#### Abstract

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.

#### 1. Introduction

Fault diagnosis (FD), including fault detection and isolation, is a very useful technique to improve the performances of control systems [1–5]. Many significant model-based FD methods have been developed, including, but not limited to, observer-based FD, filter-based FD, parity space, and parameter estimation. Among the abovementioned methods, the observer-based FD which is more practical and simpler, has become a powerful one with many results reported in the literature ([6–12] and references therein). Moreover, fault isolation technique is developed to locate the fault in the systems. Sometimes, the dynamic process of the fault and its actual value can even be obtained. Being the most powerful fault isolation technique, the fault reconstruction method has attracted much more attention than residual-based fault detection techniques ([2–4, 9, 13] and references therein). However, a number of methods in the literature are under some very restrictive assumptions which are impossible in many actual applications. For example, many FD results are proposed under the assumption that the disturbances or faults satisfy the well-known observer matching condition which is, in some sense, unrealistic for some practical systems. On the one hand, one of the major considerations in FD designs is how to deal with model uncertainties and external disturbances so that they cannot cause any negative effects on fault detection or fault reconstruction. On the other hand, it is not trivial to eliminate the negative impacts of model uncertainties or external disturbances on FD designs from the existing traditional observer-based FD methods. In fact, some FD design methods are developed, simply without considering model uncertainties and external disturbances. In recent two decades, interval observer has become popular for the systems with unknown inputs [14], and many process methods have been proposed [15–23]. The interval observer reveal only the over and under boundaries (denoted as and , respectively) of the system states instead of their asymptotic estimations. In comparison with the traditional observer design, it is much more convenient to construct an interval observer because many details about noises, disturbances, and nonlinear terms can be ignored, which is crucial in traditional observer design [4, 9, 24]. As we have known, a few researchers have investigated FD problems based on interval observers [25–32].

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 [11, 12] is adopted, viewing the measurement noise or disturbance as a system state, and then build a singular/descriptor system. Second, the observers are constructed and, based on the estimated results of the reduced-order observer, a practical fault reconstruction method is also proposed. The methods in the paper can be used in both of linear and nonlinear systems, such as electrical power systems, robot manipulator systems, and inverted pendulum systems. The major contributions of this work can be summarized as follows: (i) taking the inevitable measurement noise into account, an interval-observer-based fault detection means, which depend on the system’s actual outputs and interval output estimates, are established; (ii) a reduced-order robust interval observer is developed for the equivalent system which is augmented; (iii) an interval reconstruction method of actuator faults is proposed based on the robust observer.

The paper is organized as follows. In Section 2, some basic definitions, background knowledge, and problem statements are introduced. Section 3 proposes a full-order interval observer and an actuator fault detection method. Section 4 designs a robust reduced-order interval observer, and an actuator fault interval reconstruction method is presented based on this. The correctness and effectiveness of the developed methods are shown through two simulation examples in Section 5. The conclusion is drawn in Section 6.

Some necessary notations marked in this paper are defined here. All of the inequality between two vectors or matrices should be understood elementwise. For a constant matrix , define and , so , , and . For a square matrix , means that is a symmetric negative definite matrix. A matrix is called a Schur matrix if its spectral radius is less than one and called a nonnegative matrix if all its elements are nonnegative.

#### 2. Preliminaries and Problem Statements

Considering the following linear time-invariant discrete-time system subjected to fault and outside disturbances,where is the system state vector, is the control input vector, is the measurable output vector, and are the bounded actuator fault vector and the unknown input vector respectively, and is the measurement noise vector. All the parameter matrices are assumed with appropriate dimensions. The unknown input , which is the combination of model uncertainties and outside disturbances, is bounded by known boundaries and . Without loss of generality, we assume that and . For the goal of estimating the measurement noise , an augmented state method which was proposed in [11, 12] is used and new parameter matrices can be built as follows.

Define , , , , , , and .

System (1) will be transformed into a descriptor system’s form:

Assuming that system (2) is detectable, that is, , and , . We also assume that the initial condition is bounded by two known vectors and . It is obvious that

So, there exists a full rank matrix and a matrix so that

Then, we can obtainfrom (2). Let , and (5) becomes

Lemma 1 (see [33]). *For the following linear discrete-time system,if is nonnegative, , and the initial condition satisfies , then the system has nonnegative solutions for .*

Lemma 2 (see [16]). *For the discrete-time system,if is a Schur constant matrix, then there exists a time-varying transformation , which makes the system into a positive exponential stable system through such transformation, where , the invertible matrices satisfied for arbitrary , and is a given nonnegative constant scalar.*

Lemma 3 (see [15]). *For any constant matrix , if a vector variable satisfies for some , then one can obtain*

#### 3. Actuator Fault Detection

Generally speaking, the matrix is not Schur and non-negative. For this reason, an equivalent state transformation is used, and system (6) can be converted intowhere is an nonsingular matrix. The following interval observer can be designed:where is the observer gain matrix and and should be selected so that is a Schur matrix and a nonnegative matrix simultaneously. And, the nonsingular matrix and the gain matrix can be computed through the method which is described in Remark 1.

Theorem 1. *Suppose that there is no actuator fault occurs. If we choose and , then system (11) is an interval observer for system (10).*

*Proof. *Define the over and under errors as and , respectively. From (10) and (11), we obtainNote that , , and , soLet and , and we have by using Lemma 3. According to Lemma 1, one can obtain that and , i.e., , for arbitrary .

*Remark 1. *To find proper matrices and , Raïssi et al. [18] propose a effective means to solve such problems. Because is a nonsingular matrix, one can construct a Sylvester equation from as follows:If the coefficient matrix and the chosen matrix have no common eigenvalues, then (14) has a unique solution of and . is an arbitrary matrix and can be chosen as a Schur and nonnegative diagonal matrix here.

Notice that , the upper and lower estimations of can also be given by

*Remark 2. *We can also design the interval observer by a time-varying linear state transformation of rather than a constant one. In this way, a time-varying invertible matrix which can make into a Schur and nonnegative matrix can be found by using Lemma 2, where is calculated such that is a Schur matrix.

Then, the upper and lower estimation of , , and are , , and ; then, the output will be limited bywhich can be a fault detector. That is, when the system runs without the presence of actuator faults, its output will meet , otherwise the faults occur. It is noteworthy that since the interval is different from the artificially given threshold, the faults with small amplitudes may not all be detected. That is to say, for successful fault detection, the faults need to have enough amplitudes.

#### 4. Actuator Fault Interval Reconstruction

For , there exists an invertible matrix such that

Then, system (6) can be transformed into

Denote , , , , , and , and (18) becomes

Decompose , , and , we have

and the parameter matrices can be decomposed aswhere , , , , and . Then, (22) can be easily obtained:and make an equivalent state transformationwhere is the gain matrix of the reduced-order observer, and we have and . So, is described by the following equation:

Lemma 4 (see [9]). *If there exist a symmetric positive definite matrix and a matrix for which the following LMIsare solvable, then the observer gain can make into a Schur matrix, such that**If we denote , , , and , and notice (27); then, (24) becomes**When is not a nonnegative matrix, a time-varying transformation can be used here to handle the problem, so we obtain**From Lemma 2, it is obvious that a time-varying invertible matrix can be obtained.**Then, an interval observer for (29) is constructed as*

Theorem 2. *If LMIs (25) and (26) are solvable, let , , , , and**Then, the solutions of (29) and (30) satisfy , for . In other words, (30) is an interval observer of (29).*

*Proof. *We notice that is Schur and nonnegative matrix, so one can refer to Theorem 1 and the proof is omitted here for brevity.

Next, an interval reconstruction method is proposed to reconstruct actuator faults. There exists a nonsingular matrix such thatThen, system (6) can be rewritten asThe matrices and vector can be decomposed as follows:where , , , , , and and ; then, the dynamic for is governed byDefineand then, and are the upper and lower interval estimations of , respectively, i.e., we have . Letwhere . And, we haveNow, from (35), we obtainFrom (39), the actuator fault interval reconstruction can be built as follows:

Theorem 3. *For and defined by (40), we have , for .*

*Proof. *Notice that , , , and , and we can verify that and hold for all easily.

*Remark 3. *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.

#### 5. Numerical Simulation

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.

##### 5.1. Numerical Example

Consider a discrete-time system (1) and the parameter matrices are as follows:

Then, the parameter matrices in the rewritten system (6) are presented below:

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

The selected initial conditions are , , , has the over boundary , and the under boundary .

The matrix of and can be chosen as

To construct a Sylvester equation, we choose that

Then, substitute them into (14) and compute it, we have

Figure 1 shows the results of interval estimations when using full-order interval observer (11), and in the system, there are no actuator faults. The fault detection is plotted in Figure 2, and it reveals that the faults occur during the time period of .

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

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

Compute LMIs (25) and (26), we have

Then, we have and . The Jordan canonical form of is , and then, an interval observer in the form of (30) can be designed. Figures 3–6 show that the interval state estimations through reduced-order and full-order interval observers. By comparison of the results, the reduced-order method proposed in the paper can make estimations successfully even if there is actuator fault. It is easy to know

And, the interval reconstruction of the actuator fault is given in Figure 7.

##### 5.2. Practical System

In the second simulation, a DC motor model in [34] is considered, which has the following system dynamics:where is the motor speed and and are the armature current and voltage, respectively. In the model, the parameters , and denote the resistance, inductance, back electromotive force constant, torque constant, frictional constant, and motor inertial, respectively. The nominal parameters are given as follows:

The system dynamics are discretized by the forward Euler method with the sampling time . We have the following system parameters:

Similarly, a reduced-order interval observer can be designed for this practical system. By computing LMIs (25) and (26), we have

In the simulation, the initial value of system state is , and it is assumed that the initial values of estimated upper and lower bound are , respectively. Besides, the control input and actuator fault are set as and . The unknown input and measurement noise are assumed to be stochastic and the unknown input is bounded by . The interval estimation of the motor speed and the armature current with the existence of actuator fault are shown in Figures 8 and 9, and the interval reconstruction of actuator fault is presented in Figure 10. The estimation results of this example verify the effectiveness and practicability of the proposed method further.

#### 6. Conclusion

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.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

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

#### Acknowledgments

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.