Robust Sensor Fault Reconstruction for Lipschitz Nonlinear Systems

We extend existing theory on robust nonlinear observer design to the class of nonlinear Lipschitz systems where the systems are subject to sensor faults and disturbances. The designed observer is used for robust reconstruction of fault signals. Allowing bounded unknown disturbances to model system uncertainties, it is shown that by adjusting a design parameter we can trade off between fault reconstruction and disturbance attenuation. An LMI procedure solvable using commercially available softwares is presented. Two examples are presented to illustrate the application of the results.


Introduction
Modern control systems strongly rely on actuators, sensors, and data acquisition/interface components to ensure a proper interaction between the physical controlled system and control devices.Any faults in sensors and/or actuators may cause process performance degradation, process shutdown, or a fatal accident.For instance, in feedback control applications, faulty sensors give wrong information about the system status, which could cause disastrous results as the system may go unstable.On the other hand, even if the system is stable, inaccurate sensor values can introduce poor regulation or tracking performance, which may be highly undesirable for many high precision control applications.Similarly, faulty actuators may severely affect the overall system performance.Therefore, there is a growing demand for reliability, safety, and fault tolerance in modern control systems.To improve the reliability and safety, much effort has been made to develop model-based fault detection and isolation FDI techniques see, e.g., 1-3 , and the references therein for recent advances .One of the particular interesting techniques among all model-based techniques for FDI is an observer-based fault detection filter design.The goal has been to utilize the This paper is organized as follows.In Section 2, the sensor fault reconstruction problem is formulated.In Section 3, an easily implementable design algorithm summarizes the proposed methodology for fault reconstruction.In Section 4, this algorithm is applied to two numerical examples and simulation results are presented.Concluding remarks are given in Section 5.
The notation used in this paper is fairly standard.For a given matrix A, A T denotes its transpose.I denotes unity matrix with appropriate dimension.If A and B are symmetric matrices, A ≥ B resp., A > B denotes A − B positive semidefinite resp., positive definite and A ≤ B resp., A < B denotes A − B negative semidefinite resp., negative definite .λ min M and λ max M denote minimum and maximum eigenvalue of M, respectively.The space L 2 0, ∞ represents the set of all signals ω t which are square integrable and satisfy The following result is used in the paper.

Problem Formulation
We consider the nonlinear systems given by S: where x ∈ Ê n is the state, u ∈ Ê m is the control, y ∈ Ê p is the output, and Γ y, u, t is a known nonlinear vector function.The input w ∈ Ê l is assumed to be the unknown disturbance which can also be used to represent a general class of modeling errors.In any case, w is assumed to be an unknown exogenous disturbance/noise.Here, sensor faults are described by the vector f ∈ Ê q , assumed to be zero prior to the failure time nonzero after the fault occurrence.A, B, B φ , C, D, E and F are assumed to be known constant matrices of appropriate dimensions.
It is worth noting that the distribution matrix B φ indicates how the system 2.1 is affected by the nonlinearity φ.We assume that rank C p, rank F q, and p ≥ q.Without loss of generality, it can be assumed that the outputs of the system have been reordered and scaled if necessary so that the matrix F has a structure where F 2 ∈ Ê q×q is a nonsingular matrix.As mentioned in 16 , the assumption that only certain sensors are fault prone is a limitation.However in practical situations, some sensors may be more vulnerable to damage or may be more sensitive or delicate in terms of construction than others, and so such a situation is not unrealistic.Also certain key sensors may have backups hardware redundancy and so essentially a fault-free signal can be assumed from a certain subset of the sensors.Finally, we assume that the system 2.1 is locally Lipschitz in a region Ω containing the origin, uniformly in u, that is: for all u ∈ Ê m , for all t ∈ Ê , for all x 1 and x 2 ∈ Ω.Here, the parameter α > 0 is referred to as the Lipschitz constant and is independent of x, u, and t.Many nonlinearities are locally Lipschitz.Examples include trigonometric nonlinearities occurring in robotics, nonlinearities which are square or cubic in nature, and so forth.The function φ can also be considered as a perturbation affecting the system; see 40 for more details about nonlinear Lipschitz systems.Scaling the output y and partitioning appropriately yields where , and E 2 are appropriate matrices depending on C, D and E. The output vector has now been partitioned into nonfaulty y 1 and potentially faulty y 2 .Notice now that the subsystem 2.1 and y 1 in 2.4 makes up a fault-free system.Assume further that A, C 1 is detectable.Consider a nonlinear observer for the fault-free system defined by 2.1 and y 1 in 2.4 where x ∈ Ê n is an estimate for the state x and L ∈ Ê n× p−q is the observer gain.Define e : x − x as the state estimation error.Equations 2.1 , 2.4 , and 2.5 are combined to yield where φ φ x, u, t − φ x, u, t .A well-known result in 44 states that the error system 2.6 is asymptotically stable for all φ in 2.3 with a Lipschitz constant α if the observer gain L can be chosen in such a way that where The ratio in 2.7 is maximized when Q I 45 .The problem is then reduced to that of choosing L to satisfy α < 1 2λ max P .

2.9
As shown in 41 , using Schur's complement lemma, the inequality 2.9 is equivalent to In the following, motivated by the development in 16, 19 for the LTI systems, an approach for reconstructing the sensor fault f from the residual for Lipschitz nonlinear systems in 2.1 is proposed.Define a reconstruction for the sensor fault where ν y − y is the residual and with K 1 ∈ Ê q× p−q being a weighting matrix.It is easy to show that combining 2.1 , 2.2 , 2.11 , and 2.12 yields where e f f − f is the fault reconstruction error, and Equations 2.6 and 2.13 show the effect of the disturbance w on the quality of the fault reconstruction error, that is,

2.15
The objective now would be to minimize the effect of w on e f .To achieve unknown disturbance attenuation and fault reconstruction, the following problem can be formulated: find the gain L such that the system Σ in 2.15 be asymptotically stable and the L 2 -gain from the disturbance w to the fault reconstruction error e f is less than or equal to a prescribed H ∞ performance γ > 0, that is,

2.16
This motivates us to consider the following optimization problem.
Problem 1.Given γ > 0, find the gain L such that the error dynamic system in 2.15 be asymptotically stable and 2.17 In the next section, a solution is proposed to Problem 1 in terms of LMIs.

Fault Reconstruction
Following the lines of 41 for robust nonlinear observer design, we propose an LMI-based solution to Problem 1 that leads to a constructive algorithm for sensor fault reconstruction.
The following result summarizes the main result of this section.
Theorem 3.1.Consider the nonlinear system 2.1 .Given Lipschitz constant α > 0 and γ > 0, there exists an nth-order nonlinear observer in the form 2.5 which solves Problem 1, if there exist β > 0 and the solutions P P T > 0 and Z such that the following LMIs have a solution

3.2
Once the problem is solved Proof.Define a Lyapunov function V e T Pe, where P P T > 0 satisfies in 2.10 .From the error system 2.6 , we have V ėT Pe e T P ė where Q is given by 2.8 .The second term in the right-hand side of 3.4 can be upper bounded as follows ≤ 2α e B φ T Pe .

3.5
Using Lemma 1.1, we have where β is any positive real constant and hence from 3.4 , we have V ≤ e T Qe w T B − LE 1 T Pe e T P B − LE 1 w, 3.7 where Now, from 2.17 , it is easy to show that Therefore, a sufficient condition for J < 0 is that ∀t ∈ 0, ∞ , e f T e f − γ 2 w T w V < 0.

3.10
But, from 3.7 , we have

3.11
where Thus a sufficient condition for J < 0 is that M < 0. Using Schur's complement lemma and the change of variable Z PL, the inequality M < 0 can be replaced by 3.1 immediately.Therefore, if there exists scalars β > 0 and γ > 0 and matrices P P T > 0 and Z such that the LMIs in 3.1 have a solution, then L P −1 Z.
Using 2.5 and 2.11 , the nonlinear dynamical system for sensor fault reconstruction is given by

3.13
Thanks to Theorem 3.1, Problem 1 can be solved efficiently using the following algorithm and by reducing γ iteratively, an optimal solution is approached.Algorithm 1.Given plant 2.1 with Lipschitz constant α > 0, construct the sensor fault signal by performing the following steps.
Step 1. Choose the weighting matrix K 1 and compute K using 2.12 .
Step 3. Compute the gain L using 3.3 .
Step 4. Construct the nonlinear dynamical system Σ in 3.13 .This algorithm is constructive and can be implemented using standard scientific softwares such as Scilab 42 and Matlab 43 .
Remark 3.2.Although the main objective of this paper is sensor fault reconstruction, but the proposed method has good potential to extended to the even more interesting case of the reconstruction of actuator fault situations.It can however be an interesting topic for future research and it is under investigation.A good staring point for this research can be motivated by the developments in 46 to transform the plant 2.1 into two subsystems with one of them decoupled from the actuator fault.Then, the nonlinear observer 2.5 could be designed to provide the estimation of unmeasurable state, which are used to construct actuator fault estimation algorithm.It is worth mentioning that a constructive algorithm based on mixed H 2 /H ∞ approach is also proposed in 25 for actuator fault reconstruction for Lipschitz nonlinear systems.

Numerical Examples
To illustrate the application of the results obtained in the paper, we consider two different examples of nonlinear systems.and φ x, u, t 0.5 The sensor fault reconstruction is obtained by using Algorithm 1 where γ 0.3 and β 0.01.The LMI minimization has been performed using LMITOOL, a user-friendly Scilab package 42 .The simulation results are shown in Figures 1-3.The disturbance w 1 is set as 47 And w 2 and w 3 are white noise processes which are assumed to be zero-mean white noise processes with variance 0.05.The control signal is assumed to be u t sin t .The faulty outputs y 1 and y 2 are shown in Figure 1.As shown in Figures 2-3, the fault reconstruction scheme reconstructs the faults perfectly when sensor faults are applied in the presence of disturbance and noises which justify the proposed scheme for fault tolerant control.
Here, a comparison of the estimation capabilities of the presented approach with the descriptor system approach for Lipschitz nonlinear systems as recently proposed in 10 can be performed.In this direction, the sensor fault model 2.1 with D 0 can be denoted as where

4.4
Following the approach presented in 10 , a sensor fault estimator in the form can be constructed, where x I n 0 x and the matrices M, L, and N can be obtained through satisfying an LMI as proposed in Theorem 2 in 10 .Using LMITOOL in Scilab package, it can be shown that for the aircraft example there does not exist any solution to this LMI for all α > 0 and γ > 0, so the descriptor system approach as proposed in 10 is no longer applicable this exhibits the significance of our approach proposed in this paper.
Example 4.2.Consider the following nonlinear system 41 ẋ 0 1 where w w 1 w 2 T , where w 1 is the disturbance and w 2 is the measurement noise which is assumed to be a zero-mean white noise process with unit covariance and f is the sensor fault.The fault reconstruction scheme is performed by using Algorithm 1 in the previous section with α 1.17 and β 3. To be able to make a fair comparison between the fault reconstruction for different values of γ , the actual and estimated fault are displayed in Figures 4, 5, 6, and 7.
As shown in these figures, in order to analyze the performance of the fault reconstruction, a sensor fault f with magnitude 1 and a disturbance w 1 are applied.Figures 4-7 clearly indicate that by reducing γ , the effect of w 1 on f can be made arbitrarily small and the sensor fault f can be effectively reconstructed.Also, as shown in Figure 7, when γ is reduced to optimal value 0.38, the effect of noise will increase.This clearly shows that there is a definite tradeoff between fault reconstruction, disturbance attenuation, and noise rejection.

Conclusion
In this paper, a robust sensor fault reconstruction method for a class of Lipschitz nonlinear systems is proposed through LMI optimization in the presence of disturbances and noises.The advantage of the fault reconstruction method is that it provides a good estimate of faults, thus providing useful information for fault tolerant controller design.As shown in simulation results, by adjusting a single parameter, it becomes possible to trade off between fault reconstruction, disturbance attenuation and noise rejection.Further research work includes two aspects.The first one is that the proposed sensor fault reconstruction approach could be extended to nonlinear systems with arbitrarily large Lipschitz constant or one-sided Lipschitz systems as described in 48 .Possible extensions to   a large class of uncertain nonlinear systems as described in 49 with simultaneous actuator and sensor faults and implementation on an experimental setup similar to that in 19 could be another interesting issues.

Lemma 1 . 1
see 22 .Let D, S and F be real matrices of appropriate dimensions and F satisfying F T F ≤ I. Then for any scalar > 0 and vectors x, y ∈ Ê n , we have 2x T DFSy ≤ −1 x T DD T x y T S T Sy. 1.1

Example 4 . 1 .
Consider the plant 2.1 with the following state space matrices for an aircraft model 47