A New Adaptive Kalman Estimator for Detection and Isolation of Multiple Faults Integrated in a Fault Tolerant Control

Abstract

For sequential jumps detection, isolation, and estimation in discrete-time stochastic linear systems, Willsky and Jones (1976) have developed the Generalized Likelihood Ratio (GLR) test. After each detection and isolation of one jump, the treatment of another possible jump is obtained by a direct state estimate and covariance incrementation of the Kalman filter originally designed on the jump-free system. This paper proposes to extend this approach from a state estimator designed on a reference model directly sensitive to system changes. We will show that the obtained passive GLR test can be easily integrated in a Fault Tolerant Control System (FTCS) via a control law designed in order to asymptotically reject the effect of sequential jumps.

1. Introduction

First derived in a completely recursive form by Willsky and Jones [1], the GLR test has been used in a wide variety of applications including the detection of sensor and actuator failures [1–3], electrocardiogram analysis [4], geophysical signal processing [5], and freeways supervision [6]. For sequential jumps detection in discrete-time stochastic linear systems, the GLR test is made of the following steps:

(1)detection and isolation of one possible jump by applying a GLR detector on the innovation sequence of the Kalman filter designed on the jump-free system,(2)updating of the Kalman filter using the jump magnitude estimate given by the GLR detector,(3)go to the step 1 to detect, isolate, and estimate another possible jump from measurements immediately available after the detection time of the last jump.

The first part of this paper revisits the standard GLR test of Willsky and Jones [1] with respect to the fault detectability indexes [7] describing the time delay between the occurrence of a jump and its effect on measurements.

The second part presents a new GLR test based on the augmented state Kalman estimator designed on the least favorable reference model including all the states of hypothetical jumps. This is not a new approach in the area of an FDI in dynamic systems-based observer. The well-known fault detection filter originally developed by Beard [8] is also designed from a fixed reference model including all the possible faults. The sequential multiple decision theory is not complete and this paper has not the pretention to give a unique solution of this general problem but only a usefull solution focussed on the integration of the obtained FDI scheme in an FTCS.

Over the last two decades, growing demand for reliability, maintainability in dynamic systems has drawn significant research in the area of Fault detection and isolation (FDI) (see [9, 10] and references therein). A fault tolerant control system (FTCS) is a control system which possesses the ability to accommodate system component failures automatically. The existing methods for reconfigurable controller design can be classified as linear quadratic regulator [11, 12], eigenstructure assignment [13], multiple model [14], adaptive control [15], pseudoinverse [16], and model following [17] and recently, [18–20] have proposed a kind of controllers stabilizing for fault tolerant control. For fuzzy system [21] a new approach of FTC is developed taking into account uncertainties of system model. In this context, the last part of the paper proposes to integrate our passive GLR test in an FTCS via a control law of form 𝑢_𝑘=𝑢^𝑛_𝑘−𝐺𝜈_𝑘, where the nominal control law 𝑢^𝑛_𝑘=−𝐿𝑥_𝑘 will be corrected online by the additional input signal 𝐺𝜈_𝑘 depending on the jump magnitude prediction 𝜈_𝑘. This control law will be designed from the two-stage Kalman filter of Friedland [22] allowing the state prediction 𝑥_𝑘 of the jump-free system to be available at each step of the processing. The additional input signals 𝐺𝜈_𝑘 superimposed to the nominal control inputs will be computed so that the influence of the jumps on the regulated variables is asymptotically rejected.

2. The A Priori Assumptions

The model of the jump-free system, noted ℎ₀, is given by𝑥_𝑘+1=𝐴𝑥_𝑘+𝐵𝑢_𝑘+𝑤_𝑘,𝑦_𝑘=𝐶𝑥_𝑘+𝑣_𝑘, (1)(2) where 𝑥_𝑘∈ℜ^𝑛 is the state vector, 𝑦_𝑘∈ℜ^𝑚 the measurement vector, 𝑢_𝑘∈ℜ^𝑟 the input vector. The zero mean white Gaussian noises 𝑤_𝑘 and 𝑣_𝑘 satisfy 𝐸𝜔_𝑘𝜐_𝑘𝜔^𝑇_𝑗𝜐^𝑇_𝑗=𝛿𝑊00𝑉^𝑘,𝑗, (3) where 𝑊≥0 and 𝑉>0 and the Gaussian initial state 𝑥₀, 𝑥₀=𝐸(𝑥₀) and 𝑃₀=𝐸{(𝑥₀−𝐸(𝑥₀))(𝑥₀−𝐸(𝑥₀))^𝑇}, 𝛿_𝑘,𝑗 is the Kronecker operator. The jump hypotheses ℎ_𝑖 for 𝑖∈[1,…,𝑁] are modelized by 𝑥_𝑘+1=𝐴𝑥_𝑘+𝐵𝑢_𝑘+𝑓_𝑖(𝑘,𝑟)𝜈(𝑘,𝑟)+𝑤_𝑘,𝑦_𝑘=𝐶𝑥_𝑘+𝑣_𝑘, (4) where 𝑓_𝑖(𝑘,𝑟) is the fault distribution vector, 𝜈(𝑘,𝑟) the scalar fault magnitude, and 𝑟 the unknown time of failure occurrence. Without lost of generality, the fault magnitude 𝜈(𝑘,𝑟) is assumed to follow a constant bias model 𝜈(𝑘,𝑟)=𝜈, (𝑘≥𝑟) with step profile: 𝑓_𝑖(𝑘,𝑟)=𝑓_𝑖,(𝑘≥𝑟) and 𝑓_𝑖(𝑘,𝑟)=0, (𝑘<𝑟). Based on the results of Tanaka [23] and Caglyan [24] (not developed in this paper), the detectability and distinguishability conditions of multiple jumps are assumed to be satisfied rank[𝐶𝐴^𝜇1−1𝑓₁⋯𝐶𝐴^𝜇𝑖−1𝑓_𝑖⋯𝐶𝐴^𝜇𝑁−1𝑓_𝑁]=𝑁𝑓rank𝐼−𝐴₁⋯𝑓_𝑖⋯𝑓_𝑁𝐶0=𝑛+𝑁. (5) Let 𝜇_𝑖=min(𝑠∶𝐶𝐴^𝑠−1𝑓_𝑖≠0,𝑠=1,2,…) for 𝑖∈[1,…,𝑁].

The jump detectability indexes [7], and assume that 𝜌_𝑖<∞ signifying that the first information about the jump ℎ_𝑖, appearing at time 𝑟, will be present on measurement 𝑦_{𝑟+𝜌𝑖}.

In the case where the jumps may occur relatively infrequently, the most likelihood reference model at the beginning of the processing is the model of ℎ₀. So, under the stability and convergence conditions𝐶rang|𝑧|𝑧𝐼−𝐴=𝑛,≥1,rang−𝑒^𝑗𝑤𝐼+𝐴𝑊^1/2=𝑛,𝑤∈[0,2𝜋]. (6)(7)

The jumps may occur relatively infrequently, and we assume that the nominal control law 𝑢^𝑛_𝑘=−𝐿𝑥_𝑘 satisfying the nominal system performances has been designed on the jump-free system (1) via for example LQG approach on infinite horizon [25]. No a priori information is available concerning the jump's scenario: after the detection of one or several jumps, no information is available to focus our attention on another particular jump. The only assumption is that two jumps cannot appear simultaneously but only sequentially with a sufficient time delay to allow their real-time recursive treatment on a sliding window 𝑊=[𝑘−𝑀≤𝑟≤𝑘] of size 𝑀. In other words window 𝑊 may include occurrence time 𝑟 of one jump only. So, 𝑀 will be chosen in accordance with the maximum available computational time and with the minimum time delay between the occurrence of two jumps depending to the practical application of our works.

3. The 𝑛-Order Adaptive Kalman Estimator of Willsky and Jones

This chapter revisits the GLR test developed by Willsky and Jones [1] in relation with multihypotheses GLR test described in Willsky [26]. The a priori reference model at the beginning of the processing is the model ℎ₀. Under the stability and convergence conditions (6), (7), the Kalman filter which gives the maximum likelihood state prediction of the system is given by 𝑥_𝑘+1=𝐴𝑥_𝑘+𝐵𝑢_𝑘+𝐾_𝑘𝑦_𝑘−𝐶𝑥_𝑘,𝑃_𝑘+1=𝐴−𝐾_𝑘𝐶𝑃_𝑘𝐴−𝐾_𝑘𝐶^𝑇+𝑊+𝐾_𝑘𝑉𝐾^𝑇_𝑘,𝐻_𝑘=𝐶𝑃_𝑘𝐶^𝑇+𝑉,𝐾_𝑘=𝐴𝑃_𝑘𝐶^𝑇𝐻_𝑘⁻¹. (8)(9)(10)(11) The additive effect of jump ℎ_𝑖 on the state prediction error 𝑒_𝑘+1=𝑥_𝑘+1−𝑥_𝑘+1 and on the innovation sequence 𝛾=𝑦_𝑘−𝐶𝑥_𝑘 can be expressed as 𝑒_𝑘+1=𝑒̃_𝑘+1+𝜁_𝑖(𝑘+1,𝑟),𝛾_𝑘=𝛾_𝑘+1+𝜚_𝑖(𝑘,𝑟)𝜈, (12)(13) where 𝑒̃_𝑘+1 and 𝛾_𝑘 represent the state prediction error and the innovation sequence on the jump-free system and where 𝜁_𝑖(𝑘,𝑟) and 𝜚_𝑖(𝑘,𝑟) describe the additive effect of the jump satisfying 𝜁_𝑖(𝑘+1,𝑟)=𝐴−𝐾_𝑘𝐶𝜁_𝑖(𝑘,𝑟)+𝑓_𝑖,𝜁_𝑖()𝜚𝑟,𝑟=0,_𝑖(𝑘,𝑟)=𝐶𝜁_𝑖(𝑘,𝑟). (14)

In Willsky [26] or Basseville and Nikiforov [27], 𝜁_𝑖(𝑘,𝑟)=𝛼_𝑖(𝑘,𝑟)−𝛽_𝑖(𝑘,𝑟), where 𝛼_𝑖(𝑘,𝑟) and 𝛽_𝑖(𝑘,𝑟) describe the additive effect of the jump on the state vector of the system and on the state vector of the Kalman filter, respectively. Our formulation shows that the jump signatures (14) only depend to the state prediction error of the Kalman filter and thus are decoupled from 𝑢_𝑘 (this property will be used in the design of the FTCS). The jump hypothesis ℎ_𝑖 can be confronted to the “no-jump” hypothesis ℎ₀ as 𝐻₀∶𝐸𝛾_𝑡𝐻=0,𝑡<𝑟,_𝑖∶𝐸𝛾_𝑡=𝜚_𝑖(𝑡,𝑟)𝜈,𝑘≥𝑡≥𝑟,𝑖∈[1,…,𝑁]. (15)(16) Since 𝐸{(𝛾_𝑘−𝑡−𝐸(𝛾_𝑘−𝑡))(𝛾_𝑘−𝐸(𝛾_𝑘))^𝑇}=0forall𝑡<𝑘, the likelihood ratio between (15) and (16) gives𝜆_𝑖()=𝑃𝑘,𝑟,𝜈𝛾₀/ℎ₀⋯𝑃𝛾_{𝑟𝑖+𝜌𝑖}/ℎ_𝑖⋯𝑃𝛾_𝑘/ℎ_𝑖𝑃𝛾₀/ℎ₀⋯𝑃𝛾_{𝑟𝑖+𝜌𝑖}/ℎ₀⋯𝑃𝛾_𝑘/ℎ₀. (17) From (14), we have 𝜚_𝑖(𝑟,𝑟)=𝜚_𝑖(𝑟+1,𝑟)=⋯=𝜚_𝑖(𝑟+𝜇_𝑖−1,𝑟)=0 leading to𝑃𝛾_𝑟𝑖ℎ_𝑖=𝑃𝛾_𝑟𝑖ℎ₀,𝑃𝛾_𝑟𝑖+1ℎ_𝑖=𝑃𝛾_𝑟𝑖+1ℎ₀𝑃,…,𝛾_{𝑟𝑖+𝜇𝑖−1}ℎ_𝑖=𝑃𝛾_𝑟𝑖+𝜇_𝑖−1ℎ₀, (18) and the likelihood ratio (17) can be rewritten as𝜆_𝑖()=𝑃𝑘,𝑟,𝜈𝛾_{𝑟𝑖+𝜇𝑖}/ℎ_𝑖⋯𝑃𝛾_𝑘/ℎ_𝑖𝑃𝛾_{𝑟𝑖+𝜇𝑖}/ℎ₀⋯𝑃𝛾_𝑘/ℎ₀=exp(−1/2)∑^𝑘_{𝑡=𝑟+𝜇𝑖}𝛾_𝑡−𝜚_𝑖(𝑡,𝑟)𝜈²_𝐻𝑡⁻¹exp(−1/2)∑^𝑘_{𝑡=𝑟+𝜇𝑖}𝛾_𝑡²_𝐻𝑡⁻¹. (19)

Based on 𝛾₀,…,𝛾_𝑘, the maximum likelihood prediction of 𝜈 conditioned on a particular assumed value of 𝑟 is given by𝜈(𝑘+1,𝑟)=^𝑘_{𝑡=𝑟+𝜇𝑖}𝜚^𝑇_𝑖(𝑡,𝑟)𝐻_𝑡⁻¹𝜚_𝑖(𝑡,𝑟)^𝑘−1_{𝑡=𝑟+𝜇𝑖}𝜚^𝑇_𝑖(𝑡,𝑟)𝐻_𝑡⁻¹𝛾_𝑡, (20) and the log-likelihood ratio 𝑇_𝑖(𝑘,𝑟)=2log(𝜆_𝑖(𝑘,𝑟,𝜈(𝑘+1,𝑟))) can be written as𝑇_𝑖(𝑘,𝑟)=𝑏_𝑖(𝑘,𝑟)²𝑎_𝑖(𝑘,𝑟)⁻¹,𝑎_𝑖(𝑘,𝑟)=^𝑘_{𝑡=𝑟+𝜇𝑖}𝜚^𝑇_𝑖(𝑡,𝑟)𝐻_𝑡⁻¹𝜚_𝑖(),𝑏𝑡,𝑟_𝑖(𝑘,𝑟)=^𝑘_{𝑡=𝑟+𝜇𝑖}𝜚^𝑇_𝑖(𝑡,𝑟)𝐻_𝑡⁻¹𝛾_𝑡. (21) So, the decision rule of the GLR detector is as follows:max_{𝑖∈[1,…,𝑁],}𝑟_{∈[0,…,𝑘]}𝑇_𝑖𝑘,𝑟̃−𝜇_𝑖>𝜀, (22) where 𝜀 is the decision threshold. For a practical implementation of the GLR detector (22), the translated jump occurrence time 𝑟̃ must be constrained to belong to the sliding windows 𝑊=[𝑘−𝑀≤𝑟̃≤𝑘] of size 𝑀. In this case, (22) can be rewrittenmax_{𝑖∈[1,…,𝑁],}𝑟̃_∈𝑊𝑇_𝑖𝑘,𝑟̃−𝜇_𝑖>𝜀. (23) If max{𝑇_𝑖(𝑘,𝑟̃−𝜇_𝑖}>𝜀, then one jump is detected at time 𝑘, isolated from (𝑗,𝑟)̂̃=argmax{𝑇_𝑖(𝑘,𝑟̃−𝜇_𝑖)}, where 𝑟̂=𝑟̂̃−𝜇_𝑗 is the estimation of the jump occurrence time and where 𝜈(𝑘+1,𝑟)̂=𝑎_𝑗(𝑘,𝑟)̂⁻¹𝑏_𝑗(𝑟)̂,𝑃𝑘,_𝜈(𝑘+1,𝑟)̂=𝑎_𝑗(𝑘,𝑟)̂⁻¹ (24)(25) represent the maximum likelihood prediction of the jump magnitude based on measurements until time 𝑘 under the assumption that 𝜈 has an infinite a priori covariance.

The second step of the standard GLR test, intuitively obtained from relation (12), consists in updating the Kalman filter (8) by prediction and covariance incrementation of the Kalman filter (8) as follows:𝑥^new_𝑘+1=𝑥^old_𝑘+1+𝜁_𝑗(𝑟)̂𝜈(𝑟)̂,𝑃𝑘+1,𝑘+1,_new𝑘+1=𝑃^old_𝑘+1+𝜁_𝑗(𝑘+1,𝑟)̂𝑃𝜁_𝑗(𝑘+1,𝑟)̂^𝑇, (26) where the new state prediction (𝑥^new_𝑘+1,𝑃^new_𝑘+1) is substituted in the Kalman filter (13) (see Figure 1 in Willsky and Jones [1]). In the original version of the GLR test, the state variables of the matched filters 𝜁_𝑖(𝑘,𝑟) for 𝑖∈[1,…,𝑁] are reinitialized to zero immediately following the filter incrementation leading to consider (𝑥^new_𝑘+1,𝑃^new_𝑘+1) as a new initialization of the Kalman filter more appropriate than the initial value given at the beginning of the processing [27]. With this implementation, (24) cannot be used to improve the jump estimation from measurements available after its detection leading to choose the threshold level 𝜀 by reaching a compromise between fast detection and accurate estimation.

To avoid this tradeoff, we can modify this implementation in such a way that (𝑥^new_𝑘+1,𝑃^new_𝑘+1) is not substituted in the Kalman filter but used to generate the auxiliary innovation sequence 𝛾_𝑘=𝑦_𝑘−𝐶𝑥^new_𝑘 and its covariance matrix 𝐻_𝑘=𝐶𝑃^new_𝑘𝐶^𝑇+𝑉 expressed𝛾_𝑘=𝛾_𝑘−𝜚_𝑗(𝑟)̂𝜈(𝑟)̂,𝐻𝑘,𝑘,_𝑘=𝐻_𝑘+𝜚_𝑗(𝑘,𝑟)̂𝑃^𝜈(𝑘,𝑟)̂𝜚_𝑗(𝑘,𝑟)̂^𝑇, (27) where 𝛾_𝑘 and 𝐻_𝑘 represent the innovation sequence of the Kalman filter (8) and its covariance matrix. Now decoupled to the updating strategy, the Kalman filter, ((8), (24), and (14)) can be used to improve the jump estimation from measurements available after its detection time, and another possible jump can then be treated by the following GLR detector:max_{𝑖∈[1,…,𝑁],𝑖≠𝑗}𝑟̃_∈𝑊𝑇_𝑖𝑘−𝜌_𝑖,𝑟̃>𝜀 (28) with𝑇_𝑖(𝑘,𝑟)=𝑏̃_𝑖(𝑘,𝑟)²𝑎_𝑖(𝑘,𝑟)⁻¹,𝑎_𝑖(𝑘,𝑟)=^𝑘_{𝑡=𝑟+𝜌𝑖}𝜚^𝑇_𝑖(𝑡,𝑟)𝐻_𝑡⁻¹𝜚_𝑖(),𝑏̃𝑡,𝑟_𝑖(𝑘,𝑟)=^𝑘_{𝑡=𝑟+𝜌𝑖}𝜚^𝑇_𝑖(𝑡,𝑟)𝐻_𝑡⁻¹𝛾_𝑡, (29) where the jump signatures 𝜚_𝑖(𝑡,𝑟) are computed from (14) after having reinitialized 𝜁_𝑖(𝑘,𝑟) for 𝑖≠𝑗 to zero immediately after the detection time of the first jump.

4. The 𝑁+𝑛 Order Adaptive Kalman Estimator

Our passive GLR test will not be designed from the jump-free system but on a reference model including all the hypothetical states of jumps that we wish to detect and isolate. This model will not be considered as the true model of the faulty system since the jump hypotheses ((4), (5)) are always assumed to occur relatively infrequently. Let 𝑋_𝑘+1=𝐴𝑋_𝑘+𝐵𝑢_𝑘+Γ𝑤_𝑘,𝑦_𝑘=𝐶𝑋_𝑘+𝑣_𝑘 (30)(31) with 𝐴=^𝐴𝐹₀₁, 𝐵=^𝐵₀, Γ=^𝐼₀, 𝐶=[^𝐶0], 𝑋_𝑘=^𝑥𝑘_𝜈𝑘 the augmented state model including all jump's states 𝜈_𝑘=[𝜈¹_𝑘⋯𝜈^𝑗_𝑘⋯𝜈^𝑁_𝑘] that we wish to detect and isolate. From (30), the jump hypothesis ℎ_𝑗 described by (2) can be viewed as an impulsive abrupt change on the 𝑗^th hypothetical jump's state, modelized as𝜈^𝑗_𝑘+1=𝜈^𝑗_𝑘+Δ𝜈𝛿_𝑘𝑟,∀𝑗∈[1,…,𝑁], (32) where 𝑟 is the unknown occurrence time of impulsive abrupt change, Δ𝜈 the jump's state increment, and 𝛿_𝑘r is the Kronecker operator.

Substituting (32) in (30), we obtain the impulsive jump hypotheses, noted ℎ^Δ_𝑗, described by 𝑋_𝑘+1=𝐴𝑋_𝑘+𝐵𝑢_𝑘+𝑓^Δ_𝑗(𝑘,𝑟)Δ𝜈(𝑘,𝑟)+Γ𝑤_𝑘,𝑦_𝑘=𝐶𝑋_𝑘+𝑣_𝑘 (33) with Δ𝜈(𝑘,𝑟)=Δ𝜈𝛿_𝑘𝑟𝑖, 𝑓^Δ_𝑗(𝑘,𝑟)=𝑓^Δ_𝑗Δ𝜈(𝑘,𝑟_𝑖), and Γ=[0𝐼^𝑇_𝑗]^𝑇 where 𝐼^𝑇_𝑗=[0⋯1⋯0] has one at the 𝑗th position and zero elsewhere. Based on an approach very similar to the modified GLR test of part 3, let 𝑋_𝑘+1=𝐴𝑋_𝑘+𝐵𝑢_𝑘+𝐾_𝑘𝑦_𝑘−𝐶𝑋_𝑘Ω_𝑘+1,=𝐴Ω_𝑘𝐴^𝑇+Γ𝑊Γ^𝑇−𝐴_𝑘Ω_𝑘𝐶^𝑇𝐶Ω_𝑘𝐶^𝑇+𝑉⁻¹𝐶Ω_𝑘𝐴^𝑇𝐾_𝑘,=𝐴Ω_𝑘𝐶^𝑇𝐻_𝑘⁻¹𝐻_𝑘,=𝐶Ω_𝑘𝐶^𝑇+𝑉, (34)(35)(36)(37) the augmented state Kalman filter designed on the reference model (30), (31). The additive effect of the impulsive jump hypothese ℎ^Δ_𝑖 on the state prediction error and on the innovation sequence of the augmented state Kalman filter propagates as𝑒_𝑘+1=𝑒̃_𝑘+1+𝜁^Δ_𝑗()𝛾𝑘+1,𝑟Δ𝜈,_𝑘=𝛾_𝑘+1+𝜚^Δ_𝑗(𝑘,𝑟)Δ𝜈, (38) where 𝑒̃_𝑘+1 and 𝛾_𝑘 represent the state prediction error and the innovation sequence on the jump-free system and where 𝜁^Δ_𝑗(𝑘+1,𝑟) and 𝜚^Δ_𝑗(𝑘,𝑟) propagate as𝜁^Δ_𝑗(𝑘+1,𝑟)=𝐴−𝐾_𝑘𝐶𝜁^Δ_𝑗(𝑘,𝑟),𝜁^Δ_𝑗(𝑟,𝑟)=𝑓^Δ_𝑗,𝜚^Δ_𝑗(𝑘,𝑟)=𝐶𝜁^Δ_𝑗(𝑘,𝑟). (39) Based on the part 3, the following GLR detector is then derivedmax_{𝑗∈[1,…,𝑁],}𝑟̃_∈𝑊𝑇^Δ_𝑗𝑘,𝑟̃−𝜇_𝑗>𝜀 (40) with 𝑇^Δ_𝑗(𝑘,𝑟)=𝑏^Δ_𝑗(𝑘,𝑟)²𝑎^Δ_𝑗(𝑘,𝑟)⁻¹,𝑎^Δ_𝑗(𝑘,𝑟)=^𝑘_{𝑡=𝑟+𝜇𝑗}𝜚_𝑗^Δ𝑇(𝑡,𝑟)𝐻_𝑡⁻¹𝜚^Δ_𝑗()𝑏𝑡,𝑟,_Δ𝑗(𝑘,𝑟)=^𝑘_{𝑡=𝑟+𝜇𝑗}𝜚_𝑗^Δ𝑇(𝑡,𝑟)𝐻_𝑡⁻¹𝛾_𝑡. (41)(42)(43) If max_𝑗,𝑟{̃𝑇^Δ_𝑗(𝑘,𝑟̃−𝜇_𝑗}>𝜀 then (𝑖,𝑟)̂̃=argmax_𝑗,𝑟̃{𝑇^Δ_𝑗(𝑘,𝑟̃−𝜇_𝑗)}, and the impulsive jump ℎ^Δ_𝑖 is declared to be occurred at time, where 𝑟̂=𝑟̂̃−𝜇_𝑖 whereΔ𝜈(𝑘+1,𝑟)̂=𝑎^Δ_𝑖(𝑘,𝑟)̂⁻¹𝑏^Δ_𝑖(𝑟)̂,𝑃𝑘,_Δ𝜈(𝑘+1,𝑟)̂=𝑎^Δ_𝑖(𝑘,𝑟)̂⁻¹ (44) represent the maximum likelihood prediction of the jump increment Δ𝜈 (under the assumption that Δ𝜈 has an infinite a priori covariance). At the detection time of the first jump, the tracking ability of the augmented state Kalman filter (34) can be improved from the updating strategy of Willsky and Jones [1] as 𝑋^new_𝑘+1=𝑋^old_𝑘+1+𝜁^Δ_𝑖(𝑟)̂Δ𝜈(𝑟)̂,Ω𝑘+1,𝑘+1,_new𝑘+1=Ω^old_𝑘+1+𝜁^Δ_𝑖(𝑘+1,𝑟)̂𝑃^Δ𝜈(𝑘+1,𝑟)̂𝜁^Δ_𝑖(𝑘+1,𝑟)̂^𝑇. (45)(46)

In our case, the state of the matched filters given by 𝜁^Δ_𝑗(𝑘,𝑟) is spanned in the trajectory space of the augmented state Kalman filter's prediction errors. So, (45) and (46) substituted in the augmented state Kalman filter (34) improve its tracking ability without producing a possible instability on the resulting filter (under the stability and convergence conditions of the augmented state Kalman filter given by (6) and (7)). The treatment of another impulsive jump is then obtained by applying GLR detector (41) on the resulting filter after having reinitialized 𝜁^Δ_𝑗(𝑘,𝑟)=0, for all 𝑗∈[1,…,𝑁] immediately after the filter incrementation. The new initialization (14) allows 𝐸(𝑋^new_𝑘) to reach the true state of the system very quickly (and 𝐸(𝛾_𝑡) to reach zero for jump compensation, consequently) avoiding the detection of the same jump several times. From an inductive reasoning, the passive GLR test is then derived and consists of the following steps:(1)detection, isolation, and estimation of one impulsive jump with the GLR detector (41),(2)updating of the augmented state Kalman filter (34) with (45) to improve its tracking, (3)going to the step 1.

The sequential multiple decision theory is not complete, and the choice of the threshold level 𝜀 is not studied in this paper. However, some simulation results not presented in this paper show that only statistical tuning parameter 𝜀 can be fixed at the beginning of the processing (this is not the threshold level which is adaptive but the augmented Kalman filter).

To show that the passive GLR test is less powerful than the GLR test of Willsky and Jones [1] (for a same fixed decision threshold) a very simple proof can be given on the treatment of the first jump. The reader can easily show that 𝛿_𝑖(𝑘,𝑟)≥𝛿^Δ_𝑗(𝑘,𝑟), (47) where 𝛿_𝑖(𝑘,𝑟)=[∑^𝑘_{𝑡=𝑟+𝜇𝑗}𝜚^𝑇_𝑗(𝑡,𝑟)𝐻_𝑡⁻¹𝜚_𝑗(𝑡,𝑟)]𝜈² is the signal to noise ratio of the Willsky and Jones GLR test and where 𝛿^Δ_𝑗(𝑘,𝑟)=[∑^𝑘_{𝑡=𝑟+𝜇𝑗}𝜚_𝑗^Δ𝑇(𝑡,𝑟)𝐻_𝑡⁻¹𝜚^Δ_𝑗(𝑡,𝑟)]𝜈² is derived from (41) with Δ𝜈=𝜈.

However, our passive GLR test is more robust with respect to the nondetections and false alarms rates: on the least favorable model (30), the good postmodel of the faulty system has no any sense. Do not confuse the adaptation of the filter on its least favorable model (passive GLR test) and the adaptation of the most favorable model on the true system (not studied here).

5. Results

To illustrate the proposed approach we considered the following system described by the matrix()().𝐴=0.60.20000.20.10000.40.10000.5,𝐵=01100110,𝐶=100001000011,𝐹=10111101,𝑊=0.01∗Id4,4,𝑉=0.5∗Id3,3 (48)(49)(50)

The faults isolability is satisfied with rank[𝐶𝐴𝑓₁𝐶𝐴𝑓₂] and 𝐹=[𝑓₁𝑓₂].

The statistical variables describing the performances of the reconfigurable FTCS coincide with the statistical variables describing the performances of the statistical test. So, to simplify the Monte Carlo simulation, the proposed example will be realized in open loop.

In the field of dynamic systems, the signal-to-noise ratio 𝛿_𝑖(𝑘,𝑟) is generally more greater than the signal to noise ratio treated in the fields of electrocardiogram analysis or geophysical signal processing, and the size 𝑀 of the sliding window 𝑊=[𝑘−𝑀≤𝑟̃≤𝑘] can be generally chosen small. In our example, we take 𝑀=0 (we donnot optimize 𝑟̃ at all, and we fix the following rate of false alarms: 𝑃^𝐹=0.005 from a table of Chi-Squared distribution).

First, we suppose one that fault occurred at 350 with magnitude 2; we obtain the following results (Figures 1–20).

Figure 1 

First-state component and its estimation given by Willsky algorithm.

Figure 2 

Second-state component and its estimation given by Willsky algorithm.

Figure 3 

Third-state component and its estimation given by Willsky algorithm.

Figure 4 

Fourth-state component and its estimation given by Willsky algorithm.

Figure 5 

GLR test applied on willsky algorithm.

Figure 6 

First-state component and its estimation given by our adaptive filter algorithm.

Figure 7 

Second-state component and its estimation given by our adaptive filter algorithm.

Figure 8 

Third-state component and its estimation given by our adaptive filter algorithm.

Figure 9 

Fourth-state component and its estimation given by our adaptive filter algorithm.

Figure 10 

GLR test applied to our adaptive filter.

Figure 11 

First-state component and its estimation given by Willsky algorithm with presence of two sequential faults.

Figure 12 

Second-state component and its estimation given by Willsky algorithm with presence of two sequential faults.

Figure 13 

Third-state component and its estimation given by Willsky algorithm with presence of two sequential faults.

Figure 14 

Fourth-state component and its estimation given by Willsky algorithm with presence of two sequential faults.

Figure 15 

GLR test applied to willsky algorithm with presence of two sequential faults.

Figure 16 

First-state component and its estimation given by our adaptive algorithm with presence of two sequential faults.

Figure 17 

Second-state component and its estimation given by our adaptive algorithm with presence of two sequential faults.

Figure 18 

Third-state component and its estimation given by our adaptive algorithm with presence of two sequential faults.

Figure 19 

Fourth-state component and its estimation given by our adaptive algorithm with presence of two sequential faults.

Figure 20 

GLR test applied to adaptive algorithm with presence of two sequential faults.

In the first case, the results show that the proposed state estimation given by our filter (Figures 6–9) is more adaptive to the fault occurrence than the Willsky algorithm (Figures 1–4).

In second case, we suppose two sequential faults occurred at 350 and 400 with magnitudes 2; we obtain the following results (Figures 11–20).

In case of two sequential faults, the GLR detector applied on the passive augmented model allows to detect the first faults at 350 s and the second at 400 s (see Figure 20). The Willsky GLR detect just the first fault at 350 and cannot detect the second (see Figure 15).

5.1. Remarks and Discussion

We have also computed the rate of false alarms and the rate of good detections with 10⁵ Monte Carlo trials. We have obtained 𝑃^𝐹≃0,01, 𝑃_𝐷≃0,85 for the modified GLR test and 𝑃^𝐹≃0,0055, 𝑃^𝐷≃0,91 for the passive GLR test clearly much power. We conclude that the passive GLR test is very power when quick detections lead to bad jump estimations and thus very usefull for FTCS to maximize the rate of good decisions specially in regard to the occurrence of a big jump which may greatly affect nominal performance of the system.

We can improve this approach of detection and isolation with an active GLR based on free model which it will be augmented after each detection and isolation. The faults already detected and isolated will be considered as perturbation, and the we will update the new GLR in order to detect another fault.

The sequential multiple decision theory is not complete, and the choice of the threshold level 𝜀 is not studied in this paper. However, some simulation results not presented in this paper show that only statistical tuning parameter 𝜀 can be fixed at the beginning of the processing (this is not the threshold level which is adaptive but the augmented Kalman filter).

6. A Reconfigurable Fault Tolerant Control System

The purpose of the chapter is to show how the passive GLR test can be used in an FTCS. Our FTCS is only designed to reach the unique goal 𝐸(𝑦_𝑘)=0_𝑘→∞under𝑟<∞, (51) in order words to asymptotically reject the effect of jumps on the output of the system (the reference input will be maintained equal to zero avoiding the need of a reconfigurable feedforward control law). The proposed FTCS is based on the passive GLR test integrated via a reconfigurable control law designed on the model 𝑋_𝑘+1=𝐴𝑋_𝑘+𝐵𝑢_𝑘+Γ𝑤_𝑘,𝑦_𝑘=𝐶𝑋_𝑘+𝑣_𝑘, (52)(53) where the main problem to reach our goal is that the pair (𝐴,𝐵) has 𝑁 uncontrollable modes (under (𝐴,𝐵) controllable). The reconfigurable control law of the form 𝑢_𝑘=𝑢^𝑛_𝑘−𝐺𝜈_𝑘 will be designed in such a way that the nominal control 𝑢^𝑛_𝑘=−𝐿𝑥_𝑘 of the jump-free system (1) (obtained by a LQG approach on an infinite horizon) is reconfigured online after each detection and isolation of one impulsive jump by the additive term 𝐺𝜈_𝑘. In order to design 𝐺 in relation with the available nominal control law, we assume that the implementation of the passive GLR test is based on the two-stage Kalman filter, the only optimal filter which gives the state prediction of the jump-free system 𝑥_𝑘. So, let 𝑢_𝑘=𝑢^𝑛_𝑘−𝐺𝜈_𝑘 (54)be the control law that we wish to design for a physical rejection of jumps 𝜈_𝑘. Under the state transformation𝑥_𝑘𝜈_𝑘=𝐼𝑇𝑥0𝐼𝜈^𝑘_𝑘, (55) the system (49) with (51) can be expressed as𝑥_𝑘+1𝜈_𝑘+1=𝐴(𝐼−𝐴)𝑇+𝐹0𝐼𝑥_𝑘𝜈_𝑘+𝐵0𝑢^𝑛_𝑘−𝐺𝜈_𝑘,𝑦_𝑘=𝐶−𝐶𝑇𝑥_𝑘𝜈_𝑘, (56) and the physical rejection of jumps will be obtained if and only if 𝑇 and 𝐺 satisfy the algebraic equations(𝐼−𝐴)𝑇+𝐹=−𝐵𝐺,𝐶𝑇=0. (57) Under the existence condition of a solution of (55), (56) given byrang𝐴−𝐼𝐵−𝐹𝐶00=rang𝐼−𝐴𝐵𝐶0 (58) the gain 𝐺 of the control law (51), solution of (55) gives𝐺₁=𝐶(𝐼−𝐴)⁻¹𝐵⁻¹𝐶(𝐼−𝐴)⁻¹𝐹, (59) and 𝑇=(𝐼−𝐴)⁻¹(𝐵𝐺−𝐹) is not used. Under (55), (53) gives𝑥_𝑘+1𝜈_𝑘+1=𝐴00𝐼𝑥_𝑘𝜈_𝑘+𝐵0𝑢^𝑛_𝑘𝐼0++𝑤_𝑘,𝑦_𝑘=𝐶0𝑥_𝑘𝜈_𝑘+𝑣_𝑘, (60) where the jump-free state 𝑥_𝑘 of the system is decoupled to the jump's states 𝜈_𝑘 and where 𝑢^𝑛_𝑘=−𝐿𝑥_𝑘 can then be considered as the nominal control law assumed to be designed on the jump-free system (1). From the well-known separation principle, the reconfigurable control law, which reject the 𝑁 jump's uncontrollable modes, is then derived𝑢_𝑘=−𝐿𝑥_𝑘−𝐺𝜈_𝑘, (61) where 𝑥_𝑘 is the state prediction of the jump-free system and 𝜈_𝑘 the bias magnitude prediction given by the two-stage Kalman filter of Friedland [22] which optimally implement (34) (see Figure 21), (see the Appendix).

Figure 21 

The reconfigurable FTCS scheme based on the passive GLR test.

The last scheme resumes the control strategy to compensated automatically fault effects. The reference model used for the design of the control law GLR test coincides with the reference model used by the GLR Test. After each detection and isolation of one jump, the reference model is updated with the new state of jump and the three parts of the FTCS, that is, the GLR detector, the Kalman filter, and the control law.

7. Conclusion

Derived from the works of Willsky and Jones [1], this paper has presented the passive GLR test for sequential jumps detection in stochastic discrete-time linear systems. Based on a new adaptive augmented state filtering designed on the least favorable faulty model of the system, the passive GLR test gives a good results on detection and isolation of multiple faults, and it can be integrated very easily in a fault tolerant control System.

Appendix

The predictive form of the Freidland's two-stage Kalman filter which optimally implement the augmented state Kalman filter (34) is given by𝑥_𝑘+1=𝑥_𝑘+1+𝜁_𝑘+1𝜈_𝑘+1,𝑃_𝑘+1=𝑃_𝑘+1+𝜁_𝑘+1𝑃^𝜈_𝑘+1𝜁^𝑇_𝑘+1, (A.1) where 𝑥_𝑘+1 and 𝑃_𝑘+1 are given by the bias-free filter (1) and where 𝜈_𝑘 and 𝑃^𝜈_𝑘 are given by the bias filter𝜈_𝑘+1=𝜈_𝑘+𝐾^𝜈_𝑘𝛾_𝑘−𝜚_𝑘𝜈_𝑘,𝐾^𝜈_𝑘=𝑃^𝜈_𝑘𝜚^𝑇_𝑘𝐻_𝑘+𝜚_𝑘𝑃^𝜈_𝑘𝜚^𝑇_𝑘⁻¹,𝑃^𝜈_𝑘+1=𝑃^𝜈_𝑘−𝑃^𝜈_𝑘𝜚^𝑇_𝑘𝐻_𝑘+𝜚_𝑘𝑃^𝜈_𝑘𝜚^𝑇_𝑘⁻¹𝑃^𝜈_𝑘𝜚_𝑘. (A.2) From the coupling equation𝜁_𝑘+1=𝐴−𝐾_𝑘𝐶𝜁_𝑘𝜚+𝐹,_𝑘=𝜁_𝑘, (A.3) the initial conditions of the two-stage Kalman filter must be fixed as follows:𝑥₀𝜈₀=𝐼−𝜁₀01𝑥₀𝜈₀,𝑃₀00𝑃^𝜈₀=𝑃^𝑥₀𝑃₀^𝑥𝜈𝑃₀^𝜈𝑥𝑃^𝜈₀𝐼−𝜁₀01^𝑇 (A.4) with 𝜁₀=𝑃₀^𝑥𝜈(𝑃^𝜈₀)⁻¹.