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.

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 [

The proposed filter can play a significant role in several applications, for example, model based fault detection and isolation (FDI) problem [

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 [

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 [

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, [

The optimal filtering and robust fault diagnosis problem has been treated for stochastic systems with unknown disturbances in [

More recently, [

Later, in [

Based on the assumption that no prior knowledge about the dynamical evolution of the fault is available, the same author [

One limitation of the proposed design approach [

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

Consider the following optimization problem:

Problem (

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

The matrices

The initial state

The aim of this paper is to design a new robust proportional integral filter (RPIF) to obtain a robust fault and state estimation when

Initially, we seek to change the coordinate of system (

Let

Using the notations

Note that

Thus by defining the notation

In this section, we propose to solve equivalent system (

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.

Assume first that there are no uncertainties in

The filtered estimate

Note that problem (

Using [

We now incorporate uncertainties into

Denoting

Let

In this section we summarize the filter equations, we assume that

If

Using

Robust simultaneous fault and state estimation are as follows:

Update

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.

In this section, we propose to design a new robust proportional integral filter (RPIF) for stochastic linear uncertain system (

The proposed filter has the following structure:

Proceeding in the same manner as in the previous section, we know that the expression for

Defining the extended weight vector

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

If

Using

Robust simultaneous fault and state estimation are as follows:

Update

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 [

The presented system equations (

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

In Figure

Actual fault

Actual 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

Actual fault

Actual fault

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.

The authors declare that they have no competing interests.