Int. J. Automation and Control, Vol. 8, No. 1, 2014
1
Data-driven fault detection filter design for time-delay systems Magdi S. Mahmoud* and Haris M. Khalid Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 5067, Dhahran 31261, Saudi Arabia E-mail:
[email protected] E-mail:
[email protected] *Corresponding author Abstract: In this paper, robust fault detection (FD) problems for time-delay LTI systems with unknown inputs are studied. This paper is proposed to evaluate the robustness as well as sensitivity of residual signals to the unknown inputs as well as to the faults in terms of L2. First, the weighting matrix is selected for an appropriate design of filter, then fault detection filter design with Lyapunov-Krasovskii function (LKF) is designed with time delay. The main results include the detailed derivation of these steps followed by its implementation on an open-loop time-delay system for chemical reactor example. Keywords: fault detection; filter; Lyapunov-Krasovskii function; LKF; time-delay; quadruple tank system. Reference to this paper should be made as follows: Mahmoud, M.S. and Khalid, H.M. (2014) ‘Data-driven fault detection filter design for time-delay systems’, Int. J. Automation and Control, Vol. 8, No. 1, pp.1–16. Biographical notes: Magdi S. Mahmoud has been a Professor of Engineering since 1984. He is now a Distinguished University Professor at KFUPM, Saudi Arabia. He is the principal author of thirty (30) books, inclusive of book chapters, and the author/co-author of more than 500 peer-reviewed papers. He is the recipient of two national, one regional and four university prizes for outstanding research in engineering. He is a Fellow of the IEE, a senior member of the IEEE, the CEI (UK), and a registered Consultant Engineer of Information Engineering and Systems (Egypt). He is currently actively engaged in teaching and research in distributed control and filtering, networked-control systems, triggering mechanisms in dynamic systems, fault-tolerant systems and information technology. Haris M. Khalid obtained his MSc and PhD degrees from KFUPM, Saudi Arabia in 2010 and 2013, respectively. His is now affiliated with Saudi-Aramco on fault-tolerant project.
1
Introduction
Data-driven-based fault detection (FD) has always an area of critical interest in mission critical systems and process control industries. In such system structures, the whole Copyright © 2014 Inderscience Enterprises Ltd.
2
M.S. Mahmoud and H.M. Khalid
model or dynamics are not usually available. Due to which, the cost of system reliability goes at risk. Therefore, data is generated from the assumed model of the system, and then compared with measured data from the physical system to create residuals that relate to specific faults. For example, in fault prognosis, models based on data-driven structures are derived from statistical learning techniques, most of which originated from theories of pattern recognition. Data-driven models are usually developed by collecting the input/output data, and can process a wide variety of data types and exploit the similarities in the data that cannot be discovered by physical models. Based on this, Orchard and Vachtsevanos (2009) presented an online particle-filtering (PF)-based frame work for failure prognosis in non-linear, non-Gaussian systems. A non-linear state-space model of the plant and a PF algorithm is used to estimate the probability density function (PDF) of the state in real-time. The state PDF estimate is then used to predict the evolution in time of the fault indicator, obtaining as a result the PDF of the remaining useful life (RUL) for the faulty subsystem. HMM and HSMM belong to data-driven models. Baruah and Chinnam (2003) first pointed out that a standard HMM could be applied in the area of prognosis in machining processes. An integrated fault diagnostic and prognostic approach for bearing health monitoring and CBM was introduced in Zhang et al. (2005). The proposed scheme consists of principle component analysis, HMM, and an adaptive stochastic fault prediction model. Camci (2005) proposed an integrated diagnostics and prognostic architecture that employed support vector machine and HMM. But HMM shave some inherent limitations. One is the assumption that successive system behaviour observations are independent. The other is the Markov assumption itself that the probability in a given state at time t only depends on the state at time t1, which is sometimes untenable in practical applications. In order to cope with the inaccurate duration modelling of HMMs, some researchers have proposed hidden semi-Markov models to model explicitly the state duration (Gu et al., 1991; Falaschi, 1992). The research and application of robust FD in automated processes have received considerable attention during last decades and a great number of results have been achieved (Chen and Patton, 1999; Yoshimura et al., 1997; Zhong et al., 2003; Ding et al., 2001b). In the past three decades, many significant results concerning fault detection and isolation (FDI) problems have been developed, see e.g., Frank et al. (2000), Mangoubi and Edelmayer (2000), and Patton et al. (2000) and references therein. However, most of the achievements are for delay-free systems. On the other hand, time delays are frequently encountered in industry and are often the source of performance degradation of a system (Ding et al., 2001b). So, this paper focuses on the fault detection filter (FDF) design for time-delay LTI systems with unknown inputs. Although, time delay is an inherent characteristic of many physical systems, such as rolling mills, chemical processes, water resources, biological, economic and traffic control systems, only few researches on FDI have been carried out for them (Ding et al., 2001a; Jean-Yves and Woihida, 2000; Jiang et al., 2002a, 2002b). Jiang et al. (2002a) deal with the nominal case fault identification (without considering the influence of model uncertainty and unknown inputs), Frank et al. (2000) formulate the FDF design problem as a two-objective non-linear programming problem where no analytic solution can be constructed in general. Jiang et al. (2002b) extend the results to the discrete-time case. The authors of earlier study in Ding et al. (2001a) have also developed an FDF design approach based on H∞– filtering, but the most important and difficult issue concerning the selection of a so-called reference residual model has not been successfully
Data-driven fault detection filter design for time-delay systems
3
solved. An efficient way to tackle the FD problem for time-delay systems is as yet to be developed. This paper is organised as follows: problem formulation and the proposed scheme formulation is presented in Section 2 followed by the main results in Section 3, simulation results are followed by in Section 4. Finally, some concluding remarks are given in Section 5. Figure 1 shows the proposed implementation plan. Figure 1
2
FDF with time-delay
Problem formulation
We now concentrate our attention on the FD problems for time-delay LTI systems. The system model under consideration is given by: N
x& = Ax(t ) +
∑ A x (t − d i
xi (t )
)
i =1
(1)
L
+ Bu +
∑ B u (t − d i
ui
) + B f f + Bd d
i =1
y = Cx + Du + D f f + Dd d
(2)
where x(t) ∈ ℜn, u(t) ∈ ℜm, y(t) ∈ ℜp, are the state, input and output vectors, respectively, and Ai ∈ ℜn×n (i = 1, 2, 3, …, N), Bi ∈ ℜn×m (i = 1, 2, 3, …, L) C ∈ ℜp×n, Bf ∈ ℜn×q, Bd ∈ ℜn×s, Df ∈ ℜp×q and Dd ∈ ℜp×s are known matrices with appropriate dimensions. In addition, f ∈ ℜq and d ∈ ℜs are faults and disturbance vectors, respectively. Assume that the time-varying delays satisfy: d xi (t ) ≤ d x < ∞, dul (t ) ≤ du < ∞ d&xi (t ) ≤ mxi < 1, d&ul (t ) ≤ mul < 1
We propose to use the following FDF for the purpose of residual generation:
(3)
4
M.S. Mahmoud and H.M. Khalid x&ˆ = Axˆ (t ) +
N
∑ A xˆ ( t − d i
xi (t )
)
i =1
(4)
L
+ Bu +
∑ B u (t − d i
ui
) + H ( y − yˆ )
i =1
r = Ce + D f f + Dd d
(5)
Denoting e = x − xˆ. The dynamics of the FDF can then be expressed by: N
e& = ( A − HC )e +
∑ A e (t − d (t ) ) i
i
i =1
(6)
+ ( B f − HD f ) f + ( Bd − HDd ) d r = Ce + D f f + Dd d
(7)
Proceeding further, we introduce the following system: e& f = ( A − HC ) e f + + ( B f − HD f ) f
N
∑ Ae
i f
( t − di (t ) )
i =1
rf = Ce f + D f f
(8)
(9)
which describes the influence of the faults on the residual signals. We now formulate the problem of designing a FDF for time-delay LTI system (1–2), by similarity to the formulation given in the last section.
2.1 Selection of weighting matrix Given a performance index which describes the sensitivity of the residual signals to the faults, find an observer gain matrix, denoted by Hf, such that rf, is optimal in the sense of the given performance index. The weighting matrix is then set as: N
e& f = ( A − HC )e f + + ( B f − HD f ) f rf = Ce f + D f f
∑Ae
i f
i =1
( t − di (t ) )
(10)
(11)
2.2 Design of FDF The design problem is phrased as follows: Given constants β (> 0) as well as γ (≥ ymin) and a weighting matrix Wf(s), find an observer gain matrix H such that system (6) and (7) is stable and (3) holds.
Data-driven fault detection filter design for time-delay systems
5
2.3 Design of robust fault detection filter The design problem is phrased as follows: Given a constant γ (≥ ymin) and a weighting matrix Wf(s), find an observer gain matrix H and matrix V such that β is minimised under conditions that system (22) and (23) is stable as well as (16) holds, i.e. min H ,V β
(12)
such that: •
Observer gain:
∫
∞
0
•
reT re dt ≤ β 2
∫
∞
0
f T f dt + γ 2
∫
∞
0
d T d dt
(13)
Rate of change of error: N
e& = ( A − HC )e +
∑ A e (t − d (t ) ) i
i
i =1
(14)
+ ( B f − HD f ) f + ( Bd − HDd ) d
•
Residual: r = Ce + D f f + Dd d
(15)
We call system (6)–(7) robust fault detection filter (RFDF) for time-delay LTI system (1–2) if the observer gain matrix solves the above-defined optimisation problem.
3
Main results
In the following of this work, an LMI approach is developed to solve the above-defined RFDF design problem. To this end, following two sub-problems will be solved: •
evaluation of the influence of the faults and selection of weighting matrix and
•
solutions of the RFDF problem.
The following theorem establishes the main result.
3.1 Theorem Given > 0 μ > 0. System (6) and (7) with u(.) = 0 is delay-dependent asymptotically stable with L2-performance bound γ if there exist symmetric matrices 0 < P, 0 < Wa, 0 < Wc, 0 < Q, 0 < R, weighting matrices Na, Nc, Ns, Ma, Mc, Ms and a scalar γ > 0 satisfying the following LMI:
6
M.S. Mahmoud and H.M. Khalid ⎡ ϒ 01 ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎢⎣ ∗
ϒ 02
ϒ 03
Ma
ϒ 04
ϒ 05
ϒ Mc
Nc
∗
ϒ 06
Ms
Ns
∗
∗
− Wa
0
∗
∗
∗
− Wc
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
PΓ 0
GoT
0 0
T Gdo
0 0 −γ 2 I ∗ ∗
0 0 ΦT0
Na
0
−I ∗
PA0T W ⎤ ⎥ T PAdo W⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ PΓt0W ⎥ 0 ⎥ ⎥ − W ⎦⎥
(16)
where ϒ o1 = PAo + Aot P + Q + R + N a + N at + M a + M at ϒ o 2 = PAdo − 2 N a + N ct + M ct , W = N a + Wc ϒ o 3 = N a − M a + N st + M st
(17)
ϒ o 4 = −(1 − μ)Q − 2 N c − 2 N ct ϒ o 5 = N c − M c − 2 N st ϒ o 6 = − R + N s + N st − M s − M st
Proof: In terms of ξ (t ) = [etf (t ) etf ( t − τ (t ) ) etf (t − )]t and using the classical Leibniz rule e f (t − θ ) = e f (t ) −
∫
t
t −θ
e& f ( s )ds for any matrices Na, Nc, Ns, Ma, Mc, Ms of appropriate
dimensions, the following equations hold: ⎡ 2ξ t (t )2 N ⎢ − ⎣
⎤ e& f ( s )ds + e f (t ) − e f (t − τ ) ⎥ = 0 t − τ (t ) ⎦
∫
t
⎡ 2ξ t (t )( M − N ) ⎢ − ⎣
∫
t
t−
(18)
⎤ e& f ( s )ds + e f (t ) − e f (t − ) ⎥ = 0 ⎦
(19)
Expansion of equations (18) and (19) gives: etf (t ) [ N a + N at + M a + M at ] e f (t ) + 2etf (t ) [ −2 N a + M ct + N ct ] e f ( t − τ (t ) ) +2etf (t ) [ N a + M a + N st + M st ] e f (t − ) + 2etf ( t − τ (t ) ) [ −2 N c −2 N ct ] e f ( t − τ (t ) ) 2etf ( t − τ (t ) ) [ N c − 2 N st − M c ] e f (t − ) +2e f (t − ) [ N s + N st − M s − M st ] e f (t − ) − 2ξ t (t )2 N −2ξ t (t )( M − N )
∫
t
t − τ (t )
∫
t
t − τ (t )
(20) e& f ( s)ds
x& ( s )ds = 0
Consider now the augmented Lyapunov-Krasoviskii functional (ALKF) (Mahmoud, 2009):
Data-driven fault detection filter design for time-delay systems
7
V (t ) = Vo (t ) + Va (t ) + Vc (t ) + Vm (t ) Vo (t ) = etf Pe f (t ), Va (t ) =
0
t
−
t +s
∫ ∫ t
Vc (t ) =
∫
Vm (t ) =
∫
e&tf (α ) (Wa + Wc ) e& f (α )dα ds,
(21)
etf ( s ) Re f ( s )ds,
t− t
t − τ (t )
etf ( s )Qe f ( s )ds
where 0 < P = Pt, 0 < Wa = Wat , 0 < Wc = Wct , 0 < Q = Qt, 0 < R = Rt are the matrices of appropriate dimensions. The first term in (21) is standard to nominal system without delay. The second and fourth terms correspond to the delay-dependent conditions. The third term is introduced to compensate for the enlarged time interval from t − → t to t – τ → t. A straightforward computation gives the time-derivative of V(ef) with w(t) = 0 as: V&ot = 2etf Pe& f (t ) = 2etf (t ) P [ Ao e f (t ) + Ado e f (t − τ ) ] V&a (t ) = e&tf (Wa + Wc ) ( e& f ) (t ) −
∫
t
t−
( e& f )t ( s) (Wa + Wc ) e& f ( s)ds
(22)
Vct = etf (t ) Re f (t ) − etf (t − ) Re f (t − ) V&m (t ) = etf (t )Qe f (t ) − (1 − τ& ) etf ( t − τ (t ) ) Qe f ( t − τ (t ) ) ≤ etf (t )Qe f (t ) − (1 − μ)etf ( t − τ (t ) ) Qe f ( t − τ (t ) )
From (21)–(22) and using (19), we obtain: V& (t ) ≤ etf (t ) [ PAo + Aot + Q + R + Na + Nat + M a + M at ] e f (t ) +2etf [ PAdo − 2 Na + M ct − Nct ] etf (t − τ ) + 2etf (t ) [ Na − M a + Nst + M st ] e f (t − ) +2etf (t − τ ) [ Nc − 2 N st − M c ] e f (t − ) − etf (t − τ ) [ (1 − μ)Q + 2 Nc + 2 Nct ] e f ( t − τ (t ) ) +etf (t − ) [ − R + Ns + Nst − M s − M st ] e f (t − ) + e&tf (t ) (Wa + Wc ) e& f (t ) − −2ξ t (t )2 N
∫
t
t − τ (t )
∫
t
t−
(23)
e&tf (s) (Wa + Wc ) e& f (s)ds
e& f (s)ds − 2ξ t (t )(− N )
∫
t
t−
e& f (s)ds − 2ξ t (t )M
∫
t
t−
e& f (s)ds
Proceeding to manipulate the terms in 23, we arrive at:
∫
= −2ξ t (t )2 N = −2ξ t (t ) N
Next considering
t
t − τ (t )
∫
t
t − τ (t )
∫
e& f ( s)ds + 2ξ t (t ) N
e& f ( s)ds + 2ξ t (t ) N
t
t−
∫
e& f ( s)ds
t − τ (t )
t−
(24) e& f ( s)ds
M.S. Mahmoud and H.M. Khalid
8
∫
t
e&tf ( s ) (Wa + Wc ) e& f ( s )ds
t−
= = +
∫
t
∫
t
t−
t−
∫
e&tf ( s ) (Wa e& f ( s )ds ) + e&tf ( s ) (Wa e& f ( s )ds ) +
t − τ (t )
t−
∫
t
∫
t
t−
e&tf ( s ) (Wc ) e& f ( s )ds
t − τ (t )
e&tf ( s ) (Wc ) e& f ( s )ds
(25)
e&tf ( s )Wc e& f ( s )ds
Then V& (t ) becomes: V& (t ) ≤ ξ t ϒ o ξ (t ) −
∫
t
t−
e&tf ( s )We& f ( s )ds t
⎡ Aot ⎤ ⎡ Aot ⎤ ⎢ t ⎥ ⎢ t ⎥ t +ξ (t ) ⎢ Ado ⎥ W ⎢ Ado ⎥ ξ (t ) ⎢⎣ 0 ⎥⎦ ⎢⎣ 0 ⎥⎦ −2ξ t (t ) N
∫
t
t − τ (t )
−2ξ t (t )(− N ) −2ξ t (t ) M
∫
t
t−
∫
e& f ( s )ds
t − τ (t )
t−
(26)
e& f ( s )ds
e& f ( s )ds
Adding and subtracting the terms: ξ t (t ) [ MWa−1 M t + NWc−1 N t ] ξ
(27)
and consider the following terms: ξ t (t ) MWa−1M t ξ (t ) + ξ t (t ) NWc−1 N t ξ (t ) ξ t (t ) MWa−1M t ξ (t ) − τ (t )ξ t (t ) NWc−1 N t ξ (t ) − ( − τ (t ) ) ξ t NWc−1 N t ξ (t ) t
−2ξ t (t ) N
∫
−2ξ t (t ) M
∫
+
∫
t
t − τ (t )
t − τ (t ) t
t−
e& f ( s )ds + 2ξ t (t ) N
e& f ( s )ds +
e&tf ( s )Wc e& f ds +
∫
∫
t − τ (t )
t−
t
t−
∫
t − τ (t )
t−
e& f ( s )ds
e&tf ( s )Wc e& f ds
e&tf ( s )Wa e& f ds
After some manipulations, the terms in (28) become:
(28)
Data-driven fault detection filter design for time-delay systems
9
= ξ t (t ) [ MWa−1M t + NWc−1 N t ] ξ − − −
∫
t
∫
t − τ (t )
∫
t
t
⎡ξ t N + e&tf Wc ⎤⎦Wc−1 ⎡⎣ξ t N + e&tf Wc ⎤⎦ ds t − τ (t ) ⎣ t−
t−
(29)
t
⎡⎣ −ξ t N + e&tf Wc ⎤⎦Wc−1 ⎣⎡ −ξ t N + e& f Wc ⎦⎤ ds t
⎡⎣ξ t M + e&tf Wa ⎤⎦Wa−1 ⎣⎡ −ξ t M + e& f WA ⎦⎤ ds
Further manipulations of (26) result in V& (t ) ≤ ξ t ⎡⎣ ϒ o + MWa−1M t + τ (t ) NWc−1 N t + ( − τ (t ) ) NWc−1 N t ⎤⎦ ξ (t ) + x& (t ) (Wa + Wc ) e& f (t ) − − −
∫
t − τ (t )
∫
t
t−
t−
∫
t
t
t −1 t t ⎣⎡ξ N + e& f Wc ⎦⎤Wc ⎣⎡ξ N + e& f Wc ⎦⎤ ds t − τ (t ) t
t −1 t t ⎣⎡ −ξ N + e& f Wc ⎦⎤ Wc ⎣⎡ −ξ N + e& f Wc ⎦⎤ ds
(30)
t
⎡⎣ξ t M + e& f Wa ⎤⎦Wa−1 ⎣⎡ξ t M + e&tf Wa ⎦⎤ ds
≤ ξ t (t ) [ ϒ o + MWa−1M t + NWc−1 N t ] ξ (t ) + e&tf (Wa + Wc ) e& f (t )
In view of (16) with Go = 0, Gd = 0, Γo = 0, and Schur’s compliments, it follows from (30) that V& (t ) < 0 which establishes the internal asymptotic stability. Consider the performance measure J=
∫
∞
0
( z t ( s) z (s) − γ 2 wt (s)w(s) )ds
For any w(t) ∈ L2(0, ∞) ≠ 0 and zero initial condition x(0) = 0, we have:
∫ ≤∫
J=
∞
0 ∞
0
( z t ( s) z (s) − γ 2 wt (s) w( s) + V& ( x) )ds − V& ( x) (31)
( z ( s) z (s) − γ w (s) w( s) + V& ( x) )ds t
2
t
Proceeding further, we get: z t ( s ) z ( s ) − γ 2 wt ( s ) w( s ) + V& ( s ) = χ t ( s ) ϒχ t ( s ), χ ( s ) = ⎡⎣etf ( s ) etf ( s − τ (t ) ) etf (t − ) w( s) ⎤⎦
t
(32)
where ϒ corresponds to Γo in 16 by Schur’s Complements. It is readily seen from 16 that: z t ( s ) z ( s ) − γ 2 wt ( s ) w( s ) + V& ( s ) < 0
(33)
for arbitrary sε[t , ∞), which implies for any w(t )εL2 (0, ∞) ≠ 0 that J < 0 leading to || z (t ) ||2 < γ || w(t ) ||2 .
M.S. Mahmoud and H.M. Khalid
10
4
Application to quadruple tank system
The evaluation of the developed scheme has been made on a quadruple tank system. In the following sections, we provide the detailed implementation and simulation of the developed scheme.
4.1 System description The laboratory scale quadruple tank system has been used to collect data. The data has been collected at a sampling time of 50 ms. The different datasets have been generated for a PI controlled water level control. Different fault scenarios have also been considered for the generation of the datasets.
4.2 Experimental setup The process data has been generated through an experimental setup as shown in Figure 2. A quadruple-tank system has been used in order to collect the data with the introduction of actuator and sensor faults through the system as can be seen in the abview circuit window. An amplified voltage of 20 V has been given to eat set of tanks to handle the controller effectively for the changes/fluctuation produced in the system. So, the fault diagnosis was done here in a closed-loop setup where the controller is actually trying to suppress the faults as though they were disturbances. Figure 2
(a) The quadruple-tank system setup (b) the labview setup of the apparatus (see online version for colours)
Data-driven fault detection filter design for time-delay systems
11
4.3 Process data collection and description The process data has been collected at 50 milli-second(ms) sampling time. The main objective of the quadruple-tank system is to reach a reference height of tanks. The schematic description of the four tank system can be visualised by Figure 3. The system has two control inputs (pump throughputs) which can be manipulated to control the water level in the tanks. The two pumps are used to transfer water from a sump into four overhead tanks. By adjusting the bypass valves of the system, the proportion of the water pumped into different tanks can be changed to adjust the degree of interaction between the pump throughputs and the water levels. Thus each pump output goes to two tanks, one lower and another upper, diagonally opposite and the ratio of the split up is controlled by the position of the valve. Because of the large water distribution load, the pumps have been supplied 12 V each. The mathematical modelling of the quadruple tank process can be obtained by using Bernoulli’s law. During this process, several faults have been introduced such as the leakage faults, sensor faults, and actuator faults. The leakage faults have been introduced through the pipe clogs of the system, knobs between the tanks, etc. The sensor faults have been simulated by introducing a gain in the circuit as if there is a fault in the level sensor of the tank. Similarly, the actuator faults have been simulated by introducing a gain in the setup for the actuator that comprises of the motor and pump. A PI controller has been employed in order to reach the desired reference height. Due to the inclusion of faults, the controller was finding it difficult to reach the desired level. For this reason, the power of the motor has been increased from 10 to 20 V in order to provide it with the maximum throttle to reach the desired level. This enabled the actuator to perform well in achieving its desired level but led to the controller suppressing the faults injected into the system. So, this made the FD task rather difficult. After the data collection task was completed, techniques such as settling time, steady-state value, and coherence spectra were used to help us get an insight into the faults present in the system. Figure 3
Schematic diagram of a quadruple tank system
12
M.S. Mahmoud and H.M. Khalid
4.4 Fault model of the system Combining all the equations for the interconnected four-tank system we obtain the physical system. A fault model can then be constructed by adding extra holes to each tank. The mathematical model of the faulty quadruple tank system can be given as: dh1 a a γk d a = − 1 2 gh1 + 3 2 gh3 + 1 1 v1 + − leak1 2 gh1 dt A1 A1 A1 A1 A1 dh2 a a γk d aleak 2 = − 2 2 gh2 + 4 2 gh4 + 2 2 v2 − − 2 gh2 dt A2 A2 A2 A2 A2 (1 − γ2 ) k2 dh3 a a = − 3 2 gh3 + v2 − leak 3 2 gh3 dt A3 A3 A3
(1 − γ1 ) k1 dh4 a a = − 4 2 gh4 + v1 − leak 4 dt A4 A4 A4 dv1 v 1 = − 1 + u1 dt τ1 τ1 v 2 dv2 = − 2 + u2 τ2 τ2 dt
5
(34)
2 gh4
Simulation results
In what follows, we present simulation results for the developed FD scheme. The tasks of our LKF-based FDF scheme have been executed here with an increasing precision. Leakage faults have been considered here, that is, the leakage fault in states. Firstly, the data collected from the plant has been initialised and the parameters have been being optimised which comprises of the pre-processing and normalisation of the data. Then, the LKF-based FDF scheme is implemented detecting the correct fault profile. Following the main steps for the proposed scheme, the following are the simulation results for the system under observation as can be seen in Figures 4 to 7 and error in Figure 8. It has been shown that the estimated profiles for State 1 to State 4 which are presenting Tank 1 to Tank 4 are detecting fault, thus pointing clearly to a FDF design.
6
Conclusions
In this paper, RFDF with time-delay has been proposed using Lyapunov-Krasovskii function (LKF). Using an LMI method, the existence conditions and further solution for the optimisation problem have been derived and, based on them, an algorithm for the design of the FDFs has been proposed. The proposed scheme has been evaluated on an open-loop time-delay system for chemical reactor simulated system thus ensuring the effectiveness of the approach.
Data-driven fault detection filter design for time-delay systems Figure 4
FDF using LKF: State1 (see online version for colours)
Figure 5
FDF using LKF: State2 (see online version for colours)
13
14
M.S. Mahmoud and H.M. Khalid
Figure 6
FDF using LKF: State3 (see online version for colours)
Figure 7
FDF using LKF: State4 (see online version for colours)
Data-driven fault detection filter design for time-delay systems Figure 8
15
FDF using LKF: error (see online version for colours)
Acknowledgements The authors would like to thank the deanship for scientific research (DSR) at KFUPM for research support through project no. RG1316-1.
References Baruah, P. and Chinnam, R.B. (2003) ‘HMMs for diagnostics and prognostics in machining processes’, Proceeding of the 57th Society for Machine Failure Prevention Technology Conference, Virginia Beach, VA, April, pp.14–18. Camci, F. (2005) Process Monitoring, Diagnostics and Prognostics using Support Vector Machines and Hidden Markov Models, PhD Dissertation, Graduate school of Wayne State University, Detroit, Michigan. Chen, J. and Patton, P.R. (1999) Robust Model-based Fault Diagnosis for Dynamic Systems, Kluwer Academic Publishers, London. Ding, S.X., Zhong, M.Y. and Tang, B.Y. (2001a) ‘An LMI approach to the design of fault detection filter for time-delay LTI systems with unknown inputs’, in Proceedings of American Control Conference, Arlington, VA, pp.2137–2142. Ding, X.S., Zhong, M.Y., Tang, B.Y. and Zhang, P. (2001b) ‘An LMI approach to the design of fault detection filter for time-delay LTI systems with unknown inputs’, Proceedings of the American Control Conference Arlington, IEEE, USA, pp.2137–2142. Falaschi, A. (1992) ‘Continuously variable transition probability HMM for speech recognition’, in Laface, P. and DeMori, R. (Eds.): Speech Recognition and Understanding, pp.125–130, Springer-Verlag, Heidelberg.
16
M.S. Mahmoud and H.M. Khalid
Frank, P.M., Ding, S.X. and Koppen-Seliger, B. (2000) ‘Current developments in the theory of FDI’, in Proc. SAFEPROCESS2000, Budapest, Hungary, pp.16–27. Gu, H.Y., Tseng, C.Y. and Lee, L.S. (1991) ‘Isolated-utterance speech recognition using hidden Markov models with bounded state duration’, IEEE Trans. Signal Processing, Vol. 39, No. 8, pp.1743–1751. Jean-Yves, K. and Woihida, A. (2000) ‘Robust fault isolation observer for discrete-times systems with time delay’, in Proc. SAFEPROCESS 2000, Budapest, Hungary, June, pp.390–395. Jiang, B., Staroswiecki, M. and Cocquempot, V. (2002a) ‘Fault identification for a class of time-delay systems’, in Proceedings of American Control Conference, Anchorage, USA, May, pp.8–10. Jiang, B., Staroswiecki, M. and Cocquempot, V. (2002b) ‘H∞ fault detection filter for a class of discrete-time systems with multiple time delays’, in Proceedings 15th IFAC World Congress, Barcelona, Spain, CDROM, July. Mahmoud, M.S. (2009) ‘Improved robust stability criteria for delay systems’, Proc. IME Part-I-Journal of Systems and Control Engineering, June, Vol. 233, No. 14, pp.745–752. Mangoubi, R.S. and Edelmayer, A.M. (2000) ‘Model based fault detection: the optimal past, the robust present and a few thoughts on the future’, in Proc. SAFEPROCESS2000, Budapest, Hungary, pp.64–75. Orchard, M.E. and Vachtsevanos, G.J. (2009) ‘A particle-filtering approach for on-line fault diagnosis and failure prognosis’, Trans. the Institute of Measurement and Control, Vol. 31, No. 4, pp.221–246. Patton, R.J., Frank, P.M. and Clark, R.N. (2000) Issues of Fault Diagnosis for Dynamic Systems, Springer-Verlag Ltd., London. Yoshimura, M., Frank, P.M. and Ding, X. (1997) ‘Survey of robust residual generation and evaluation methods in observer-based fault detection systems’, Journal of Process Control, Vol. 7, No. 6, pp.403–424. Zhang, X.D., Xu, R., Chiman, K., Liang, S.Y., Xie, Q.L. and Haynes, L. (2005) ‘An integrated approach to bearing fault diagnostics and prognostics’, Proc. the 2005 American Control Conference, Jun 810, Portland, OR, Vol. 4, pp.2750–2755. Zhong, M.Y., Ding, X.S., Lam, J. and Wang, H.B. (2003) ‘An LMI approach to design robust fault detection filter for uncertain LTI systems’, Automatica, Vol. 39, No. 3, pp.543–550.
