Fault Diagnosis for Linear Discrete Systems Based on an Adaptive Observer

Abstract

This paper presents a fault diagnosis algorithm to estimate the fault for a class of linear discrete systems based on an adaptive fault estimation observer. And observer gain matrix and adaptive adjusting rule of the fault estimator are designed. Furthermore, the adaptive regulating algorithm can guarantee the first-order difference of a Lyapunov discrete function to be negative, so that the observer is ensured to be stable and fault estimation errors are convergent. Finally, simulation results of an aircraft F-16 illustrate the advantages of the theoretic results that are obtained in this paper.

1. Introduction

As the scale and complexity of modern control systems are increasing, the requirements on system reliability are also increasing. Therefore, the design and analysis of fault detection and diagnosis (FDD) algorithms have received considerable attention during the past three decades. The development of FDD has been addressed by more and more authors and fruitful results have been obtained; see [1–3].

The observer technology is one of the important methods for FDD and fault-tolerant control [4, 5]. Commonly used observer-based fault estimation methods include sliding mode observers [6], unknown input observers [7], 𝐻_∞ filtering methods [8], neural networks observers [9], and adaptive observers [10, 11]. But most of them focus on the continuous systems, and only few results have been reported on the fault estimation design in discrete-time systems. The discrete-time is widely used in practical implementations, for example, computer control systems, networked control systems, and so forth [12]. Reference [13] designed an 𝐻_∞ FD filter for a class of linear discrete-time systems in a networked environment. Reference [14] used an adaptive fault observer to deal with discrete-time systems, but it did not involve the issue of fault estimation. Reference [15] also dealt with discrete-time nonlinear systems, but its inequality functions were complex and it was difficult to get solutions. On the other hand, aircraft flight control systems are good examples for applications of fault accommodation/active reliable control, and the fault observers have been employed to detect faults [11].

In this paper, a novel discrete-time actuator fault estimation scheme is proposed to deal with abrupt actuator failures. The proposed actuator fault estimation scheme is then applied by using the Lyapunov method. Simulation results of a numerical example are also given.

The paper is organized as follows. Section 2 describes the mathematical preliminaries and problem formulation. In Section 3, concerning the theoretical results of the proposed fault diagnosis scheme, a fault estimation scheme is proposed to deal with the actuator failures of discrete-time system. In Section 4, an example of aircraft flight control system is given to illustrate the performance of the proposed scheme. The concluding remarks are given in Section 5.

2. System Description and Preliminaries

Consider a discrete-time linear system 𝑥()(𝑘)(𝑘)(𝑘),𝑦(𝑘)(𝑘),𝑘+1=𝐴𝑥+𝐵𝑢+𝐸𝑓=𝐶𝑥 (1) where 𝑥(𝑘)∈𝑅^𝑛 is the state, 𝑢∈𝑅^𝑚 is the control input, 𝑓(𝑘)∈𝑅^𝑞 with 𝑞≤𝑚 is the function to model the actuator faults, and 𝑦(𝑘)∈𝑅^𝑟 is the measurable output. Matrices 𝐴, 𝐵, 𝐶, and 𝐸 are real matrices of appropriate dimensions. Matrix 𝐸 is of full column rank; that is, rank (𝐸)=𝑞.

For estimating the actuator fault 𝑓(𝑘), an adaptive observer is constructed as follows: 𝑥()𝑥(𝑘)(𝑘)𝑓(𝑘)𝑦(𝑘)(𝑘,𝑦(𝑘)𝑥(𝑘),𝑘+1=𝐴+𝐵𝑢+𝐸−𝐿−𝑦)=𝐶 (2) where 𝑥(𝑘)∈𝑅^𝑛 is the observer state vector, 𝑦(𝑘)∈𝑅^𝑟 is the observer output vector, 𝑓(𝑘)∈𝑅^𝑞 is the estimate of the actuator fault 𝑓(𝑘), 𝐿∈𝑅^𝑛×𝑟 is the observer gain to be designed.

Denoting that 𝑒_𝑥=𝑥(𝑘)−𝑥(𝑘),𝑒_𝑦=𝑦(𝑘)(𝑘),𝑒−𝑦_𝑓=𝑓(𝑘)−𝑓(𝑘), (3) then the estimation error dynamics is modeled as follows: 𝑒_𝑥(𝑘+1)=(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)+𝐸𝑒_𝑓(𝑘),𝑒_𝑦(𝑘)=𝐶𝑒_𝑥(𝑘). (4)

3. Main Results

3.1. Modified Adaptive Fault Estimation Algorithm

Theorem 1. For constant actuator fault 𝑓(𝑘), if there exist matrices 𝑃>0 and 𝑄>0 and positive scalars 𝜀 and defined Γ=2𝜀𝐸𝐸^𝑇+𝑄⁻¹ (5) such that the following condition holds: ()Ξ=𝐴−𝐿𝐶^𝑇𝑃(𝐴−𝐿𝐶)−𝑃(𝐴−𝐿𝐶)^𝑇𝑃𝐸∗𝐸^𝑇−Γ^𝑇𝑄Γ+𝑃𝐸<0, (6) then the algorithm Δ𝑓(𝑘+1)=−𝜀𝐸^𝑇Γ𝑒_𝑥(𝑘+1)−(𝐴−𝐿𝐶)𝑒_𝑥(𝑘) (7) and the fault estimation 𝑓(𝑘+1)=𝑓(𝑘)+Δ𝑓(𝑘+1)=𝑓(0)+^𝑘+1_𝑖=1Δ𝑓(𝑖) (8) can realize estimation error of both the state and fault uniformly bounded for the entire time period.

Proof. From system (4), one gets 𝐸𝑒_𝑓(𝑘)=𝑒_𝑥(𝑘+1)−(𝐴−𝐿𝐶)𝑒_𝑥(𝑘), so the algorithm (7) becomes Δ𝑓(𝑘+1)=−𝜀𝐸^𝑇Γ𝑒_𝑥(𝑘+1)−(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)=−𝜀𝐸^𝑇Γ𝐸𝑒_𝑓(𝑘). (9) Then, 𝑒_𝑓()=𝑓()()=𝑓()−𝑓(𝑘)+𝑓(𝑘)().𝑘+1𝑘+1−𝑓𝑘+1𝑘+1−𝑓𝑘+1 (10) Because 𝑓(𝑘) is constant, 𝑓(𝑘+1)=𝑓(𝑘) and Δ𝑓(𝑘+1)=𝑓(𝑘+1)−𝑓(𝑘); then one gets 𝑒_𝑓(𝑘+1)=Δ𝑓(𝑘+1)+𝑒_𝑓(𝑘)=𝐼−𝜀𝐸^𝑇Γ𝐸𝑒_𝑓(𝑘). (11) Based on (4) and (11), we can obtain the following augmented system: 𝑒_𝑥()𝑒𝑘+1_𝑓()=𝑘+1𝐴−𝐿𝐶𝐸0𝐼−𝜀𝐸^𝑇𝑒Γ𝐸^𝑥(𝑘)𝑒_𝑓(𝑘). (12) Consider the following Lyapunov function: 𝑉(𝑘)=𝑒^𝑇_𝑥(𝑘)𝑃𝑒_𝑥(𝑘)+𝜀⁻¹𝑒^𝑇_𝑓(𝑘)𝑒_𝑓(𝑘). (13) Then, Δ𝑉(𝑘)=𝑒^𝑇_𝑥(𝑘+1)𝑃𝑒_𝑥(𝑘+1)−𝑒^𝑇_𝑥(𝑘)𝑃𝑒_𝑥(𝑘)+𝜀⁻¹𝑒^𝑇_𝑓(𝑘+1)𝑒_𝑓(𝑘+1)−𝜀⁻¹𝑒^𝑇_𝑓(𝑘)𝑒_𝑓(𝑘)=𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)−𝑃𝑒_𝑥(𝑘)+2𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃𝐸𝑒_𝑓(𝑘)+𝑒^𝑇_𝑓(𝑘)𝐸^𝑇𝑃𝐸𝑒_𝑓(𝑘)+𝜀⁻¹𝑒^𝑇_𝑓(𝑘)𝐼−𝐸^𝑇Γ𝐸^𝑇𝐼−𝐸^𝑇Γ𝐸𝑒_𝑓(𝑘)−𝜀⁻¹𝑒^𝑇_𝑓(𝑘)𝑒_𝑓(𝑘)=𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃(𝐴−𝐿𝐶)−𝑃𝑒_𝑥(𝑘)+2𝑒^𝑇_𝑥(𝑘)(𝐴−𝐿𝐶)^𝑇𝑃𝐸𝑒_𝑓(𝑘)+𝑒^𝑇_𝑓(𝑘)𝐸^𝑇−Γ^𝑇𝑄Γ+𝑃𝐸𝑒_𝑓(𝑘). (14) Denoting that 𝑒(𝑘)=[𝑒_𝑥(𝑘)𝑒_𝑓(𝑘)]^𝑇, then Δ𝑉(𝑘)=𝑒(𝑘)^𝑇Ξ𝑒(𝑘)<0, and Ξ is defined as (6).

Remark 2. The pair (𝐴,𝐶) is observable, and define the observe matrix 𝑈=[𝐶^𝑇(𝐶𝐴)^𝑇…(𝐶𝐴^𝑙−1)^𝑇]^𝑇. So, it gets that the rank of 𝑈 is 𝑛. When 𝐶 is full column rank,we have Δ𝑓(𝑘+1)=−𝜀𝐸^𝑇Γ𝑒_𝑥(𝑘+1)−(𝐴−𝐿𝐶)𝑒_𝑥(𝑘)=−𝜀𝐸^𝑇Γ𝐶⁻¹𝑒_𝑦(𝑘+1)−𝐴𝐶⁻¹−𝐿𝑒_𝑦(𝑘). (15) At the first step, the online fault estimation is as follows: 𝑓(1)=𝑓(0)𝑓(1)=𝑓(0)+Δ−𝜀𝐸^𝑇Γ𝐶⁻¹𝑒_𝑦(1)−𝐴𝐶⁻¹−𝐿𝑒_𝑦(0). (16) And at the 𝑘th step, fault estimation is as (8).
If 𝐶 is not full column rank, then more system outputs and observer estimations should be used to get the state estimation.
When the output 𝑦(𝑘) satisfies Lipschitz condition, define 𝑙_min=argmax_𝑙[rank(𝑈)], and define 𝐹=(𝐴−𝐿𝐶)⁻¹ and 𝑈_𝐹=[𝐶^𝑇(𝐶𝐹)^𝑇⋯(𝐶𝐹^𝑙min−1)^𝑇]^𝑇; then one can get Δ𝑓(𝑘+1)=−𝜀𝐸^𝑇Γ𝑈_𝐹⁻¹𝑒_𝑦(𝑘+1)^𝑇⋯𝑒_𝑦(𝑘+2−𝑙min)^𝑇𝑇−𝐴𝑈_𝐹⁻¹𝑒_𝑦(𝑘)^𝑇⋯𝑒_𝑦(𝑘+1−𝑙min)^𝑇+𝐿𝑒_𝑦(𝑘). (17) Then, fault estimation is 𝑓(𝑘+1)=𝑓(𝑘)+Δ𝑓(𝑘+1)=𝑓(0)+∑^𝑘+1_𝑖=1Δ𝑓(𝑖).

Remark 3. Equation (6) is a nonlinear matrix inequality about matrices Γ and 𝑄, and it is not easy to be calculated, so let 𝑄=𝜀𝐸𝐸^𝑇+𝛿tr(𝐸𝐸^𝑇). And after we choose 0<𝛿<𝜀, the solution can be found easily.

Remark 4. If the fault 𝑓(𝑘) is not constant but is a linear function, such as 𝑓(𝑘+1)=𝐻𝑓(𝑘), then 𝑒_𝑓(𝑘+1)=𝐻−𝐸^𝑇Γ𝐸𝑒_𝑓(𝑘). (18) Consider the same Lyapunov function 𝑉(𝑘)=𝑒^𝑇_𝑥(𝑘)𝑃𝑒_𝑥(𝑘)+𝜀⁻¹𝑒^𝑇_𝑓(𝑘)𝑒_𝑓(𝑘). We can also get ()Ξ=𝐴−𝐿𝐶^𝑇𝑃(𝐴−𝐿𝐶)−𝑃(𝐴−𝐿𝐶)^𝑇𝑃𝐸∗𝐸^𝑇−𝜀Γ^𝑇𝑄Γ+𝑃𝐸+𝜀𝐻^𝑇𝐻−𝐼<0. (19) The proof is smilar to that of Theorem 1 and it is omitted here for brevity.

4. Simulation Results

In this section, the fault estimation algorithm is applied to a model of the vertical dynamics of an F-16 aircraft. The model is taken from [16]. The signals and their generation in the simulations are summarized in Table 1, where size means the variance for the inputs and constant magnitude for the faults, respectively.

Table 1 

Signals in the F-16 simulation study.

We have the following numerical values in (1): ,,,,𝑙𝐴=10.00140.11330.0004−0.099700.9945−0.0171−0.00050.007000.000310.0957−0.004900.006100.9130−0.09660−0.02860.00020.10040.9879𝐵=−0.007800.0003−0.01150.099700.02120−0.00810.41500.0003−0.15890.1794−0.0014−0.0158𝐸=−0.007800.0003−0.01150.099700.02120−0.0810.41500.0003−0.15890.1794−0.0014−0.0158𝐶=100000100000100_min=2,𝑈_𝐹=𝐶^𝑇(𝐶𝐹)^𝑇𝑇. (20) By solving conditions in Theorem 1, one can obtain the following solutions after iterations: 𝐿 = [0.0001 0 0; 0 0.0001 0; 0 0 0.0002; 0.0001 0.0001 0; 0 0.0001 0.0001]; 𝜀 = 1, 𝛿 = 0.1. Then, one can take the learning rate Γ = [189923  −147.6  646.2  2443  1727.8;  −147.6  100.6  1.31  −0.58  1.24;  646.2  1.31 9365  −465.8  −1.27;   2443  −0.589  −465.8  71.3  −5.26;   1727.8  1.24  −1.27  −5.26 87.5].

In this simulation, it is assumed that three kinds of actuator faults 𝑓_𝑖(𝑘) are, respectively, created as 𝑓₁(𝑘)=𝑓0,𝑘<100,0.5,𝑘>100,²(𝑘)𝑓isfree,₃(𝑘)=.0,𝑘<20001,𝑘>2000 (21) The fault 𝑓₁(𝑘) estimation result is shown in Figure 1, while Figure 2 illustrates the estimation of 𝑓₂(𝑘), and Figure 3 illustrates the estimation of 𝑓₃(𝑘).

Figure 1 

Estimation of the fault 𝑓1(𝑘).

Figure 2 

Estimation of the fault 𝑓2(𝑘).

Figure 3 

Estimation of the fault 𝑓3(𝑘).

5. Conclusions

In this paper, a fault estimation algorithm is established for linear discrete systems with actuator faults. The algorithm can enhance the performance of fault estimation. And simulation results show that using the algorithm, the accuracy of fault estimation can be improved evidently. Extension of the proposed fault estimation method to more general nonlinear systems is an interesting issue, which will be investigated in our future research work.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 61273171, 61304112, and 61104020), the Natural Science Foundation of Jiangsu Province (BK20131364), and the Doctoral Fund of Ministry of Education of China (no. 20113218110011).