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For the problem of the actuator fault diagnosis in the control systems, this paper presents a novel method by using an interval estimation approach to detect the faults and reconstruct them. In order to make estimations of the unavoidable measurement noise, a descriptor system form is built. Firstly, a full-order interval observer is developed to detect actuator faults for its sensitiveness to them. Then, a reduced-order one, which is robust to actuator faults, is presented. This method does not need the boundary information of faults; thus, the design condition is more relaxed. In order to make the interval observer stable and cooperative, linear matrix inequalities and a time-varying transformation are employed to ensure the error system matrix to be Schur and nonnegative. Based on the interval estimation results of the aforementioned method, an interval reconstruction method of actuator faults is proposed. Finally, results of the two simulation examples verify the proposed methods are effective and accurate.

Fault diagnosis (FD), including fault detection and isolation, is a very useful technique to improve the performances of control systems [

Considering the above background, our purpose is using interval observers to solve FD problems by constructing both of full-order and robust reduced-order interval observers. First, to eliminate the influence of measurement noise or disturbance, an augmented state method which was proposed by [

The paper is organized as follows. In Section

Some necessary notations marked in this paper are defined here. All of the inequality between two vectors

Considering the following linear time-invariant discrete-time system subjected to fault and outside disturbances,

Define

System (

Assuming that system (

So, there exists a full rank matrix

Then, we can obtain

For the following linear discrete-time system,

For the discrete-time system,

For any constant matrix

Generally speaking, the matrix

Suppose that there is no actuator fault occurs. If we choose

Define the over and under errors as

Note that

Let

To find proper matrices

If the coefficient matrix

Notice that

We can also design the interval observer by a time-varying linear state transformation of

Then, the upper and lower estimation of

For

Then, system (

Denote

Decompose

and the parameter matrices can be decomposed as

If there exist a symmetric positive definite matrix

If we denote

When

From Lemma

Then, an interval observer for (

If LMIs (

Then, the solutions of (

We notice that

Next, an interval reconstruction method is proposed to reconstruct actuator faults. There exists a nonsingular matrix

Then, system (

The matrices and vector can be decomposed as follows:

Define

Now, from (

From (

For

Notice that

Generally speaking, the basic idea of an interval state observer design is to construct a couple of new systems which can produce interval estimations for the original system states by using the information of the boundaries of unknown inputs together with, if necessary, the information of the measured outputs of the original system. So, designing an interval state observer for the system with unknown inputs, the knowledge of the boundaries of the unknown inputs is usually crucial. In the present paper, on the one hand, with reduced-order observer design techniques, an interval reduced-order observer is developed without knowing the boundary information of the actuator faults, which can actually be regarded as unknown inputs. On the other hand, based on the partial known information, an interval reconstruction method of the actuator fault is developed.

In this section, the details of the design process of the developed methods are given through a numerical example and a practical system; then, the effectiveness is illustrated.

Consider a discrete-time system (

Then, the parameter matrices in the rewritten system (

We assume that control input, unknown disturbances, and actuator fault are

The selected initial conditions are

The matrix of

To construct a Sylvester equation, we choose that

Then, substitute them into (

Figure

Interval estimation without actuator fault, i.e.,

Fault detection.

Next is the design process of the robust interval observer. The invertible matrix is

Compute LMIs (

Then, we have

And, the interval reconstruction of the actuator fault is given in Figure

Actuator fault interval reconstruction.

In the second simulation, a DC motor model in [

The system dynamics are discretized by the forward Euler method with the sampling time

Similarly, a reduced-order interval observer can be designed for this practical system. By computing LMIs (

In the simulation, the initial value of system state is

Actuator fault interval reconstruction.

This paper concerns the descriptor system form used in the interval estimations. A full-order interval observer is developed as the initial actuator fault detector for linear discrete-time systems when there exist actuator faults and unknown disturbances. A reduced-order one is devised by minimizing the effects of actuator faults when systems are running. Based on its estimation results, an interval reconstruction method for actuator faults is given. At last, two simulation examples illustrate the proposed full-order interval observers can serve as fault detectors and the reduced-order one can produce a reconstruction of actuator faults effectively.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was funded by the Headquarters Science and Technology Guide Project of SGCC Grant no. 523HQ200054 and National Natural Science Foundation of China Grant no. 61304104 and also funded by Chongqing Technology Innovation and Application Special Key Project under Grant no. cstc 2019jscx-mbdxX0015.