Design of Sliding Mode Observers for TS Fuzzy Systems with Application to Disturbance and Actuator Fault Estimation Patrick Gerland, Dominic Groß, Horst Schulte and Andreas Kroll Abstract— In this paper an observer-based fault detection approach for a class of nonlinear systems, which can be modeled by Takagi-Sugeno (TS) fuzzy models, is presented. We propose a sliding mode fuzzy observer that deals with bounded uncertainties in the plant and allows fault estimation based on an equivalent output error injection approach. Necessary conditions for the existence of the robust observer are derived from sliding mode theory [11]. Stability is ensured by linear matrix inequality (LMI) based sufficient conditions. An illustrative example demonstrates the effectiveness of the proposed scheme.
I. INTRODUCTION Fault detection and isolation (FDI) hve been an important research topic in control systems for many years. A fault is deemed to occur when a system experiences an abnormal condition, such as component malfunction. The purpose of an FDI system is to immediately generate an alarm and identify the location of the fault. Different approaches to FDI have been proposed (e.g. [1],[2],[3],[4],[5]). A wellknown method is to generate residuals by comparing outputs of an observer and measured system outputs [2]. Different schemes [1] are employed that use multiple observers and a subsequent evaluation logic to isolate faults. Furthermore, the residuals either have to be decoupled from uncertainties [1] or (adaptive) thresholds have to be used to avoid false alarms ([2],[6]). This results in a quite involved design process. Recently, an approach using the feedback of all available measurements has been discussed. Edwards et al. [7] utilized sliding mode observers (SMO) for nominal linear systems that are designed to track the system outputs even in the presence of faults. Faults are detected by manipulating the so-called equivalent output injection signal which represents the effort necessary to maintain the motion on a sliding surface [8]. Based on a system description including faults and uncertainty, this signal can be used directly for fault isolation, estimation and even reconstruction ([7],[9]). While the SMO presented in ([7],[11]) can effectively deal with model uncertainties, the resulting fault sensitivity might be poor in the case of large uncertainties. Therefore, nonlinear systems require the application of nonlinear observers to avoid linearization errors and ensure good performance of the FDI system. Yan & Edwards [12] proposed an observer similar to [7] for a class of nonlinear systems, however, the P. Gerland and A. Kroll are with the Institute of Measurement and Control, Dept. of Mechanical Engineering, University of Kassel (Germany) {p.gerland,
[email protected]} D. Groß is with the Institute of Control and System Theory, Dept. of Electrical Engineering, University of Kassel (Germany) {
[email protected]} H. Schulte is with the Faculty of Electrical Engineering and Automation, University of Applied Science Berlin (Germany) {
[email protected]}
system inputs are assumed to be linear and the observer design is mostly based on linear dynamics of the system. A well-known nonlinear observer design are TS fuzzy observers based on the PDC concept [13]. Gao et al. [10] proposed an approach to observer based sensor fault reconstruction using a TS fuzzy observer based on an augmented fuzzy descriptor system. Based on the feedback of all measurements, estimates of the fault-free system states and the output disturbances are obtained. However, modeling uncertainty and uncertain inputs might cause false and missed alarms which may make the FDI system unreliable. To achieve robustness TS fuzzy observers have been combined with sliding mode ([14],[15]). Bergsten et al. [14] proposed two different TS SM observers. A TS fuzzy observer [13] incorporating a switched term to account for uncertainties and a second observer design applying the SMO design from [11] to a special form of TS fuzzy system called dominant linear system. Furthermore approximation errors from classical TS modeling and the difficulties they pose for observer design are considered. However in [14], the application to FDI is not considered. Therefore, fault sensitivity and isolation are not discussed. In this paper, the "sector nonlinearity approach" [13] is used for TS modeling to avoid approximation errors of classical TS modeling. By extending the sliding mode methodology from ([11],[12]) to a class of TS systems, a robust observer is obtained and the equivalent output injection signal can be used for fault estimation. In contrast to ([11],[12]), the new observer design is based on a fully nonlinear model. Compared to ([14],[15]), no approximation errors are introduced and the FDI framework of [12] is used. This leads to a robust nonlinear observer that allows dealing with multiple faults and bounded uncertainty in a well posed theoretical framework, including fault estimation and possibly reconstruction. This paper is organized as follows. In Sec. II, a TS model for nonlinear systems including faults and uncertainties is presented. Based on this model, a TS-SMO and a fault reconstruction method using the equivalent output injection signal is developed in Sec. III. A simulation study in Sec. IV is used to compare the proposed observer to [14] and demonstrate the effectiveness of the proposed FDI scheme. II. S YSTEM M ODEL IN TS F UZZY S YSTEM F ORM Consider an uncertain nonlinear MIMO system subject to actuator faults and only perturbed by input disturbances ˙ =f (x(t))+g(x(t))u(t)+d(x(t))ξ(t)+ea (x(t))fa (t) x(t) y(t) =Cx(t)
(1)
d(x(t)) : Rn → Rq and ea (x(t)) : Rn → Ra are nonlinear functions that are Lipschitz with respect to x(t) ∈ Rn , and the unknown function ξ(t) ∈ Rq represents all modeling uncertainties and disturbances experienced by the system except for actuator faults represented by fa (t) ∈ Ra . The output matrix C ∈ Rp×n is assumed to be full row rank. It is assumed that the nonlinear model (1) can be represented by a TS fuzzy system [16] consisting of a fuzzy rule base where each rule i is of the form Model rule i: IF (α1 (t) is Mi1 ) and . . . and (αl (t) is Mil ) ( ˙ x(t) = Ai x(t) + Bi u(t) + Di ξ(t) + Ei fa (t) THEN y(t) = Cx(t) where αj (t) is the jth entry of the vector of premise variables α(t) and Mij is the fuzzy set for the ith model rule and the jth premise variable. The matrices Ai ∈ Rn×n , Bi ∈ Rn×m , Di ∈ Rn×q and Ei ∈ Rn×a represent the system dynamics, input gains and gains of the uncertain inputs. The final fuzzy system with the membership functions hi (α(t)) and nr rules is inferred as follows nr X ˙ hi (α(t))[Ai x(t) + Bi u(t) + Di ξ(t) + Ei fa (t)] x(t) = i=1
y(t) =Cx(t)
(2)
There are two major methods to obtain a TS fuzzy model. One is system identification using input-output data. The other one is TS fuzzy model construction based on a rigorous model. In the latter case the system matrices can either be derived by linearization of the nonlinear system (1) resulting in affine models [17] that can be interpreted locally. However, the resulting model is an approximation of the original system. Another way to construct a TS fuzzy model is the so-called "sector nonlinearity approach" proposed in [13]. This method allows to derive an exact representation (globally or at least semi-globally) of a nonlinear system if the nonlinearities fk (α(t)) can be replaced by sector nonlinearities zk (α(t)). The main idea is to derive the TS model by replacing the nl nonlinear terms of the system by the sum of two linear models weighted by nonlinear sector functions wk,1 (α(t)) and wk,2 (α(t)), k = 1, . . . , nl : fk (α(t)) = f k
zk (α(t)) − f k f¯k − zk (α(t)) + f¯k ¯ fk − f k f¯k − f k | {z } | {z } =wk,1 (α(t))
(3)
=wk,2 (α(t))
where f k := min(zk (α(t))) and f¯k := max(zk (α(t))). FurP2 thermore, the sector functions satisfy g=1 wk,g (α(t)) = 1 and 0 ≤ wk,g (α(t)) ≤ 1 for g = 1, 2. In a last step, aggregated membership functions hi (α(t)) can be calculated from all combinations of the nl membership functions wk,g (α(t)) [13]. This results in a TS fuzzy model with nr = 2nl linear models (rules) of the form (2). Therein, each aggregated model (Ai , Bi , C) represents the system at one combination of the sector Pnr bounds and the membership functions hi (α(t)) satisfy i=1 hi (α(t)) = 1, 0 ≤ hi (α(t)) ≤ 1. Using
LMI conditions to ensure stability of a TS fuzzy observer, a large number of models results in a large number of LMIs. Thus, LMI conditions for stability become more conservative ([13],[18]). For this and other reasons, a TS system should be constructed such that the number of linear models nr is as small as possible. III. S LIDING M ODE F UZZY O BSERVER Edwards & Spurgeon ([7],[11]) propose a SMO and design ˙ [19] but procedure which is similar to that of Walcott and Zak avoids the use of symbolic manipulation and always tracks the system outputs. The SMO concept involves the design of a discontinous feedback ensuring that a sliding surface is reached in finite time and a sliding motion is maintained. ˆ (t)−x(t) During sliding motion the estimation error e(t) = x remains close to zero, ensuring robust estimation for bounded uncertainty. A drawback of this SMO is that only nominal linear systems are considered. Therefore, we propose a similar SMO based on concepts from TS fuzzy systems (Sec. II) for a class of nonlinear systems. In the following some assumptions will be imposed on (2): Assumption 1: The input uncertainty and actuator faults are unknown but bounded by the Euclidean norm: k[ξ T (t) faT (t)]T k ≤ Ξ. Furthermore, it is assumed that individual bounds kξ(t)k ≤ Ξξ and kfa (t)k ≤ Ξfa exist and that the system states x(t) and the inputs u(t) are bounded. Assumption 2: rank(C[Di Ei ]) = rank([Di Ei ]) and rank([Di Ei ]) = q˜. This implies q˜ ≤ p.
Assumption 3: All invariant zeros of (Ai , [Di Ei ], C) lie in C− . Consider a TS-SMO for the uncertain system (2) with ˆ (t) − y(t) that is very similar to [11]: ey (t) = y ˆ˙ (t) = x
nr X i=1
ˆ ˆ (t) + Bi u(t) − Gl,i ey (t) hi (α(t))[A ix
+ Gn,i ν(t)]
(4)
ˆ (t) = Cˆ y x(t) where Gl,i , Gn,i ∈ Rn×p are appropriate gain matrices and ν(t) ∈ Rp represents a discontinuous feedback which is necessary to induce a sliding motion. Note that the general case of using estimated states as premise variables such that ˆ ˆ (t) is considered below. α(t) might contain x A. A canonical form for TS SM observer (TS-SMO) Consider a nonlinear change of coordinates obtained by combining linear coordinate transformations for each linear system of the TS fuzzy model (2) z(t) = T(α(t))x(t) =
nr X
hi (α)Ti x(t)
(5)
i=1
The linear coordinate transformations are obtained from a ˜c T ˜ DE,i T ¯ obs T ¯ L,i . Under series of transformations Ti = T i assumption 1 and 2, there always exists a
˜ c and T ˜ DE,i such that C = [0 Ip ], D = [0 D2 ]T and T E = [0 E2 ]T [12]. Thus, the uncertainty and the faults only affect the measured states. ¯ obs that puts Ai into observability canonical form • T i [11]. ¯ L,i these transformations can be easily obtained Exept for T by applying the numerical method from [11] to each linear system. In the observability canonical form, Ai has the form ¯ obs ¯ 12,i A A 11,i ¯ obs ¯ 221,i ˜ i Ai T ˜ −1 = A ¯ obs = A A T (6) i 211,i i ¯ 212,i A ¯ 222,i A •
where As22 is a stable design matrix and the discontinuous vector ν(t) is defined by ( P e (t) −ρ kP22 eyy (t)k if ey (t) 6= 0 (13) ν(t) = 0 otherwise where ρ is a positive scalar and P2 ∈ Rp×p is the unique symmetric positive definite (s.p.d.) solution to the Lyapunov equation with the s.p.d. design matrix Q2 ∈ Rp×p : P2 As22 + As22 T P2 = −Q2
(14)
˜i = T ˜c T ˜ DE,i T ¯ obs . Define a final nonsingular where T i transformation similar to that from Edwards & Spurgeon [11] I(n−p) [Li 0(n−p)טq ] ¯ L,i = (7) T ˜T 0p×(n−p) Y i
Note that As22 does not depend on the premise variables, therefore it is identical for all nr submodels. The observer sliding surface is given by the hyperplane [11]:
˜ i is obtained in previous steps (see [11]) and where Y appropiate Li can be obtained using LMIs as discussed in the next section. The system matrix in the final coordinates is obtained as follows A11,i A12,i Ai = Ti Ai T−1 = (8) i A21,i A22,i
ˆx (t) − zx (t), ey (t) = z ˆy (t) − zy (t) and Letting ezx (t) = z T T T ez (t) = ezx (t) ey (t) the following state estimation error dynamics are obtained from (11) - (12) (see [14])
where A11,i ∈ R(n−p)×(n−p) . Also notice that the disturbance matrices in the final coordinates are given by 0 0 Ti [Di Ei ] = (9) D 2,i E 2,i As proposed by Edwards & Spurgeon [11], it is straightforward to show from (7) and (8) that ¯ obs + Li A ¯ obs A11,i = A (10) 11,i 211,i If assumption 3 holds, a Li exists such that A11,i is stable [11]. Applying the nonlinear change of coordinates (5), a system of the following form is obtained nr X z˙ x (t) = hi (α(t))[A11,i zx (t) + A12,i zy (t) + B1,i u(t)] i=1
z˙ y (t) =
nr X
hi (α(t))[A21,i zx (t) + B2,i u(t) + D 2,i ξ(t)
i=1
+ E 2,i fa (t)] y(t) = Cz(t)
(11)
where the measured states zy (t) ∈ R are separated from the states zx (t) ∈ R(n−p) that have to be estimated such that z(t) = [zTx (t) zTy (t)]T . The system in canonical form (11) is used for the design of a TS SM observer. Consider an observer of the form nr X ˆ˙ x (t) = ˆ ˆx (t) + A12,i z ˆy (t) + B1,i u(t) z hi (α(t))[A 11,i z p
i=1
− A12,i ey (t)] nr X ˆ˙ y (t) = ˆ ˆx (t) + A22,i z ˆy (t) + B2,i u(t) z hi (α(t))[A 21,i z i=1
− (A22,i − As22 ey (t) + ν(t)]
ˆ (t) = Cˆ y z(t)
(12)
S = {ez (t) ∈ Rn : Cez (t) = 0}
e˙ zx (t) = e˙ y (t) =
nr X i=1 nr X
ˆ hi (α(t))[A 11,i ezx (t)] + ∆x
(15)
(16a)
s ˆ hi (α(t))[A 21,i ezx (t) + A22 ey (t) + ν(t)
i=1
− D 2,i ξ(t) − E 2,i fa (t)] + ∆y
(16b)
wherein the arguments of the functions ∆x and ∆y ˆ ∆x (α(t), α(t), z(t), u(t)) =
nr X ˆ (hi (α(t)) − hi (α)(t)) i=1
· [A11,i zx (t) + A12,i zy (t) + B1,i u(t)]
(17)
ˆ ∆y (α(t), α(t), z(t), u(t), ξ(t), fa (t)) = n r X ˆ (hi (α(t)) − hi (α(t)))[A21,i zx (t) + A22,i zy (t) (18) i=1
+ B2,i u(t) + D 2,i ξ(t) + E 2,i fa (t)] are ∆ = T for notational convenience. Note that Tomitted T ∆x ∆Ty → 0 when ez (t) = [ezx (t) ey (t)] → 0. Bergsten et al. [14] proposed to treat ∆ as an unstructured vanishing perturbation that is growth-bounded by k∆x k ≤ Lgx kez (t)k with a Lipschitz gain Lgx > 0. There exists a constant ϑ such that k∆x k ≤ ϑkex (t)k and from the nonlinear changeof coordinates we obtain −1 nr P ˜ ik ¯ L,i k−1 kez (t)k such ˆ kex (t)k ≤ hi (α(t))k T kT i=1
ˆ that kex (t)k ≤ κkez (t)k where κ is maximal for some α(t) and Li = 0(n−p)×(q) . Therefore, a Lgx is known before computing Li . Furthermore, we assume that scalars K , G exist such that k∆y k ≤ K and k∆x k ≤ G . Consider a Lyapunov function candidate for the estimation error ezx (t) T
Ve (ezx (t)) = ezx (t) P1 ezx (t)
(19)
where P1 is the unique s.p.d. solution of the following Lyapunov equation with decay rate β P1 A11,i + AT11,i P1 ≤ −2βP1
(20)
with the derivative of (19) along the system trajectory V˙ e =
nr X
h i T ˆ hi (α(t)) ezx (t) P1 A11,i + AT11,i P1 ezx
i=1
+ 2∆Tx P1 ezx (t)
(21)
Denoting Q = 2βP1 and substituting into (21) yields T V˙ e ≤ −ezx (t) Qezx (t) + 2∆Tx P1 ezx (t)
(22)
Using the well-known inequality 2xT y ≤ 1 xT x+yT y and choosing = 1 it follows that 1 T T V˙ e ≤−ezx (t) Qezx (t)+ ∆Tx ∆x + (P1 ezx (t)) P1 ezx (t) T T V˙ e ≤−ezx (t) Qezx (t) + k∆x k2 + ezx (t) P1 P1 ezx (t) T 2 2 V˙ e ≤−ez (t) (Q − P1 P1 )ez (t) + Lgx kez k (23) x
x
Assuming that the system is already on the sliding surface (ey (t) = 0), implies that T 2 V˙ e ≤ −ezx (t) (Q − Lgx − P1 P1 )ezx (t)
(24)
2 − P1 P1 ) Hence, (16a) is asymptotically stable if (Q − Lgx 2 2 is positive definite. With Lgx = γ, the condition (Q−Lgx − P1 P1 ) > 0 can be expressed as LMI [20] " # Q − γ P1 >0 (25) P1 I
Substituting (10) in (20) and introducing a change of variables Ni = P1 Li [20], yields the LMI ¯ obs +Ni A ¯ obs + A ¯ obs T P1 + A ¯ obs Ni T ≤ −2βP1 P1 A 11,i 211,i 11,i 211,i (26) Li can be obtained from Li = P−1 N if the LMIs (25) and i 1 (26) are feasible for a given γ and β and a stable observer is obtained. Alternatively, γ could be maximized for a given β [14]. Consider a second quadratic Lyapunov function candidate T
Vs (ey (t)) = ey (t) P2 ey (t)
(27)
Derivating along the trajectory yields T V˙ s = 2ey (t) P2
nr X
hi (α(t)) ˆ A21,i ezx (t) + As22 ey (t)
i=1
+ν(t) − [D 2,i E 2,i ]
ξ(t) fa (t)
T
+ 2ey (t) P2 ∆y
(28)
From (13) we obtain T
2ey (t) P2 ν(t) ≥ −2kP2 ey (t)kρ
(29)
and considering (14), it holds that T
2ey (t) P2 As22 ey (t) = −ey (t)Q2 ey (t) ≤ 0
(30)
An upper bound kezx (t)k ≤ ζ exists because in (16a) P nr ˆ 11,i is stable and ∆x is bounded. Applying i=1 hi (α(t))A further manipulations, substituting from (29) and (30) as well as considering assumption 1, yields nr X V˙s ≤2kP2 ey (t)k hi (α(t)) ˆ [kA21,i kζ (31) i=1 + k[D 2,i E 2,i ]k Ξ + k∆y k − ρ]
Because each matrix norm is maximal for some i it holds that V˙s ≤2kP2 ey (t)k [kA21,max kζ (32) + k[D 2,max E 2,max ]k Ξ + K − ρ]
which is negative definite by large enough choice of ρ. The observer gains in the original coordinates (4) are given by [11] A12,i −1 −1 0(n−p)×p Gl,i = Ti , Gn,i = Ti (33) A22,i − As22 Ip B. Estimation of actuator faults via equivalent injection principle The method for estimating actuator faults presented in the following is directly based on the equivalent output injection concept [7]. Assume that an observer with the structure given in (12) has been designed. During the sliding motion ey (t) = 0 and e˙ y (t) = 0 holds. From (24) it follows that ezx (t) → 0. This implies ∆y → 0 and thus (16b) becomes nr X ˆ hi (α(t))[ν(t) − D 2,i ξ(t) − E 2,i fa (t)] (34) 0= i=1
Pnr ˆ Denote D(α(t)) = 2,i and E (α(t)) = i=1 hi (α(t))D Pnr ˆ h ( α(t))E . Considering ν(t) ≈ ν eq (t) yields i 2,i i=1 ξ(t) ν eq (t) ≈ [D(α(t)) E (α(t))] (35) fa (t) ν eq (t) is the so-called equivalent output injection signal which represents the average behaviour of the discontinuous component ν(t) and the effort necessary to maintain the motion on the sliding surface. According to [11], ν eq (t) can be computed online by using the continuous approximation ( P ey (t) if ey (t) 6= 0 −ρ kP2 e2y (t)k+δ (36) ν eq (t) = 0 otherwise where δ is a small positive scalar chosen to remove chattering in the sliding motion. ν eq (t) can be used for fault detection within the scope of FDI (see [7],[11],[12]). Depending on the disturbance and actuator fault gain matrices, faults and uncertainties can be reconstructed by ˆ ξ(t) + (37) ˆfa (t) = [D(α(t)) E (α(t))] ν eq (t) where (·)+ denotes a left pseudo-inverse that has to sat+ isfy [D(α(t)) E (α(t))] [D(α(t)) E (α(t))] = I. If no such pseudo inverse exists a pseudo inverse E (α(t))+ can be used for fault estimation with the threshold kE (α(t))+ D(α(t))kΞξ (t). The task of finding E (α(t))+ such that the threshold is minimal can be posed as a LMI optimization problem [12].
IV. S IMULATION E XAMPLE The following example deals with a pendulum on a cart ˙ (Fig. 1) with the states x(t) = [Θ(t) Θ(t) xc (t) x˙ c (t)]T considered in [14]. The nonlinear dynamics are given by x2 (t) 2 −g sin(x1 (t))−mlax2 (t) sin(2x1 (t))/2 4l/3−mla cos2 (x1 (t)) ˙ x(t) = x4 (t) mag sin(2x1 (t))/2+alx2 (t)2 sin(x1 (t))4m/3 4/3−ma cos2 (x1 (t))
0
+
−a cos(x1 ) 4l/3−mla cos2 (x1 (t))
(38)
(u(t) − fc (t))
0
4a/3 4/3−ma cos2 (x1 (t))
where a = 1/ (m + M ), m = 2 (kg), M = 8 (kg), l = 0.5 (m), g = 9.81 sm2 and fc (t) = µc FN sign (x4 (t)) (N) is the Coulomb friction between the cart and the surface with the coefficient of friction µc = 0.05 and normal force FN = g/a (N). Compared to [14] the friction force fc (t) has been modified. A TS fuzzy model is obtained using the concept of sector nonlinearity. The TS fuzzy model (39) exactly represents the dynamics of (38) under the assumption ˙ that Θ(t) is bounded such that x2 (t) ∈ [−7π ; 7π](rad/s). ˙ x(t) =
2 2 X 2 X 2 X X
w1,j w2,k w3,r w4,n
j=1 k=1 r=1 n=1
· +
0 1 − gl bj ck −mabj dr en 0 0 4 magbj dr en mlab j dr 3 0 − al bj en (u(t) − fc (t)) 0 4 ab j 3
0 0 0 0 x(t) 0 1 0 0
(39)
with the following sector functions w1,1 (α1 (t)) =
1 4/3−ma cos2 (x1 (t))
( w2,1 (α1 (t)) =
b1 − b2 sin(x1 (t))−c2 x1 (t) (c1 −c2 )x1 (t)
1
− b2 if x1 (t) 6= 0
otherwise x2 (t) sin (x1 (t)) − d2 w3,1 (α1 (t), α2 (t)) = d1 − d2 cos (x1 (t)) − e2 w4,1 (α1 (t)) = e1 − e2
(40)
1 and the sector bounds b1 = 4/3−ma , b2 = 3/4, c1 = −1, c2 = −0.22, d1 = x2max , d2 = x2min , e1 = 1 and e2 = −1. Note wk,2 (α(t)) = 1 − wk,1 (α(t)) and α(t) = [x1 (t) x2 (t)]T . By aggregating the membership functions hi (α(t)) and corresponding matrices Ai and Bi , (39) can be rewritten as a combination of nr = 24 = 16 linear submodels as follows nr X ˙ x(t) = hi (α(t)) [Ai x(t) + Bi (u(t) − fc (t))] (41) i=1
y(t) =Cx(t)
Fig. 1.
Pendulum on a cart
wherein the constant output matrix C is used to allow a direct comparison to [14]. 1 0 0 0 C= 0 0 1 0 (42) 0 0 0 1 Letting Di = Bi , the observer in the form (4) can be designed. From (42) it follows that the estimate x ˆ2 (t) has ˆ to be used for α(t). In this case study, the discontinuous control ν(t) is replaced by its continuous approximation ν eq (t) from (36). With suitable Lipschitz gain Lgx = 0.05 and the design parameters δ = 10−4.3 , ρ = 28.39, the poles p = [−10 −10 −10] of As22 and a decay rate β = 1 the LMI synthesis presented in Sec. III-A yields the gain matrices in the original coordinates. The intended use of the TS-SMO (4) is FDI. Due to the structure of the disturbance, actuator faults have to be larger than the disturbance (Ξfa > Ξξ ) to be detectable. Hence we reconstruct the (unknown) friction force based on (35) !+ nr X ˆfc (t) = ˆ hi (α(t))D ν eq (t) (43) 2,i i=1
In a simulation study, the TS-SMO (4), FSM (fuzzy sliding mode) and DLSM (dominant linear fuzzy sliding mode) observers from [14] are applied to the nonlinear uncertain system (38) with the initial conditions x(0) = [0.18, 0, 0, 0]T ˆ (0) = [0.38, 0.2, 1, 0.1]T . The FSM and DLSM are and x based on 3 linear models for x1 (t) = [− π4 , 0, + π4 ]. Figure 2 shows the input u(t), the simulated states and the friction force fc (t). The state estimation errors are shown in Fig. 3. The TS-SMO (4) reconstructs the states accurately despite the unknown friction force fc (t). While the DLSM observer performs well in its limited designated operating range the FSM observer has some difficulty tracking the system states. As expected the DLSM and FSM observers are not suitable for a larger operating range (Fig. 2-3, t > 4s). Furthermore, the estimation error efc (t) = fc (t) − ˆfc (t) obtained from the TS-SMO is shown in Fig. 3. After reaching the sliding surface (t = 1s), the TS-SMO accurately reconstructs the friction force fc (t). V. CONCLUSIONS AND FUTURE WORKS This paper presented a sliding mode fuzzy observer for disturbance and actuator fault estimation in the presence of bounded uncertainties. Based on sliding mode theory [11], necessary conditions for the existence of a robust observer were derived. An explicit design procedure using numeric methods was given. The performance of this observer was evaluated in a simulation study dealing with a pendulum
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Fig. 3. Error signals for TS-SMO (4) (solid) as well as FSM (dotted) and DLSM (dash-dotted) observers [14] applied to the nonlinear system (38). ˆ Only the TS-SMO provides the estimate . fc (t).
on a cart. Furthermore the proposed observer was compared with two observers presented in [14]. Future work will deal with the extension of the proposed observer FDI approach to sensor fault estimation and application to a hydrostatic transmission of a wheel loader [21]. In this application scenario model, uncertainties strongly depend on the operating phase of the system, hence a detection of the operating phase [22] should be included in the FDI system. R EFERENCES [1] J. Chen and R.J. Patton, Robust model-based fault diagnosis for dynamic systems, Kluwer Academic Publishers, Norwell, MA; 1999. [2] P.M. Frank, Fault Diagnosis in Dynamic Systems Using Analytical and Knowledge-based Redundancy - A Survey and Some New Results, Automatica, 26(3), 1990, pp. 459-474. [3] S.X. Ding, Model-based Fault Diagnosis Techniques - Design Schemes, Algorithms, and Tools, Springer, Berlin; 2008. [4] R. Isermann, Fault-Diagnosis Systems - An Introduction from Fault Detection to Fault Tolerance, Springer, Berlin; 2006.
[5] M. Blanke, M. Kinnaert, J. Lunze and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, 2nd. ed, Springer, Heidelberg; 2006. [6] S. Montes de Oca and V. Puig, "Adaptive Threshold Generation for Passive Robust Fault Detection Using Interval Observers", in European Control Conference (ECC), Budapest, Hungary, 2009, pp. 3094-3099. [7] C. Edwards, S.K. Spurgeon, and R.J. Patton, Sliding mode observers for fault detection and isolation. Automatica, 36(4), 2000, pp.: 541553. [8] V.U. Utkin, Sliding Modes in Control and Optimization, Springer, Berlin; 1992. [9] R.J. Patton, S. Klinkhieo, and D. Putra, "A Fault-Tolerant Control Approach to Friction Compensation", in European Control Conference (ECC), Budapest, Hungary, 2009, pp. 3677-3682. [10] Z. Gao, X. Shi, and S.X. Ding, Fuzzy State/Disturbance Observer Design for TS Fuzzy Systems With Application to Sensor Fault Estimation, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38(3), June 2008, pp. 875-880. [11] C. Edwards and S.K. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor and Francis, London; 1998. [12] X.G. Yan and C. Edwards, Nonlinear robust fault reconstruction and estimation using a sliding mode observer, Automatica, 43(9), 2007, pp. 1605-1614. [13] K. Tanaka and H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, John Wiley & Sons, Inc.; 2001. [14] P. Bergsten, R. Palm, and D. Driankov, Observers for Takagi-Sugeno fuzzy systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 32(1), August 2002, pp. 114-121. [15] A. Akhenak, M. Chadli, J. Ragot, and D. Maquin, "Design of sliding mode unknown input observer for uncertain Takagi-Sugeno model", in Mediterranean Conference on Control and Automation, Athens, Greece, 2007. [16] T. Takagi and M. Sugeno, Fuzzy Identification of Systems and its Applications to Modeling and Control, in IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 15(1), 1985, pp. 116-132. [17] T.A. Johansen, K.J. Hunt, P.J. Gawthrop, and H. Fritz, Off-equilibrium linearisation and design of gain-scheduled control with application to vehicle speed control, Control Engineering Practice, vol. 6, 1998, pp. 167-180. [18] A. Sala, T. M. Guerra, and R. Babuska, Perspectives of fuzzy systems and control, Fuzzy Sets & Systems (special issue: 40th Anniversary of Fuzzy Sets), 156(3), 2005, pp.432-444. ˙ [19] B. Walcott and S.H. Zak, State observation of nonlinear uncertain dynamical systems, IEEE Transactions on Automatic Control, 32(2), February 1987, pp. 166-170. [20] S.P. Boyd, L.E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System & Control Theory, SIAM, Philadelphia, 1994. [21] P. Gerland, H. Schulte, and A. Kroll, "Comparative study of nonlinear observer concepts for fault diagnosis using the example of hydrostatic drive trains" (in German), in Proc. 19. Workshop Computational Intelligence, vol. 19, 2009, pp. 61-74. [22] P. Gerland, H. Schulte, and A. Kroll, "Probability-based global state detection of complex technical systems and application to mobile working machines", in European Control Conference (ECC), Budapest, Hungary, 2009, pp. 1269-1274.
