- Email: [email protected]
Design of Robust FDI Systems Using the Generalized Structured Singular Value
Copyright @ IFAC Fault Detection, Supervision and Safety for Technical Processes, Budapest, Hungary. 2000
DESIGN OF ROBUST FDI SYSTEMS USING THE GENERALIZED STRUCTURED SINGULAR VALUE. D. Henry A. Zolghadri M. Monsion
Laboratoire d'Automatique et de Productique Universite Bordeaux I 351 cours de la Liberation 33405 Talence cedex, France. Email: [email protected] Tel: (33)556842400 - Fax: (33)556846644
Abstract: The purpose of this paper is to apply the concept of frequency-domain model invalidation to integrity monitoring of multi variable uncertain systems. The approach is based on the concept of the generalized structured singular value. New formulations of robustness objectives in fault diagnosis are explored and experimental results applied to a hydraulic process demonstrate the potential of the proposed method. Copyright @20001FAC Keywords: Failure detection, failure isolation, robustness, model invalidation, generalized structured singular value.
a large class of uncertain systems, i.e. parametric uncertainties, negliged dynamics and small nonlinearities can be considered. The purpose of this paper is the design of a robust FDI scheme based on the frequency-domain model invalidation technique. The concept of the generalized structured singular value /1-9 is used here to robustly detect and isolate faults. It should be pointed out that, due to its implementation in the frequency domain, the tool provided by the proposed approach is not intended to serve as an on-line FDI unit, but as an off-line integrity monitoring system. This paper is organized as follows: In section 2, the review of /1-9 is considered. The theorical backgrounds for model invalidation in the frequency domain are presented. Section 3 is dedicated to the application of /1-9 framework to integrity monitoring of multivariable uncertain systems. Finally, experimental results applied to a hydraulic process demonstrate the potential of the proposed approach.
1. INTRODUCTION.
Different approaches to fault detection and isolation (FDI) using mathematical models have been developed over last two decades (see (Patton, 1997) for a survey). The main objective of the FDI design procedure is the achievement of a low missed-alarm and a low false-alarm rates. So robustness to all unknown inputs (modeling errors and disturbances) and sensitivity to faults are the main challenge when designing a robust FDI system. On the other hand, in recent years there has been a renewed interest for frequency-domain model invalidation techniques in robust control design (Newlin and Smith, 1998). The model invalidation problem is concerned with determining whether a mathematical model is consistent (or covers) a collection of experimental data. The Linear Fractional Transformation (LFT) is also used to describe the mathematical model. It consists of a nominal plant model and uncertainties described by bounded perturbations acting on the nominal model. This formalism has the advantage to account for 635
Notations In dealing with vectors, the Euclidean norm is always used and is written without a subscript; for example Ilxll . Similarly in the matrix case, we used the induced vector norm: 110411 = 0'(04) where 0'(.4) denotes the maximum singular value of o4. Signals, w(t) or w , are assumed to be of bounded energy (Lebesgue square integral, denoted by L2), and their norm is denoted by Ilw112. We will assume that all systems are real rational operators with bounded induced L2 norm . Uncertain systems are represented by the linear fractionnal representation of the matrix .6. posed on P , which is referred to as the star product
Fig. 1. The generic structure of model invalidation problem on the plant P. The unknown inputs are supposed to be of bounded energy. The measured output is represented by y and is also known. The signals;; and v are internal to the model. It is also assumed that the model is robustly stable, i.e. sUPw J.L~(Pl1 (jw)) < Ih. According to figure 1, the robust model is defined by
(.6. * P) Alternative, equivalent notations found in the literature are Fu(P,.6.) and S(.6., P). .6. is modeled, without loss of generality, by a prescribed block diagonal structure and represents multiple perturbations . .6. is used to describe the set of all perturbations of a prescribed structure as
y=Wn n+(.6.*P)
(1)
with
(.6. * P)
.6. = {block diag(bT h" ... , O~Jk~r' of h O~,Jkmdm , ,.6.f, ... , .6.·~c)' or E R,of E c,.6.f E C} where o[h"i = 1, .. . ,mr, O'jh~dj , j = 1, .. . ,mc and .6.f, I = 1, ..., me are known respectively as the md "
[~]
= P21 .6.(I - P ll .6.) -1 [P12 P 13 ]
+[P22 P23 ]
... ,
(2)
Model invalidation frequency domain approach is formulated as follows (Newlin and Smith, 1998):
"repeated real scalar" blocks , the "repeated complex scalar" blocks and the "full complex" blocks.
Problem 1. : Let P be a robustly stable plant model with uncertainty structure .6. E .6.. Given measurements (y, u), does there exist .6. : 11.6.1100 < Ih and signals d : IIdl1 2 ::; Ih and n : IInl12 ::; Ih, such that
2. REVIEW OF FREQUENCY-DOMAIN MODEL INVALIDATION.
y
In the interest of brevity, throughout this section an earnest attempt will be made to avoid duplicating material presented in (Newlin and Smith , 1998). Towards this end, the focus of this section will lie wholly with the results summarized by theorem 3. Thus, we will assume the reader is familiar with the developments in (Newlin and Smith, 1998), and will often refer the reader to this paper for necessary backgrounds and proofs. The model invalidation problem is concerned with determining whether a mathematical model is consistent with (or covers) a collection of experimental data. The mathematical model consists of a nominal LTI model and uncertainties described by bounded perturbations, acting on the nominal model and denoted .6.(s): .6.(s) E .6. : 11.6.1100 = suPw O'(.6.(jw)) ::; Ih·
= Wnn + (.6. * P) [ ~]
?
The model invalidation problem is also to find (.6., d, n) as small as possible according to the used norm, subject to the above conditions. If no (.6.,d,n) exists, then the model is said to be invalidated. To solve the problem 1, define the matrix P as
P= (
P 12 P13U(S) ) 0 0 1 W;l P21 W;l Pn W;l(P23U(S) - y(s))
Pll
and introduce the structure
3.
(3)
in the following manner:
.6. J = {diag(.6.,.6. d ) 1.6. E .6.,.6. d E Cdim(d)xl} .6. K = {.6. n 1 .6. n E clxdim(n)} 3. = diag(.6. J , .6. K ) (4)
P = (~JJ ~JK) P KJ P KK
Figure 1 corresponds to a general interconnection for a robust model which is suitable for model invalidation .
where PKK = W,:;-1(P23U(S) - y(s)) . .6. d and .6. n are fictitious blocks related respectively to the exogenous signal d and the measurements n. They are defined so that
Referring to figure 1, known inputs are denoted u. Unknown inputs are sensor noise, denoted by n which is explicitly weighted by Wn (assumed to be invertible), and exogenous disturbances denoted by d acting directly
d(s)
= .6. d (s)
X
1
1 = .6. n (s) x (-n(s)) 636
(5)
These fictitious blocks are introduced in order to transform the L2 norm constraints (i.e. IIdl1 2 ::; 1/1, IInl12 ::; 1/1) , into Loo norm constraints (i.e II~dlloo ::; 1/1, II~nlloo
(LMI) optimization problem, while a lower bound algorithm seeks to optimize t::.J and ~K explicitly.
2,)·
According to the previous definitions of P and .6. , it is shown in (Newlin and Smith, 1998) that the initial problem 1 can be transformed into a problem which looks like a robust performance analysis problem (which has a similar block diagram, i.e. looped structure Ai -~) as depicted in the figure 2.
U)
3. APPLICATION OF /1e FRAMEWORK TO INTEGRITY MONITORING OF MULTIVARIABLE UNCERTAIN SYSTEMS. In this section the generalized structured singular value is used as a tool for robust fault detection and isolation. The concept was first introduced in (Eich and Oehler, 1997). The contribution here, is the design of a systematic scheme for robust FDI.
U)
3.1 Fault detection . The robust fault detection task consists of the negation of theorem 3, i.e. a fault occurs in the supervised system if (Henry et al., 1999b)
Fig. 2. Similar block diagram as a /1 problem. The difference between the two problems is that, in the model invalidation problem, we are not looking for solutions with the largest gain from input to output . Rather, we want the norm of n to be small, and therefore the norm of ~n to be large (see equation (5)). Thus a generalization of the structured singular value /1 is introduced to solve the model invalidation problem 1: the generalized structured singular value Jig.
(6)
3.2 Fault isolation. • The most straightforward idea for the fault isolation task, is to examine the frequency behavior of /1g. As it is noted in (Eich and Oehler, 1997), for fault classification one can distinguish either different values of " or the shapes of /1g plot above the frequency axis, or finally the results for the ~J and ~K structures which are obtained from the calculation of the /1g lower bound.
Definition 2. (Newlin and Smith, 1998): Consider the matrix P partitioned in accordance with .6. as in (3), (4), (5) . The positive function /1g~(p(jw)) of the complex constant matrix P(jw), is defined on a domain dom(/1g) by /1 A (P(jw)) L
~
max {,: IIvll =1
IlvJlh::; Ilzjll,Vj E J IIvkl12 Ilzkll" Vk E K
• Another possibility is the construction of a set of dedicated /1g. The basic idea of this approach is the same as one used in the Generalized Observer Scheme approach (Patton, 1997). The fault isolation procedure consists of designing a bank of dedicated /1g functions, such that the kth function is sensitive to all faults (multiplicative and/or additive), and insensitive to the kth fault. For a definition of multiplicative and additive faults, see (Isermann, 1997) .
}
The domain of definition of /1g denoted by dom(/1g) , is given by
PE dom(/1g)
iff PKKVK = 0
=> VK = O.
So the generalized structured singular value can be seen as a measure of the smallest ~J E ~J and the biggest ~ K E .6. K such that there exists a sol u tion of the looped equations depicted by the figure 2. The following theorem gives the solution to the model invalidation problem 1 in the /1g framework .
To formalize this concept, consider the case of the kth multiplicative fault affecting the supervised system. A possible state space model of the faulty system can be i: = A(B,B{)x { y = C(B,Bj)x
Theorem 3. (Newlin and Smith, 1998) : Let P and .6. be defined as in (3) and (4). Then infw /1g~(P(jw)) 2 , if and only if the robustly stable plant model P and measurements are ,-consistent.
+ B(B,Bj)u + Eld + D(B,Bj)u + n
(7)
where x, u, y, d, n denote respectively the state vector, the input vector, the output vector, the disturbances vector (distributed via a constant matrix El) and the measurement noise vector. B is a polynomial vector; its variations round the nominal value Bo model the effects of modeling mismatches. It is assumed that B is norm bounded, i.e. IB;! ::; f3;, i = 1, ... , dim(O).
Because /1g is not easily computable in general, it is necessary to develop computable bounds. An upper bound of /19 can be formulated as a Linear Matrix Inequality 637
a bank of dedicated /-1g, generating "nI" /-1g functions (corresponding to "n 1" faults to be isolated), where the kth function is sensitive to all faults excepted to the kth fault. The "nI" /-1g functions are then processed by a suitable decision logic in order to isolate the faults uniquely. Figure 3 illustrates this method.
Bj is a polynomial vector; its abnormal variations (supposed bounded, i.e.IBjil::; TJf,i = 1, ... ,dim(Bj)) represent the effect of the eh multiplicative fault. When no fault occurs, it is assumed that the polynomial vector Bj is also bounded, such that IBji I ::; (l~ ::; TJf, i = 1, ... , dim(Bj). Introducing weighted functions for IBd, i = 1, .. . , dim (B) , IBjil,i = 1, ... ,dim(Bj), IIdll2 and IIn1l2' the state space model (7) can be rewritten as the LFT (~~ * p::') : 1I~~lIoo ::; 1/1, which is suitable for model invalidation (see (Cockburn and Morton, 1997) for a systematic method to obtain the LFT representation).
faults
faults
faults
ref
Applying the model invalidation theorem 3 to the robust model p~ leads to inf/-1 l:,.k (p~,(jw)) 2:, {:} w 9_ rn
3~~ : 1I~~,lIoo ::; 1/1, IIdl1 2 such that y - k
::;
1/1, IInll2 ::; 1/1 (8)
= W n n + (~~ * p!) ( ~ )
-
where ~m and P::' are defined in the same manner as (4) and (3). This relation outlines that the /-1g function is not affected by the kth multiplicative fault .
Fig. 3. Dedicated /-1g approach for fault isolation
Now, consider the case of the kth additive fault affecting the supervised system. The state space model of the faulty system is given by
X = A(B)x { y = C(B)x
+ B(B)u + Eld + Kf fk + D(B)u + Kffk + n
• The third method for fault isolation, is to measure the sensitivity of /-1g with respect to each element of .i I, where .i I is related to the faulty system. Consider that "nI" (multiplicatives and additives) faults can affect the supervised system. A possible state space model of the faulty system can be
(9)
where x, u, y, d, n , El and B are defined as above. The vector fk represents the kth additive fault, distributed via constant matrices K~ and Kg. If no fault occurs , fk = O. Moreover, the kth additive fault is supposed to be of bounded energy, i.e. IIfk 112 ::;
where x , u, y, d, n and El are defined as in (7). Bd is a polynomial vector; its variations round the nominal value Bdo model the effects of parametric uncertainties. It is assumed that Bd is norm bounded, i.e. IBdi I ::; E;, i = 1, ... , dim(Bd). Note that, due to its definition, Bd ~ B, where B is defined as in (7) . Thus, it is obvious that E; E {;Jj},i = 1...dim(B d),j = 1...dim(B). BI is a polynomial vector; its abnormal variations (supposed bounded, i.e. IBlil ::; p;,i = 1, ... ,dim(BI )) represent the effect of all multiplicative faults. Note that, due to its definition, Bj ~ BI, where Bj is defined as in (7) . Thus, it is clear that TJf E {pJ},i = 1...dim(Bj),j = 1.. .dim( Bf). When no fault occurs, it is assumed that the polynomial vector Bf is also bounded, such that IB li I ::; (; ::; p;,i = 1, ... ,dim(B I ). The vector f represents all additive faults affecting the system. If no fault occurs, it is assumed that f = O. Moreover, additive faults are supposed to be of bounded energy, i.e. IIfll2 ::; x. Introducing weighted functions for IBdi I, i = 1, .. . , dim(B d), IBli I, i = 1, ... , dim(B I ), IId1l 2, IIflb and IIn1l2 ' the state space model (11) can be rewritten as a LFT (~I *
Applying the model invalidation theorem 3 to the robust yields model
P:
inf/-1 l:,.k(P!(jW)) 2:, {:} w 9_ a
3~~ : 1I~~lIoo ::; 1/1, lI~tll2 ::; 1/1, IInll2 ::; 1/1 (10) such that y = Wnn k
TT
+ (~~ * P;) (
!)
-k-
where Q = (d T fk) and where ~ and P: are defined in the same manner as (4) and (3) . The relation (10) outlines that the /-1g function is insensitive to the kth additive fault. So, the proposed fault isolation procedure results in 638
Pt) : II~f 1100 ::; 1/1 which is suitable for model invalidation (see again (Cockburn and Morton, 1997) to obtain (~f * Pt))· Now, define the matrix Ft and the structure I::. t , related to the faulty system, as in (4) , i.e.: AtJ
= diag(~t, D..d, ~~), ~~ E Cdim(f)xl A/K = D.. n At = diag(AtJ> A/K)
Lemma 6. A monotonic function defined on an interval has a finite derivative almost everywhere on the interval.
o Remark 7. Because II~tlloo, Ildlb , 111112 and Ilnllz are, in practice, normalized (i.e. IIAtlloo ::; 1 {::} I = 1), the /-lg-sensitivity functions are defined on a " relative basis". In other words, the /-lg-sensitivity functions give a measure of /-lg, in percentage, with respect to each element of I::. t J •
(12)
Ft = (Ft JJ FfJK) P hJ PhK Left multiply each element of I::. I J by a scalar 'l/Ji, i 1...dim(l::.tJ, where 'l/J is real positive. Then
Remark 8. A generalized definition of (15) can be considered:
(13) where 'l/J = diag( 'l/JJr.) and ri is the dimension of the ith uncertainty I::./J,. Absorb the matrix 'l/J into FI to give
This definition is an ext.ension of (15) for the general case of MIMO systems. In this case, no /-lg continuity result is established, and thus, the existence of 8/-l 9i can not be assured.
(14)
In the next section, the proposed fault diagnosis approach is applied to a hydraulic process. Fault isolation task is based on a bank of dedicated /-lg functions.
Define Ft Cl/Ji) to be equal to Ft (1/;) for 1/;i such that 'l/J j = 1, 'rj j =I- i. Now consider the matrix 'l/J) relative to Ft('l/J) defined for the kth row of the output vector y. The " /-lg-sensitivity" functions
Ft (
-k
-k
4. APPLICATION TO A HYDRAULIC PROCESS /-lgz:"f (Pt ('l/Ji)) - J-lgz:"f (Pt ('l/Ji - 8'l/Ji)) 8 /-lg, = lim J: (15)
£::,
Proposition 4. Existence of 8k /-lg,: The" flg-Sensitivity" functions 8k /-lg, are well-defined and finite for all block
I::. t · Proof: The proof of the existence of 8k /-lg, is deduced from the two following lemmas:
Lemma 5. In the case of MISO (Multi Input Single Output) systems, the positive definite /-lg function can be refined in terms of a /-l function as (Newlin and Smith, 1998): /-l gZ:,.(F) = /-lZ:,.(F) where I::. = diag(~J,~I) and where defined as in (4) . P is given by
F= (F j j
1::., ~J
Fig. 4. Physical structure of the hydraulic process. The plant consists of three cylinders T J , T 2 , T3 with cross section A. These are connected serially with one another cylindrical pipes with a cross section Sn. The out-flowing liquid (usually distilled water) is collected in a reservoir, which supplies the pumps 1 and 2. Here the circle is closed. QJ and Q2 are the flown rates of the pumps 1 and 2. The three water levels h l , h2 and h3 are measured via piezoresistive pressure sensors. For the purpose of simulating clogging and operating errors, the connecting pipes are equipped with manually adjustable
and ~K are
-F:~~~KkFKJ FJl!~Kk)
-PKKPKJ PKK Observing that ~K is "full complex block", on can deduce that /-l~ (F) is continuous in P (Packard and Pandey, 1993) . 639
ball valves, which allow the corresponding pipe to be closed. Sensor and actuator faults can also be generated \'ia hardware and software.
1 'r/w). Moreover, it can be seen from the figures that the effects of faults (low frequency for leak and high frequency for increasing noise) is also transmitted to the /kg function.
The FDI objective is to detect and isolate a leak affecting the second tank and abnormal increasing noise of the first sensor . After collecting data, the /kg model invalidation tool is applied to the robust model P,~ and P; , where P;" and P; are robust models, designed such that /kg(P!'J is insensitive to the leak, and /kg(P;) not affected by the sensor fault . Furthermore, uncertainty, perturbation and noise are normalized (i.e "y = 1). To obtain the robust models P,~ and P; and their associated uncertainty blocks and fl ~, the interested l1 reader can refer to (Henry et al., 1999a). Figures 4 and .) illustrate the /kg behavior in both faulty situations.
5. CONCLUSIONS. This paper investigates the application of frequency domain model invalidation techniques to integrity monitoring of multi-variable uncertain systems. The generalized structured singular value, /kg was considered as a tool for robust FDI system design. Three methods are proposed to isolate faults: the frequency behavior of /kg analysis, bank of dedicated /kg functions, and /kgsensitivity functions . A more general benefit of b/k9i analysis for complex systems, is to understand the implications of certain sub-system failures of the overall system performance (for instance, post-analysis of flight data). The advantage of the proposed monitoring system is that the uncertainty modeling is quite general, and a priori information about the frequency behavior of faults can be included, allowing a more reliable fault diagnosis.
fl:
, 10
10'
10'
10'
tr9QU9ncy 10 rad/s
10'
6. REFERENCES
10'
,. 10'
10
Cockburn, J.C. and B.G . Morton (1997). Linear fractional representations of uncertain systems. A utomatica 33(7), 1263- 1271. Eich, J . and R. Oehler (1997). On the application of the generalised structured singular value to robust fdi system design. In: Proceedings of SAFEPROCESS '97. IFAC. Hull - England. pp . 885-890. Henry, D., A. Zolghadri and M. Monsion (1999a). Detection et localisation robustes de defauts par invalidation de modele: etude d'un cas .. In: loumees doctorales lDA '99. Nancy, France. pp. 257-260. Henry, D., A. Zolghadri and M. Monsion (1999b). Integrity monitoring of induction motors using frequency-domain model invalidation techniques. In: European Control Conference ECC'99. Karlsruhe, Germany. pp. Paper F 1056- 6. Isermann, R. (1997). Supervision, fault detection and fault diagnosis methods - an introduction. Control Eng. Practice 5(5) , 639-652 . Newlin, M .P . and R.S. Smith (1998) . A generalization of the structured singular value and its application to model validation .. IEEE Transactions on A utomatic Control 43, 901- 907. Packard, A. and P. Pandey (1993). Continuity properties of the reall complex structured singular value. IEEE Transactions on Automatic Control 38(3) , 415- 428. Patton, R. (1997) . Fault-tolerant control: the 1997 situation . In: Proceedings of SAFEPROCESS'97. IFAC. Hull - England. pp. 1033- 1055.
:
10. 2
10 - '
10'
10 '
10'
Irequency in radls
Fig. 5. Behavior of J.ig(P;") (top) and J.ig(pJ) (bottom ) - Leak in tank 2.
10
10- 2
10-'
10'
10'
10'
frequency in radls 10'
.:1."'10'
10'
IO '~
10'
10 IreqlAnCy in nldls
Fig. 6. Behavior of J.ig(P;") (top) and J.ig(pD (bottom) Increasing noise on first sensor.
Using a simple logic decision, faults are successfully detected and isolated (in the case of the leak :lw : /kg(P;) < 1 whereas /kg(P:"') :2 1 'r/w, and in the case of the sensor fault, :lw : /kg(P:"') < 1 whereas /kg(P;) :2 640
DESIGN OF ROBUST FDI SYSTEMS USING THE GENERALIZED STRUCTURED SINGULAR VALUE. D. Henry A. Zolghadri M. Monsion
Laboratoire d'Automatique et de Productique Universite Bordeaux I 351 cours de la Liberation 33405 Talence cedex, France. Email: [email protected] Tel: (33)556842400 - Fax: (33)556846644
Abstract: The purpose of this paper is to apply the concept of frequency-domain model invalidation to integrity monitoring of multi variable uncertain systems. The approach is based on the concept of the generalized structured singular value. New formulations of robustness objectives in fault diagnosis are explored and experimental results applied to a hydraulic process demonstrate the potential of the proposed method. Copyright @20001FAC Keywords: Failure detection, failure isolation, robustness, model invalidation, generalized structured singular value.
a large class of uncertain systems, i.e. parametric uncertainties, negliged dynamics and small nonlinearities can be considered. The purpose of this paper is the design of a robust FDI scheme based on the frequency-domain model invalidation technique. The concept of the generalized structured singular value /1-9 is used here to robustly detect and isolate faults. It should be pointed out that, due to its implementation in the frequency domain, the tool provided by the proposed approach is not intended to serve as an on-line FDI unit, but as an off-line integrity monitoring system. This paper is organized as follows: In section 2, the review of /1-9 is considered. The theorical backgrounds for model invalidation in the frequency domain are presented. Section 3 is dedicated to the application of /1-9 framework to integrity monitoring of multivariable uncertain systems. Finally, experimental results applied to a hydraulic process demonstrate the potential of the proposed approach.
1. INTRODUCTION.
Different approaches to fault detection and isolation (FDI) using mathematical models have been developed over last two decades (see (Patton, 1997) for a survey). The main objective of the FDI design procedure is the achievement of a low missed-alarm and a low false-alarm rates. So robustness to all unknown inputs (modeling errors and disturbances) and sensitivity to faults are the main challenge when designing a robust FDI system. On the other hand, in recent years there has been a renewed interest for frequency-domain model invalidation techniques in robust control design (Newlin and Smith, 1998). The model invalidation problem is concerned with determining whether a mathematical model is consistent (or covers) a collection of experimental data. The Linear Fractional Transformation (LFT) is also used to describe the mathematical model. It consists of a nominal plant model and uncertainties described by bounded perturbations acting on the nominal model. This formalism has the advantage to account for 635
Notations In dealing with vectors, the Euclidean norm is always used and is written without a subscript; for example Ilxll . Similarly in the matrix case, we used the induced vector norm: 110411 = 0'(04) where 0'(.4) denotes the maximum singular value of o4. Signals, w(t) or w , are assumed to be of bounded energy (Lebesgue square integral, denoted by L2), and their norm is denoted by Ilw112. We will assume that all systems are real rational operators with bounded induced L2 norm . Uncertain systems are represented by the linear fractionnal representation of the matrix .6. posed on P , which is referred to as the star product
Fig. 1. The generic structure of model invalidation problem on the plant P. The unknown inputs are supposed to be of bounded energy. The measured output is represented by y and is also known. The signals;; and v are internal to the model. It is also assumed that the model is robustly stable, i.e. sUPw J.L~(Pl1 (jw)) < Ih. According to figure 1, the robust model is defined by
(.6. * P) Alternative, equivalent notations found in the literature are Fu(P,.6.) and S(.6., P). .6. is modeled, without loss of generality, by a prescribed block diagonal structure and represents multiple perturbations . .6. is used to describe the set of all perturbations of a prescribed structure as
y=Wn n+(.6.*P)
(1)
with
(.6. * P)
.6. = {block diag(bT h" ... , O~Jk~r' of h O~,Jkmdm , ,.6.f, ... , .6.·~c)' or E R,of E c,.6.f E C} where o[h"i = 1, .. . ,mr, O'jh~dj , j = 1, .. . ,mc and .6.f, I = 1, ..., me are known respectively as the md "
[~]
= P21 .6.(I - P ll .6.) -1 [P12 P 13 ]
+[P22 P23 ]
... ,
(2)
Model invalidation frequency domain approach is formulated as follows (Newlin and Smith, 1998):
"repeated real scalar" blocks , the "repeated complex scalar" blocks and the "full complex" blocks.
Problem 1. : Let P be a robustly stable plant model with uncertainty structure .6. E .6.. Given measurements (y, u), does there exist .6. : 11.6.1100 < Ih and signals d : IIdl1 2 ::; Ih and n : IInl12 ::; Ih, such that
2. REVIEW OF FREQUENCY-DOMAIN MODEL INVALIDATION.
y
In the interest of brevity, throughout this section an earnest attempt will be made to avoid duplicating material presented in (Newlin and Smith , 1998). Towards this end, the focus of this section will lie wholly with the results summarized by theorem 3. Thus, we will assume the reader is familiar with the developments in (Newlin and Smith, 1998), and will often refer the reader to this paper for necessary backgrounds and proofs. The model invalidation problem is concerned with determining whether a mathematical model is consistent with (or covers) a collection of experimental data. The mathematical model consists of a nominal LTI model and uncertainties described by bounded perturbations, acting on the nominal model and denoted .6.(s): .6.(s) E .6. : 11.6.1100 = suPw O'(.6.(jw)) ::; Ih·
= Wnn + (.6. * P) [ ~]
?
The model invalidation problem is also to find (.6., d, n) as small as possible according to the used norm, subject to the above conditions. If no (.6.,d,n) exists, then the model is said to be invalidated. To solve the problem 1, define the matrix P as
P= (
P 12 P13U(S) ) 0 0 1 W;l P21 W;l Pn W;l(P23U(S) - y(s))
Pll
and introduce the structure
3.
(3)
in the following manner:
.6. J = {diag(.6.,.6. d ) 1.6. E .6.,.6. d E Cdim(d)xl} .6. K = {.6. n 1 .6. n E clxdim(n)} 3. = diag(.6. J , .6. K ) (4)
P = (~JJ ~JK) P KJ P KK
Figure 1 corresponds to a general interconnection for a robust model which is suitable for model invalidation .
where PKK = W,:;-1(P23U(S) - y(s)) . .6. d and .6. n are fictitious blocks related respectively to the exogenous signal d and the measurements n. They are defined so that
Referring to figure 1, known inputs are denoted u. Unknown inputs are sensor noise, denoted by n which is explicitly weighted by Wn (assumed to be invertible), and exogenous disturbances denoted by d acting directly
d(s)
= .6. d (s)
X
1
1 = .6. n (s) x (-n(s)) 636
(5)
These fictitious blocks are introduced in order to transform the L2 norm constraints (i.e. IIdl1 2 ::; 1/1, IInl12 ::; 1/1) , into Loo norm constraints (i.e II~dlloo ::; 1/1, II~nlloo
(LMI) optimization problem, while a lower bound algorithm seeks to optimize t::.J and ~K explicitly.
2,)·
According to the previous definitions of P and .6. , it is shown in (Newlin and Smith, 1998) that the initial problem 1 can be transformed into a problem which looks like a robust performance analysis problem (which has a similar block diagram, i.e. looped structure Ai -~) as depicted in the figure 2.
U)
3. APPLICATION OF /1e FRAMEWORK TO INTEGRITY MONITORING OF MULTIVARIABLE UNCERTAIN SYSTEMS. In this section the generalized structured singular value is used as a tool for robust fault detection and isolation. The concept was first introduced in (Eich and Oehler, 1997). The contribution here, is the design of a systematic scheme for robust FDI.
U)
3.1 Fault detection . The robust fault detection task consists of the negation of theorem 3, i.e. a fault occurs in the supervised system if (Henry et al., 1999b)
Fig. 2. Similar block diagram as a /1 problem. The difference between the two problems is that, in the model invalidation problem, we are not looking for solutions with the largest gain from input to output . Rather, we want the norm of n to be small, and therefore the norm of ~n to be large (see equation (5)). Thus a generalization of the structured singular value /1 is introduced to solve the model invalidation problem 1: the generalized structured singular value Jig.
(6)
3.2 Fault isolation. • The most straightforward idea for the fault isolation task, is to examine the frequency behavior of /1g. As it is noted in (Eich and Oehler, 1997), for fault classification one can distinguish either different values of " or the shapes of /1g plot above the frequency axis, or finally the results for the ~J and ~K structures which are obtained from the calculation of the /1g lower bound.
Definition 2. (Newlin and Smith, 1998): Consider the matrix P partitioned in accordance with .6. as in (3), (4), (5) . The positive function /1g~(p(jw)) of the complex constant matrix P(jw), is defined on a domain dom(/1g) by /1 A (P(jw)) L
~
max {,: IIvll =1
IlvJlh::; Ilzjll,Vj E J IIvkl12 Ilzkll" Vk E K
• Another possibility is the construction of a set of dedicated /1g. The basic idea of this approach is the same as one used in the Generalized Observer Scheme approach (Patton, 1997). The fault isolation procedure consists of designing a bank of dedicated /1g functions, such that the kth function is sensitive to all faults (multiplicative and/or additive), and insensitive to the kth fault. For a definition of multiplicative and additive faults, see (Isermann, 1997) .
}
The domain of definition of /1g denoted by dom(/1g) , is given by
PE dom(/1g)
iff PKKVK = 0
=> VK = O.
So the generalized structured singular value can be seen as a measure of the smallest ~J E ~J and the biggest ~ K E .6. K such that there exists a sol u tion of the looped equations depicted by the figure 2. The following theorem gives the solution to the model invalidation problem 1 in the /1g framework .
To formalize this concept, consider the case of the kth multiplicative fault affecting the supervised system. A possible state space model of the faulty system can be i: = A(B,B{)x { y = C(B,Bj)x
Theorem 3. (Newlin and Smith, 1998) : Let P and .6. be defined as in (3) and (4). Then infw /1g~(P(jw)) 2 , if and only if the robustly stable plant model P and measurements are ,-consistent.
+ B(B,Bj)u + Eld + D(B,Bj)u + n
(7)
where x, u, y, d, n denote respectively the state vector, the input vector, the output vector, the disturbances vector (distributed via a constant matrix El) and the measurement noise vector. B is a polynomial vector; its variations round the nominal value Bo model the effects of modeling mismatches. It is assumed that B is norm bounded, i.e. IB;! ::; f3;, i = 1, ... , dim(O).
Because /1g is not easily computable in general, it is necessary to develop computable bounds. An upper bound of /19 can be formulated as a Linear Matrix Inequality 637
a bank of dedicated /-1g, generating "nI" /-1g functions (corresponding to "n 1" faults to be isolated), where the kth function is sensitive to all faults excepted to the kth fault. The "nI" /-1g functions are then processed by a suitable decision logic in order to isolate the faults uniquely. Figure 3 illustrates this method.
Bj is a polynomial vector; its abnormal variations (supposed bounded, i.e.IBjil::; TJf,i = 1, ... ,dim(Bj)) represent the effect of the eh multiplicative fault. When no fault occurs, it is assumed that the polynomial vector Bj is also bounded, such that IBji I ::; (l~ ::; TJf, i = 1, ... , dim(Bj). Introducing weighted functions for IBd, i = 1, .. . , dim (B) , IBjil,i = 1, ... ,dim(Bj), IIdll2 and IIn1l2' the state space model (7) can be rewritten as the LFT (~~ * p::') : 1I~~lIoo ::; 1/1, which is suitable for model invalidation (see (Cockburn and Morton, 1997) for a systematic method to obtain the LFT representation).
faults
faults
faults
ref
Applying the model invalidation theorem 3 to the robust model p~ leads to inf/-1 l:,.k (p~,(jw)) 2:, {:} w 9_ rn
3~~ : 1I~~,lIoo ::; 1/1, IIdl1 2 such that y - k
::;
1/1, IInll2 ::; 1/1 (8)
= W n n + (~~ * p!) ( ~ )
-
where ~m and P::' are defined in the same manner as (4) and (3). This relation outlines that the /-1g function is not affected by the kth multiplicative fault .
Fig. 3. Dedicated /-1g approach for fault isolation
Now, consider the case of the kth additive fault affecting the supervised system. The state space model of the faulty system is given by
X = A(B)x { y = C(B)x
+ B(B)u + Eld + Kf fk + D(B)u + Kffk + n
• The third method for fault isolation, is to measure the sensitivity of /-1g with respect to each element of .i I, where .i I is related to the faulty system. Consider that "nI" (multiplicatives and additives) faults can affect the supervised system. A possible state space model of the faulty system can be
(9)
where x, u, y, d, n , El and B are defined as above. The vector fk represents the kth additive fault, distributed via constant matrices K~ and Kg. If no fault occurs , fk = O. Moreover, the kth additive fault is supposed to be of bounded energy, i.e. IIfk 112 ::;
where x , u, y, d, n and El are defined as in (7). Bd is a polynomial vector; its variations round the nominal value Bdo model the effects of parametric uncertainties. It is assumed that Bd is norm bounded, i.e. IBdi I ::; E;, i = 1, ... , dim(Bd). Note that, due to its definition, Bd ~ B, where B is defined as in (7) . Thus, it is obvious that E; E {;Jj},i = 1...dim(B d),j = 1...dim(B). BI is a polynomial vector; its abnormal variations (supposed bounded, i.e. IBlil ::; p;,i = 1, ... ,dim(BI )) represent the effect of all multiplicative faults. Note that, due to its definition, Bj ~ BI, where Bj is defined as in (7) . Thus, it is clear that TJf E {pJ},i = 1...dim(Bj),j = 1.. .dim( Bf). When no fault occurs, it is assumed that the polynomial vector Bf is also bounded, such that IB li I ::; (; ::; p;,i = 1, ... ,dim(B I ). The vector f represents all additive faults affecting the system. If no fault occurs, it is assumed that f = O. Moreover, additive faults are supposed to be of bounded energy, i.e. IIfll2 ::; x. Introducing weighted functions for IBdi I, i = 1, .. . , dim(B d), IBli I, i = 1, ... , dim(B I ), IId1l 2, IIflb and IIn1l2 ' the state space model (11) can be rewritten as a LFT (~I *
Applying the model invalidation theorem 3 to the robust yields model
P:
inf/-1 l:,.k(P!(jW)) 2:, {:} w 9_ a
3~~ : 1I~~lIoo ::; 1/1, lI~tll2 ::; 1/1, IInll2 ::; 1/1 (10) such that y = Wnn k
TT
+ (~~ * P;) (
!)
-k-
where Q = (d T fk) and where ~ and P: are defined in the same manner as (4) and (3) . The relation (10) outlines that the /-1g function is insensitive to the kth additive fault. So, the proposed fault isolation procedure results in 638
Pt) : II~f 1100 ::; 1/1 which is suitable for model invalidation (see again (Cockburn and Morton, 1997) to obtain (~f * Pt))· Now, define the matrix Ft and the structure I::. t , related to the faulty system, as in (4) , i.e.: AtJ
= diag(~t, D..d, ~~), ~~ E Cdim(f)xl A/K = D.. n At = diag(AtJ> A/K)
Lemma 6. A monotonic function defined on an interval has a finite derivative almost everywhere on the interval.
o Remark 7. Because II~tlloo, Ildlb , 111112 and Ilnllz are, in practice, normalized (i.e. IIAtlloo ::; 1 {::} I = 1), the /-lg-sensitivity functions are defined on a " relative basis". In other words, the /-lg-sensitivity functions give a measure of /-lg, in percentage, with respect to each element of I::. t J •
(12)
Ft = (Ft JJ FfJK) P hJ PhK Left multiply each element of I::. I J by a scalar 'l/Ji, i 1...dim(l::.tJ, where 'l/J is real positive. Then
Remark 8. A generalized definition of (15) can be considered:
(13) where 'l/J = diag( 'l/JJr.) and ri is the dimension of the ith uncertainty I::./J,. Absorb the matrix 'l/J into FI to give
This definition is an ext.ension of (15) for the general case of MIMO systems. In this case, no /-lg continuity result is established, and thus, the existence of 8/-l 9i can not be assured.
(14)
In the next section, the proposed fault diagnosis approach is applied to a hydraulic process. Fault isolation task is based on a bank of dedicated /-lg functions.
Define Ft Cl/Ji) to be equal to Ft (1/;) for 1/;i such that 'l/J j = 1, 'rj j =I- i. Now consider the matrix 'l/J) relative to Ft('l/J) defined for the kth row of the output vector y. The " /-lg-sensitivity" functions
Ft (
-k
-k
4. APPLICATION TO A HYDRAULIC PROCESS /-lgz:"f (Pt ('l/Ji)) - J-lgz:"f (Pt ('l/Ji - 8'l/Ji)) 8 /-lg, = lim J: (15)
£::,
Proposition 4. Existence of 8k /-lg,: The" flg-Sensitivity" functions 8k /-lg, are well-defined and finite for all block
I::. t · Proof: The proof of the existence of 8k /-lg, is deduced from the two following lemmas:
Lemma 5. In the case of MISO (Multi Input Single Output) systems, the positive definite /-lg function can be refined in terms of a /-l function as (Newlin and Smith, 1998): /-l gZ:,.(F) = /-lZ:,.(F) where I::. = diag(~J,~I) and where defined as in (4) . P is given by
F= (F j j
1::., ~J
Fig. 4. Physical structure of the hydraulic process. The plant consists of three cylinders T J , T 2 , T3 with cross section A. These are connected serially with one another cylindrical pipes with a cross section Sn. The out-flowing liquid (usually distilled water) is collected in a reservoir, which supplies the pumps 1 and 2. Here the circle is closed. QJ and Q2 are the flown rates of the pumps 1 and 2. The three water levels h l , h2 and h3 are measured via piezoresistive pressure sensors. For the purpose of simulating clogging and operating errors, the connecting pipes are equipped with manually adjustable
and ~K are
-F:~~~KkFKJ FJl!~Kk)
-PKKPKJ PKK Observing that ~K is "full complex block", on can deduce that /-l~ (F) is continuous in P (Packard and Pandey, 1993) . 639
ball valves, which allow the corresponding pipe to be closed. Sensor and actuator faults can also be generated \'ia hardware and software.
1 'r/w). Moreover, it can be seen from the figures that the effects of faults (low frequency for leak and high frequency for increasing noise) is also transmitted to the /kg function.
The FDI objective is to detect and isolate a leak affecting the second tank and abnormal increasing noise of the first sensor . After collecting data, the /kg model invalidation tool is applied to the robust model P,~ and P; , where P;" and P; are robust models, designed such that /kg(P!'J is insensitive to the leak, and /kg(P;) not affected by the sensor fault . Furthermore, uncertainty, perturbation and noise are normalized (i.e "y = 1). To obtain the robust models P,~ and P; and their associated uncertainty blocks and fl ~, the interested l1 reader can refer to (Henry et al., 1999a). Figures 4 and .) illustrate the /kg behavior in both faulty situations.
5. CONCLUSIONS. This paper investigates the application of frequency domain model invalidation techniques to integrity monitoring of multi-variable uncertain systems. The generalized structured singular value, /kg was considered as a tool for robust FDI system design. Three methods are proposed to isolate faults: the frequency behavior of /kg analysis, bank of dedicated /kg functions, and /kgsensitivity functions . A more general benefit of b/k9i analysis for complex systems, is to understand the implications of certain sub-system failures of the overall system performance (for instance, post-analysis of flight data). The advantage of the proposed monitoring system is that the uncertainty modeling is quite general, and a priori information about the frequency behavior of faults can be included, allowing a more reliable fault diagnosis.
fl:
, 10
10'
10'
10'
tr9QU9ncy 10 rad/s
10'
6. REFERENCES
10'
,. 10'
10
Cockburn, J.C. and B.G . Morton (1997). Linear fractional representations of uncertain systems. A utomatica 33(7), 1263- 1271. Eich, J . and R. Oehler (1997). On the application of the generalised structured singular value to robust fdi system design. In: Proceedings of SAFEPROCESS '97. IFAC. Hull - England. pp . 885-890. Henry, D., A. Zolghadri and M. Monsion (1999a). Detection et localisation robustes de defauts par invalidation de modele: etude d'un cas .. In: loumees doctorales lDA '99. Nancy, France. pp. 257-260. Henry, D., A. Zolghadri and M. Monsion (1999b). Integrity monitoring of induction motors using frequency-domain model invalidation techniques. In: European Control Conference ECC'99. Karlsruhe, Germany. pp. Paper F 1056- 6. Isermann, R. (1997). Supervision, fault detection and fault diagnosis methods - an introduction. Control Eng. Practice 5(5) , 639-652 . Newlin, M .P . and R.S. Smith (1998) . A generalization of the structured singular value and its application to model validation .. IEEE Transactions on A utomatic Control 43, 901- 907. Packard, A. and P. Pandey (1993). Continuity properties of the reall complex structured singular value. IEEE Transactions on Automatic Control 38(3) , 415- 428. Patton, R. (1997) . Fault-tolerant control: the 1997 situation . In: Proceedings of SAFEPROCESS'97. IFAC. Hull - England. pp. 1033- 1055.
:
10. 2
10 - '
10'
10 '
10'
Irequency in radls
Fig. 5. Behavior of J.ig(P;") (top) and J.ig(pJ) (bottom ) - Leak in tank 2.
10
10- 2
10-'
10'
10'
10'
frequency in radls 10'
.:1."'10'
10'
IO '~
10'
10 IreqlAnCy in nldls
Fig. 6. Behavior of J.ig(P;") (top) and J.ig(pD (bottom) Increasing noise on first sensor.
Using a simple logic decision, faults are successfully detected and isolated (in the case of the leak :lw : /kg(P;) < 1 whereas /kg(P:"') :2 1 'r/w, and in the case of the sensor fault, :lw : /kg(P:"') < 1 whereas /kg(P;) :2 640