LMI-Based Sliding Mode Observers for Incipient Faults Detection in Nonlinear System

Abstract

This paper presents a diagnosis scheme based on a linear matrix inequality (LMI) algorithm for incipient faults in a nonlinear system class with unknown input disturbances. First, the nonlinear system is transformed into two subsystems, one of which is unrelated to the disturbances. Second, for the subsystem that is free from disturbances, a Luenberger observer is constructed; a sliding mode observer is then constructed for the subsystem which is subjected to disturbances, so that the effect of the unknown input disturbances is eliminated. Together, the entire system achieves both robustness to disturbances and sensitivity to incipient faults. Finally, the effectiveness and feasibility of the proposed method are verified through a numerical example using a single-link robotic arm.

1. Introduction

An electronic system is structurally complex, involving a number of electronic components. It is therefore difficult to accurately analyze the relationship between the input and output for each component and the cause of any faults within, as each fault may exhibit diverse manifestations. Incipient faults are important in electronic systems, and some common manifestations include zero drift, reduced precision, delayed response, and equipment aging. If these potential incipient faults cannot be detected in time, the long-term stable operation of the devices will be affected and, more importantly, they will cause a decline in production capacity and an increase in the cost of production, or even accidents. Therefore, the study of incipient faults in electronic devices has practical significance.

The sliding mode variable structure control algorithm has the advantages of simplicity, high robustness, and high reliability and is extensively used in motion control. The sliding mode observer has attracted attention for fault diagnosis since the sliding mode variable structure control enables the system to be robust against unknown input disturbances, especially against external disturbances [1, 2]. Observer technology is one of the most important methods for fault diagnosis and detection, estimation and fault-tolerant control [3–5]. A robust fault diagnosis method for the sliding mode observer has been proposed [6]. This scheme adopts the Walcoot-Zak sliding mode observer, which makes system state error robust against disturbances, and is maintained on the sliding mode surface. Much progress has been made in measuring the deviation in the trajectory of the motion of the system by using a sliding mode surface [5, 7, 8]. However, the current fault diagnosis based on sliding mode observers is mostly used for mutant or intermittent faults that have significant numerical changes. There have been few reports on slow faults with a small initial value. This is because the sliding mode control is essentially a type of continuous nonlinear control, and buffeting is inevitable for the system output, which causes the incipient fault signals to be submerged in the buffeting signals for a long period of time after their generation, thus they cannot be detected.

Chen and Chowdhury [9] have proposed a fault detection scheme for actuator’s incipient faults in linear systems. Building on the work of Chen and Chowdhury [9], this paper presents an LMI-based fault diagnosis scheme, which is designed for incipient fault detection in nonlinear systems with unknown input disturbances. The adoption of the LMI algorithm relaxes the selection criteria for key parameters in the design, making it easier to obtain these parameters.

2. System Descriptions

For a nonlinear systems with disturbances, 𝑥̇(𝑡)(𝑡)(𝑡)()(),𝑦(𝑡)(𝑡),=𝐴𝑥+𝐵𝑢+𝑓𝑥,𝑢,𝑡+𝐷𝑑𝑥,𝑢,𝑡=𝐶𝑥 (2.1) where 𝑥(𝑡)∈𝑅^𝑛 is unmeasurable state vector; 𝑢(𝑡)∈𝑅^𝑚 is measurable input vector; 𝑦(𝑡)∈𝑅^𝑝 is the measurable output vector; 𝑓(𝑥,𝑢,𝑡)∈𝑅^𝑛 is the known nonlinear function; 𝑑(𝑥,𝑢,𝑡)∈𝑅^𝑞 is unknown bounded nonlinear function, representing the unknown input disturbances and modeling errors, which are collectively referred to as input disturbances; 𝐷∈𝑅^𝑛×𝑞 is known disturbance distribution matrix. 𝐴, 𝐵, and 𝐶 are known matrices, where 𝐴∈𝑅^𝑛×𝑛,??𝐵∈𝑅^𝑛×𝑚,??𝐶∈𝑅^𝑝×𝑛,??𝑛>𝑝>𝑞.

Assumption 2.1. (𝐴,𝐶) is observable.

Assumption 2.2. 𝐷 is full column rank and rank(𝐶𝐷)=rank(𝐷).

Assumption 2.3. 𝑑 is a bounded disturbance such that ‖𝑑‖≤𝛾₁, where 𝛾₁ is a known function greater than 0.

3. Design of Fault Diagnosis Scheme

3.1. Coordinate Transformation

Coordinate transformation is used, under certain geometric conditions, to decouple the unknown input disturbance and faults [10]. Assumption 2.2 implies that there are two transformation matrixes, 𝑇 and 𝑆 [9]. The system (2.1) can be decomposed into the following two subsystems (the details of the transformation process are given elsewhere [9]): 𝑧̇₁=𝐴₁₁𝑧₁+𝐴₁₂𝑧₂+𝐵₁𝑢+𝑓₁,𝑣₁=𝐶₁₁𝑧₁,𝑧̇₂=𝐴₂₁𝑧₁+𝐴₂₂𝑧₂+𝐵₂𝑢+𝑓₂+𝐷₂𝑣𝑑,₂=𝐶₂₂𝑧₂, (3.1)(3.2) where 𝑧₁∈𝑅^𝑛−𝑞, 𝑧₂∈𝑅^𝑞, 𝐴₁₁∈𝑅^{(𝑛−𝑞)×(𝑛−𝑞)}, 𝐴₁₂∈𝑅^{(𝑛−𝑞)×𝑞}, 𝐴₂₁∈𝑅^{𝑞×(𝑛−𝑞)}, 𝐴₂₂∈𝑅^𝑞×𝑞, 𝑣₁∈𝑅^𝑝−𝑞,??𝑣₂∈𝑅^𝑞,??𝐶₁₁∈𝑅^{(𝑝−𝑞)×(𝑛−𝑞)},??𝐶₂₂∈𝑅^𝑞×𝑞.

The matrix transformation and definition in systems (3.1) and (3.2) are as follows: 𝑇𝐴𝑇⁻¹=𝐴=𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂,𝑇𝐵=𝐵=𝐵₁𝐵₂,𝑇𝑓=𝑓=𝑓₁𝑓₂,𝑇𝐷=0𝐷=𝐷₂,𝑆𝐶𝑇⁻¹=𝐶₁₁00𝐶₂₂, (3.3) where 𝐶₂₂ is an invertible matrix.

The transformation matrix 𝑇 is 𝐼𝑇=^𝑛−𝑞−𝐷₁𝐷₂⁻¹0𝐼_𝑞. (3.4)

3.2. Observer Design

The following assumptions are made for transformed systems (3.1) and (3.2).

Assumption 3.1. (𝐴₁₁,𝐶₁₁) and (𝐴₂₂,𝐶₂₂) are observable.
Assumption 3.1 implies that there are matrixes 𝐿₁ and 𝐿₂, which enable 𝐴₀₁ and 𝐴₀₂ to be stable matrices: 𝐴₀₁=𝐴₁₁−𝐿₁𝐶₁₁,𝐴₀₂=𝐴₂₂−𝐿₂𝐶₂₂. (3.5)
There are also two Lyapunov equations: 𝐴^T₀₁𝑃₁+𝑃₁𝐴₀₁=−𝑄₁,𝐴^T₀₂𝑃₂+𝑃₂𝐴₀₂=−𝑄₂, (3.6) where 𝑃₁,??𝑄₁,??𝑃₂, and 𝑄₂ are all symmetric positive definite (SPD) matrices.

Assumption 3.2. The function 𝑓 satisfies the Lipschitz condition: 𝑓(𝑥,𝑢,𝑡)−𝑓(𝑥,𝑢,𝑡)≤𝛾‖𝑥−𝑥‖, (3.7) where 𝛾‖𝑥−𝑥‖=𝛾‖𝑇⁻¹‖‖𝑧−𝑧‖=𝛾‖𝑧−𝑧‖, and 𝛾 is the Lipschitz constant.

A Luenberger observer is constructed for subsystem (3.1): 𝑧̇₁=𝐴₁₁𝑧₁+𝐴₁₂𝑧₂+𝐵₁𝑢+𝑓₁𝑇⁻¹𝑧,𝑢,𝑡+𝐿₁𝑣₁−𝑣₁,𝑣₁=𝐶₁₁𝑧₁, (3.8) where superscript “Λ” indicates estimate value.

A sliding mode variable structure observer is constructed for subsystem (3.2): 𝑧̇₂=𝐴₂₁𝑧₁+𝐴₂₂𝑧₂+𝐵₂𝑢+𝑓₂𝑇⁻¹𝑧,𝑢,𝑡+𝐷₂𝑤+𝐿₂𝑣₂−𝑣₂,𝑣₂=𝐶₂₂𝑧₂, (3.9) where 𝑤 is the input signal of sliding mode variable structure, expressed as 𝐹𝑣𝑤=−𝜌²−𝑣₂𝐹𝑣₂−𝑣₂𝑣+𝛿if₂−𝑣₂𝑣≠00if₂−𝑣₂=0, (3.10) where 𝐹∈𝑅^𝑞×𝑞 is the matrix to be designed; 𝜌 is the scalar function to be designed, 𝜌≥𝛾₁; 𝛿 is a positive constant of small value.

Assumption 3.3. The matrices 𝑃₂ and 𝐹 have to be chosen such that 𝑃₂𝐷₂=𝐶^T₂₂𝐹^T.
Define 𝑒₁=𝑧₁−𝑧₁,??𝑒₂=𝑧₂−𝑧₂,??𝑒=^𝑒1_𝑒2 as the state estimation errors and 𝑒_𝑣1=𝑣₁−𝑣₁=𝐶₁₁𝑒₁,??𝑒_𝑣2=𝑣₂−𝑣₂=𝐶₂₂𝑒₂ as the output estimation errors.

Based on (3.1), (3.2), (3.8), and (3.9), the corresponding observation-error dynamic equations are given by: 𝑒₁̇=𝐴₁₁−𝐿₁𝐶₁₁𝑒₁+𝐴₁₂𝑒₂+𝑓₁𝑇⁻¹𝑧,𝑢,𝑡−𝑓₁𝑇⁻¹,𝑒̇𝑧,𝑢,𝑡₂=𝐴₂₂−𝐿₂𝐶₂₂𝑒₂+𝐴₂₁𝑒₁+𝑓₂𝑇⁻¹𝑧,𝑢,𝑡−𝑓₂𝑇⁻¹𝑧,𝑢,𝑡+𝐷₂𝑑−𝑤₂. (3.11)

Lemma 3.4 (see [11]). If 𝑔(𝑥,𝑢,𝑡) satisfies the Lipschitz condition, then there will be an SPD matrix 𝑃 that satisfies the following equation: 2𝜀^𝑇𝑃𝑔𝑥₁,𝑢,𝑡−𝑔𝑥₂,𝑢,𝑡≤𝑘²𝜀^𝑇𝑃𝑃𝜀+𝜀^𝑇𝜀, (3.12) where 𝜀=𝑥₁−𝑥₂ and 𝑘 is the Lipschitz constant.

Lemma 3.5 (Schur Complement [12]). For a given symmetric matrix 𝑆=^𝑆11𝑆12_𝑆21𝑆22, where 𝑆₁₁∈𝑅^𝑟×𝑟. The following three conditions are equivalent:(1)𝑆<0;(2)𝑆₁₁<0,??𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0;(3)𝑆₂₂<0,??𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

Theorem 3.6. Under Assumptions 2.1–2.3 and 3.1–3.3, if the following LMI holds, 𝐻₁𝐴^𝑇₂₁𝑃₂+𝑃₁𝐴₁₂𝑃₁0𝐴^𝑇₁₂𝑃₁+𝑃₂𝐴₂₁𝐻₂0𝑃₂𝑃₁10−𝛾²𝐼00𝑃₂10−𝛾²𝐼<0, (3.13) where 𝐻₁=𝐴^T₁₁𝑃₁+𝑃₁𝐴₁₁−𝑌^T₁−𝑌₁+𝐼,??𝐻₂=𝐴^T₂₂𝑃₂+𝑃₂𝐴₂₂−𝑌^T₂−𝑌₂+𝐼,??𝑌₁=𝑃₁𝐿₁𝐶₁₁ and 𝑌₂=𝑃₂𝐿₂𝐶₂₂, then 𝑒₁ and 𝑒₂ will converge to the zero point.

Proof. Consider the following Lyapunov function: 𝑉=𝑒^T₁𝑃₁𝑒₁+𝑒^T₂𝑃₂𝑒₂. (3.14) The derivative of the Lyapunov function with respect to time is 𝑉̇=𝑒^T₁𝐴^T₀₁𝑃₁+𝑃₁𝐴₀₁𝑒₁+2𝑒^T₁𝑃₁𝐴₁₂𝑒₂+𝑓₁(𝑧,𝑢,𝑡)−𝑓₁(𝑧,𝑢,𝑡)+𝑒^T₂𝐴^T₀₂𝑃₂+𝑃₂𝐴₀₂𝑒₂+2𝑒^T₂𝑃₂𝐴₂₁𝑒₁+𝑓₂(𝑧,𝑢,𝑡)−𝑓₂(𝑧,𝑢,𝑡)+𝐷₂(𝑑−𝑤). (3.15) Let 𝐴=𝐴₁₁−𝐿₁𝐶₁₁𝐴₁₂𝐴₂₁𝐴₂₂−𝐿₂𝐶₂₂. (3.16) then 𝑉̇=𝑒^T𝐴^T𝑃+𝑃𝐴𝑒+2𝑒^T𝑃𝑓(𝑧,𝑢,𝑡)−𝑓(𝑧,𝑢,𝑡)+2𝑒^T₂𝐶^T₂₂𝐹^T𝑑−𝜌𝐹𝑒_𝑣2𝐹𝑒_𝑣2≤𝑒^T𝐴^T𝑃+𝑃𝐴𝑒+2𝑒^T𝑃𝑓(𝑧,𝑢,𝑡)−𝑓(𝑧,𝑢,𝑡)−2𝐹𝑒_𝑣2𝜌−𝛾₁. (3.17) From Lemma 3.4 we find that 𝑉̇≤𝑒^T𝐴^T𝑃+𝑃𝐴+𝛾²𝑃𝑃+𝐼𝑒. (3.18)
According to Lemma 3.5, when the following inequality is satisfied, 𝐴^T1𝑃+𝑃𝐴+𝐼𝑃𝑃−𝛾²𝐼<0 (3.19) then 𝑉̇<0.
Since 𝐴=^{𝐴11−𝐿1𝐶11𝐴12}_{𝐴21𝐴22−𝐿2𝐶22}, 𝑃=^𝑃10_0𝑃2, 𝐿=^𝐿10_0𝐿2, the above LMI (3.19) can be transform into inequality (3.13), which is listed in Theorem 3.6.
This completes the proof.

The main advantage of the method presented in this paper is that the complex equations for obtaining parameters given by Chen and Chowdhury [9] are transformed into inequality (3.13), a standard LMI. Then the solution becomes much easier when the LMI toolbox is used in MATLAB.

3.3. Fault Diagnosis

When a fault occurs, the systems (3.1) and (3.2) are transformed into the following equations: 𝑧̇₁=𝐴₁₁𝑧₁+𝐴₁₂𝑧₂+𝐵₁𝑢+𝑓₁+𝐸₁𝑓_𝑎(𝑡)𝑣₁=𝐶₁₁𝑧₁𝑧̇₂=𝐴₂₁𝑧₁+𝐴₂₂𝑧₂+𝐵₂𝑢+𝑓₂+𝐷₂𝑑+𝐸₂𝑓_𝑎(𝑡),𝑣₂=𝐶₂₂𝑧₂, (3.20) where 𝐸₁ is a nonzero matrix, 𝐸₁∈𝑅^𝑛−𝑞, 𝑇𝐸=^𝐸1_𝐸2, 𝐸∈𝑅^𝑛×𝑟 is a known fault distribution matrix. 𝑓_𝑎(𝑡)∈𝑅^𝑟 is an unknown bounded nonlinear function, which represents incipient faults of the system.

The dynamic equations of 𝑒₁ and 𝑒₂ of observer can be obtained from (3.8), (3.9), and(3.20), thus errors 𝑒_𝑣1,??𝑒_𝑣2 are written in the following forms: 𝑒₁̇̃=𝐴₁₁−𝐿₁𝐶₁₁𝑒₁+𝐴₁₂𝑒₂+𝑓₁𝑇⁻¹𝑧,𝑢,𝑡−𝑓₁𝑇⁻¹𝑧,𝑢,𝑡+𝐸₁𝑓_𝑎(𝑡),𝑒̃_𝑣1=𝐶₁₁𝑒₁̃,𝑒₂̇̃=𝐴₂₂−𝐿₂𝐶₂₂𝑒₂+𝐴₂₁𝑒₁+𝑓₂𝑇⁻¹𝑧,𝑢,𝑡−𝑓₂𝑇⁻¹𝑧,𝑢,𝑡+𝐷₂𝑑−𝑤₂+𝐸₂𝑓_𝑎(𝑡),𝑒̃_𝑣2=𝐶₁₁𝑒₂̃. (3.21)(3.22)

It can be concluded from Theorem 3.6 that, in spite of the unknown input disturbance, 𝑒_𝑣1 and 𝑒_𝑣2 still converge to zero field when there is no fault, or deviate from zero when the faults occur. In (3.22), which is an explicit of both faults and the unknown input disturbance, a sliding mode observer is designed for weakening the impact resulting from the disturbance. However, the inherent chattering phenomenon of slide mode variable structure control makes it difficult to distinguish the early incipient faults signals from the chattering signals until the faults develop to be serious enough. Thus, 𝑅₂(𝑡)=𝑒̃_𝑣2 is unsuitable to be the faults detecting residual. By contrast, the 𝑒̃_𝑣1 in (3.21) is designed with no sliding mode observer in order to avoid the impact from chattering; therefore, 𝑅₁(𝑡) can be used as the residual for incipient fault detecting. The simulation results in Section 4 will verify the above statements.

4. Simulation Study

Consider a nonlinear system, a single-link robotic arm with a revolute elastic joint rotating in a vertical plane whose motion equations are [13] 𝐽_𝑙𝑞̈₁+𝐹_𝑙𝑞̇₁+𝑘𝑞₁−𝑞₂+𝑚𝑔𝑙sin𝑞₁𝐽=0,_𝑚𝑞̈₂+𝐹_𝑚𝑞̇₂−𝑘𝑞₁−𝑞₂=𝑢, (4.1) where 𝑞₁ and 𝑞₂ are the link displacement and the rotor displacement, respectively. The link inertia 𝐽_𝑙, the motor rotor inertia 𝐽_𝑚, the elastic constant 𝑘, the link mass 𝑚, the gravity constant 𝑔, the center of mass 𝑙, and the viscous friction coefficients 𝐹_𝑙, 𝐹_𝑚??are all positive constant parameters. The control 𝑢 is the torque delivered by the motor.

The state variables are chosen as 𝑥₁=𝑞₁,??𝑥₂=𝑞̇₁,??𝑥₃=𝑞₂,??𝑥₄=𝑞̇₂.

When handling different objects, the load was carried by the manipulator changes. In addition, the friction coefficient of the joint and other parameters also varies over time. All these factors are uniformly classified as unknown input disturbances, denoted by 𝑑. Suppose a fault 𝑓_𝑎 occurs to the manipulator, then the single-link robotic arm model with unknown input disturbances and faults can be expressed as the following fourth-order nonlinear state equation: 𝑥̇₁𝑥̇₂𝑥̇₃𝑥̇₄=0100−𝑘𝐽_𝑙−𝐹_𝑙𝐽_𝑙𝑘𝐽_𝑙0𝑘0001𝐽_𝑚0−𝑘𝐽_𝑚−𝐹_𝑚𝐽_𝑚𝑥₁𝑥₂𝑥₃𝑥₄+0−𝑚𝑔𝑙𝐽_𝑙sin𝑥₁00+0001𝐽_𝑚𝑢+𝐸𝑓_𝑎𝑥+𝐷𝑑,𝑦=𝐶₁𝑥₂𝑥₃𝑥₄^𝑇. (4.2)

The selected manipulator parameters are 𝑘=2?Nm/rad, 𝐹_𝑚=1, 𝐹_𝑙=0.5?Nm/(rad/s), 𝐽_𝑚=1?Nm², 𝐽_𝑙=2?Nm², 𝑚=0.15?kg, 𝑔=9.8, 𝑙=0.3?m.

Corresponding to system (2.1), the parameter matrices for each equation are, respectively, 0001,𝑓()=0𝐴=0100−1−0.2510000120−2−1,𝐵=,𝐶=100000100001𝑥,𝑢,𝑡−0.2205sin𝑥₁00,𝐷=0100.5^T,𝐸=2.9293.81441^T. (4.3)

The selected transformation matrices 𝑇 and 𝑆 are, respectively, 𝐼𝑇=^𝑛−𝑞−𝐷₁𝐷₂⁻¹0𝐼_𝑞=,.1000010−200100001𝑆=100010001 (4.4)

Using the transformation matrices described above, system (4.2) is decomposed into the following two subsystems: 𝑧̇₁=𝑧̇₁₁𝑧̇₁₂𝑧̇₁₃=010𝑧−5−0.255000¹¹𝑧₁₂𝑧₁₃+211.5𝑧₂+0−0.2205sin𝑧₁₁0+4𝑓2.9291.814_𝑎+00𝑣−2𝑢,₁=𝑧100001¹¹𝑧₁₂𝑧₁₃,𝑧̇₂=𝑧20−2𝑧¹¹₁₂𝑧₁₃−𝑧₂+0+𝑓_𝑎𝑣+0.5𝑑+𝑢,₂=𝑧₂. (4.5)(4.6)

After the coordinate transformation, a reduced-order subsystem, as shown in (4.5), is decomposed from the system (4.2). The subsystem only contains fault 𝑓_𝑎; it does not contain the unknown input disturbances 𝑑. The impact of 𝑑 on subsystem (4.5) is delivered by the state variables 𝑧₁ and 𝑧₂. Using the above observer, the impact of 𝑑 on the system is eliminated, and the unknown input disturbance and the fault are decoupled.

The LMI toolbox in MATLAB is used to solve the inequalities. 𝐿₁, 𝐿₂, and 𝐹 are selected to satisfy Assumptions 3.1 and 3.3. 𝐿₁=6.60740.55460.38042.69500.04831.4015,𝐿₂=[2.7052],𝐹=[0.3428]. (4.7)

It is assumed that the incipient fault 𝑓_𝑎(𝑡) begins at time instant 30?s, and 𝑓_𝑎(𝑡)=0, when 𝑡<30s. The input signal is 𝑢=sin(𝑡/3) and the simulation parameters are 𝜌=220, 𝛿=0.01.

With the above simulation parameters, we use three kinds of faults to verify the effectiveness of the proposed method. In the first case, an incipient fault is selected as 𝑓_𝑎(𝑡)=0.002exp(0.0667𝑡), and a high frequency disturbance 𝑑(𝑡)=sin(200𝑡) is used. As shown in Figure 1, the output estimation error from the sliding mode observer 𝑒̃_𝑣2 vibrates at a high frequency after the fault occurs, though its amplitude is small. This indicates that it is insensitive to the fault. However, the output estimation error 𝑒̃_𝑣1 increases rapidly after the fault occurs. Thus, the error 𝑒̃_𝑣1 is defined as the fault diagnosis residual 𝑅₁(𝑡), and a fault alarm can be sent immediately after a time of 30?s.

(a) 𝑒̃𝑣1

(b) 𝑒̃𝑣2

(c) 𝑒̃𝑣2 at 29.5–30.5 s

Figure 1 

Output estimation error in case 1 using a high frequency disturbance.

Another low frequency disturbance 𝑑(𝑡)=sin(0.1𝑡) is used to verify the proposed method. The simulation results are shown in Figure 2. It can be seen that output estimation error from the Luenberger observer is sensitive to the fault, but that from the sliding mode observer is still insensitive.

(a)  𝑒̃𝑣1

(b) 𝑒̃𝑣2

(c) 𝑒̃𝑣2 at 29.5–30.5 s

Figure 2 

Output estimation error in case 1 using a low frequency disturbance.

In the second case, a low-frequency sinusoidal signal is selected to illustrate that the fault detection is sensitive to incipient faults, that is, 𝑓_𝑎(𝑡)=sin(𝑡). A high frequency disturbance 𝑑(𝑡)=sin(200𝑡) and a low frequency disturbance 𝑑(𝑡)=sin(0.1𝑡) are used, respectively. The associated simulation results in Figures 3 and 4 verify that the proposed approach can be applied to detect incipient fault rapidly.

Figure 3 

Output estimation error in case 2 using a high frequency disturbance.

Figure 4 

Output estimation error in case 2 using a low frequency disturbance.

In the third case, 𝑓_𝑎(𝑡)=tanh(𝑡). A high frequency disturbance 𝑑(𝑡)=sin(200𝑡) and a low frequency disturbance 𝑑(𝑡)=sin(0.1𝑡) are used, respectively. The associated simulations are shown in Figures 5 and 6. Through the above verifications, we can see that the proposed method can effectively detect incipient faults.

Figure 5 

Output estimation error in case 3 using a high frequency disturbance.

Figure 6 

Output estimation error in case 3 using a low frequency disturbance.

5. Conclusion

This paper has introduced a fault diagnosis scheme which combines a sliding mode observer and a Luenberger observer for nonlinear systems. The proposed method makes full use of the complementarity between the sliding mode observer and the Luenberger observer and transforms the system, using a coordinate transformation, into two subsystems. The input estimation error from the Luenberger observer, which is constructed for the unrelated subsystem with unknown input disturbances, is defined as the fault detection residual in incipient fault diagnosis. A sliding mode observer is constructed to eliminate the impact of faults and disturbances in the other subsystem. An LMI-based approach is employed to design the observer, which makes it easy to obtain its parameters. The simulation results verify the validity of the proposed method.

Acknowledgments

This work is supported by the Natural Science Foundation of China (61273157 and 61104024), Hunan Province Education Department (12A040), and Construct Program of the Key Discipline in Hunan Province.