 6 A =.. := A R A 2 n n 7 C = R n 8 Clearly, the matrix pair A,A 2 is observable and thus there exists a matrix L R n such that A LA 2 is stable. For system 5 6, introduce a further linear coordinate transformation z = T 2 w with In L T 2 := 9 Let T = T 2 T. It is obvious that T : x z is a diffeomorphism definedinadomainoftheorigin. Itfollows that in the new coordinate system z defined by z = Tx, system 2 has the following form A ż = T 2 T A 2 2 z + T 2 Gy,u + T2 Φy,uθ +T 2 Ψy,uft y = Cz since CT 2 = C. The system above can be further decomposed as ż = A LA 2 z + A LA 2 Lz2 + g y,u +φ y,uθ + ψ y,uft 2 ż 2 = A 2 z + A 2 Lz 2 + g 2 y,u + φ 2 y,uθ +ψ 2 y,uft 3 y = z 2 4 where z = colz,z 2 with z R n and z 2 R and g y,u g 2 y,u φ y,u φ 2 y,u ψ y,u ψ 2 y,u := T 2 Gy,u = Tx x Gx,u Tx := T 2 Φy,u = x Φx,u Tx := T 2 Ψy,u = x Ψx,u 5 x=t z 6 x=t z 7 x=t z with g, φ, ψ R n and g 2, φ 2, ψ 2 R. Remark. Since T is available in Marino & Tomei 995 and the matrix T 2 has been given in 9, the transformation T is available. Therefore, the vectors g i, φ i and ψ i for i =,2 can be obtained directly using the coordinate transformation z = Tx and thus system 2 4 is well defined and can be obtained from 2. Assumption 3 The functions φ 2 and ψ 2 satisfy φ 2 and ψ 2 in the domain Y U. Furthermore ψ is 2 assumed to be bounded in Y U. 897

3 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 28 Remark 2. Assumption 3 is a limitation on the fault and uncertain parameter distributions. It should be noted that φ 2 and ψ 2 are necessary for the parameter θ to be identified and the fault f to be estimated, since in system 2 3, only z 2 is measurable and z is not available. 3. ADAPTIVE SLIDING MODE OBSERVER DESIGN In this section, as in Fu & Liao 99, it is assumed that ẏ is measurable. A sliding mode observer will now be proposed employing an adaptive law. For system 2 4, consider the dynamical observer system ẑ = A LA 2 ẑ + A LA 2 Ly + g y,u +φ y,uˆθ 8 ẑ 2 = A 2 ẑ + A 2 Ly Λ ẑ 2 y + g 2 y,u +φ 2 y,uˆθ + ν 9 ŷ = ẑ 2 2 where ŷ R is the observer output, Λ > is a design scalar, and the estimate of θ denoted by ˆθ is given by the following adaptive law ˆθ = φ 2 φ 2 y,uˆθ + A 2 ẑ + A 2 Ly + g 2 y,u ẏ 2 The output error injection term ν is defined by ν = k t,y,u sgny ŷ 22 where sgn denotes the usual signum function. The gain k is to be determined later and only depends on known information. Let e z = z ẑ, e y = y ŷ and e θ = θ ˆθ. It follows from 2 4 and 8 2 that the dynamical error system is described by ė z = A LA 2 e z + φ y,ue θ + ψ y,uft 23 ė θ = φ 2 y,ua 2 e z φ 2 2y,ue θ φ 2 y,uψ 2 y,uft 24 ė y = A 2 e z Λe y + φ 2 y,ue θ + ψ 2 y,uft ν 25 The following conclusion is now ready to be presented: Theorem. Assume that there exists a positive definite matrix P and a positive constant ε such that the matrix W W W := 2 W2 T W 22 is positive definite for y,u Y U where W := A LA 2 T P + PA LA 2 + ψ 2 εψ2 2 P 2 W 2 := P φ y,u φ 2 y,u + P ψ y,u εψ 2 y,u + AT 2 W 22 := 2 ε Then, under Assumptions 2 and 3, the trajectories of will enter a bounded domain of the origin in finite time if ψ 2 y,uρt is bounded in Y U R + and γ := inf y,u Y U {λ min Wy,u} >. Otherwise the approach proposed in Levant 998 can be employed to compute the time derivative of the system output. Proof: Consider a candidate Lyapunov function V for the error dynamics as follows V e z,e θ = e T z Pe z + e 2 θ 26 Then the time derivative of V along the trajectories of system are given by V = e T z A LA 2 T P + PA LA 2 T e z + 2e T z Pφ e θ +2e T z Pψ y,uft 2φ 2 y,ue θ A 2 e z 2φ 2 2y,ue 2 θ 2φ 2 y,uψ 2 y,ue θ ft 27 From Young s inequality and ψ 2, it follows that for any positive constant ε 2e T z Pψ y,uft 2φ 2 y,uψ 2 y,ue θ ft ε et z P ψ 2 y,u ψ 2 y,u φ 2y,ue θ + ε ψ 2 y,uft 2 = ψ y,u 2 ε et z ψ2 2y,u P 2 e z + ε φ2 2y,ue 2 θ 2 φ 2y,u εψ 2 y,u et z Pψ y,ue θ + ε ψ 2 y,uft 2 28 Substituting 28 into 27, it follows that V = e T z A LA 2 T P + PA LA 2 + ψ 2 εψ2 2 P 2 e z 2e T z P φ y,u φ 2 y,ue θ φ 2 y,u + P ψ y,u εψ 2 y,u + AT 2 2 φ 2 ε 2y,ue 2 θ + ε ψ 2 y,uft 2 = e T z φ 2 e θ W e z + ε ψ φ 2 e 2 ft 2 29 θ Since by assumption ψ 2 y,uρt is bounded in Y U R +, suppose that ψ 2 y,uρt 2 M. Then, it follows from 29 that when e z 2 + φ 2 2y,ue 2 θ > εm γ γ with < γ <, V γ e z 2 + φ 2 2y,ue 2 θ + ε ψ2 y,uft 2 γ e z 2 + φ 2 2y,ue 2 θ + εm < γ e z 2 + φ 2 2 e 2 θ + γ γ e z 2 + φ 2 2 e 2 θ = γγ e T z e z + φ 2 2y,ue 2 θ Since φ 2 2 in the considered domain, it follows that the trajectories of system will enter a bounded domain in finite time Khalil 22. Hence the conclusion follows. # Remark 3. Theorem shows that under certain conditions both e z and e θ are bounded. Throughout the paper, it is assumed that for any t,y,u R + Y U there exist χ and χ 2 such that e z t χ, e θ χ 2 3 Now, for the error dynamical equation 23 25, consider the following sliding surface S = {e z,e θ,e y e y = } 3 898

4 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 28 The reduced-order sliding mode associated with the surface in 3 can be described by Theorem has shown thatundercertain conditions, whenthe sliding motion takes place, the trajectories of the sliding mode dynamics will enter a bounded domain in finite time. A reachability condition will be developed so that the error dynamical system is driven to the sliding surface 3 in finite time and a sliding motion maintained thereafter. Theorem 2. Assume that 3 holds. Thenthe trajectories of the dynamical system can be driven to the sliding surface 3 in finite time if the gain kt,y,u in 22 satisfies k χ + Λ e y + φ 2 y,u χ 2 + ψ 2 y,u ρt + η 32 for some η >. Proof: Substituting 22 into 25, it follows that e y ė y = e y A 2 e z e y Λe y + e y φ 2 y,ue θ + e y ψ 2 y,uft e y k sgne y From 3, 3 and the fact that e y sgne y = e y, e y ė y e y A 2 χ + Λ e 2 y + φ 2 y,u e y χ 2 + e y ψ 2 y,u ρt k e y = χ +Λ e y + φ 2 χ 2 + ψ 2 ρ k e y 33 In establishing 33 the fact that A 2 = has been used. Then, using 32, it is easy to see that 33 becomes e y ė y η e y This shows that the traditional reachability condition Utkin 992 is satisfied and thus e y in finite time. Hence the conclusion follows. # Remark 4 From sliding mode theory, Theorems and 2 show that system 8 2 is an observer for the system 2 4 and the estimation error enters a bounded domain in finite time. 4. FAULT ESTIMATION In this section, it is assumed that an adaptive sliding mode observer 8 2 has been properly designed and that the conditions in Theorems and 2 are satisfied. When a sliding motion takes place, e y t =, and ė y t = 34 and in this case the outputerror injection signalν in 25 can be replaced by an equivalent output error injection signal ν eq Utkin 992. By applying 34 to 25, it follows that when the sliding motion takes place A 2 e z + φ 2 y,ue θ + ψ 2 y,uft ν eq = 35 From Assumption 3, ψ 2 y,u. Then, from 35 the fault signal satisfies ft = ψ 2 y,u A 2e z φ 2 y,ue θ + ν eq 36 Inorderto estimate the fault ft, it is necessaryto recover the equivalent output error injection signal ν eq. In the work described in Utkin 992, it was obtained using a lowpass filter. Here a modification to the approach given in Edwards et al. 2 will be employed. From 22, the equivalent output error injection signal ν eq in 36 can be approximated to any accuracy by ν σ := k t,y,u ζe y 37 where k satisfies 32 and e y ζe y = e y + δ exp{ δ 2 t} where δ and δ 2 are positive design constants. Construct a signal ˆft = 38 ψ 2 y,u ν σ 39 where the function ν σ is defined by 37. It will be shown that ˆft is a reasonable estimate for the fault ft. Define Z := A 2 e z + φ 2 y,ue θ 4 Then, the following conclusion is ready to be presented: Theorem 3. Consider the system described in 2. Assume that the conditions of Theorem are satisfied and that the matrix W W2 W := W 22 W T 2 defined by W := A LA 2 T P + PA LA 2 + ψ y,u 2 εψ2 2y,u P 2 + A T 2 A 2 W 2 := P φ y,u φ 2 y,u + P ψ y,u εψ 2 y,u W 22 := ε is semi-positive definite where ε is a positive scalar. Then, i Z 2 V +ε ψ 2 y,uft 2 with V and Z defined by 26 and 4 respectively; ii the L 2 norm of the estimation error ft ˆft where ˆft is defined by 39, satisfies ψ 2 ft ˆft 2 V + ε ψ 2 ft 2 + ε 4 where ε is an arbitrary small positive constant. Proof: From the definition of Z in 4, it follows that Z 2 = e T z A T 2 A 2 e z + 2e T z A T 2 φ 2 y,ue θ + φ 2 2y,ue 2 θ 42 Then, from the proof of Theorem and 29, V e T z φ 2 y,ue θ W e z φ 2 y,ue θ Z 2 + ε ψ 2 y,uft 2 43 Since W is semi-positive, it follows from 43 that for t R + t t V t V Z 2 dt + ε ψ 2 y,uft 2 dt 899

5 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 28 Because V t >, t Z 2 dt V + ε Hence conclusion i follows. t ψ 2 y,uft 2 dt Since ν σ can approximate ν eq to any accuracy, and from Assumption 3 ψ 2 is bounded in Y U, it follows that for any ε > there exist σ and σ 2 such that ν eq ν σ 2 < ε Consequently, from 36, 39 and conclusion i, ψ 2 ft ˆft 2 V + Z + ν eq ν σ 2 V + ε ψ 2 ft 2 + ε 44 Hence, conclusion ii follows. # Remark 5. Theorem 3 shows that ˆf is an estimate for the fault f and the estimation error is given by 4. Clearly, the conservatism of the estimation error ψ 2 f ˆf 2 can be reduced by optimizing ε. Remark 6. From the proof of Theorems 2 and 3, it is easy to see that if W is positive definite then this implies that W is positive definite. However, the fact that W is semipositive definite does not imply W is positive definite. In the generic case it is hard to give a general approach for optimising the parameter ε. However, it is possible to do this for some special cases. Aninteresting case is when φ y,u = d φ 2 y,u, ψ y,u = d 2 ψ 2 y,u 45 where d,d 2 R n are constant vectors. In this case the matrices W and W have the following forms respectively Γ Ξ + A W = T 2 Ξ T + A 2 2 ε Γ A T W = 2 A 2 Ξ Ξ T ε where Γ := A LA 2 T P + PA LA 2 + d 2 2 P 2 ε Ξ := P d d 2 ε In order to get the least conservative bound for the fault estimation error f ˆf, one approach is to minimize ε such that W is positive definite and W is semi-positive definite. 5. SIMULATION EXAMPLE Consider a DC-to-DC boost power converter Ortega, Schaft, Maschke & G. 22. The dynamical equation describing this system is given by ẋ D = ẋ2 D R L L x E C x + V 2 in 46 where x is the induction flux, x 2 is the charge in the capacitor and V in represents the input voltage. The scalar C is the capacitance, L is the inductance and the constant D < D < is the duty ratio. R L is the output load resistance and E is the DC voltage. The system in 46 can be expressed in the form of 2 as follows ẋ D C = x 2 ẋ2 D L x R x LC 2 Fx y = C D x Eu + Gx,u k + k 2 Φx,u E θ + k f Ψx,u where u := V in is the input and y = C D x is the output. The parameter θ is assumed to be a constant disturbance affecting the system and ft is an actuator fault added to illustrate the results obtained in this paper and is not a feature of Ortega et al. 22. By direct computation, it follows that Assumptions and 2 are both satisfied. Then introduce the coordinate transformations w = x 2 R L D x w 2 = C D x and z = l w where z := colz,z 2. Then the system can be described in the coordinates z in the canonical form 2 4 as ż = lz l 2 z 2 + l D2 R L L y C + lc E R L D u g +k 2 k R L D + Cl D k θ + lc E R L D k f ft 47 φ ψ ż 2 = z + lz C D k θ R L C y CE D u g 2 φ 2 + C D Ek f ft 48 ψ 2 y = z 2 49 From 8 9, the corresponding candidate observer for system is given by ẑ = lẑ l 2 y + l R L D2 L y C + lc R L E D u +k 2 k R L D + Cl D k ˆθ 5 ẑ 2 = ẑ + ly Λẑ 2 y + R L C y CE D u + C D k ˆθ + ν 5 ŷ = ẑ 2 52 where ν is defined by 22 and ˆθ is given by ˆθ = C D k C D k ˆθ + ẑ + ly R L C y CE D u ẏ 53 As in Sira-Ramirez, Perez-Moreno, Ortega & Garcia- Esteban 997, the parameters are chosen as: C = 2µF, L = 2mH, R L =.5Ω, D =.5 and E = 5V. If k =.2, k 2 =, k f =.2 and the design parameters f 9

6 7th IFAC World Congress IFAC'8 Seoul, Korea, July 6-, 28 are chosen as l =, P =, Λ =.8, η = 2.5 and ε =., by direct computation, it follows that 2. W = and.99 W 9 =.99 Clearly, both matrices are positive definite and thus 5 53 is a sliding mode observer of the system in The corresponding estimates for x and x 2 are given by ˆx = D C ẑ2 and ˆx 2 = ẑ + L R ẑ LC 2 respectively. For simulation purposes, choose θ =.5, x = 8, x 2 = 4, ˆx = 5, ˆx 2 = 2 and ˆθ = 5. A simple feedback controller u =.33.77x has been introduced to stabilise the nominal system for demonstration purposes. The simulation results in Figures and 2 show that the proposed approach is effective. 2 θ estimation of θ time sec 3 2 x estimation of x time sec 2 x 2 estimation of x time sec Fig.. The responses of the system and the corresponding observer with the adaptive estimation fault signal estimate signal time sec Fig. 2. The fault signal and its estimation 6. CONCLUSION A class of single output nonlinear systems has been considered in this paper. By using a geometric approach and sliding mode techniques, a slidingmode observer has been established in the presence of a fault and an uncertain parameter. Using the observer, a fault estimation signal has been proposed using the output estimation error injection signal. An approach based on the L 2 norm has been suggested as a measure of the fault estimation error. REFERENCES Chen, W. & Saif, M. 27. Adaptive actuator fault detection, isolation and accommodation in uncertain systems, Int. J. Control 8: Edwards, C., Spurgeon, S. K. & Patton, R. J. 2. Sliding mode observers for fault detection and isolation, Automatica 364: Floquet, T., Barbot, J. P., Perruquetti, W. & Djemai, M. 24. On the robust fault detection via a sliding mode disturbance observer, Int. J. Control 777: Fu, L. & Liao, T. 99. Globally stable robust tracking of nonlinear systems using variable structure control and with an application to a robotic manipulator, IEEE Trans. on Automat. Control 352: Isidori, A Nonlinear Control Systems ThirdEdition, Great Britain: Springer-Verlag London Limited. Jiang, B., Staroswiecki, M. & Cocquempot, V. 24. Fault estimation in nonlinear uncertain systems using robust sliding-mode observers, IEE Proc. Part D: Control Theory Appl. 5: Jo, N. H. & Seo, J. H. 22. Observer design for non-linear systems that are not uniformly observable, Int. J. Control 755: Khalil, H. K. 22. Nonlinear Systems Third Edition, New Jersey: Prentice Hall, Inc. Levant, A Robust exact differentiation via sliding mode technique, Automatica 343: Marino, R. & Tomei, P Nonlinear Control Design, Englewood Cliff: Prentice Hall International. Marino, R. & Tomei, P. 2. Adaptive output feedback tracking with almost disturbance decoupling for a class of nonlinear systems, Automatica 362: Ortega, R., Schaft, A., Maschke, B. & G., E. 22. Interconnection and damping assignment passivity-based control of portcontrolled Hamiltonian systems, Automatica 38: Persis, D. C. & Isidori, A. 2. Ageometric approach to nonlinear fault detection and isolation, IEEE Trans. on Automat. Control 466: Shafai, C., Pi, O., Bas, S. N. & Linder, S. 2. A general purpose observer architecture with application to failure detection and isolation, Proc. of theamerican Control Conference, Arlington, VA., pp Sira-Ramirez, H., Perez-Moreno, R. A., Ortega, R. & Garcia-Esteban 997. Passivity-based controllers for the stabilization of DCto-DC converters, Automatica 334: Tan, C. P. & Edwards, C. 22. Sliding mode observers for detection and reconstruction of sensor faults, Automatica 385: Tan, C. P. & Edwards, C. 23. Sliding mode observers for robust detection and reconstruction of actuator sensor faults, Int. J. Robust Nonlinear Control 35: Utkin, V. I Sliding modes in control optimization, Berlin: Springer-Verlag. Wang, H. & Daley, S Actuator fault diagnosis: an adaptive observer-based technique, IEEE Trans. on Automat. Control 4: Xu, A. & Zhang, Q. 24. Residual generation for fault diagnosis in linear time-varying systems, IEEE Trans. on Automat. Control 495: Yan, X. & Edwards, C. 27. Nonlinear robust fault reconstruction and estimation using a sliding mode observer, Automatica 43: Zhang, X., Polycarpou, M. M. & Parisini, T. 24. Adaptive faulttolerant control of nonlinear uncertain systems: an informationbased diagnostic approach, IEEE Trans. on Automat. Control 498: