Adaptive Sliding-Mode-Observer-Based Fault ... - Semantic Scholar
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
4029
Adaptive Sliding-Mode-Observer-Based Fault Reconstruction for Nonlinear Systems With Parametric Uncertainties Xing-Gang Yan and Christopher Edwards, Member, IEEE
Abstract—In this paper, a class of nonlinear systems with uncertain parameters is considered. A novel adaptive law is designed to identify unknown parameters under the assumption that the time derivative of some of the outputs is measurable. Then, a sliding-mode observer is proposed to estimate the system state variables. By using the inherent features of sliding-mode observers, a fault-reconstruction scheme is proposed which can be implemented online. The proposed reconstruction signal can approximate the fault signal to any required accuracy even in the presence of uncertain parameters. A simulation example for a magnetic-levitation system is given to illustrate the feasibility and effectiveness of the proposed scheme. Index Terms—Adaptive systems, fault diagnosis, nonlinear estimation, sliding modes.
I. I NTRODUCTION
W
ITH the advancement of modern technology, more complicated automation schemes are being applied to real systems. The complexity comes from strong nonlinearities and uncertainties which exist in real systems. Such classes of systems are vulnerable to faults because human-operator interaction is reduced/removed. The effects of faults can escalate and may corrupt the system to such an extent that the automated system becomes useless. Therefore, prompt detection and isolation of faults has become important. This has motivated much research in the field (see, e.g., [1], [2], [7], [18], [20]). Compared with linear systems, nonlinear systems are much more complicated, which makes their study more difficult. Due to the variety of nonlinear features, it is impossible to study nonlinear systems as systematically as linear systems, but it is possible and useful to study certain classes of nonlinear systems. In recent decades, some control-inspired approaches, for instance, sliding-mode techniques [5], modern differential geometric approaches [21], and adaptive control [13], [30], have been successfully incorporated within the observer-based Fault Detection and Isolation (FDI) approach. In particular, modern geometric approaches have been shown to be effective in the investigation of a class of nonlinear systems [11], [19]. When a system has unknown parameters, adaptive control techManuscript received January 15, 2008; revised July 22, 2008. First published August 19, 2008; current version published October 31, 2008. The authors are with the Control Systems Research Group, Department of Engineering, University of Leicester, Leciester, LE1 7RH, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2008.2003367
niques can be employed to estimate the unknown parameters [19], [27], [30]. It is well known that sliding-mode control exhibits high robustness to matched uncertainty [10]. Moreover, sliding-mode observer-based approaches have been widely applied to real systems, see [16] and [26] for example. Recently, sliding-mode techniques have been successfully used for fault detection and isolation [3], [5], [6], [8], [22], [28], and it has been proven to be an effective way to estimate/reconstruct system faults. A “precise” fault reconstruction approach is proposed in [5] based on the equivalent output-error-injection concept. Based on the work in [5], a sensor fault-reconstruction scheme has been proposed in [22]. It should be noted that, in [5] and [22], only linear systems are considered, and uncertainty is not included in the formulation. Later, a fault-estimation approach was proposed for linear systems with uncertainty by Tan and Edwards [23] which focused on minimizing the L2 gain between the uncertainty and the fault-reconstruction signal using linear matrix inequalities. More recently, a robust actuator faultreconstruction scheme has been presented in [28] using the characteristics of the uncertain structure and fault distribution. Jiang et al. [12] proposed a fault-estimation scheme for a class of systems with uncertainty, and a robust fault-detection method for nonlinear systems with disturbances was considered in [6]. However, in these papers involving uncertainty, nearly all are concerned with fault estimation instead of “precise” reconstruction, and all require that the uncertainties are bounded with known bounds. Although a “precise” reconstruction scheme is proposed in [28], it is assumed that the structural matrix for the uncertainty satisfies a strong structural condition. In this paper, fault reconstruction is considered for a class of nonlinear systems with uncertain parameters, where the faultdistribution vector and the structure matrix of the uncertainty are allowed to be functions of the system output and input. Under the assumption that the time derivative of a subset of the system outputs is available, a novel adaptive update law is given. The unknown parameters are estimated, and a sliding-mode observer is developed using the estimated parameters. Based on the sliding-mode observer, a fault-reconstruction scheme, which can be implemented online, is proposed to reconstruct the fault signal. The reconstruction signal can approximate the fault signal to any required accuracy. Furthermore, the structural condition for the distribution of the uncertainties is removed. The uncertain parameters can be arbitrarily large, and it is not necessary to know the bounds on them a priori. This is an improvement as compared with the existing work in
0278-0046/$25.00 © 2008 IEEE
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4030
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
[6], [12], [23], and [28], where the uncertain bounds are used in the observer design and the fault-estimation/reconstruction scheme. A simulation demonstrates the practicality of the proposed approach. Notation: The notation is broadly as used in [19]. In addition, for a square matrix A, λmax (A) denotes the maximum eigenvalue of A, and A > 0 denotes a symmetric positive definite matrix. The symbol In represents the nth-order unit matrix, and R+ represents the set of nonnegative real numbers. The set of n × m matrices with elements in R will be denoted by Rn×m . A function f (x1 , x2 , . . . , xn1 , y1 , y2 , . . . , yn2 ) is also written as f (x, y), where x = [x1 , x2 , . . . , xn1 ]T ∈ Rn1 and y = [y1 , y2 , . . . , yn2 ]T ∈ Rn2 . The symbol · denotes the Euclidean norm or its induced norm. II. P RELIMINARIES The following results will be used in the later analysis. Lemma 1: Assume that x = col(x1 , x2 , . . . , xN ), where xi ∈ Rni for i = 1, 2, . . . , N . Then, the following expressions hold: 1) x ≤ xT sgn(x). N N 2 2) i=1 (di xi ) ≤ ( i=1 di )x, where di is a positive constant for i = 1, 2, . . . , N . Proof: 1) Let xi = col(xi1 , xi2 , . . . , xini ). It follows that for i = 1, 2, . . . , N
xT i sgn(xi )
= [ xi1
xi2
···
⎡ sgn(x ) i1 ⎢ sgn(xi2 ) xini ] ⎢ .. ⎣ .
⎤
Therefore,
xT 2
···
⎤ ⎥ ⎥ ⎦
sgn(xN ) T T = xT 1 sgnx1 + x2 sgnx2 + xN sgnxN
≥ x1 + x2 + · · · + xN ≥ x. Hence, conclusion 1) follows. 2) Taking p = q = 1/2 in Hölder’s inequality [15], p1 N q1 N N
1 1 p q + =1 |ai bi | ≤ |ai | |bi | , p q i=1 i=1 i=1 it follows that for di ≥ 0 N 12 N 12
N
N
2 2 di xi ≤ di xi = d2i x i=1
i=1
i=1
and conclusion 2) follows.
x˙ i = Fi xi + Gi (y, u) + Φi (y, u)θ + Ψi (y, u)f (t)
(1)
yi = Ci xi ,
(2)
i = 1, 2, . . . , p
where x = col(x1 , . . . , xp ) ∈ Rn and xi := col(xi1 , xi2 , . . . , xini ) ∈ Rni are system state variables; u ∈ U ⊂ Rm and col(y1 , y2 , . . . , yp ) := y ∈ Y ⊂ Rp are the inputs and the outputs, respectively. Assume the pair (Fi , Ci ) has Brunovsky standard form [19] as follows: ⎡
0 0 ⎢1 0 ⎢. . Fi = ⎢ ⎢ .. .. ⎣0 0 0 0
··· ··· .. .
0 0 .. .
0 0 .. .
··· ···
1 0
0 1
⎤ 0 0⎥ .. ⎥ ⎥ := .⎥ 0⎦
Ai1 Ai2
0 0
ni ×ni
0 (3)
Ci = [ 0
···
0 1 ]1×ni
(4)
where Gi (x, u) ∈ Rni , Φi (x, u) ∈ Rni ×q , and Ψi (x, u) ∈ Rni ×r have smooth elements with p ≥ q + r for i = 1, 2, . . . , p. The symbol θ ∈ Rq denotes unknown constant parameters. The unknown function vector f (t) ∈ Rr represents faults experienced by the system and is assumed to satisfy (5)
where ρ(·) is known. Remark 1: The limitation that (Fi , Ci ) has Brunovsky form can be replaced by the condition that (Fi , Ci ) is observable, since a simple coordinate transformation can be used to transfer the system (1) and (2) to Brunovsky form if (Fi , Ci ) is observable. Remark 2: It should also be noted that the approach developed in this paper can be applied to a class of nonlinear systems which can be transformed to the system representation (1) and (2). Geometric conditions have been proposed in [19], under which more general nonlinear systems can be transformed to the structure in (1) and (2). It is obvious that the matrix pair (Fi , Ci ) is observable, and ˜ i Ci is stable. ˜ i such that Fi − K thus, there exists a matrix K ˜ Therefore, for any Qi > 0, the Lyapunov equation
= |xi1 | + |xi2 | + · · · + |xini | ≥ xi .
xT sgn(x) = [ xT 1
Consider a nonlinear system described by
f (t) ≤ ρ(t)
⎥ ⎥ ⎦
sgn(xini )
⎡ sgn(x ) 1 sgn(x2 ) ⎢ ⎢ xT .. N ]⎣ .
III. P ROBLEM F ORMULATION
i=1
˜ i Ci )T P˜i + P˜i (Fi − K ˜ i Ci ) = −Q ˜i (Fi − K
(6)
has a unique solution P˜i > 0 for i = 1, 2, . . . , p. Assumption 1: There exist functions Γi (y, u) with i = 1, 2, . . . , p such that (Ψi (y, u))T P˜i = Γi (y, u)Ci ,
i = 1, 2, . . . , p
(7)
where Ψi (y, u) is given by (1). Remark 3: Assumption 1 is a structural constraint on the fault distribution Ψi (·). However, it should be noted that there
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
YAN AND EDWARDS: ADAPTIVE SLIDING-MODE-OBSERVER-BASED FAULT RECONSTRUCTION
are no structural constraints on the distribution associated with the parametric uncertainties. This is in comparison with the work in [28] for example. In particular, if the ith subsystem is first order, Assumption 1 is satisfied automatically. Partition P˜i as ˜ Pi1 P˜i = ˜ T P
P˜i2 P˜i3
i2
Ti :=
Ini −1
−1 ˜ Pi2 −P˜i1
0
1
i = 1, 2, . . . , p.
(8)
Then, in the new coordinate system z, system (1) and (2) has the following form: z˙i = Ti
Ai1 Ai2
0 T −1 z + Ti Gi (y, u) 0 i i
+ Ti Φi (y, u)θ + Ti Ψi (y, u)f (t) yi = Ci zi ,
⎡
0(p−q−r)×q ⎢ ΨII (y, u) ] = ⎣ N1 (y, u) 0r×q
,
system (11) and (12) is not unique. A transformation has been provided in (8). Assumption 2: There exists a nonsingular matrix W ∈ Rp×p such that
W [ ΦII (y, u)
where P˜i1 ∈ R(ni −1)×(ni −1) and P˜i3 ∈ R. For system (1) and (2), introduce a coordinate transformation z = T x, where T := diag{T1 , T2 , . . . , Tp } and
4031
i = 1, 2, . . . , p
(9) (10)
where z = col(z1 , z2 , . . . , zp ) with zi := (zi1 , zi2 , . . . , zini ). −1 ˜ Pi2 Ai2 is stable. Let It follows from [29] that Ai1 − P˜i1 −1 Ki := P˜i1 P˜i2 . By reordering the state variables, system (9) and (10) can be further described as z˙I = (A1 − KA2 )zI + (A1 − KA2 )Ky + GI (y, u) + ΦI (y, u)θ
(11)
y˙ = A2 zI + A2 Ky + GII (y, u) + ΦII (y, u)θ + ΨII (y, u)f (t)
⎤ 0(p−q−r)×r ⎥ 0q×r ⎦ N2 (y, u)
where N1 (·) ∈ Rq×q and N2 (·) ∈ Rr×r are both nonsingular in Y × U. Remark 5: Assumption 2 implies that the uncertain parameters and the faults are separable by applying a further nonsingular transformation to system (11) and (12). The limitation in Assumption 2 can be interpreted as, the matrix [ΦII (y, u) ΨII (y, u)] must be block-diagonalizable by elementary row transformations. Remark 6: It should be noted that the analysis earlier requires that ni > 1. If, for example, nk = 1, then, xk = yk in (1) is a scalar, and in this case, the kth subsystem will be directly merged with subsystem (12), and does not contribute to the differential equations in (11). All the analysis which follows and the results obtained are still valid by appropriate modifications to the expressions. The objective of this paper is to design a sliding-mode observer for system (1) and (2). Then, based on the sliding observer, an approach is proposed to reconstruct the fault signal f (·) precisely. A novel adaptive law is given to estimate the parameters. System (1) and (2) is equivalent to system (11) and (12), i.e., the former can be transformed to the latter by a nonsingular linear transformation, but the form and specific structure in (11) and (12) will be shown to be useful for slidingmode-observer design. IV. A DAPTIVE S LIDING -M ODE -O BSERVER D ESIGN
(12)
where zI = col(z11 , . . . , z1(n1 −1) , . . . , zp1 , . . . , zp(np −1) ) ∈ Rn−p , y = col(z1n1 , . . . , zpnp ), A1 = diag{A11 , . . . , Ap1 }, A2 = diag{A12 , . . . , Ap2 }, and K = diag{K1 , K2 , . . . , Kp }. Obviously, A1 − KA2 = diag{A11 − K1 A12 , . . . , Ap1 − Kp Ap2 }
In this section, a sliding-mode observer will be designed based on a novel adaptive law. Let zy = W y, where W is defined in Assumption 2. The analysis in the previous section shows that, under Assumptions 1 and 2, the system (1) and (2) can be described by z˙I = (A1 − KA2 )zI + (A1 − KA2 )KW −1 zy + GI (y, u) + ΦI (y, u)θ
is stable, and thus, for Q > 0(Q ∈ R Lyapunov equation
(n−p)×(n−p)
(A1 − KA2 )T P + P (A1 − KA2 ) = −Q
), the
(13)
has a unique solution P > 0. Remark 4: Since the matrix T has been given in (8), the functions GI (·), GII (·), ΦII (·), ΨI (·), and ΨII (·) can be obtained directly using the coordinate transformation z = T (x), and thus, the system (11) and (12) is well defined and can be obtained from (1) and (2) directly. It should be noted that the transformation matrix T which transforms system (1) and (2) to
z˙y = W A2 zI + W A2 KW −1 zy + W GII (y, u) ⎡ ⎤ ⎡ ⎤ 0 0 ⎦ f (t) + ⎣ N1 (y, u) ⎦ θ + ⎣ 0 0 N2 (y, u) y = W −1 zy
(14)
(15) (16)
where the matrix W and the matrix functions N1 (·) and N2 (·) are determined by Assumption 2. In order to further exploit the system structure, let zy = col(zyo , zyθ , zyf ), where zyo ∈ Rp−q−r , zyθ ∈ Rq ,
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4032
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
and zyf ∈ Rr , and partition matrices W , W A2 , and W A2 K in a compatible way with zy as follows: ⎡
⎤ W11 W := ⎣ W21 ⎦ W31
⎤ W12 W A2 := ⎣ W22 ⎦ W32 ⎡
⎤ W13 W A2 K := ⎣ W23 ⎦ . W33 (17) ⎡
Then, system (14)–(16) can be further rewritten as z˙I = (A1 − KA2 )zI + (A1 − KA2 )KW −1 zy + GI (y, u) + ΦI (y, u)θ (18) z˙yo = W12 zI + W13 y + W11 GII (y, u) (19) z˙yθ = W22 zI + W23 y + W21 GII (y, u) + N1 (y, u)θ (20) z˙yf = W32 zI + W33 y + W31 GII (y, u) + N2 (y, u)f (21) ⎡ ⎤ zyo y = W −1 ⎣ zyθ ⎦ . (22) zyf Now, consider a dynamical system zI + (A1 − KA2 ) Ky zˆ˙ I = (A1 − KA2 )ˆ + GI (y, u) + ΦI (y, u)θˆ
(23)
zˆ˙ yo = W12 zˆI + W13 y − Λo W11 (y − yˆ) + W11 GI I (y, u) + νo zˆ˙ yθ = W22 zˆI + W23 y − Λθ W21 (y − yˆ) + W21 GI I (y, u) + N1 (y, u)θˆ + νθ zˆ˙ yf = W32 zˆI + W33 y − Λf W31 (y − yˆ)
(24) (25)
+ W31 GI I (y, u) + νf ⎡ ⎤ zˆyo yˆ = W −1 zˆy := W −1 ⎣ zˆyθ ⎦ zˆyf
(26) (27)
where yˆ ∈ Rp is the output of the dynamical system with zˆyo ∈ Rp−q−r , zˆyθ ∈ Rq , and zˆyf ∈ Rr ; the matrices Λo ∈ R(p−q−r)×(p−q−r) , Λθ ∈ Rq×q , and Λf ∈ Rr×r are parameters which by design are stable. The estimate of θ, ˆ is given by the following adaptive law: denoted by θ, ˙ θˆ = −β(y, u) N1 (y, u)θˆ + W22 zˆI + W23 y + W21 GII (y, u) − z˙yθ
(28)
where β(·) ∈ Rq×q is a matrix design parameter and col(νo , νθ , νf ) := ν is the output error injection term defined by νo = ko (y, yˆ)sgn(zo − zˆo ) ˆ νθ = kθ (y, yˆ, u, θ)sgn(z ˆθ ) θ −z νf = kf (t, y, yˆ, u)sgn(zf − zˆf )
(29) (30) (31)
where sgn denotes the usual sign vector function and scalars ko , kθ (·), and kf (·) are to be determined. Remark 7: The adaptive law (28) requires that z˙yθ is available. Obviously, z˙yθ is measurable if y˙ is measurable, which is an assumption that has been employed by many authors (see, for example, [9]). Furthermore, the exact differentiator
proposed in [17] can be used to estimate the time derivative of the system output in finite time if y˙ is not available. ˆ eyo = zyo − zˆyo , eyθ = Define ez = zI − zˆI , eθ = θ − θ, zyθ − zˆyθ , and eyf = zyf − zˆyf . Since eyo = W11 (y− yˆ),
eyθ = W12 (y− yˆ),
eyf = W13 (y − yˆ)
it follows from (14)–(16) and (23)–(26) that the error dynamical equation is described by e˙ z e˙ θ e˙ yo e˙ yθ e˙ yf
= (A1 − KA2 )ez + Φ1 (y, u)eθ = − β(y, u)W22 ez − β(y, u)N1 (y, u)eθ = W12 ez + Λo eyo − νo = W22 ez + Λθ eyθ + N1 (y, u)θˆ − νθ = W32 ez + Λf eyf + N2 (y, u)f − νf .
(32) (33) (34) (35) (36)
The error systems (32) and (33) will be considered first. Theorem 1: Consider systems (32) and (33) and assume that (13) holds. Then, systems (32) and (33) are asymptotically stable if, for any (y, u) ∈ Y × U, the matrix M = Q − (ΦI (y, u))T P + β(y, u)W22 − P ΦI (y, u) + (β(y, u)W22 )T β(y, u)N1 (y, u) + N1T (y, u)β T (y, u) is positive definite, where P and Q satisfy (13) and β(·) ∈ Rq×q is the design parameter in (28). Proof: Consider a candidate Lyapunov function for system (32) and (33) as T V (ez , eθ ) = eT z P ez + eθ eθ .
Then, the time derivative of V along the trajectories of system (32) and (33) is given by T T V˙ |(32)−(33) = eT ez z (A1 −KA2 ) P +P (A1 −KA2 ) T + 2eT z P Φ1 (y, u)eθ −2eθ β(y, u)W22 ez T T − eT θ β(y, u)N1 (y, u)eθ −eθ (β(y, u)N1 (y, u)) eθ T T eθ ≤ −eT z Qez + 2ez P Φ1 −(βW22 ) T T eθ −eT θ βN1 + N1 β ez = −[ eT eT . z θ ]M e θ
Hence, the conclusion follows from the assumption that M is positive definite in ∈ Y × U. Remark 8: Theorem 1 shows that, under certain conditions, the error dynamical system (32) and (33) is asymptotically stable, which means that limt→∞ ez (t) = 0 and limt→∞ eθ (t) = 0. Consequently, the errors ez and eθ are bounded so that W12 ez (t) ≤ χ1 ,
W22 ez ≤ χ2 ,
W32 ez ≤ χ3 (37)
where χi for i = 1, 2, and 3 are positive constants. Remark 9: A necessary condition for matrix M in Theorem 1 to be positive definite is that the function matrix β(·) in (28) must be designed such that β(y, u)N1 (y, u) + N1T (y, u)β T (y, u) is positive definite in (y, u) ∈ Y × U.
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
YAN AND EDWARDS: ADAPTIVE SLIDING-MODE-OBSERVER-BASED FAULT RECONSTRUCTION
Next, for the dynamical error equations (32)–(36), consider the following sliding surface: S = {col(ez , eθ , eyo , eyθ , eyf )|eyo = 0, eyθ = 0, eyf = 0} . (38) The reduced-order sliding dynamics associated with the sliding surface (38) can be described by (32) and (33). Theorem 1 has shown that, under certain conditions, the sliding-mode dynamics are asymptotically stable when the sliding motion takes place. In what follows, the objective is to study reachability issues. Specifically, a reachability condition will be developed such that the error dynamics in (32)–(36) can be driven to the sliding surface (38) in finite time, and a sliding motion maintained on it thereafter. Theorem 2: Assume that (37) holds. Then, the trajectories of the dynamical error system (32)–(36) can be driven to the sliding surface (38) in finite time if the gains ko , kθ (·), and kf (·) in (29)–(31) satisfy ko ≥ χ1 + η1 ˆ ≥ χ2 + kθ (y, u, θ) N1 (y, u)θˆ + η2
(39)
kf (t, y, u) ≥ χ3 + N2 (y, u) ρ(t) + η3
(41)
(40)
for some positive constants η1 , η2 , and η3 , where χi for i = 1, 2, and 3 satisfy (37). Proof: Let ey = col(eyo , eyθ , eyf ). It follows from (34)–(36) that T T T eT y e˙ y = eyo e˙ yo + eyθ e˙ yθ + eyf e˙ yf T T T = eT yo W12 ez + eyo Λo eyo − eyo νo + eyθ W22 ez + eT Λθ eyθ + eT N1 (y, u)θˆ − eT νθ + eT W32 ez
+
yθ eT yf Λf eyf
+
yθ yθ yf T T eyf N2 (y, u)f (t) − eyf νf .
(42)
Since the design parameters Λo , Λθ , and Λf are all chosen as negative definite, it follows that
Then, substituting (29)–(31) into (42) T T T eT y e˙ y ≤ eyo W12 ez − ko eyo sgn(eyo ) + eyθ W22 ez T T ˆ + eT yθ N1 (y, u)θ − kθ eyθ sgn(eyθ ) + eyf W32 ez
(43)
From (5), (37), and (43), and from part 1) of Lemma 1, it follows that T eT y e˙ y ≤ eyo χ1 − ko eyo sgn(eyo ) + eyθ χ2 ˆ − kθ (y, u, θ)e ˆ T sgn(eyθ ) + eyθ N1 (y, u)θ yθ + eyf χ3 + eyf N2 ρ(t) − kf eT yf sgn(eyf ) ˆ eyθ = − (ko − χ1 )ey0 − kθ (·) − χ2 − N1 (·)θ
− (kf (t, y, u) − χ3 − N2 (y, u)ρ(t)) eyf .
Then, from (39)–(41) and part 2) of Lemma 1, it follows that eT y e˙ y ≤ − η1 ey0 − η2 eyθ − η3 eyf ≤ − (η12 + η22 + η32 )ey := −ηey . This shows that a traditional reachability condition [4], [24] is satisfied, and thus, the error system can be driven to the sliding surface in finite time and a sliding motion maintained thereafter. Hence, the conclusion follows. From sliding-mode theory, Theorems 1 and 2 have shown that system (23)–(27) is an asymptotic observer for the system (14)–(16). V. F AULT R ECONSTRUCTION In this section, it is assumed that an adaptive sliding-mode observer as proposed in Section IV has been designed. Then, by using a modified equivalent output error injection [5], the fault f (t) will be reconstructed based on the observer proposed in the last section. An important feature of the sliding-mode observer is that the error dynamical system can be driven to the sliding surface in finite time and a sliding motion maintained thereafter. When the sliding motion takes place ey (t) = 0,
e˙ y (t) = 0
and in particular eyf (t) = 0,
e˙ yf (t) = 0.
(44)
During the sliding motion, the output-error-injection signal νf in (36) can theoretically be replaced by an equivalent outputerror-injection signal νf eq . By applying (44) to (36), it follows that when a sliding motion takes place W32 ez + N2 (y, u)f (t) − νf eq = 0.
(45)
From Assumption 2, N2 (y, u) is nonsingular in Y × U, and it follows from (45) that f (t) = −N2−1 (y, u)(W32 ez − νf eq ).
T T eT yo Λo eyo + eyθ Λθ eyθ + eyf Λf eyf ≤ 0.
T + eT yf N2 (y, u)f (t) − kf eyf sgn(eyf ).
4033
(46)
In order to reconstruct the fault f (t), it is necessary to recover the equivalent output-error-injection signal νf eq . In the work described in [24], it was obtained using a low-pass filter. In this paper, a modification to the approach given in [5] will be employed. From (31), the equivalent output-error-injection signal νf eq in (46) can be approximated to any accuracy by νf σ := kf (t, y, u)ζ(eyf )
(47)
where kf (·) satisfies (41) and ⎡ ⎢ ⎢ ζ(eyf ) := ⎢ ⎢ ⎣
eyf 1 |eyf 1 |+δ eyf 2 |eyf 2 |+δ
.. .
eyf r |eyf r |+δ
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(48)
where col(eyf 1 , eyf 2 , . . . , eyf r ) := eyf and δ usually is a “small” positive design constant. The choice of the constant δ
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4034
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
determines the degree of approximation to ideal sliding which is attained. In practice, the choice of kf (·) in (47), which must satisfy (41), also affects the approximation to ideal sliding and the accuracy to which eyf approaches zero. The following conclusion is now ready to be presented. Theorem 3: Consider system (1) and (2). Under Assumptions 1 and 2, the signal fˆ(t) := N2−1 (y, u)νf σ
(49)
with νf σ defined by (47), is a reconstruction of the fault f (t). Furthermore, it is a precise reconstruction since lim f (t) − fˆ(t) = 0
t→∞
if N2−1 (y, u), as defined in Assumption 4, is bounded in the domain Y × U. Proof: The earlier analysis has shown that (46) is true. It follows from (46) and (49) that f (t)− fˆ(t) = N2−1 (y, u)W32 ez +N2−1 (y, u) (νf eq −νf σ ) ≤N −1 (·)W32 ez + N −1 (·) νf eq −νf σ . 2
2
(50) From Theorems 1 and 2, it follows that lim ez (t) = 0.
t→∞
Since νf σ can approximate νf eq to any accuracy and N2 (y, u) is bounded in the domain Y × U, it follows that lim f (t) − fˆ(t) = 0. t→∞
Hence, the conclusion follows. The formulas (47) and (48) show that νf σ depends only on known system information: time t, the system output y, the system input u, and the residual signal eyf . From (49), the reconstruction signal fˆ(·) is only dependent on available information, and thus, the fault f (t) can be reconstructed online. This shows that the FDI scheme is practical for real implementation. Remark 10: In Theorem 3, it is required that N2−1 (y, u) is bounded in the domain Y × U. If Y × U is a compact set and N2−1 (y, u) is continuous in Y × U, then N2−1 (y, u) will be bounded. Therefore, in the local case, this condition is not strong. Remark 11: From (49), it is clear that the proposed fault reconstruction fˆ is independent of the unknown parameter θ ˆ The proposed approach allows the and its estimated value θ. unknown parameter to be arbitrarily large from a theoretical point of view. VI. S IMULATION Consider a magnetic-levitation system consisting of an iron ball in a vertical magnetic field created by a single electromag-
net [14]. The model of the system is described in [14] by w˙ 1 = w2 μ(w1 ) 2 w˙ 2 = w3 − g m R 1 w3 + u w˙ 3 = − L(w1 ) L(w1 )
(51) (52) (53)
where w1 , w2 , and w3 represent the (upward) displacement of the ball from the nominal position, the velocity of the ball, and the current through the electromagnet. The parameter m is the mass of the ball, g is the acceleration of gravity, L(w1 ) is the electromagnet inductance, R is the resistance, u(t) is the supplied voltage source, and μ(·) is a known strictly positive function. It is assumed that w1 and w3 are measurable. Here, in order to illustrate the results obtained in this paper, it is assumed that there is a fault f (t) in the input channel. This represents a fault in the circuit associated with the electromagnet—possibly a sudden change in the inductance or the resistance. The system (51)–(53) can be described by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ w2 0 0 1 1) 2 ⎦ ⎣ ⎦ ⎣ 0 ⎦f ⎦ + ⎣ μ(w 0 w˙ = ⎣ m w3 −g + 1 θ + R 1 1 u − L(w1 ) w3 0 L(w1 ) L(w1 ) (54) w1 y= (55) w3 where the term associated with an unknown constant θ has been added to demonstrate the approach given in this paper, and is not a feature of the real magnetic-levitation system. Introduce a transformation (w1 , w2 ) → (w2 , w1 ) := (x11 , x12 ) = x1 and x2 = w3 . Then, in the x coordinates, the system in (54) and (55) can be described in the form of (1) and (2) with μ(y ) 1 2 0 0 m y2 − g F1 = , G1 (·) = 1 0 0 1 , Ψ1 (·) = 0 Φ1 (·) = 1 R 1 y2 + F2 = 0, G2 (·) = L(y1 ) L(y1 ) 1 Φ2 (·) = 0, Ψ2 (·) = L(y1 ) C1 = [0 1], C2 = 1. It is clear that Assumption 1 is satisfied, since Ψ1 (·) = 0 and the second subsystem is first order. Let zI = x11 − x12 . The system is then in the form of (11) and (12). By direct computation, it follows that Assumption 2 is satisfied with W = I2 , N1 = 1, and N2 = 1/L(y1 ). Let zyθ = y1 and zyf = y2 , and then the system can be rewritten in the form of (18)–(21) as follows: z˙I = − zI − zyθ +
μ(zyθ ) 2 zyf − g m
z˙yθ = zI + zyθ + θ 1 R 1 zyf + u+ f (t) z˙yf = − L(y1 ) L(y1 ) L(y1 ) zyθ y= . zyf
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
(56) (57) (58) (59)
YAN AND EDWARDS: ADAPTIVE SLIDING-MODE-OBSERVER-BASED FAULT RECONSTRUCTION
Now, consider the dynamic system μ(zyθ ) 2 zˆ˙ I = − zˆI − zyθ + zyf − g m zˆ˙ yθ = zˆI + zyθ − Λθ (zyθ − zˆyθ ) + θˆ + νθ R 1 zyf + u − Λf (zyf − zˆyf ) + νf zˆ˙ yf = − L(y1 ) L(y1 ) zˆyθ yˆ = zˆyf
(60) (61) (62) (63)
where θˆ is given by the following adaptive law: ˙ θˆ = −β(θˆ + zˆI + zyθ − z˙yθ ).
(64)
The signals νθ and νf are output-error-injection terms given by νθ = kθ sgn(zyθ − zˆyθ ) νf = kf sgn(zyf − zˆyf )
(65) (66)
Fig. 1. Time responses of the switching functions.
where kθ (·) and kf (·) satisfy (40) and (41) and Λθ , Λf , β are design parameters. Let Q = 4, then P = 2. Since ΦI (·) = 0 and W22 = 1, by direct computation, it follows that M=
4 β
β 2β
and, thus, from Theorems 1 and 2, system (60)–(62) is a sliding-mode observer of system (56)–(58) if β is chosen such that 0 < β < 8. Furthermore, from Theorem 3, it follows that fˆ(t) = L(y1 )νf σ where νf σ defined by (47) and (48) is a reconstruction of the fault f (t). If, in (64), z˙yθ is not available, then the Levant second-order differentiator [17] can be applied to estimate z˙yθ : specifically create the system
Fig. 2. Adaptive estimation of the unknown parameter θ.
v˙ 1 = v2 1
v2 = v3 − 1 |v1 − zyθ | 2 sgn(v1 − zyθ ) v˙ 3 = 2 sgn(v1 − zyθ ) where 1 and 2 are positive constants, then v2 is the optimal estimate of z˙yθ [17]. In the following simulation, let β = 7.5, 1 = 1.5, 2 = 1.2, Λθ = Λf = −0.5, g = 9.8 m/s2 , m = 1.5 kg, R = 2.55 Ω, and μ(x1 ) = 1/(1 + x21 ). For simulation purposes, according to [25], L(·) is chosen as L(y1 ) = 2 + (2/(1 + 0.5|y1 |)). Fig. 1 shows the time responses of the switching functions. Fig. 2 shows that the adaptive law gives a good estimate for the parameter θ, and Fig. 3 shows that the fault signal can be reconstructed faithfully. Remark 12: It should be noted that the design parameters in (61) and (62) are scalars Λθ and Λf , which should be negative (stable). The output-injection term is given in (65) and (66), where the parameters kθ and kf should satisfy (40) and (41).
Fig. 3. Fault reconstruction for the magnetic-levitation system.
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4035
4036
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
VII. C ONCLUSION In this paper, a class of nonlinear multi-output systems with uncertain parameters has been considered. An adaptive law has been given to identify the parameters. Based on slidingmode techniques, an observer has been developed to estimate the system state variables asymptotically even in the presence of uncertain parameters and faults. Furthermore, the fault signals have been reconstructed from the equivalent output-errorinjection signals associated with the observer. The results have good robustness, since the estimate of the uncertain parameters can be used in the observer design. A magnetic-levitation system has been introduced to illustrate the approach proposed in this paper. Simulations show that the results which have been obtained are effective. R EFERENCES [1] J. Chen and R. J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems. Boston, MA: Kluwer, 1999. [2] W. Chen and M. Saif, “A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonlinear systems,” Int. J. Syst. Sci., vol. 38, no. 12, pp. 943–955, Jan. 2007. [3] J. Davila, A. Fridman, and L. Poznyak, “Observation and identification of mechanical systems via second order sliding modes,” Int. J. Control, vol. 79, no. 10, pp. 1251–1262, Oct. 2006. [4] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and Applications. London, U.K.: Taylor & Francis, 1998. [5] C. Edwards, S. K. Spurgeon, and R. J. Patton, “Sliding mode observers for fault detection and isolation,” Automatica, vol. 36, no. 4, pp. 541–553, Apr. 2000. [6] T. Floquet, J. P. Barbot, W. Perruquetti, and M. Djemai, “On the robust fault detection via a sliding mode disturbance observer,” Int. J. Control, vol. 77, no. 7, pp. 622–629, May 2004. [7] P. M. Frank and X. Ding, “Survey of robust residual generation and evaluation methods in observer-based fault detection systems,” J. Process Control, vol. 7, no. 6, pp. 403–424, Dec. 1997. [8] L. Fridman, Y. Shtessel, C. Edwards, and X. G. Yan, “Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems,” Int. J. Robust Nonlinear Control, vol. 18, no. 4/5, pp. 399–412, Mar. 2008. [9] L. Fu and T. Liao, “Globally stable robust tracking of nonlinear systems using variable structure control and with an application to a robotic manipulator,” IEEE Trans. Autom. Control, vol. 35, no. 12, pp. 1345–1350, Dec. 1990. [10] J. Y. Hung, W. Gao, and J. C. Hung, “Variable structure control: A survey,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 2–22, Feb. 1993. [11] A. Isidori, Nonlinear Control Systems, 3rd ed. London, U.K.: SpringerVerlag, 1995. [12] B. Jiang, M. Staroswiecki, and V. Cocquempot, “Fault estimation in nonlinear uncertain systems using robust sliding-mode observers,” Proc. Inst. Elect. Eng.—Part D: Control Theory Appl., vol. 151, no. 1, pp. 29–37, Jan. 2004. [13] B. Jiang, M. Staroswiecki, and V. Cocquempot, “Fault accommodation for nonlinear dynamic systems,” IEEE Trans. Autom. Control, vol. 51, no. 9, pp. 1578–1583, Sep. 2006. [14] D. Karagiannis, A. Astolfi, and R. Ortega, “Two results for adaptive output feedback stabilization of nonlinear systems,” Automatica, vol. 39, no. 5, pp. 857–866, May 2003. [15] P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. London, U.K.: Academic, 1985. [16] C. Lascu and G.-D. Andreescu, “Sliding-mode observer and improved integrator with DC-offset compensation for flux estimation in sensorlesscontrolled induction motors,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 785–794, Jun. 2006. [17] A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, Mar. 1998. [18] K. A. Loparo, M. L. Adams, W. Lin, M. F. Abdel-Magied, and N. Afshari, “Fault detection and diagnosis of rotating machinery,” IEEE Trans. Ind. Electron., vol. 47, no. 5, pp. 1005–1014, Oct. 2000. [19] R. Marino and P. Tomei, Nonlinear Control Design. Englewood Cliffs, NJ: Prentice-Hall, 1995.
[20] O. Moseler and R. Isermann, “Application of model-based fault detection to a brushless DC motor,” IEEE Trans. Ind. Electron., vol. 47, no. 5, pp. 1015–1020, Oct. 2000. [21] C. De Persis and A. Isidori, “A geometric approach to nonlinear fault detection and isolation,” IEEE Trans. Autom. Control, vol. 46, no. 6, pp. 853–865, Jun. 2001. [22] C. P. Tan and C. Edwards, “Sliding mode observers for detection and reconstruction of sensor faults,” Automatica, vol. 38, no. 10, pp. 1815– 1821, Oct. 2002. [23] C. P. Tan and C. Edwards, “Sliding mode observers for robust detection and reconstruction of actuator and sensor faults,” Int. J. Robust Nonlinear Control, vol. 13, no. 5, pp. 443–463, Apr. 2003. [24] V. I. Utkin, Sliding Modes in Control Optimization. Berlin, Germany: Springer-Verlag, 1992. [25] H. H. Woodson and J. R. Melcher, Electromechanical Dynamics. Malabar, FL: Krieger, 1990. [26] W. F. Xie, “Sliding-mode-observer-based adaptive control for servo actuator with friction,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp. 1517– 1527, Jun. 2007. [27] A. Xu and Q. Zhang, “Residual generation for fault diagnosis in linear time-varying systems,” IEEE Trans. Autom. Control, vol. 49, no. 5, pp. 767–772, May 2004. [28] X. G. Yan and C. Edwards, “Nonlinear robust fault reconstruction and estimation using a sliding mode observer,” Automatica, vol. 43, no. 9, pp. 1605–1614, Sep. 2007. [29] X. G. Yan and L. Xie, “Reduced-order control for a class of nonlinear similar interconnected systems with mismatched uncertainty,” Automatica, vol. 39, no. 1, pp. 91–99, Jan. 2003. [30] X. Zhang, M. M. Polycarpou, and T. Parisini, “A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems,” IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 576–593, Apr. 2002.
Xing-Gang Yan received the B.Sc. degree from Shanxi Teachers University, Xi’an, China, in 1985, the M.Sc. degree from Qufu Teachers University, Qufu, China, in 1991, and the Ph.D. degree in engineering from Northeastern University, Shenyang, China, in 1997. From 1985 to 1988, he was a Teaching Assistant and, then, a Lecturer with the Department of Mathematics, Weinan Educational University, Weinan, China. From 1991 to 1994, he was a Lecturer with the Department of Applied Mathematics, Qingdao University, Qingdao, China. From 1998 to 1999, he was a Postdoctoral Fellow with Northwestern Polytechnical University, Xi’an, China. During this period, he was also a Research Associate with the University of Hong Kong, Hong Kong, for eight months. From 1999 to 2001, he was a Research Fellow with Nanyang Technological University, Singapore. He is currently a Research Associate with the Control Systems Research Group, Department of Engineering, University of Leicester, Leicester, U.K. His research interests include decentralized control, nonlinear control, robust control, sliding-mode control, fault detection and isolation, and applications in practical systems.
Christopher Edwards (M’00) was born in Swansea, U.K. He received the B.Sc. degree in mathematics from Warwick University, Coventry, U.K., in 1987, and the Ph.D. degree supported by a British Gas Research Scholarship from Leicester University, Leicester, U.K., in 1995. From 1987 to 1991, he was a Research Officer with British Steel Technical, Port Talbot, where he worked in the R&D Division. He was a Lecturer in 1996, a Senior Lecturer in 2004, and has been a Reader since 2008 with the Control Systems Research Group, Department of Engineering, Leicester University. He is the coauthor of 170 refereed papers in the area of sliding-mode control and observation, fault detection and isolation, and fault-tolerant control. He is also the coauthor of two books on sliding-mode control.
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4029
Adaptive Sliding-Mode-Observer-Based Fault Reconstruction for Nonlinear Systems With Parametric Uncertainties Xing-Gang Yan and Christopher Edwards, Member, IEEE
Abstract—In this paper, a class of nonlinear systems with uncertain parameters is considered. A novel adaptive law is designed to identify unknown parameters under the assumption that the time derivative of some of the outputs is measurable. Then, a sliding-mode observer is proposed to estimate the system state variables. By using the inherent features of sliding-mode observers, a fault-reconstruction scheme is proposed which can be implemented online. The proposed reconstruction signal can approximate the fault signal to any required accuracy even in the presence of uncertain parameters. A simulation example for a magnetic-levitation system is given to illustrate the feasibility and effectiveness of the proposed scheme. Index Terms—Adaptive systems, fault diagnosis, nonlinear estimation, sliding modes.
I. I NTRODUCTION
W
ITH the advancement of modern technology, more complicated automation schemes are being applied to real systems. The complexity comes from strong nonlinearities and uncertainties which exist in real systems. Such classes of systems are vulnerable to faults because human-operator interaction is reduced/removed. The effects of faults can escalate and may corrupt the system to such an extent that the automated system becomes useless. Therefore, prompt detection and isolation of faults has become important. This has motivated much research in the field (see, e.g., [1], [2], [7], [18], [20]). Compared with linear systems, nonlinear systems are much more complicated, which makes their study more difficult. Due to the variety of nonlinear features, it is impossible to study nonlinear systems as systematically as linear systems, but it is possible and useful to study certain classes of nonlinear systems. In recent decades, some control-inspired approaches, for instance, sliding-mode techniques [5], modern differential geometric approaches [21], and adaptive control [13], [30], have been successfully incorporated within the observer-based Fault Detection and Isolation (FDI) approach. In particular, modern geometric approaches have been shown to be effective in the investigation of a class of nonlinear systems [11], [19]. When a system has unknown parameters, adaptive control techManuscript received January 15, 2008; revised July 22, 2008. First published August 19, 2008; current version published October 31, 2008. The authors are with the Control Systems Research Group, Department of Engineering, University of Leicester, Leciester, LE1 7RH, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2008.2003367
niques can be employed to estimate the unknown parameters [19], [27], [30]. It is well known that sliding-mode control exhibits high robustness to matched uncertainty [10]. Moreover, sliding-mode observer-based approaches have been widely applied to real systems, see [16] and [26] for example. Recently, sliding-mode techniques have been successfully used for fault detection and isolation [3], [5], [6], [8], [22], [28], and it has been proven to be an effective way to estimate/reconstruct system faults. A “precise” fault reconstruction approach is proposed in [5] based on the equivalent output-error-injection concept. Based on the work in [5], a sensor fault-reconstruction scheme has been proposed in [22]. It should be noted that, in [5] and [22], only linear systems are considered, and uncertainty is not included in the formulation. Later, a fault-estimation approach was proposed for linear systems with uncertainty by Tan and Edwards [23] which focused on minimizing the L2 gain between the uncertainty and the fault-reconstruction signal using linear matrix inequalities. More recently, a robust actuator faultreconstruction scheme has been presented in [28] using the characteristics of the uncertain structure and fault distribution. Jiang et al. [12] proposed a fault-estimation scheme for a class of systems with uncertainty, and a robust fault-detection method for nonlinear systems with disturbances was considered in [6]. However, in these papers involving uncertainty, nearly all are concerned with fault estimation instead of “precise” reconstruction, and all require that the uncertainties are bounded with known bounds. Although a “precise” reconstruction scheme is proposed in [28], it is assumed that the structural matrix for the uncertainty satisfies a strong structural condition. In this paper, fault reconstruction is considered for a class of nonlinear systems with uncertain parameters, where the faultdistribution vector and the structure matrix of the uncertainty are allowed to be functions of the system output and input. Under the assumption that the time derivative of a subset of the system outputs is available, a novel adaptive update law is given. The unknown parameters are estimated, and a sliding-mode observer is developed using the estimated parameters. Based on the sliding-mode observer, a fault-reconstruction scheme, which can be implemented online, is proposed to reconstruct the fault signal. The reconstruction signal can approximate the fault signal to any required accuracy. Furthermore, the structural condition for the distribution of the uncertainties is removed. The uncertain parameters can be arbitrarily large, and it is not necessary to know the bounds on them a priori. This is an improvement as compared with the existing work in
0278-0046/$25.00 © 2008 IEEE
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4030
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
[6], [12], [23], and [28], where the uncertain bounds are used in the observer design and the fault-estimation/reconstruction scheme. A simulation demonstrates the practicality of the proposed approach. Notation: The notation is broadly as used in [19]. In addition, for a square matrix A, λmax (A) denotes the maximum eigenvalue of A, and A > 0 denotes a symmetric positive definite matrix. The symbol In represents the nth-order unit matrix, and R+ represents the set of nonnegative real numbers. The set of n × m matrices with elements in R will be denoted by Rn×m . A function f (x1 , x2 , . . . , xn1 , y1 , y2 , . . . , yn2 ) is also written as f (x, y), where x = [x1 , x2 , . . . , xn1 ]T ∈ Rn1 and y = [y1 , y2 , . . . , yn2 ]T ∈ Rn2 . The symbol · denotes the Euclidean norm or its induced norm. II. P RELIMINARIES The following results will be used in the later analysis. Lemma 1: Assume that x = col(x1 , x2 , . . . , xN ), where xi ∈ Rni for i = 1, 2, . . . , N . Then, the following expressions hold: 1) x ≤ xT sgn(x). N N 2 2) i=1 (di xi ) ≤ ( i=1 di )x, where di is a positive constant for i = 1, 2, . . . , N . Proof: 1) Let xi = col(xi1 , xi2 , . . . , xini ). It follows that for i = 1, 2, . . . , N
xT i sgn(xi )
= [ xi1
xi2
···
⎡ sgn(x ) i1 ⎢ sgn(xi2 ) xini ] ⎢ .. ⎣ .
⎤
Therefore,
xT 2
···
⎤ ⎥ ⎥ ⎦
sgn(xN ) T T = xT 1 sgnx1 + x2 sgnx2 + xN sgnxN
≥ x1 + x2 + · · · + xN ≥ x. Hence, conclusion 1) follows. 2) Taking p = q = 1/2 in Hölder’s inequality [15], p1 N q1 N N
1 1 p q + =1 |ai bi | ≤ |ai | |bi | , p q i=1 i=1 i=1 it follows that for di ≥ 0 N 12 N 12
N
N
2 2 di xi ≤ di xi = d2i x i=1
i=1
i=1
and conclusion 2) follows.
x˙ i = Fi xi + Gi (y, u) + Φi (y, u)θ + Ψi (y, u)f (t)
(1)
yi = Ci xi ,
(2)
i = 1, 2, . . . , p
where x = col(x1 , . . . , xp ) ∈ Rn and xi := col(xi1 , xi2 , . . . , xini ) ∈ Rni are system state variables; u ∈ U ⊂ Rm and col(y1 , y2 , . . . , yp ) := y ∈ Y ⊂ Rp are the inputs and the outputs, respectively. Assume the pair (Fi , Ci ) has Brunovsky standard form [19] as follows: ⎡
0 0 ⎢1 0 ⎢. . Fi = ⎢ ⎢ .. .. ⎣0 0 0 0
··· ··· .. .
0 0 .. .
0 0 .. .
··· ···
1 0
0 1
⎤ 0 0⎥ .. ⎥ ⎥ := .⎥ 0⎦
Ai1 Ai2
0 0
ni ×ni
0 (3)
Ci = [ 0
···
0 1 ]1×ni
(4)
where Gi (x, u) ∈ Rni , Φi (x, u) ∈ Rni ×q , and Ψi (x, u) ∈ Rni ×r have smooth elements with p ≥ q + r for i = 1, 2, . . . , p. The symbol θ ∈ Rq denotes unknown constant parameters. The unknown function vector f (t) ∈ Rr represents faults experienced by the system and is assumed to satisfy (5)
where ρ(·) is known. Remark 1: The limitation that (Fi , Ci ) has Brunovsky form can be replaced by the condition that (Fi , Ci ) is observable, since a simple coordinate transformation can be used to transfer the system (1) and (2) to Brunovsky form if (Fi , Ci ) is observable. Remark 2: It should also be noted that the approach developed in this paper can be applied to a class of nonlinear systems which can be transformed to the system representation (1) and (2). Geometric conditions have been proposed in [19], under which more general nonlinear systems can be transformed to the structure in (1) and (2). It is obvious that the matrix pair (Fi , Ci ) is observable, and ˜ i Ci is stable. ˜ i such that Fi − K thus, there exists a matrix K ˜ Therefore, for any Qi > 0, the Lyapunov equation
= |xi1 | + |xi2 | + · · · + |xini | ≥ xi .
xT sgn(x) = [ xT 1
Consider a nonlinear system described by
f (t) ≤ ρ(t)
⎥ ⎥ ⎦
sgn(xini )
⎡ sgn(x ) 1 sgn(x2 ) ⎢ ⎢ xT .. N ]⎣ .
III. P ROBLEM F ORMULATION
i=1
˜ i Ci )T P˜i + P˜i (Fi − K ˜ i Ci ) = −Q ˜i (Fi − K
(6)
has a unique solution P˜i > 0 for i = 1, 2, . . . , p. Assumption 1: There exist functions Γi (y, u) with i = 1, 2, . . . , p such that (Ψi (y, u))T P˜i = Γi (y, u)Ci ,
i = 1, 2, . . . , p
(7)
where Ψi (y, u) is given by (1). Remark 3: Assumption 1 is a structural constraint on the fault distribution Ψi (·). However, it should be noted that there
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
YAN AND EDWARDS: ADAPTIVE SLIDING-MODE-OBSERVER-BASED FAULT RECONSTRUCTION
are no structural constraints on the distribution associated with the parametric uncertainties. This is in comparison with the work in [28] for example. In particular, if the ith subsystem is first order, Assumption 1 is satisfied automatically. Partition P˜i as ˜ Pi1 P˜i = ˜ T P
P˜i2 P˜i3
i2
Ti :=
Ini −1
−1 ˜ Pi2 −P˜i1
0
1
i = 1, 2, . . . , p.
(8)
Then, in the new coordinate system z, system (1) and (2) has the following form: z˙i = Ti
Ai1 Ai2
0 T −1 z + Ti Gi (y, u) 0 i i
+ Ti Φi (y, u)θ + Ti Ψi (y, u)f (t) yi = Ci zi ,
⎡
0(p−q−r)×q ⎢ ΨII (y, u) ] = ⎣ N1 (y, u) 0r×q
,
system (11) and (12) is not unique. A transformation has been provided in (8). Assumption 2: There exists a nonsingular matrix W ∈ Rp×p such that
W [ ΦII (y, u)
where P˜i1 ∈ R(ni −1)×(ni −1) and P˜i3 ∈ R. For system (1) and (2), introduce a coordinate transformation z = T x, where T := diag{T1 , T2 , . . . , Tp } and
4031
i = 1, 2, . . . , p
(9) (10)
where z = col(z1 , z2 , . . . , zp ) with zi := (zi1 , zi2 , . . . , zini ). −1 ˜ Pi2 Ai2 is stable. Let It follows from [29] that Ai1 − P˜i1 −1 Ki := P˜i1 P˜i2 . By reordering the state variables, system (9) and (10) can be further described as z˙I = (A1 − KA2 )zI + (A1 − KA2 )Ky + GI (y, u) + ΦI (y, u)θ
(11)
y˙ = A2 zI + A2 Ky + GII (y, u) + ΦII (y, u)θ + ΨII (y, u)f (t)
⎤ 0(p−q−r)×r ⎥ 0q×r ⎦ N2 (y, u)
where N1 (·) ∈ Rq×q and N2 (·) ∈ Rr×r are both nonsingular in Y × U. Remark 5: Assumption 2 implies that the uncertain parameters and the faults are separable by applying a further nonsingular transformation to system (11) and (12). The limitation in Assumption 2 can be interpreted as, the matrix [ΦII (y, u) ΨII (y, u)] must be block-diagonalizable by elementary row transformations. Remark 6: It should be noted that the analysis earlier requires that ni > 1. If, for example, nk = 1, then, xk = yk in (1) is a scalar, and in this case, the kth subsystem will be directly merged with subsystem (12), and does not contribute to the differential equations in (11). All the analysis which follows and the results obtained are still valid by appropriate modifications to the expressions. The objective of this paper is to design a sliding-mode observer for system (1) and (2). Then, based on the sliding observer, an approach is proposed to reconstruct the fault signal f (·) precisely. A novel adaptive law is given to estimate the parameters. System (1) and (2) is equivalent to system (11) and (12), i.e., the former can be transformed to the latter by a nonsingular linear transformation, but the form and specific structure in (11) and (12) will be shown to be useful for slidingmode-observer design. IV. A DAPTIVE S LIDING -M ODE -O BSERVER D ESIGN
(12)
where zI = col(z11 , . . . , z1(n1 −1) , . . . , zp1 , . . . , zp(np −1) ) ∈ Rn−p , y = col(z1n1 , . . . , zpnp ), A1 = diag{A11 , . . . , Ap1 }, A2 = diag{A12 , . . . , Ap2 }, and K = diag{K1 , K2 , . . . , Kp }. Obviously, A1 − KA2 = diag{A11 − K1 A12 , . . . , Ap1 − Kp Ap2 }
In this section, a sliding-mode observer will be designed based on a novel adaptive law. Let zy = W y, where W is defined in Assumption 2. The analysis in the previous section shows that, under Assumptions 1 and 2, the system (1) and (2) can be described by z˙I = (A1 − KA2 )zI + (A1 − KA2 )KW −1 zy + GI (y, u) + ΦI (y, u)θ
is stable, and thus, for Q > 0(Q ∈ R Lyapunov equation
(n−p)×(n−p)
(A1 − KA2 )T P + P (A1 − KA2 ) = −Q
), the
(13)
has a unique solution P > 0. Remark 4: Since the matrix T has been given in (8), the functions GI (·), GII (·), ΦII (·), ΨI (·), and ΨII (·) can be obtained directly using the coordinate transformation z = T (x), and thus, the system (11) and (12) is well defined and can be obtained from (1) and (2) directly. It should be noted that the transformation matrix T which transforms system (1) and (2) to
z˙y = W A2 zI + W A2 KW −1 zy + W GII (y, u) ⎡ ⎤ ⎡ ⎤ 0 0 ⎦ f (t) + ⎣ N1 (y, u) ⎦ θ + ⎣ 0 0 N2 (y, u) y = W −1 zy
(14)
(15) (16)
where the matrix W and the matrix functions N1 (·) and N2 (·) are determined by Assumption 2. In order to further exploit the system structure, let zy = col(zyo , zyθ , zyf ), where zyo ∈ Rp−q−r , zyθ ∈ Rq ,
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4032
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
and zyf ∈ Rr , and partition matrices W , W A2 , and W A2 K in a compatible way with zy as follows: ⎡
⎤ W11 W := ⎣ W21 ⎦ W31
⎤ W12 W A2 := ⎣ W22 ⎦ W32 ⎡
⎤ W13 W A2 K := ⎣ W23 ⎦ . W33 (17) ⎡
Then, system (14)–(16) can be further rewritten as z˙I = (A1 − KA2 )zI + (A1 − KA2 )KW −1 zy + GI (y, u) + ΦI (y, u)θ (18) z˙yo = W12 zI + W13 y + W11 GII (y, u) (19) z˙yθ = W22 zI + W23 y + W21 GII (y, u) + N1 (y, u)θ (20) z˙yf = W32 zI + W33 y + W31 GII (y, u) + N2 (y, u)f (21) ⎡ ⎤ zyo y = W −1 ⎣ zyθ ⎦ . (22) zyf Now, consider a dynamical system zI + (A1 − KA2 ) Ky zˆ˙ I = (A1 − KA2 )ˆ + GI (y, u) + ΦI (y, u)θˆ
(23)
zˆ˙ yo = W12 zˆI + W13 y − Λo W11 (y − yˆ) + W11 GI I (y, u) + νo zˆ˙ yθ = W22 zˆI + W23 y − Λθ W21 (y − yˆ) + W21 GI I (y, u) + N1 (y, u)θˆ + νθ zˆ˙ yf = W32 zˆI + W33 y − Λf W31 (y − yˆ)
(24) (25)
+ W31 GI I (y, u) + νf ⎡ ⎤ zˆyo yˆ = W −1 zˆy := W −1 ⎣ zˆyθ ⎦ zˆyf
(26) (27)
where yˆ ∈ Rp is the output of the dynamical system with zˆyo ∈ Rp−q−r , zˆyθ ∈ Rq , and zˆyf ∈ Rr ; the matrices Λo ∈ R(p−q−r)×(p−q−r) , Λθ ∈ Rq×q , and Λf ∈ Rr×r are parameters which by design are stable. The estimate of θ, ˆ is given by the following adaptive law: denoted by θ, ˙ θˆ = −β(y, u) N1 (y, u)θˆ + W22 zˆI + W23 y + W21 GII (y, u) − z˙yθ
(28)
where β(·) ∈ Rq×q is a matrix design parameter and col(νo , νθ , νf ) := ν is the output error injection term defined by νo = ko (y, yˆ)sgn(zo − zˆo ) ˆ νθ = kθ (y, yˆ, u, θ)sgn(z ˆθ ) θ −z νf = kf (t, y, yˆ, u)sgn(zf − zˆf )
(29) (30) (31)
where sgn denotes the usual sign vector function and scalars ko , kθ (·), and kf (·) are to be determined. Remark 7: The adaptive law (28) requires that z˙yθ is available. Obviously, z˙yθ is measurable if y˙ is measurable, which is an assumption that has been employed by many authors (see, for example, [9]). Furthermore, the exact differentiator
proposed in [17] can be used to estimate the time derivative of the system output in finite time if y˙ is not available. ˆ eyo = zyo − zˆyo , eyθ = Define ez = zI − zˆI , eθ = θ − θ, zyθ − zˆyθ , and eyf = zyf − zˆyf . Since eyo = W11 (y− yˆ),
eyθ = W12 (y− yˆ),
eyf = W13 (y − yˆ)
it follows from (14)–(16) and (23)–(26) that the error dynamical equation is described by e˙ z e˙ θ e˙ yo e˙ yθ e˙ yf
= (A1 − KA2 )ez + Φ1 (y, u)eθ = − β(y, u)W22 ez − β(y, u)N1 (y, u)eθ = W12 ez + Λo eyo − νo = W22 ez + Λθ eyθ + N1 (y, u)θˆ − νθ = W32 ez + Λf eyf + N2 (y, u)f − νf .
(32) (33) (34) (35) (36)
The error systems (32) and (33) will be considered first. Theorem 1: Consider systems (32) and (33) and assume that (13) holds. Then, systems (32) and (33) are asymptotically stable if, for any (y, u) ∈ Y × U, the matrix M = Q − (ΦI (y, u))T P + β(y, u)W22 − P ΦI (y, u) + (β(y, u)W22 )T β(y, u)N1 (y, u) + N1T (y, u)β T (y, u) is positive definite, where P and Q satisfy (13) and β(·) ∈ Rq×q is the design parameter in (28). Proof: Consider a candidate Lyapunov function for system (32) and (33) as T V (ez , eθ ) = eT z P ez + eθ eθ .
Then, the time derivative of V along the trajectories of system (32) and (33) is given by T T V˙ |(32)−(33) = eT ez z (A1 −KA2 ) P +P (A1 −KA2 ) T + 2eT z P Φ1 (y, u)eθ −2eθ β(y, u)W22 ez T T − eT θ β(y, u)N1 (y, u)eθ −eθ (β(y, u)N1 (y, u)) eθ T T eθ ≤ −eT z Qez + 2ez P Φ1 −(βW22 ) T T eθ −eT θ βN1 + N1 β ez = −[ eT eT . z θ ]M e θ
Hence, the conclusion follows from the assumption that M is positive definite in ∈ Y × U. Remark 8: Theorem 1 shows that, under certain conditions, the error dynamical system (32) and (33) is asymptotically stable, which means that limt→∞ ez (t) = 0 and limt→∞ eθ (t) = 0. Consequently, the errors ez and eθ are bounded so that W12 ez (t) ≤ χ1 ,
W22 ez ≤ χ2 ,
W32 ez ≤ χ3 (37)
where χi for i = 1, 2, and 3 are positive constants. Remark 9: A necessary condition for matrix M in Theorem 1 to be positive definite is that the function matrix β(·) in (28) must be designed such that β(y, u)N1 (y, u) + N1T (y, u)β T (y, u) is positive definite in (y, u) ∈ Y × U.
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
YAN AND EDWARDS: ADAPTIVE SLIDING-MODE-OBSERVER-BASED FAULT RECONSTRUCTION
Next, for the dynamical error equations (32)–(36), consider the following sliding surface: S = {col(ez , eθ , eyo , eyθ , eyf )|eyo = 0, eyθ = 0, eyf = 0} . (38) The reduced-order sliding dynamics associated with the sliding surface (38) can be described by (32) and (33). Theorem 1 has shown that, under certain conditions, the sliding-mode dynamics are asymptotically stable when the sliding motion takes place. In what follows, the objective is to study reachability issues. Specifically, a reachability condition will be developed such that the error dynamics in (32)–(36) can be driven to the sliding surface (38) in finite time, and a sliding motion maintained on it thereafter. Theorem 2: Assume that (37) holds. Then, the trajectories of the dynamical error system (32)–(36) can be driven to the sliding surface (38) in finite time if the gains ko , kθ (·), and kf (·) in (29)–(31) satisfy ko ≥ χ1 + η1 ˆ ≥ χ2 + kθ (y, u, θ) N1 (y, u)θˆ + η2
(39)
kf (t, y, u) ≥ χ3 + N2 (y, u) ρ(t) + η3
(41)
(40)
for some positive constants η1 , η2 , and η3 , where χi for i = 1, 2, and 3 satisfy (37). Proof: Let ey = col(eyo , eyθ , eyf ). It follows from (34)–(36) that T T T eT y e˙ y = eyo e˙ yo + eyθ e˙ yθ + eyf e˙ yf T T T = eT yo W12 ez + eyo Λo eyo − eyo νo + eyθ W22 ez + eT Λθ eyθ + eT N1 (y, u)θˆ − eT νθ + eT W32 ez
+
yθ eT yf Λf eyf
+
yθ yθ yf T T eyf N2 (y, u)f (t) − eyf νf .
(42)
Since the design parameters Λo , Λθ , and Λf are all chosen as negative definite, it follows that
Then, substituting (29)–(31) into (42) T T T eT y e˙ y ≤ eyo W12 ez − ko eyo sgn(eyo ) + eyθ W22 ez T T ˆ + eT yθ N1 (y, u)θ − kθ eyθ sgn(eyθ ) + eyf W32 ez
(43)
From (5), (37), and (43), and from part 1) of Lemma 1, it follows that T eT y e˙ y ≤ eyo χ1 − ko eyo sgn(eyo ) + eyθ χ2 ˆ − kθ (y, u, θ)e ˆ T sgn(eyθ ) + eyθ N1 (y, u)θ yθ + eyf χ3 + eyf N2 ρ(t) − kf eT yf sgn(eyf ) ˆ eyθ = − (ko − χ1 )ey0 − kθ (·) − χ2 − N1 (·)θ
− (kf (t, y, u) − χ3 − N2 (y, u)ρ(t)) eyf .
Then, from (39)–(41) and part 2) of Lemma 1, it follows that eT y e˙ y ≤ − η1 ey0 − η2 eyθ − η3 eyf ≤ − (η12 + η22 + η32 )ey := −ηey . This shows that a traditional reachability condition [4], [24] is satisfied, and thus, the error system can be driven to the sliding surface in finite time and a sliding motion maintained thereafter. Hence, the conclusion follows. From sliding-mode theory, Theorems 1 and 2 have shown that system (23)–(27) is an asymptotic observer for the system (14)–(16). V. F AULT R ECONSTRUCTION In this section, it is assumed that an adaptive sliding-mode observer as proposed in Section IV has been designed. Then, by using a modified equivalent output error injection [5], the fault f (t) will be reconstructed based on the observer proposed in the last section. An important feature of the sliding-mode observer is that the error dynamical system can be driven to the sliding surface in finite time and a sliding motion maintained thereafter. When the sliding motion takes place ey (t) = 0,
e˙ y (t) = 0
and in particular eyf (t) = 0,
e˙ yf (t) = 0.
(44)
During the sliding motion, the output-error-injection signal νf in (36) can theoretically be replaced by an equivalent outputerror-injection signal νf eq . By applying (44) to (36), it follows that when a sliding motion takes place W32 ez + N2 (y, u)f (t) − νf eq = 0.
(45)
From Assumption 2, N2 (y, u) is nonsingular in Y × U, and it follows from (45) that f (t) = −N2−1 (y, u)(W32 ez − νf eq ).
T T eT yo Λo eyo + eyθ Λθ eyθ + eyf Λf eyf ≤ 0.
T + eT yf N2 (y, u)f (t) − kf eyf sgn(eyf ).
4033
(46)
In order to reconstruct the fault f (t), it is necessary to recover the equivalent output-error-injection signal νf eq . In the work described in [24], it was obtained using a low-pass filter. In this paper, a modification to the approach given in [5] will be employed. From (31), the equivalent output-error-injection signal νf eq in (46) can be approximated to any accuracy by νf σ := kf (t, y, u)ζ(eyf )
(47)
where kf (·) satisfies (41) and ⎡ ⎢ ⎢ ζ(eyf ) := ⎢ ⎢ ⎣
eyf 1 |eyf 1 |+δ eyf 2 |eyf 2 |+δ
.. .
eyf r |eyf r |+δ
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(48)
where col(eyf 1 , eyf 2 , . . . , eyf r ) := eyf and δ usually is a “small” positive design constant. The choice of the constant δ
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4034
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
determines the degree of approximation to ideal sliding which is attained. In practice, the choice of kf (·) in (47), which must satisfy (41), also affects the approximation to ideal sliding and the accuracy to which eyf approaches zero. The following conclusion is now ready to be presented. Theorem 3: Consider system (1) and (2). Under Assumptions 1 and 2, the signal fˆ(t) := N2−1 (y, u)νf σ
(49)
with νf σ defined by (47), is a reconstruction of the fault f (t). Furthermore, it is a precise reconstruction since lim f (t) − fˆ(t) = 0
t→∞
if N2−1 (y, u), as defined in Assumption 4, is bounded in the domain Y × U. Proof: The earlier analysis has shown that (46) is true. It follows from (46) and (49) that f (t)− fˆ(t) = N2−1 (y, u)W32 ez +N2−1 (y, u) (νf eq −νf σ ) ≤N −1 (·)W32 ez + N −1 (·) νf eq −νf σ . 2
2
(50) From Theorems 1 and 2, it follows that lim ez (t) = 0.
t→∞
Since νf σ can approximate νf eq to any accuracy and N2 (y, u) is bounded in the domain Y × U, it follows that lim f (t) − fˆ(t) = 0. t→∞
Hence, the conclusion follows. The formulas (47) and (48) show that νf σ depends only on known system information: time t, the system output y, the system input u, and the residual signal eyf . From (49), the reconstruction signal fˆ(·) is only dependent on available information, and thus, the fault f (t) can be reconstructed online. This shows that the FDI scheme is practical for real implementation. Remark 10: In Theorem 3, it is required that N2−1 (y, u) is bounded in the domain Y × U. If Y × U is a compact set and N2−1 (y, u) is continuous in Y × U, then N2−1 (y, u) will be bounded. Therefore, in the local case, this condition is not strong. Remark 11: From (49), it is clear that the proposed fault reconstruction fˆ is independent of the unknown parameter θ ˆ The proposed approach allows the and its estimated value θ. unknown parameter to be arbitrarily large from a theoretical point of view. VI. S IMULATION Consider a magnetic-levitation system consisting of an iron ball in a vertical magnetic field created by a single electromag-
net [14]. The model of the system is described in [14] by w˙ 1 = w2 μ(w1 ) 2 w˙ 2 = w3 − g m R 1 w3 + u w˙ 3 = − L(w1 ) L(w1 )
(51) (52) (53)
where w1 , w2 , and w3 represent the (upward) displacement of the ball from the nominal position, the velocity of the ball, and the current through the electromagnet. The parameter m is the mass of the ball, g is the acceleration of gravity, L(w1 ) is the electromagnet inductance, R is the resistance, u(t) is the supplied voltage source, and μ(·) is a known strictly positive function. It is assumed that w1 and w3 are measurable. Here, in order to illustrate the results obtained in this paper, it is assumed that there is a fault f (t) in the input channel. This represents a fault in the circuit associated with the electromagnet—possibly a sudden change in the inductance or the resistance. The system (51)–(53) can be described by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ w2 0 0 1 1) 2 ⎦ ⎣ ⎦ ⎣ 0 ⎦f ⎦ + ⎣ μ(w 0 w˙ = ⎣ m w3 −g + 1 θ + R 1 1 u − L(w1 ) w3 0 L(w1 ) L(w1 ) (54) w1 y= (55) w3 where the term associated with an unknown constant θ has been added to demonstrate the approach given in this paper, and is not a feature of the real magnetic-levitation system. Introduce a transformation (w1 , w2 ) → (w2 , w1 ) := (x11 , x12 ) = x1 and x2 = w3 . Then, in the x coordinates, the system in (54) and (55) can be described in the form of (1) and (2) with μ(y ) 1 2 0 0 m y2 − g F1 = , G1 (·) = 1 0 0 1 , Ψ1 (·) = 0 Φ1 (·) = 1 R 1 y2 + F2 = 0, G2 (·) = L(y1 ) L(y1 ) 1 Φ2 (·) = 0, Ψ2 (·) = L(y1 ) C1 = [0 1], C2 = 1. It is clear that Assumption 1 is satisfied, since Ψ1 (·) = 0 and the second subsystem is first order. Let zI = x11 − x12 . The system is then in the form of (11) and (12). By direct computation, it follows that Assumption 2 is satisfied with W = I2 , N1 = 1, and N2 = 1/L(y1 ). Let zyθ = y1 and zyf = y2 , and then the system can be rewritten in the form of (18)–(21) as follows: z˙I = − zI − zyθ +
μ(zyθ ) 2 zyf − g m
z˙yθ = zI + zyθ + θ 1 R 1 zyf + u+ f (t) z˙yf = − L(y1 ) L(y1 ) L(y1 ) zyθ y= . zyf
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
(56) (57) (58) (59)
YAN AND EDWARDS: ADAPTIVE SLIDING-MODE-OBSERVER-BASED FAULT RECONSTRUCTION
Now, consider the dynamic system μ(zyθ ) 2 zˆ˙ I = − zˆI − zyθ + zyf − g m zˆ˙ yθ = zˆI + zyθ − Λθ (zyθ − zˆyθ ) + θˆ + νθ R 1 zyf + u − Λf (zyf − zˆyf ) + νf zˆ˙ yf = − L(y1 ) L(y1 ) zˆyθ yˆ = zˆyf
(60) (61) (62) (63)
where θˆ is given by the following adaptive law: ˙ θˆ = −β(θˆ + zˆI + zyθ − z˙yθ ).
(64)
The signals νθ and νf are output-error-injection terms given by νθ = kθ sgn(zyθ − zˆyθ ) νf = kf sgn(zyf − zˆyf )
(65) (66)
Fig. 1. Time responses of the switching functions.
where kθ (·) and kf (·) satisfy (40) and (41) and Λθ , Λf , β are design parameters. Let Q = 4, then P = 2. Since ΦI (·) = 0 and W22 = 1, by direct computation, it follows that M=
4 β
β 2β
and, thus, from Theorems 1 and 2, system (60)–(62) is a sliding-mode observer of system (56)–(58) if β is chosen such that 0 < β < 8. Furthermore, from Theorem 3, it follows that fˆ(t) = L(y1 )νf σ where νf σ defined by (47) and (48) is a reconstruction of the fault f (t). If, in (64), z˙yθ is not available, then the Levant second-order differentiator [17] can be applied to estimate z˙yθ : specifically create the system
Fig. 2. Adaptive estimation of the unknown parameter θ.
v˙ 1 = v2 1
v2 = v3 − 1 |v1 − zyθ | 2 sgn(v1 − zyθ ) v˙ 3 = 2 sgn(v1 − zyθ ) where 1 and 2 are positive constants, then v2 is the optimal estimate of z˙yθ [17]. In the following simulation, let β = 7.5, 1 = 1.5, 2 = 1.2, Λθ = Λf = −0.5, g = 9.8 m/s2 , m = 1.5 kg, R = 2.55 Ω, and μ(x1 ) = 1/(1 + x21 ). For simulation purposes, according to [25], L(·) is chosen as L(y1 ) = 2 + (2/(1 + 0.5|y1 |)). Fig. 1 shows the time responses of the switching functions. Fig. 2 shows that the adaptive law gives a good estimate for the parameter θ, and Fig. 3 shows that the fault signal can be reconstructed faithfully. Remark 12: It should be noted that the design parameters in (61) and (62) are scalars Λθ and Λf , which should be negative (stable). The output-injection term is given in (65) and (66), where the parameters kθ and kf should satisfy (40) and (41).
Fig. 3. Fault reconstruction for the magnetic-levitation system.
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
4035
4036
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008
VII. C ONCLUSION In this paper, a class of nonlinear multi-output systems with uncertain parameters has been considered. An adaptive law has been given to identify the parameters. Based on slidingmode techniques, an observer has been developed to estimate the system state variables asymptotically even in the presence of uncertain parameters and faults. Furthermore, the fault signals have been reconstructed from the equivalent output-errorinjection signals associated with the observer. The results have good robustness, since the estimate of the uncertain parameters can be used in the observer design. A magnetic-levitation system has been introduced to illustrate the approach proposed in this paper. Simulations show that the results which have been obtained are effective. R EFERENCES [1] J. Chen and R. J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems. Boston, MA: Kluwer, 1999. [2] W. Chen and M. Saif, “A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonlinear systems,” Int. J. Syst. Sci., vol. 38, no. 12, pp. 943–955, Jan. 2007. [3] J. Davila, A. Fridman, and L. Poznyak, “Observation and identification of mechanical systems via second order sliding modes,” Int. J. Control, vol. 79, no. 10, pp. 1251–1262, Oct. 2006. [4] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and Applications. London, U.K.: Taylor & Francis, 1998. [5] C. Edwards, S. K. Spurgeon, and R. J. Patton, “Sliding mode observers for fault detection and isolation,” Automatica, vol. 36, no. 4, pp. 541–553, Apr. 2000. [6] T. Floquet, J. P. Barbot, W. Perruquetti, and M. Djemai, “On the robust fault detection via a sliding mode disturbance observer,” Int. J. Control, vol. 77, no. 7, pp. 622–629, May 2004. [7] P. M. Frank and X. Ding, “Survey of robust residual generation and evaluation methods in observer-based fault detection systems,” J. Process Control, vol. 7, no. 6, pp. 403–424, Dec. 1997. [8] L. Fridman, Y. Shtessel, C. Edwards, and X. G. Yan, “Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems,” Int. J. Robust Nonlinear Control, vol. 18, no. 4/5, pp. 399–412, Mar. 2008. [9] L. Fu and T. Liao, “Globally stable robust tracking of nonlinear systems using variable structure control and with an application to a robotic manipulator,” IEEE Trans. Autom. Control, vol. 35, no. 12, pp. 1345–1350, Dec. 1990. [10] J. Y. Hung, W. Gao, and J. C. Hung, “Variable structure control: A survey,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 2–22, Feb. 1993. [11] A. Isidori, Nonlinear Control Systems, 3rd ed. London, U.K.: SpringerVerlag, 1995. [12] B. Jiang, M. Staroswiecki, and V. Cocquempot, “Fault estimation in nonlinear uncertain systems using robust sliding-mode observers,” Proc. Inst. Elect. Eng.—Part D: Control Theory Appl., vol. 151, no. 1, pp. 29–37, Jan. 2004. [13] B. Jiang, M. Staroswiecki, and V. Cocquempot, “Fault accommodation for nonlinear dynamic systems,” IEEE Trans. Autom. Control, vol. 51, no. 9, pp. 1578–1583, Sep. 2006. [14] D. Karagiannis, A. Astolfi, and R. Ortega, “Two results for adaptive output feedback stabilization of nonlinear systems,” Automatica, vol. 39, no. 5, pp. 857–866, May 2003. [15] P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. London, U.K.: Academic, 1985. [16] C. Lascu and G.-D. Andreescu, “Sliding-mode observer and improved integrator with DC-offset compensation for flux estimation in sensorlesscontrolled induction motors,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 785–794, Jun. 2006. [17] A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, Mar. 1998. [18] K. A. Loparo, M. L. Adams, W. Lin, M. F. Abdel-Magied, and N. Afshari, “Fault detection and diagnosis of rotating machinery,” IEEE Trans. Ind. Electron., vol. 47, no. 5, pp. 1005–1014, Oct. 2000. [19] R. Marino and P. Tomei, Nonlinear Control Design. Englewood Cliffs, NJ: Prentice-Hall, 1995.
[20] O. Moseler and R. Isermann, “Application of model-based fault detection to a brushless DC motor,” IEEE Trans. Ind. Electron., vol. 47, no. 5, pp. 1015–1020, Oct. 2000. [21] C. De Persis and A. Isidori, “A geometric approach to nonlinear fault detection and isolation,” IEEE Trans. Autom. Control, vol. 46, no. 6, pp. 853–865, Jun. 2001. [22] C. P. Tan and C. Edwards, “Sliding mode observers for detection and reconstruction of sensor faults,” Automatica, vol. 38, no. 10, pp. 1815– 1821, Oct. 2002. [23] C. P. Tan and C. Edwards, “Sliding mode observers for robust detection and reconstruction of actuator and sensor faults,” Int. J. Robust Nonlinear Control, vol. 13, no. 5, pp. 443–463, Apr. 2003. [24] V. I. Utkin, Sliding Modes in Control Optimization. Berlin, Germany: Springer-Verlag, 1992. [25] H. H. Woodson and J. R. Melcher, Electromechanical Dynamics. Malabar, FL: Krieger, 1990. [26] W. F. Xie, “Sliding-mode-observer-based adaptive control for servo actuator with friction,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp. 1517– 1527, Jun. 2007. [27] A. Xu and Q. Zhang, “Residual generation for fault diagnosis in linear time-varying systems,” IEEE Trans. Autom. Control, vol. 49, no. 5, pp. 767–772, May 2004. [28] X. G. Yan and C. Edwards, “Nonlinear robust fault reconstruction and estimation using a sliding mode observer,” Automatica, vol. 43, no. 9, pp. 1605–1614, Sep. 2007. [29] X. G. Yan and L. Xie, “Reduced-order control for a class of nonlinear similar interconnected systems with mismatched uncertainty,” Automatica, vol. 39, no. 1, pp. 91–99, Jan. 2003. [30] X. Zhang, M. M. Polycarpou, and T. Parisini, “A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems,” IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 576–593, Apr. 2002.
Xing-Gang Yan received the B.Sc. degree from Shanxi Teachers University, Xi’an, China, in 1985, the M.Sc. degree from Qufu Teachers University, Qufu, China, in 1991, and the Ph.D. degree in engineering from Northeastern University, Shenyang, China, in 1997. From 1985 to 1988, he was a Teaching Assistant and, then, a Lecturer with the Department of Mathematics, Weinan Educational University, Weinan, China. From 1991 to 1994, he was a Lecturer with the Department of Applied Mathematics, Qingdao University, Qingdao, China. From 1998 to 1999, he was a Postdoctoral Fellow with Northwestern Polytechnical University, Xi’an, China. During this period, he was also a Research Associate with the University of Hong Kong, Hong Kong, for eight months. From 1999 to 2001, he was a Research Fellow with Nanyang Technological University, Singapore. He is currently a Research Associate with the Control Systems Research Group, Department of Engineering, University of Leicester, Leicester, U.K. His research interests include decentralized control, nonlinear control, robust control, sliding-mode control, fault detection and isolation, and applications in practical systems.
Christopher Edwards (M’00) was born in Swansea, U.K. He received the B.Sc. degree in mathematics from Warwick University, Coventry, U.K., in 1987, and the Ph.D. degree supported by a British Gas Research Scholarship from Leicester University, Leicester, U.K., in 1995. From 1987 to 1991, he was a Research Officer with British Steel Technical, Port Talbot, where he worked in the R&D Division. He was a Lecturer in 1996, a Senior Lecturer in 2004, and has been a Reader since 2008 with the Control Systems Research Group, Department of Engineering, Leicester University. He is the coauthor of 170 refereed papers in the area of sliding-mode control and observation, fault detection and isolation, and fault-tolerant control. He is also the coauthor of two books on sliding-mode control.
Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 08:26 from IEEE Xplore. Restrictions apply.
