MULTIOBJECTIVE DESIGN OF FAULT DETECTION FILTERS S. X. Ding∗ , P. Zhang∗ , E. L. Ding∗∗ , P. M. Frank∗ and M. Sader‡ ∗Institute for Automatic Control and Complex systems University of Duisburg-Essen, Bismarckstrasse 81 BB, 47048 Duisburg, Germany Email:
[email protected] ∗∗Dept. of Physical Engineering, University of Applied Sciences Gelsenkirchen 45877 Gelsenkirchen, Germany ‡Department of IEM, University of Applied Sciences Lausitz, Germany Keywords: Fault detection; Observer based approach; Multiobjective optimization; Robustness.
A2.
A − jωI C
A3. sup d Abstract In this paper, problems related to the design of fault detection filters (FDF) for dynamic systems with unknown inputs are studied. Core of our study is to consider the design problems under two different residual evaluation functions, the H2 -norm and the peak amplitude of the residual signals. The background of this study is the fault detection strategy of using multi-residual evaluation functions, which is widely adopted in practice. Based on the above-mentioned two residual evaluation functions, different design schemes for FDF are formulated as multiobjective optimization problems, which are then solved using the well-established LMI-technique.
1
Introduction and background
2
Ed Fd
has full row rank for all ω
≤ δ d,2 , sup d(t) ≤ δ d,peak , where t≥0
d
2
∞
=
1/2
d (t)d(t)dt
0 1/2
kd
d(t)
d2i (t)
= i=1
and δ d,2 , δ d,peak are bounds on the unknown inputs. A typical FD system consists of a residual generator and a residual evaluation stage [1], [4], [5], [6], [7], [10]. For the purpose of residual generation, we use the so-called fault detection filter (FDF) [1], [7], [10] of the form dˆ x dt r
= Aˆ x + Bu + L(y − C x ˆ − Du)
(3)
= V (y − yˆ) = V (y − C x ˆ − Du)
(4)
k In this contribution, we consider fault detection (FD) where r ∈ R r is the residual signal and L, V are the design parameter matrices. problems for linear time-invariant (LTI) dynamic systems described by Let e = x − x ˆ, then we have
x˙ = Ax + Bu + Ed d + Ef f y = Cx + Du + Fd d + Ff f
(1) (2)
¯f f ¯ +E ¯d d + E (5) e˙ = Ae (6) r = V (Ce + Fd d + Ff f ) ¯f = Ef − LFf A¯ = A − LC, E¯d = Ed − LFd , E
where u(t) ∈ Rku and y(t) ∈ Rm denote the process input and output vectors, f(t) ∈ Rkf fault vector that has to be detected and d(t) ∈ Rkd vector of unknown To evaluate residual signal r, the H2 -norm of r is often and bounded inputs, matrices A, B, C, D, Ed , Ef , Fd and used: 1/2 ∞ Ff are known and of appropriate dimensions. r (t)r(t)dt (7) r 2= 0
For our purpose, the following assumptions are made In the past, we learnt from different industrial applithroughout the paper: cations that it is very popular in practice to evaluate residual signals using multi-evaluation functions in orA1. (C, A) is detectable; der to reduce missing detection and false alarm rate. A
typical combination is the evaluation of energy change optimized FD performance [2], [6]. The main objecand peak amplitude of the residual signals. The latter tive of this paper is to approach the integrated design of FDF (3)-(4) under consideration of residual evaluacan be described by tion functions (7)-(8). To this end, we first introduce 1/2 kr an approach to the integrated design of FDF under H2 ri2 (t) (8) evaluation function and then give an algorithm for the r peak := sup r(t) , r(t) = t≥0 i=1 calculation of Jth,peak . The main attention of this paIn this contribution, we are going to study residual eval- per will be devoted to the formulation of different FDF uation schemes based on a combined use of r 2 and design problems and their solutions. r peak evaluation functions. The last step to a successful fault detection is the establishment of a decision unit. For this purpose, we first introduce two thresholds corresponding the above-defined two residual evaluation functions: Jth,2
=
sup r
2
f =0,d
Jth,peak
=
sup r
peak
2
An integrated design of FDF under H2 evaluation function
(9) In this section, we shall briefly describe the so-called unified solution that is used to design an FDF with H2 (10) norm as residual evaluation function.
f =0,d
Consider the following optimization problem: Given sysHaving introduced two different residual evaluation tem (1)-(2) and FDF (3)-(4), find L, V such that for all functions, we can form different decision logic depend- ω and σ (V (F + C(jωI − A) ¯ −1 E¯f )) = 0 i f ing on the requirements on the performance of fault de¯ −1 E ¯d )) tection systems, in order to improve the system perforV (Fd + C(sI − A) ∞ (11) J → min ,J = mance from the viewpoint of a suitable trade-off between ¯ ¯f ) σi (V (Ff + C(jωI − A)−1 E the false alarm rate and missing detection rate, for inwhere σi (·) denoting a non-zero singular value of a transstance, fer function matrix. The following theorem provides a solution to the above-defined optimization problem Logic 1 : if r 2 > Jth,2 and r peak > Jth,peak ,then whose proof is given in [2]. alarm (a fault is detected), otherwise no alarm (no fault) Theorem 1 Given system (1)-(2) and suppose AsLogic 2: if r 2 > Jth,2 or r peak > Jth,peak ,then sumptions A1-A2 hold, then alarm (a fault is detected), otherwise no alarm (no (12) L∗ = (Ed Fd + Y C )Q−1 fault) V ∗ = Q−1/2 (13) Logic 3: if w1 r 2 + w2 r peak > w1 Jth,2 + alarm (a fault is de- solve optimization problem (11), where Q = F F and w2 Jth,peak ,then d d tected),otherwise no alarm (no fault), where Y ≥ O is a solution of Riccati equation w1 > 0 and w2 > 0 are known weighting factors. A˜T Y + Y A˜ − Y C Q−1 CY + Ed REdT = 0 A˜ = A − Ed Fd Q−1 C It is evident that Logic 1 ensures a higher robustness to the unknown inputs and thus reduces the false alarm R = I − Fd Q−1 Fd rate on the one side but increase the missing detection rate on the other side, while using Logic 2 the FD system The optimal value is given by becomes more sensitive to the faults and thus has a lower 1 ∗ missing detection rate but at cost of a higher false alarm σi (V (Ff + C(jωI − A + L∗ C)−1 (Ef − L∗ Ff )) rate. By a suitable selection of weighting factors w1 and w2 , we are able to use Logic 3 to achieve a suitable tradeRemarks off between the false alarm rate and missing detection rate, as desired by the application. • The transfer function matrix Considering that a fault detection system consists of V ∗ (Fd + C(sI − A + L∗ C)−1 (Ed − L∗ Fd )) both the residual generator and the residual evaluator including the residual evaluation functions and the is a co-inner matrix [16]. Thus, Jth,2 = sup r 2 = thresholds, an integrated design of the residual generaf =0,d δ d,2 . tor and the residual evaluator is needed to achieve an
• The above solution not only gives an optimal so- iff there exists a symmetric matrix P satisfying the follution to the optimization problem defined by (11) lowing two LMI’s but also provides an optimal trade-off between the ¯d A¯T P + P A¯ P E < 0 (14) missing detection rate and the false alarm rate. ¯T P E −I d • The above solution is called unified solution, beP CT V T > 0 (15) cause the well known H∞/∞ - and H∞/ min - optiVC αI mization problems [1], [5], [6] are only two special cases of the optimization problem defined by (11). To get G r1 d g , we can solve the following optimization This result is the basis of the following study. problem in an iterative way:
Determination of Jth,peak
3
min α subject to (14) − (15) Using this solution we are able to set the threshold Jth,peak as follows: Jth,peak
In this section, an LMI algorithm will be derived for the calculation of Jth,peak defined by (10).
¯ −1 (E ¯d d + E ¯f f ) = V C(sI − A) = V Fd d + V Ff f
=
r
peak
= sup r(t) ≤ sup r1 (t) + sup r2 (t) t≥0
t≥0
t≥0
= sup r1 (t) + sup V Fd d(t) t≥0
t≥0
= sup r1 (t) + σ max (F¯d )δ d,peak t≥0
4 4.1
It follows from (8) that for f = 0
sup r
peak
f =0,d
For our purpose, we first decompose system (5)-(6) into two parts: r1 r2
=
Gr1 d
g
δ d,2 + σmax (F¯d )δ d,peak
FDF design Basic idea and problem formulation
The basic idea of the below study can be described as follows. Considering that the unified solution presented in Section 2 leads to an optimal trade-off between the false alarm rate and missing detection rate if H2 -norm is used as residual evaluation function, we shall start from this solution and modify it (slightly) such that the requirements on the residual evaluation function r peak could also be satisfied but without strongly affecting the system performance achieved by using the unified solution.
Considering that r1 describes the dynamic part of the FDF, it is thus reasonable to evaluate the influence of the unknown inputs at a certain energy level instead of evaluating the influence of the peak amplitude of d on r1 . Under this consideration, we now introduce the socalled generalized H2 -norm for our purpose: For a given Suppose that L∗ , V ∗ are the solution of the optimal FDF system y(s) = Gw (s)w(s) with xw denoting the state with H2 -norm as residual evaluation function, as given vector, the generalized H2 -norm is defined by [14] in Theorem 1. Let L = L∗ + ∆L. Then we have ˜f f (16) e˙ = (A¯ − ∆LC)e + E˜d d + E y(T ) : xw (0) = 0, T ≥ 0 T ¯ ¯ ¯ Gw g := sup (17) r = Ce + Fd d + Ff f w(t) 2 dt ≤ 1 ∗ ∗ ∗ ¯ ¯ ¯ A = A − L C, C = V C, Fd = V Fd 0 ¯ ¯d − ∆LFd , E ˜f = E ¯f − ∆LFf Ff = V ∗ Ff , E˜d = E According to this definition, we have ∗ ∗ ¯f = Ef − L Ff E¯d = Ed − L Fd , E sup r1 (t) = Gr1 d g δ d,2 t≥0 Note that ¯ −1 E ¯d Gr1 d (s) = V C(sI − A) ¯ + ∆LC −1 E ˜d + F¯d C¯ sI − A −1 In order to calculate Gr1 d g , we now introduce the fol¯ sI − A˜ ¯ Gr1 d (s) + F¯d = I −C ∆L lowing theorem [14] which provides us with an effective algorithm for the calculation of Gr1 d g and thus ¯ − A¯ + ∆LC)−1 E ˜f + F¯f CsI Jth,peak . −1 ¯ sI − A˜ ¯ Gr f (s) + F¯f = I −C ∆L 1 Theorem 2 Given system Gr1 d (s) and a constant α (> ¯ ∆L ¯ = ∆LV ∗−1 A˜ = A¯ − ∆LC, 0), then 2 ¯ −1 E¯f Gr1 f (s) = V ∗ C(sI − A) Gr1 d g < α
It leads to
r
¯ sI − A˜ I −C
−1
¯ r∗ (s) ∆L
r
=
∗
= F¯d d + F¯f f + C¯ sI − A¯
−1
(18)
• Design problem 2: For a given constant α2 > 0, ¯ such that the residual generator (16)-(17) find ∆L is stable and C¯ sI − A˜
¯d d + E ¯f f E
where r∗ is the residual signal delivered by the optimal FDF with H2 -norm as residual evaluation function. Remember that the objective of FDF design consists in the selection of the residual generator parameters that results in a suitable compromise between the evaluation functions (7) and (8). This may be well approached if we design the residual generator so that the generated residual signals r(s) is not strongly different from r∗ (s) on the one side and the threshold established based on the evaluation function (8), Jth,peak , is smaller than a certain value on the other side. Following this idea, we formulate the following design problems:
C¯ sI − A˜
−1
−1
¯ ∆L ∞
→ min subject to 2
¯d − ∆ L ¯ F¯d ) (E
g
≤ α2
• Design problem 3: For a given constant 0 < α1 0, w2 > 0, r − r∗ 2 ¯ such that the residual generator (16)-(17) find ∆L ≤ α1 (19) ∗ r 2 is stable and Jth,peak ≤ α ˆ2 (20) w1 Jth,2 + w2 Jth,peak → min Note that r − r∗ r∗ 2
Note that 2
−1
≤ α1 ⇔ C¯ sI − A˜ Gr1 d
g
¯ ∆L
Jth,2 = sup r
≤ α1
∞
Jth,peak ≤ α ˆ2 ⇒ δ d,2 + σ max (F¯d )δ d,peak ≤ α ˆ2
⇔ C¯ sI − A˜
−1
=
g 1/2 α2
C¯ sI − A˜ C¯ sI − A˜
−1
¯ ∆L ∞
≤ α1
¯d − ∆ L ¯ F¯d ) (E
δ d,2 ∞ −1
¯d − ∆L ¯ F¯d ) (E
g
w1 Jth,2 + w2 Jth,peak → min ⇒ w1 I − C¯ sI − A˜ ¯ sI − A˜ w2 C
2 g
¯ ∆L
Thus,
(21) ≤ α2
−1
δ d,2 + σmax (F¯d )δ d,peak
Thus, this design problem can be re-formulated as find¯ such that ing ∆L −1
I − C¯ sI − A˜
Jth,peak = C¯ sI − A˜
¯d − ∆ L ¯ F¯d ) (E
≤ α ˆ 2 − σ max (F¯d )δ d,peak /δ d,2 =
2
f =0,d
(22)
−1
−1
¯ ∆L
+
(23)
∞
¯d − ∆L ¯ F¯d ) (E
g
→ min
It is evident that reducing the threshold under fault detection Logic 3 will increase the system sensitivity to the It is evident that (19) describes the requirement on faults and thus reduce the missing detection rate. the residual generator under consideration of H2 -norm evaluation function, while (20) indicates that threshold Jth,peak should be limited to a desired value.
4.2
Solutions
Depending on the requirements in applications, Design problem 1 can also be modified and re-formulated into In this sub-section, solutions will be derived for the design problems formulated above. For this purpose, the the following two problems.
results of Theorem 2 and the following well-known rela- Solution of design problem 3 is given by solving the following optimization problem: finding matrices P and tionship are needed. Y such that Given system Gw (s) = Dw + Cw (sI − Aw )−1 Bw , then Gw is stable and satisfies Gw ∞ < α(> 0) iff there min α2 subject to exists a symmetric matrix P with T Q Y C¯ T T Aw P + P Aw P Bw Cw Y T −α1 I O 0 (24) Bw P −αI Dw ¯ C O −α1 I Cw Dw −αI ¯d − Y F¯d Q PE 0 ¯ Following Theorem 2 and relation (24), Design problem C α2 I ¯ 1 described by (21)-(22) can be re-formulated as find ∆L such that there exist symmetric matrices P1 , P2 with Similar to the solutions given above, the solution of the T ¯ ¯T following optimization problem provides us with a soluA˜ P1 + P1 A˜ P1 ∆L C ¯ T P1 tion to the Design problem 4 described by (23): Finding ∆L −α1 I O < 0, ¯ matrices P and Y such that C O −α1 I T ˜ ˜ ˜ A P2 + P2 A P2 Ed 0, P1 > 0 ¯ α2 I Y T −α1 I C −I < 0 ¯ C −I −α1 I Set ¯d − Y F¯d Q PE ¯ = P −1 Y, P1 = P2 = P 0 ¯T C¯ α2 I Q Y C Y T −α1 I (26) O 0 (28) ¯ α2 I C In this contribution, we have studied the problems of Q = A¯T P − C¯ T Y T + P A¯ − Y C¯ designing FDF under two different residual evaluation
As a result, we claim that Design problem 1 can be functions. The background of this study is the observasolved if there exist matrices P and Y solving LMI’s tion that in practice the fault detection strategy of using multi-residual evaluation functions is widely used. The ¯ is given by (25). (26)-(28). The solution for ∆L two residual evaluation functions considered in this paSolutions for Design problems 2 and 3 follow directly per are the H2 -norm, which evaluates the energy level of the residual signal, and the peak amplitude of the residfrom the above discussion. ual signal. Based on the two residual evaluation funcSolution of design problem 2 is given by solving the tions mentioned above, different design schemes for FDF following optimization problem: finding matrices P and are formulated as multiobjective optimization problems, Y such that which are then solved using the LMI-technique. min α1 subject to ¯T Q Y C Y T −α1 I O 0 ¯ α2 I C
Different academic examples of designing FDF have been successfully solved using the methods presented in this paper. As next step, these methods will be used in different laboratory systems.
