Sliding Mode Observer-Based Fault Reconstruction for Uncertain Linear Systems

Problem statement: Fault reconstruction scheme is different from the m ajority of Fault Detection and Isolation (FDI) methods described in the literature in the sense that it not only detect s and isolates the fault, but provides an estimate of the fault. This approach is very useful for incipient faults and slow drifts, which are very difficult to detect . Also, this approach is very useful for Fault Tolerant Control (FTC) systems in the sense that in stead of reconfiguration of the control system, the faulty sensors or actuators can be corrected and th e simple control method can still be effectively used. Motivated by these useful features of fault, we are int rested in performing observer-based fault reconstruction scheme for uncertain linear systems. Approach: In this study we present a scheme to design robust sliding mode observer for linear syst em where both faults and uncertainties are considered. The objective is to derive a sufficient condition using Linear Matrix Inequalities (LMIs) for the stability of the observer. The so-called eq uivalent output error injection is discussed for fa ult reconstruction. Results: we get a simple sliding mode observer design for de tection and reconstruction of faults for uncertain linear systems. Conclusion: With the real model of the seventh-order aircraft we show that the methods provided by present paper hav e good performances .


INTRODUCTION
Fault Detection and Isolation (FDI) has received considerable attention during the last three decades, both in research and in application. A fault is defined as an abnormal condition in a system with component malfunction or variation in operating condition. Fault in a dynamics system may occur in all possible location, such as actuators, sensors and system's parameters. The main function of an FDI scheme is to generate an alarm when a fault occurs (Fault Detection) and then to determine the location of the fault (fault isolation). There is a large variety of FDI approaches (Patton and Frank, 2000;Chen and Patton, 1999;Gertler, 1998;Prasannamoorthy and Devarajan, 2011;Chatchanayuenyong, 2009). The most effective methods for model based FDI are based on observers where the measured plant output is compared to the output of an observer and the output error are used to form a residual (Patton and Frank, 2000;Chen and Patton, 1999). The residual is then examined for the likelihood of faults by using a fixed or adaptive threshold.
Another approach different to residual generation is fault estimation or fault reconstruction which can determine the size, location and dynamics behavior of the fault. However, most fault reconstruction schemes are designed about a model, which usually possess uncertainties. These uncertainties could corrupt the reconstruction and could produce a false alarm. In this context, the robustness problem in fault reconstruction is essential. Edwards et al. (2000) and Chamsai et al. (2010) proposed an approach based the equivalent output error injection where the sliding mode was maintained even in the presence of faults which can be reconstructed under certain conditions. Later it was extended by Tan and Edwards (2002) where sensor faults were considered. However, uncertainty was not considered in these early papers. A FDI scheme for a class of linear systems with uncertainty was proposed by Tan and Edwards (2003) who focused on minimizing the L 2 gain between the uncertainty and the fault reconstruction signal using Linear Matrix Inequalities (LMIs).
In this study, we develop a robust sliding mode observer design method for a class of uncertain linear systems which completely eliminate the effect of uncertainty on state estimation and fault reconstruction. A LMIs based sufficient condition is presented for the existence and stability of this observer. The design method will applied to reconstruct actuator faults using the equivalent output error injection concept.

Notation:
n ℝ : Denotes the n-dimensional real Euclidean space. For a square matrix λ min (A): Denotes the minimum eigenvalue of A I n : Represents an n th order identity matrix for all complex number s with nonnegative real part. As described in Yan and Edwards (2007), under Assumptions 3-4, there exists a non singular linear x ∈ ℝ such that the matrices (A, E, D, C) from (1) in the new coordinates are given by Eq. 6: 1 2 ( n p) xr (n p) xq where, where, is of full rank. By construction, the pair (A 22 , A 31 ) is completely observable and the eigenvalues of A 11 are the invariant zeros of the triple (A, R, C) (Edwards and Spurgeon, 1998).
A sliding mode observer for the system (1) is Eq. 8: where n x ∈ ℝ is the estimated of the state x, p y ∈ ℝ is the estimated of the output y. The matrices G 1 n p n G × ∈ ℝ are observer gains that are to be designed. In particular, G n has the structure Eq. 9: (y y) ( y, u, t) if y y 0 P (y y) : where, p p 2 P × ∈ ℝ is a symmetric positive definite matrix which will be formally defined later. The scalar function (y, u, t) η must be upper bound on the magnitude of the uncertainty plus the fault and satisfies Eq. 11: where 0 0 η > is a small positive scalar.
Subtracting observer Eq. 8 from system Eq. 1 results in the system error dynamics Eq 12: is the state estimate error and yê : y y = − is the output estimation error.
Firstly, two lemmas will be introduced to provide the stability of sliding motion.
For system (12), consider the sliding surface Eq. 13: y {e e Ce 0} = = = S (13) Lemma 1: If an ideal sliding motion takes place on S in finite time, then the sliding dynamics are given by the system matrix A 1 + L 1 A 31 .
Proof: Assume that an ideal sliding mode exists then the error system (12) will become Eq. 14: In order for a unique equivalent output injection to existe n det(CG ) 0 ≠ , it follows from (14) that Eq. 15: A sliding motion insensitive to the faults and uncertainties is governed then by Eq. 16: Hence the reduced order sliding motion is governed by A 1 + L 1 A 31 .
Lemma 2: under assumption 4, there exists a matrix (n p) p L − × ∈ ℝ from structure (9) such that the matrix A 1 + LA 3 is stable.
Proof: Using the fact that (A 22 , A 31 ) is observable if follows that there exists a matrix (n p l) (p k ) 12 L R − − × − ∈ such that A 22 + L 12 A 31 is stable. Partition L from (9) as Eq. 17: Then, from the partition in (7), it follows that Eq. 18: is stable from the stability of A 11 and A 22 + L 12 A 31 .
Reduced-order Sliding mode observer design: One way to identify the reduced order sliding motion is to perform a further change of coordinates according to the nonsingular matrix Eq. 19: Hence, the matrices (A, E, D, C) from (6) and G n from (9) are transformed to be Eq. 20: where 1 and choose l G so that l,1 A is a stable design matrix. For simplicity in the subsequent analysis, it will be assumed that s A is symmetric negative definite. This is not a stringent assumption since s A is a design parameter. Partitioning the state estimation error (12) conformably with (20)

MATERIALS AND METHODS
To show the stability of sliding motion we analyze the dynamic performance of error system (22) using Lyapunov method and derive a sufficient condition for the asymptotically stability of the sliding dynamics via Linear Matrix Inequalities techniques.

RESULTS AND DISCUSSION
where, Considering inequality (11) and substituting the υ given in (10) into (28), we have Eq. 29: T T T y 2 2 2 y 2 2 2 a y 2 2 y T 2 y 2 2 2 2 a 2 y 2 y 2 y 2 2 0 2 2 2 y 2e P C E d 2e P C D f 2e P P e 2 P e ( C E d C D f ) 2 (.)(P e ) P e 2 P e ( C E d C D (.)) 2 0 P e Combining Eq. 29 into Eq. 28 yields: If the condition (11) Where :   T  T  1  1  3  1  3  1  2 2 3  T  2 2 3 2 s s 2 P (A LA ) (A LA ) P (P C A ) P C A P P If Ω<0 then V 0 < ɺ . Thus, the observer error dynamics (22) is asymptotically stable. Notice that the Inequality (32) is not linear because of the product P 1 L. This problem can be solved by using the changes of variables Y = P 1 L. Therefore, Inequality (23) can then be obtained.
Remark 1: Note that the problem of finding symmetric positive matrices P 1 , P 2 and a matrix Y to satisfy (23) is a standard LMI feasibility problem which can be solved using the standard LMI algorithm (Gahinet et al., 1995).

Reachability of the sliding motion:
In order to ensure the stability of the observer it is only required to prove that the error system (22) can be driven to the sliding surface S in finite time by choosing an appropriate gain (y, u, t) η in (12).

Theorem 2:
Under Assumptions 1-4, the error system (22) is driven to the sliding surface S and remains on it if the gain (y, u, t) η in (12) is chosen to satisfy Eq. 33: where, η 1 is a positive scalar, d 0 and ρ are the upper bounds of d and f a , respectively.
Proof: Consider the following Lyapunov function candidate Eq. 34: The derivative along the trajectory is given by Eq. 35: T T T T y 2 s s 2 y y 2 2 3 1 y 2 2 2 T T y 2 2 2 a y 2 V e (P P )e 2e P C A e 2e P C E d 2e p C D f 2e P Using the fact that as s A is stable design matrix, it follows that Eq. 36: By applying (2), (3) and (33), we obtain Eq. 37: 2 y 2 3 1 2 2 2 2 a 2 y T 2 y 2 y 2 y 2 3 1 2 2 0 2 2 V 2 P e ( C A e C E d 2 C D f ) P e 2 (y,u, t)(P e ) P e 2 P e ( C A e C E d C D (y,u, t)) From Theorem 1, the error e 1 (t) is asymptotically stable. Thus, there exist an instant t f and a positive scalar k such That Eq. 38: Therefore from the definition of η(y, u, t) in (33), it follows that ∀t≥t f : Integrating the last deferential inequality, it follows that an ideal sliding motion is achieved and maintained after some finite time (Utkin, 1992).

Design of the sliding motion system matrix:
This section considers the sliding motion design problem and shows how additional LMI constraints can be augmented to tune the sliding mode performance. One approach is to achieve pole placement of 1 1 3 A LA = + A in regions of the complex plane.
Definition 1: Chilali and Gahinet (1996)  The matrix A is called D-stable if all its poles lie in A sub region of the complex left-half plane. As an example, the poles may be required to lie in the following sectors: • Disk of radius r and center (q, 0) • Conic sector centred at (0, 0) with inner angle θ • Vertical strip -h 1 < x < -h 2 < 0 Chilali and Gahinet (1996) have proven that these sectors are an LMI region described by the following inequalities: T T 1 1 1 1 1 1 1 1 T T 1 1 1 1 1 1 1 1 ( P P )sin ( P P )cos 0 (P P ) cos ( P P )sin T 1 1 1 1 2 1 T 1 1 1 1 1 1 P P 2h P 0 (P P ) 2h P 0 Therefore, the dynamics of the sliding motion are designed by solved the LMIs (23) and (41) Consequently, the error dynamics for e y in sliding mode is given by Eq. 45: where, υ eq denotes the equivalent output error injection and represents the average behavior of the discontinuous function υ(t) defined by (10), which is necessary to maintain an ideal sliding motion Applying the structural properties of E 2 and D 2 in (7), yields Eq. 46: where, (k q ) r 1 H − × ∈ ℝ and q q 2 H × ∈ ℝ is nonsingular. This condition guarantees that the fault can be completely decoupled from uncertainty.
Multiplying both sides of (46) by Y yields Eq. 48: Let W 2 denote the last q rows of W. It follows from (48) that Eq. 49: and since H 2 is non singular, yields Eq. 50: Now, it is required to recover the equivalent output error injection υ eq . Two practical approaches can be used. The first approach is to pass the output of the injection term υ eq through a low-pass filter. The second approach is to use a boundary layer to smooth out the discontinuous υ eq . Here the approach given in (Edwards and Spurgeon, 1998) will be employed to produce the υ eq . From (10), the equivalent output error injection signal υ eq can be approximated by Eq. 51: 2 y 2 y P e (y, u, t) P e σ υ = η + σ where, σ is a small positive scalar. Define a would-be actuator fault reconstruction: Then from (50) and (52) Eq. 53: Since 1 t lime (t) 0 →∞ = and since eq σ υ − υ can be made arbitrarily small by choice of σ then â f (t) defined by (52) is a reconstruction for the actuator fault f a (t).

An illustrative example:
The method proposed in this paper will now be demonstrated with an example, which is a seventh-order model of an aircraft (Yan and Edwards, 2007). The states are: 7 r a bank angle (rad ) r yaw rate (rad / s) p roll rate (rad / s) x sideslip angle (rad) x w ashout filter state. rudder deflection (rad) aileron deflection (rad ) The inputs are: rc ac rudder com mand (rad ) u aileron com mand (rad ) The outputs are: 2 a 2 a 7 r roll acceleration (rad / s ) p yaw acceleration (rad / s ) y bank angle (rad ) x w ashout filter state In the notation of (1), the matrices A, B and C can be obtained from (Tan and Edwards, 2001), whereas the matrices D and E are: It can be easily verified that the Assumptions 3 and 4 are satisfied. Hence, the method proposed in this paper can be used. Performing the co-ordinate transformation T to obtain the canonical form described in (6)-(7) yields the following matrices: It can be verified that the system (A, E, D, C) has an invariant zero at -2.0722 and the pair (A 22 , A 31 ) is completely observable.
Observer design: Suppose that wish to assign the eigenvalues of the sliding mode represented by the system matrix 1 A to lie in the intersection of the following regions: • A circle of centre (0,0) and radius 5 • A vertical upper bound at x = -2 • A conic sector symmetric about the real axis, with inner angle θ = 40 When we refer to synthesis procedure and the imposing of the constraints (23) and (41)-(43), we deduce that the Matlab's LMI toolbox in (Gahinet et al., 1995) returns the values of P 1 , P 2 and L: The poles of matrix s A are located at -2.2, -2.4, -2.6 and -2.8, respectively. Consequently, the associated gain matrices from the observer representation in (8)

Robust state estimation:
In the simulations that follow, the scalar function η(y, u, t) from (10) was chosen to be 200 and σ was chosen to be 10 −5 . Furthermore, the system was assumed to have an initial condition: of x(0) = [-0.01 -0.2 0.3 0.1 0 -0.6 0.3] and the observer was assumed to have zero initial condition. A disturbance d = 3sin(0.5t) is applied to the system from t = 0. Both actuators were assumed to be faulty. The fault on the first actuator is a ramp signal applied at t = 15s and settles at t = 20s and the fault on the second actuator starts at t = 10s, settles at t = 15s and switches at t = 20s. Figure 1 show the trajectories of system states x as well as the estimates x provided by the scheme in this paper. It shows that the estimates converge very quickly to the actual states.
Then, for any fault a f (t) , the signal â f (t) obtained from (52) is a reconstruction of the fault. Figure 2 shows the faults that are applied to the actuators as well as their reconstructions. It is clear that the sliding mode observer faithfully reconstructing faults simultaneously occurring in both actuators despite the effect of the uncertainty and initial condition of the system. Figure 3 considers the case when the sensor signals were subject to white noise of standard deviation of 10-4. It shows satisfactory actuator fault reconstruction in the presence of noise.

CONCLUSION
This study has proposed a method for robust actuator and sensor faults reconstruction in uncertain linear systems using sliding mode observer. Compared to existing works, the observer in this study eliminates completely the effect of uncertainty on the state estimation and fault reconstruction. This method is initially formulated to solve the problem of actuator faults reconstruction. It is extended to the case of sensor faults by the introduction of an appropriate filter. The simulation for a real model of the seventh-order aircraft shows that the method provided by present paper has good performances.