Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 9251031, 12 pages
https://doi.org/10.1155/2017/9251031

Research Article

Simultaneous Robust Fault and State Estimation for Linear Discrete-Time Uncertain Systems

Feten Gannouni and Fayçal Ben Hmida

National Higher School of Engineering of Tunis (ENSIT), Laboratoire d’Ingenierie des Systèmes Industriels et des Energies Renouvelables (LISIER), University of Tunis, 5 Taha Hussein Street, BP 56, 1008 Tunis, Tunisia

Correspondence should be addressed to Feten Gannouni; fetengannouni@yahoo.fr

Received 30 July 2016; Revised 8 October 2016; Accepted 24 October 2016; Published 15 January 2017

Academic Editor: Alberto Borboni

Copyright © 2017 Feten Gannouni and Fayçal Ben Hmida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the problem of robust simultaneous fault and state estimation for linear uncertain discrete-time systems with unknown faults which affect both the state and the observation matrices. Using transformation of the original system, a new robust proportional integral filter (RPIF) having an error variance with an optimized guaranteed upper bound for any allowed uncertainty is proposed to improve robust estimation of unknown time-varying faults and to improve robustness against uncertainties. In this study, the minimization problem of the upper bound of the estimation error variance is formulated as a convex optimization problem subject to linear matrix inequalities (LMI) for all admissible uncertainties. The proportional and the integral gains are optimally chosen by solving the convex optimization problem. Simulation results are given in order to illustrate the performance of the proposed filter, in particular to solve the problem of joint fault and state estimation.

1. Introduction

This paper is concerned with the problem of joint fault and state estimation of linear discrete-time uncertain systems under convex bounded parametric uncertainty. This problem is solved by using a robust filtering approach to produce a robust fault and state estimation [1–6].

The proposed filter can play a significant role in several applications, for example, model based fault detection and isolation (FDI) problem [1–7] and fault tolerant control (FTC) problem [7, 8].

In the past three decades, the problem of robust state estimation in the presence of uncertainties has attracted the interests of many researchers. This problem is largely treated in the literature by different approaches: the guaranteed cost design [9–12], the filtering [13–15], and the set-valued estimation [16, 17]. One limitation of the different design approach for online filter operation is that they require continuous testing of a certain existence condition. When the condition fails at any particular iteration, the proposed filters can diverge.

From the point of view of minimizing the worst possible regularized residual norm over the class of admissible uncertainties, new robust filters are designed for linear uncertain systems by [18, 19]. Compared with the aforementioned robust formulations the developed filters perform data regularization rather than deregularization which represent an important property for online operation. The proposed filters in [19] are based in data regularization solution. This filter guarantees an error variance with an optimized guaranteed upper bound for any allowed uncertainty. To improve robustness against uncertainties such as disturbances and modeling errors, [20] introduced a proportional integral Kalman filter (PIKF). The proportional and the integral Kalman gains were obtained from the solution of Riccati equation leading to minimum error variance. Later, [21] developed a new robust proportional integral Kalman filter for stochastic linear uncertain systems. The filtering problem is converted into a convex optimization problem for continuous time systems with polytopic uncertainties and the filter parameters are optimally chosen by solving this problem.

The problem of robust Kalman filtering and optimal filtering in the presence of unknown inputs and unknown faults has received considerable attention in the last two decades due to its significations role in many applications, for example, geophysical and environmental applications, fault detection and isolation (FDI) problems, and fault tolerant control (FTC) problems.

The FDI (fault detection and isolation) problem for linear systems with unknown disturbances is largely studied in the literature by different approaches; see, for example, [2–7] and [22, 23]. By using the error innovation technique, a robust fault detection and isolation filter in continuous time is developed in [22] to generate unbiased white residuals signals. In [23] a new method is developed for linear time-invariant (LTI) stochastic discrete-time systems with unknown inputs. This method is important to detect and isolate multiple faults appearing simultaneously or sequentially in linear time-invariant (LTI) systems.

The optimal filtering and robust fault diagnosis problem has been treated for stochastic systems with unknown disturbances in [6, 7]. An optimal observer is proposed for linear time-varying systems. This observer can produce disturbances decoupled state estimation with minimum-variance. The output estimation error with disturbance decoupling [6, 7] is used as a residual signal. After that, a statistical testing procedure is applied to examine the residual and to diagnose faults. Nevertheless, the simultaneous actuator and sensor faults problem is not considered in [6, 7].

More recently, [3] presents a new recursive filter to joint fault and state estimation of linear time-varying discrete-time systems in the presence of unknown disturbances. The method is based on the assumption that no prior knowledge about the dynamical evolution of the fault and the unknown disturbances is available. The filter considers an arbitrary direct feedthrough matrix of the fault and it permits a multiple faults estimations. However, the obtained filter may in certain cases suffer from poor quality fault estimation.

Later, in [24] the problem of joint fault and state estimation of linear systems in the presence of unknown input with uncertain noise covariances was presented. This problem was solved by using the proportional integral three-stage Kalman filter (PI-ThSKF) to estimate the state and the fault of stochastic discrete-time systems with unknown inputs. However, this approach assumes that the models for the dynamical evolution of the fault and the unknown inputs are available.

Based on the assumption that no prior knowledge about the dynamical evolution of the fault is available, the same author [25] was proposed a new recursive optimal filter structure with transformation of the original system. A new recursive optimal unbiased minimum-variance filter has been developed when the direct feedthrough matrix of the fault has an arbitrary rank. However, the filtering algorithm requires the knowledge of a perfect dynamic model. Thus the developed filter may not be robust against modeling uncertainty in the state and the output matrices.

One limitation of the proposed design approach [3, 25] is that they require testing of a certain existence conditions. When the condition fails at any particular iteration, the desired performance is lost and the filter can diverge. In addition the disadvantages of the existing approaches [3, 4, 24, 25] are that the filter lost its optimality in the presence of uncertainties in the state and the output matrices.

In this paper, we consider the problem of robust joint fault and state estimation for linear discrete-time systems with norm bounded uncertainties in both the state and output matrices. The problem addressed is the design of robust linear filters that bound the state covariance matrix for all admissible uncertainties. It is shown that a robust proportional filter (RPF) is developed using transformation of the original system. This transformation is based on the singular value decomposition of the direct feedthrough matrix distribution of the fault which assumed to be arbitrary rank. The proposed filter guarantees that the variance of the estimation error is not more than an optimized upper bound for all admissible uncertainties. The minimization problem of the upper bound on the estimation error variance is formulated as a convex optimization problem subject to linear matrix inequalities and the filter parameters are optimally chosen by solving this problem. To improve robustness against uncertainties and to improve robust estimation of unknown time-varying fault, a new robust proportional integral filter (RPIF) is proposed. The proportional and the integral gains are optimally chosen by solving a convex optimization problem. So the resulted filter will be applied to solve a simultaneous actuator and sensor faults estimation problem.

The remainder of this paper is organized as follows. In Section 2 we set up the robust regularized least square problem for models with data uncertainties. Section 3 states the problem of interest. In Section 4 we design the robust proportional filter. Next in Section 5 we propose a design approach for the robust proportional integral filter (RPIF). Finally, in Section 6, the estimation performance of the proposed filters is demonstrated through an illustrative example.

2. Preliminaries

Consider the following optimization problem:

where is the data matrix, is the measurement vector which is assumed to be known,is the unknown vector,and are given weighting matrices, andare uncertainties assumed to satisfy a model of the form:

whereis an arbitrary contraction,, and are known quantities of appropriate dimension.

Problem (1) and (2) has a unique solution that is given by [17]:

where the modified weighting matrices are defined by

and is a nonnegative scalar parameter obtained by the following optimization problem:

where

3. Problem Statement

Consider the linear stochastic uncertain discrete-time system with unknown additive fault in the form:

where is the state vector, is the observation vector, is the unknown additive fault vector,and are uncorrelated white noise sequences of zero-mean and with covariances matricesand, respectively.

The matrices , , , , and are known and have appropriate dimensions.

The initial state is a Gaussian random variable that is uncorrelated withfor all with and, where denote the expectation operator.

and are unknown matrices which represent time-varying parameter uncertainties. These uncertainties are assumed to be of the following structure:

where, , andare known time-varying matrices of appropriate dimensions, while is an unknown time-varying matrix satisfying arbitrary contraction,.

The aim of this paper is to design a new robust proportional integral filter (RPIF) to obtain a robust fault and state estimation whenin spite of the presence of parametric uncertainties.

Initially, we seek to change the coordinate of system (7) by using the same technique developed in [26].

Let, and then the singular value decomposition of the matrixis given by

where , and

and are unitary matrices.

Using the notations

whereand , we obtain the following equivalent system of the original system (7).

whereis of full-column rank due to (11).

Note thatin (12) may not have full-column rank and the unknown faultmay not be estimable. However it can be solved by finding a full-rank factorization of, that is, , where is full-column rank matrix andisfull-row rank matrix [27].

Thus by defining the notation

(12) becomes

4. Robust Proportional Filter Design

In this section, we propose to solve equivalent system (13), (15) for , and , such thatis minimized, where is the state estimation error.

Here, we adopt a robust least-squares estimation approach to obtain a robust estimate for the state variable and the unknown faults by following a two-step procedure.

4.1. Bounded Uncertainties inAlone

Assume first that there are no uncertainties in ; we will incorporate the uncertainties in later. With bounded uncertainties in the output matrix alone and by using the robust least-squares estimation procedure, we can transform the original system (13), (15) into the following augmented output equation (AOE):

where

where . Here is the estimate of the fault and is zero-mean Gaussian and independent of and

The filtered estimate , the unknown fault , and the delayed unknown fault can be obtained by solving the following robust least-squares problem:

where

Here andwill be defined later.

Note that problem (19) can be written more compactly in the form (1) and (2) with the identifications

From (1)–(3) the solution to (19) is given by

where

Moreover, is the minimizing argument in the interval of the corresponding scalar-valued function in (6) constructed with identification (21).

Using [28, Prop. 2.8.7], we find that the inverse in (22) can be written as

Note that, , andcan be identified as the covariance matrices of , , and; that is,

where the inverse of,, andare given by

Finally substituting (24) in (22) yields

where

Furthermore, we find that and are given, respectively, by

Finally, it follows from (17) and (30) that

whereand are defined in terms of the parameter as

Note that these expressions for and are defined linearly in terms of and have been determined by assuming uncertainties in alone. We will incorporate the uncertainties in in the next section.

4.2. Bounded Uncertainties inand

We now incorporate uncertainties into . That is, we consider norm bounded uncertainties in andas in (13) and (15). We will move to select the parameter by assuming uncertainties inalone and one that meets robustness criterion (19) when there are uncertainties in .

Denoting , we define the extended weight vector. Then in the absence of uncertainties in , we find thatsatisfies

where

The covariance matrix ofsatisfies

where

where

Now observe that the expressions forare parameterized linearly in terms of the parameter . We shall choose by minimizing an upper bound of in the absence of uncertainties in .

Letbe a scalar such that and letbe a scalar such that Bearing in mind (45), we can see thatis bounded by

We choose , by solving

Or equivalently

Since inequality (51) is affine in , thus found will ensure minimum error covariance over all possiblein the bounded domain. Therefore the desired robust proportional filter is given by (40)–(41) and (31)–(38). So initializing for and a scalar , the resulting filter is listed in the following section.

4.3. Summary of the RPF

In this section we summarize the filter equations, we assume thatis the estimate of the initial state a zero-mean and has known variance .

Initial Condition. , whereand.

Step 1. If , then set Otherwise, construct of (6) with identification (21) and determineby minimizingover the interval

Step 2. Using, computeby solving (49) subject to inequality (51), where andare given, respectively, by (43) and (46).

Step 3. Robust simultaneous fault and state estimation are as follows:

where,, and are given by (33)–(37).
Updatetoas

where andare given by (41).

Remark 1. Note that the robust filter (RPF) developed in the previous section gives a better estimation of the state and the fault; however, when the unknown fault is time-varying, the performances of the robust filter can deteriorate. So we will extend the RPF to further propose a new robust proportional integral filter (RPIF) structure, in which the integral action is believed to improve robust estimation of the unknown time-varying faults and to improve robustness against uncertainties.

5. Robust Proportional Integral Filter Design

In this section, we propose to design a new robust proportional integral filter (RPIF) for stochastic linear uncertain system (13) and (15) to improve the estimation of unknown time-varying faults and to improve robustness against uncertainties such as disturbances and modeling errors.

The proposed filter has the following structure:

where the matrices and represent a proportional gain and an integral gain, respectively. The variableis related to the weighted integral of the output estimation error. The constant valuestands for a fading effect coefficient that regulates the transient response. The matrixis an integral effect coefficient. The two design parameters are assumed to be preselected by designers. Theses expressions for ,,,, andhave been determined by assuming uncertainties in alone. We now move on to select the parameterby assuming uncertainties in alone. By doing so, we will arrive at a filter that minimizes a bound on the state error covariance matrix when there are uncertainties inalone.

Proceeding in the same manner as in the previous section, we know that the expression for can be parameterized linearly in terms of the parameter .

Defining the extended weight vector . Ignoring the uncertainties in , we find that satisfies

where

Suppose thatsatisfies

where

Letbe a scalar such that and letbe a scalar such that Bearing in mind (59), we can see thatis bounded by

We choose , by solving

Or equivalently

Therefore the desired robust proportional integral filter (RPIF) is given by (55) whereis the positive-definite solution of (66) subject to (67) with the initial condition for. Note that there always exists a solution to (66)-(67). The resulting filter is listed in following section.

Summary of the RPIF. In this section we summarize the filter equations; we assume thatis the estimate of the initial state a zero-mean and has known variance .

The initialization step of the filter is then given as follows.

Initial Condition. , whereand.

Step 1. If , then set Otherwise, construct of (6) with the identification (21) and determineby minimizing over the interval

Step 2. Using, compute by solving (66) subject to inequality (67), where andare given by (57) and (60)–(62).

Step 3. Robust simultaneous fault and state estimation are as follows:

where , , and are given by (33)–(35).
Update to as

where andare given by (41).

6. Illustrative Example

Robust estimation of simultaneous actuator and sensor faults is as follows.

In this section, we propose the use of the resulting filters RPF and RPIF to solve the robust estimation of simultaneous actuator and sensor faults problem.

We consider the same numerical example used in (Chen and Patton [5, 6]). The linearized model of a simplified longitudinal flight control system is as follows:

where the state variables are pitch angle , pitch rate , and normal velocity, the control inputis the elevator control signal, andandare the matrices distribution of the actuator faultand sensor fault

The presented system equations (73) can be rewritten as follows:

where and are the matrices injection of the faults vector in the state and the measurement equations.

The system parameter matrices are

whereand represent parametric uncertainties in the state matrix satisfying

We inject simultaneously two faults in the system:

whereis the unit-step function. The first fault occurs in the actuator and the second faultoccurs in the sensor for .

The matrices injection of the fault and the unknown disturbances is taken as follows:

In the simulation we set, and.

In Figure 1, we have plotted the actual and the estimated value of the first element and the second element of the fault vector, respectively, using the RPF and the RPIF. Figure 1 presents the simulation results for the worst case ( and ). In Figure 2, we have plotted the actual and the estimated value of the state vector for the two worst cases ( and ).

Figure 1: Actual fault and estimated fault .

Figure 2: Actual stateand estimated state .

According to the simulation results, it can be seen that both the proposed filters RPF and RPIF give a better estimation of the state and the faults. Mainly, we focus on the simultaneous estimation of the actuator and the sensor faults in spite of norm bounded uncertainties in the state and the observation matrix.

Figures 3 and 4 present, respectively, the actual and the estimated value of the first element of the fault vector using the RPF and the RPIF. We can conclude that in the case of time-varying faults the RPIF gives a better estimation results than the RPF. So the integral action is believed to improve robustness against uncertainties and to improve the estimation of time-varying faults.

Figure 3: Actual fault and estimated fault (using RPF).

Figure 4: Actual fault and estimated fault (using RPIF).

7. Conclusion

In this paper, we have derived a new robust filter for linear time-varying discrete-time systems with unknown faults that affect both the system and the output. By including the unknown fault vector as a part of the augmented system state and using transformation of the original system, a new robust proportional filter (RPF) is developed to solve the robust estimation problem, further, and show how to enforce certain minimum error variance property. We design robust filters that bound the state error covariance matrix for all admissible uncertainties. The robustness criterion used is based on robust square estimation approach. We have extended the developed filter to further propose a new robust proportional integral filter (RPIF) to improve robust estimation of unknown faults and to improve robustness against uncertainties. The design procedure is through the solution of a robust weighted recursive least square problem and it enforces a minimum state error variance property. The advantages of this filter are especially important in the case when the direct feedthrough matrix of the fault has an arbitrary rank and when we do not have any prior information about the fault. An application of the proposed filter has been shown by an illustrative example. The proposed robust filter is able to obtain a robust state and fault estimation in spite of the presence of norm bounded uncertainties. In the next step, we can consider nonlinear systems and the extended Kalman filter can be used to solve the problem of simultaneous fault and state estimation.

Competing Interests

The authors declare that they have no competing interests.

References

P. K. Kitanidis, “Unbiased minimum-variance linear state estimation,” Automatica, vol. 23, no. 6, pp. 775–778, 1987.

F. Gannouni and F. Ben Hmida, “Robust fault and state-space estimation for linear uncertain systems: an RLS approach,” in Proceedings of the 1st International Conference on Electrical Engineering and Software Applications (ICEESA '13), Hammamet, Tunisia, March 2013.

F. Ben Hmida, K. Khémiri, J. Ragot, and M. Gossa, “Unbiased minimum-variance filter for state and fault estimation of linear time-varying systems with unknown disturbances,” Mathematical Problems in Engineering, vol. 2010, Article ID 343586, 17 pages, 2010.

F. Ben Hmida, K. Khémiri, J. Ragot, and M. Gossa, “Robust filtering for state and fault estimation of linear stochastic systems with unknown disturbance,” Mathematical Problems in Engineering, vol. 2010, Article ID 591639, 24 pages, 2010.

D. Krokavec, A. Filasová, and P. Liščinský, “Unitary approximations in fault detection filter design,” Mathematical Problems in Engineering, vol. 2016, Article ID 7249803, 14 pages, 2016.

J. Chen and R. Patton, “Optimal filtering and robust fault diagnosis of stochastic system with unknown disturbances,” IEE Proceeding-Control Theory Application, vol. 43, pp. 31–36, 1996.

J. Chen and R. Patton, Robust model-based fault diagnosis for dynamic systems, Kluwer Academic Publishers, 1999.

M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, Springer, Berlin, Germany, 2006.

L. Xie, Y. C. Soh, and C. E. de Souza, “Robust Kalman filtering for uncertain discrete-time systems,” IEEE Transactions on Automatic Control, vol. 39, no. 6, pp. 1310–1314, 1994.

M. Fu, E. de Souza, and L. Xie, “ estimation for discrete-time linear uncertain systems,” International Journal of Robust and Nonlinear Control, vol. 2, pp. 87–105, 1991.

X. Zhu, Y. C. Soh, and L. Xie, “Design and analysis of discrete-time robust Kalman filters,” Automatica, vol. 38, no. 6, pp. 1069–1077, 2002.

M. Fu, C. E. de Souza, and Z.-Q. Luo, “Finite-horizon robust Kalman filter design,” IEEE Transactions on Signal Processing, vol. 49, no. 9, pp. 2103–2112, 2001.

Y. S. Hung and F. Yang, “Robust filtering with error variance constraints for discrete time-varying systems with uncertainty,” Automatica, vol. 39, no. 7, pp. 1185–1194, 2003.

L. Xie, L. Lu, D. Zhang, and H. Zhang, “Improved Robust and filtering for uncertain Discrete-time Systems,” Automatica, vol. 40, pp. 873–880, 2004.

Z. Duan, J. Zhang, C. Zhang, and E. Mosca, “Robust and filtering for uncertain linear systems,” Automatica, vol. 42, no. 11, pp. 1919–1926, 2006.

D. P. Bertsekas and I. B. Rhodes, “Recursive state estimation for a set-membership description of uncertainty,” IEEE Transactions on Automatic Control, vol. AC-16, pp. 117–128, 1971.

B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Upper Saddle River, NJ, USA, 1979.

A. H. Sayed, “A framework for state-space estimation with uncertain models,” IEEE Transactions on Automatic Control, vol. 46, no. 7, pp. 998–1013, 2001.

A. Subramanian and A. H. Sayed, “Regularized robust filters for time-varying uncertain discrete-time systems,” IEEE Transactions on Automatic Control, vol. 49, no. 6, pp. 970–976, 2004.

O. Y. Bas, B. Shafai, and S. P. Linder, “Design of optimal gains for the proportional integral Kalman filter with application to single particle tracking,” in Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99), pp. 4567–4571, Phoenix, Ariz, USA, December 1999.

J. Jung, S. Han, and K. Huh, “Robust proportional-integral Kalman filter design using a convex optimization method,” Journal of Mechanical Science and Technology, vol. 22, no. 5, pp. 879–886, 2008.

R. Nikoukhah, “Innovations generation in the presence of unknown inputs: application to robust failure detection,” Automatica, vol. 30, no. 12, pp. 1851–1867, 1994.

J. Keller, “Fault isolation filter design for linear stochastic systems with unknown inputs,” in Proceedings of the 37th IEEE Conference on Dicision and Control, Tampa, Fla, USA, December 1998.

K. Khémiri, F. Gannouni, F. Ben Hmida, J. Ragot, and M. Gossa, “Robust fault and state estimation for linear discrete-time systems with unknown disturbances using PI three-stage Kalman filter,” in Proceedings of the International Conference on Communications, Computing and Control Application (CCCA '11), Hammamet, Tunisia, March 2011.

K. Khémiri, F. Ben Hmida, J. Ragot, and M. Gossa, “Novel optimal recursive filter for state and fault estimation of linear stochastic systems with unknown disturbances,” International Journal of Applied Mathematics and Computer Science, vol. 21, no. 4, pp. 629–637, 2011.

C. S. Hsieh, “Optimal filtering for systems with unknown inputs via descriptor Kalman filtering,” in Proceedings of the 8th International Conference on Control and Automation, pp. 655–660, Xiamen, China, 2010.

R. Nikoukhah, A. S. Willsky, and B. C. Levy, “Kalman filtering and Riccati equations for descriptor systems,” IEEE Transactions on Automatic Control, vol. 37, no. 9, pp. 1325–1342, 1992.

D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, NJ, USA, 2005.