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Fault diagnosis based on analytical models for linear and nonlinear systems - a tutorial
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Copyright (;> IFAC Fault Detection, Supervision and Safety of Technical Processes, Washi ngton, D.C., USA, 2003
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FAULT DIAGNOSIS BASED ON ANALYTICAL MODELS FOR LINEAR AND NONLINEAR SYSTEMS - A TUTORIAL Michel Kinnaert Dpt. of Control Engineering and System Analysis, Universite Libre de Bruxelles. CP 165/55, 50 Av. F. D. Roosevelt, B-1050 Brussels, BELGIUM
Abstract: The diagnosis systems considered in this paper rely on the inconsistency between the actual process behaviour and its expected behaviour as described by an analytical model. The inconsistency is exhibited in signals called residuals. Two methods for residual generation are presented in a tutorial way: the parity space and the observer based approaches. Linear and nonlinear models are successively considered as a basis for the design of the residual generators. Copyright © 2003 IFAC Keywords: Fault detection and isolation, linear and nonlinear systems, observer, analytical redundancy, residual generation, parity space method.
1. INTRODUCTION
it increases the weight which is a critical issue in some applications, and it is quite costly.
For the complex highly automated systems encountered in process industries, in aeronautics or power plants for instance, it is fundamental to be able to monitor the health of the installation at each time instant in order to detect incipient faults and to locate the deteriorated components. Indeed, for safety critical systems such as nuclear power plants and airplanes, the total failure of a component can cause hazard to personnel or to the population and hence it must be avoided at any cost. Moreover the early detection of a malfunction allows one to plan the required maintenance actions and decreases the number of emergency shutdowns of a process, which are often very costly.
Most industrial monitoring systems that are not based on hardware redundancy rely on the comparison of measured signals to specified thresholds. They do not exploit the correlation existing between the different measured signals. Therefore they only allow the detection of important deviations from normal operating conditions and not incipient changes. These drawbacks motivate the development of diagnosis systems based on analytical models. The main idea behind such systems is to check the consistency between the measurements of different variables taken on the supervised process, and the expected behavior of this process as described by an analytical model. As shown in figure 1, typical diagnosis systems are made of two parts: a residual generator and a decision making module. The residuals are signals that are generated by processing the measured plant inputs and outputs. In the absence of fault, they deviate from zero only due to modelling uncertainties and measurement
For safety critical systems, the present monitoring tools heavily rely on hardware redundancy. For instance three to four sensors are often used to measure the same quantity, and a voting scheme determines its most likely value. This requires additional space to accommodate the equipment,
37
noise. When a fault occurs, the deviation from zero is such that the new condition can be distinguished from the faultless working mode. The role of the decision module is to determine whether the residuals differ significantly from zero, and, from the pattern of zero and non-zero residuals, to decide which are the most likely faulty components.
Mcaaurcd in
U
11
where 9 and h are smooth 1 nonlinear functions of their arguments, x(t) is the state vector, u(t) the vector of known inputs (the manipulated inputs for instance), y(t) the vector of measured outputs, d(t) the vector of unknown inputs and f(t) the vector of faults. The dimensions of the different vectors are respectively n, rn, p, nd, and nl' In faultless working mode, f(t) is zero for all t. The unknown inputs correspond to unmeasured disturbances such as waves acting on a ship or wind gusts on an airplane for instance. When system (1), (2) is working around nominal operating conditions that are characterized by an equilibrium state, its behaviour can be described by a linear time invariant (LTI) model of the form
±(t) = Ax(t) + Bu(t) + Edd(t) + EJf(t) (3) y(t) = Cx(t) + Du(t) + Gdd(t) + GJf(t) (4) Fig. 1. Structure of a diagnosis system In this paper the main focus is on the design of residual generators for systems described by deterministic analytical models. The first part contains a tutorial presentation of both parity space and observer based approaches to the design of residual generators for linear systems. A very limited knowledge of system theory is expected from the reader for this part (sections 2 to 4). As most processes are inherently nonlinear, the residual generators based on linear models work properly provided the process is running in the operating region for which the linear approximation is valid. To relax this condition, it is necessary either to account for modelling uncertainties due to the linear approximation or to exploit the nonlinear model of the process instead of its linear approximation around an equilibrium point. After a discussion of different ways to handle modelling errors in section 5, various classes of nonlinear models are considered and the ways to generalize either the parity space method or the observer based method for residual generation are presented (see sections 6 and 7). Some open problems are pointed out in the conclusion.
2. MODEL OF THE SUPERVISED PROCESS
The fault f(t) appears as an additional input in the linear model, and hence it is called an additive fa.ult (as opposed to a multiplicative fault which consists of a. change in the entries of the matrices A,B,C, ... ) . It might correspond to a sensor bias, the jamming of an actuator or a leak for instance. Indeed, for a sensor bias on the first measurement, occurring at time to, it suffices to take El = 0, GI = (1 0 ···0 )T, and f(t) = al(t - to) where "a" is the bias magnitude and l(t) is the unit step function (l(t) = 0 for t < 0, and let) = 1 for t > 0). The jamming of the first actuator can be modelled with El = B .,1> G I = 0, and f(t) = HI - UI(t), where B.,1 denotes the first column of matrix B, UI(t), the first component of vector u(t), and HI is the value at which the actuator is stuck. A leak in an hydraulic system can be modelled by a supplementary input in a model, namely the flow of the leaking fluid. It can thus be expressed in the form of model (3},(4) when a linear model is used. Multiplicative faults are not considered in this paper. The reader is referred to (Isermann, 1993) and (Zhang et al., 1994) for an introduction to this topic.
Using physical laws, a large class of engineering systems can be modelled by differential equations of the form
±(t) = g(x(t), u(t), d(t), f(t» yet) = h(x(t), u(t), d(t), f(t»
Although it is not indicated explicitly, it is assumed that x(t), u(t), d(t), f(t) and yet) represent deviations from their value at equilibrium in (3), (4). The linear model is obtained by performing a Taylor series expansion of the functions g(x(t), u(t), d(t), J(t» and h(x(t), u(t), d(t), f(t» around the equilibrium point, and keeping the first order terms only.
(1)
1 A smooth function haa continuous partial derivative8 of any order with respect to its arguments.
(2)
38
(8) y=Cx +Du y=Cx +Du= CAx+ CBu+ Du (9) ii = CA 2x + CABu + CBu + Dil (10)
3. PARIT Y SPACE METH OD FOR LTI SYSTEMS
3.1 Introduction A fundamental notion on which diagnosis sy~ tems based on analyti cal models rely is analytic al redundancy. Analytical redund ancy relations are equations that are deduced from an analytical model and which solely involve measured variables. They must be fulfilled in the absence of fault. As a very simple example, consider the relation
between the velocity and the acceleration of a mobile device: v(t)
= a(t)
y(.)
+CBu (·-l)
(11)
f,
(5)
Y.(t) = O.x(t)
+ T;'U.(t)
(12)
Assuming that the derivatives of u(t) and y(t) can be obtaine d from the available measurements (see section 3.4 for this issue), the only unknown in (12) is the state vector x(t). When 8 is chosen sufficiently large, the left null space of matrix 0., namely the space generat ed by the row vectors w. which fulfil
(6)
may be used to detect sensor faults (the index m indicates measurement). r(t) is zero for all t in the absence of fault and it becomes nonzero upon occurrence of sensor faults. Indeed, let vm(t) = v(t) + I,,(t) denote the measured velocity, where Iv(t) is non-zero upon occurrence of a sensor fault, and consider a similar expression for the acceleration: am(t) = a(t)+ 10 (t). Substit uting these expressions in (6) yields
r(t) = j,,(t) -/a(t)
+ Du(·)
where the time argume nt has been omitted for compactness of the notatio ns. Letting Y.(t)(U .(t» denote the vector obtaine d by stacking y(t) (u(t» and its derivatives up to order 8, T,'U be the lower triangu lar block Toeplitz 3 matrix with first column [DT (CB)T ... (CA(· -l)B)T f, and O. = equatio ns (8)-(11) can [CT (CAf ... (CAa)T as form t be written in a compac
If both variables are measured perfectly, and the derivative of the velocity can be determined exactly, the quantit y
r(t) = vm(t) - am(t)
= CA·x + CAa-1 Bu + ...
w.O.
= 0,
(13)
is non-zero. Multiplying (12) on the left by one such vector yields (14) w.(y.(t ) - T;'U.( t» = 0
(14) is an analytical redund ancy relation, namely a relation that only involves known variables and derivatives of known variables. This motivates the definition of the quantit y r .(t) as
(7)
which takes at least tempor arily non-zero values when Iv(t) and/or la(t) is non-zero 2 • r(t) is said to be a residual that allows detection of faults on the velocity and acceleration sensors and (5) is an analytical redund ancy relation.
r.(t) = w.(y.(t ) - T;'U.( t»
(15)
Obviously, r.(t) is null in the absence of fault. To determ ine its value in faulty operati ng condition, the equatio n corresponding to (12) in the presence of fault is used:
3.2 Analytical redundancy relations and parity
Y.(t) = O.x(t)
+ T;'U.(t) + T,' F.(t)
(16)
junctions where T,' is the lower triangu lar block Toeplitz T matrix with first column and F.(t) is a E,)T] l) (CA(·[Gr (CE,)T ... up to order ives derivat its and vector made of I (t) yields (15) into Y.(t) for (16) 8. Substit uting
Our aim is now to determ ine how to generate analytical redund ancy relations to monitor systems described by a model of the form (3), (4). To this end let us consider the successive derivatives of the output y(t). For the sake of simplicity, the case where there are no unknown input (Ed = 0) and no fault (/(t) = 0) is first considered. From (3) and (4), one deduces
r.(t) =
3
ate 2 The patholog ical caae where both fault. compens each other is not consider ed 88 it is improba ble in practice.
the matrix [
w.r! F.(t)
~Z CA(a-l) B ...
39
(17)
As long as
w:r/ i- 0,
Table 1. Effects of the faults on the residuals - "diagonal" coding set
(18)
1
r,(t) is an indicator of the presence of a fault; it is called a residual. (15), (16) and (17) define a parity function or parity check, r,{Y,{t), U,{t», namely a function of yet), u(t) and their derivatives that is equal to zero in the absence of fault and is non-zero in the presence of a fault. The set of all parity functions involving signal derivatives up to order 8 can be obtained by linear combinations of the entries of the parity vector~ f,(t) defined as f,(t) = n,(y'(t) - 'P.U,(t»
o
o
(19)
n,
+ 'P.U,(t) + 7,d D,(t)
w,(7,' F,(t) + 7,d D,(t»
1
A diagnosis system should not only detect faults but also isolate them, namely determine which fault(s) occurred. To this end, residuals that are sensitive to certain faults and insensitive to others must be designed. Assume for instance that n I faults have to be isolated. If residuals can be designed in such a way that the ith residual is only affected by the ith fault, then fault isolation can be achieved easily. The corresponding situation is depicted in table 1 in the case of three faults. A 1 indicates that the fault in the corresponding column affects the residual of the corresponding row. Each column of the table defines a code associated to the corresponding fault. The set of fault codes forms a coding set.
(20)
where the new notations 7,d and D,(t) are defined in a similar way as 7,' and F,(t) . Substituting this expression into (15) now yields r ,et) =
o
3.3 Fault isolation
When unknown inputs are present, it is necessary to assure that the residual is not affected by such inputs. With Ed. i- 0, (16) becomes
+7,' F,(t)
o
1
time systems. In that framework, they are defined as relations between the measured inputs and outputs over a fixed time horizon, that must be fulfilled in the absence of fault. Here we have chosen to work with Continuous-time models to ease the generalization to nonlinear systems. The parity space approach to residual generation was also generalized to transfer function models, in which case the residual may depend on the whole set of prior data, and not only on the data over a fixed time horizon (Massoumnia and Vander Velde, 1988), (Gertler, 1993), (Gertler, 1998)
where n, denotes a basis 5 for the left null space of 0,. The dimension of this subspace is np = (8+ l)p - rankO,. 8 must thus be chosen sufficiently large to assure that np is positive. Any suitable row vector w, has the form w, = v,n .. where v, is an np-dimensional row vector chosen so that v,n,T/ i- o. As n, is not unique, there exists an infinite number of parity vectors f,. The space generated by these vectors is called the parity space.
Y,(t) = O,x(t)
o
o
(21)
For the coding set of table 1 to be achievable, the number of faults to be isolated cannot be larger than the number of measured outputs, as will be clear from the next paragraph. If this condition is not met, it might still be possible to perform fault isolation, but under the hypothesis that only a single fault can occur at a time. In this case, residuals may be designed so that the ith residual is affected by all faults, except the ith one (see table 2). Clearly, when simultaneous faults occur, all residuals become non-zero for this type of coding set, which prevents isolation 6 • More generally, isolation of single faults can be achieved by choosing a coding set with no identical oodes. Yet, for avoiding erroneous isolation when the reaction of one residual is not as expected upon occurrence of a fault (due to noise, model uncertainty or time history of the fault signal), some coding sets are
To assure that r,(t) is not affected by d(t), the condition (22)
n,
must be imposed on w, besides (13) and (18). A necessary condition for the existence of a row vector w, that meets the three conditions is that the number of unknown inputs ne/. be lower than the number of measured outputs p. Remark Analytical redundancy relations were presented in (Chow and Willsky, 1984) for linear discrete 4 It is assumed that there is no state variable that is exclusively controlla.ble from the fault (see (Nyberg, 1999),
An alternative approach to handle multiple faults might be to define a different code for each fault combination but this would lead to large residual aeta.
page 2(9).
11
Such a basis can be computed by performing a singular value decomposition of matrix 0 •.
&
40
Table 2. Effects of the faults on the residuals - isolation of single faults
o
1
o
4. OBSERVER BASED APPROACH FOR LTI SYSTEMS
4.1 Refresher on Luenberyer state observer
1 1
o
preferable to others as discussed in (Gertler and Singer, 1990), (Gertler and Kunwer, 1993). Residuals which are designed according to a specific coding set are called structured residuals. The method to design residuals according to a given coding set is a straightforward extension of the results presented in the previous section. Indeed, for a given residual, the fault vector f(t) can be separated into two parts: fG(t) the faults that should affect the residual, and f no. (t) the faults that should not affect the residual. The method of the previous section can then be applied to design an appropriate residual by substituting respectively fG(t) and [d(t)T fno(t)Tf for the vectors f(t) and d(t) in section 3.2 . From the existence condition stated at the end of that section, it is seen that the dimension of [d(t)T fno(t)Tf must be lower than the number of outputs for a solution to exist.
Let us consider a LTI system with no unknown input and in faultless operation. It is described by the state space model
:t(t) = Ax(t) + Bu(t)
(24)
yet) = Cx(t) + Du(t)
(25)
A state observer is a filter that receives as inputs the signals u(t) and yet) and generates an estimate of the state x(t) denoted x(t). A Luenberger state observer is described by the following differential equation: ~(t) = Ax(t)
X(O)
+ Bu(t) + L(y(t) -
•
r.(s)
~~,--
- (s + {3)-
= (A -
LC)ez(t)
(27)
ez(O) = Xo - Xo When the system is observable 8 , matrix L can be determined so that the eigenvalues of (A - LC) take fixed values. Imposing that all eigenvalues have negative real part ensures that ez(t) asymptotically decays to zero. From (25), an estimate yet) of the output is deduced as
Due to the presence of derivatives in (15), r.(t) cannot be generated as such from yet) and u(t). Only a filtered version, say r{ (t), of this signal can be obtained. As an example, using Laplace transform expressions, r{ (8) can be generated by
r J (s) -
(26)
where L is a n x p constant matrix of which the entries are chosen in such a way that the transient in the observation error ez(t) = x(t) - x(t) decays sufficiently fast to zero. The dynamics of the error is obtained by subtracting (26) from (24) and substituting (25) for yet). It yields
ez(t) 3.4 Implementation of parity checks
Cx(t) - Du(t»
= Xo
yet) = Cx(t)
(23)
+ Du(t)
(28)
The output estimation error where /3 E m.+ is chosen in such a way that the effect of noise is sufficiently filtered 7. It has been shown that the same signal r{ (t) can be generated using an observer based approach (Magni and Mouyon, 1994). Although this equivalence is well established for linear systems, it is not the case for nonlinear systems. As the parity space and the observer based methods will both be presented for the design of residual generators from nonlinear models, an introduction is given to observer based residual generation for LTI systems first.
el/(t)
= yet) -yet)
(29)
is directly linked to ez(t), since eu(t) = Cez(t).
4.2 State observer for fault detection and isolation A LTI system without unknown input is considered. In faulty working mode, the system is described by (3), (4) with Ed = 0 and Gd = 0, and the error equations (21), (29) become 8 A system is obeervable if the output trajectories corresponding to distinct initial states differ for at least some time instant.
7 A bold character has been UII6d here for the Laplace operator in order to avoid any confusion with the order of the derivatives and the order of the filter.
41
e:l:(t) = (A - LC)e:l:(t)
+ (E, -
LG I )/(t)(30)
e:l:(O) = Xo - Xo el/(t) = Ce:l:(t) + G II(t)
(31)
The transfer function between I and el/ is obtained by taking the Laplace transform of (30) , (31); it yields:
with !fi(t) substituted for yet). Computing the estimation error as in (30), (31), one deduces that the ith output estimation error is only affected by the ith fault, hence the possibility to isolate sensor faults by analyzing the pattern of zero and nonzero residuals. This type of diagnosis system is called the dedicated observer scheme (Frank and Wunnenberg, 1989).
H'e.(s) = C(sI - A + LC)-l(E,- LGf) + Gf. As long as none ofthe columns of H/e,,(s) is equal to zero, any non zero component of I(t) will affect at least temporarily el/(t). Thus after the transient due to initial conditions has vanished, ev(t) can be used to detect the occurrence of a fault in the system. Indeed, in the absence of fault ev(t) is close to zero and it becomes distinguishably different from zero when I(t) deviates significantly from zero. ev(t) thus Qualifies as a residual signal and (26), (28), (29) is the associated residual generator. Notice that el/(t) now depends on the whole time history of y(r), u(r), for r E IO,tj.
There are two approaches to achieve fault is0lation using Luenberger observers. The first one amounts to choosing the observer gain matrix L in such a way that the trajectories induced by each type of faults are confined to independent subspaces of the el/-space. By projecting ev(t) into each of these subspaces and analysing the magnitude oC each projection, one can achieve fault isolation (Massoumnia, 1986), (White and Speyer, 1987). We shall not go along this line here, as the second approach, based on a bank of observers, offers more flexibility. We present it in the particular case oC sensor faults. The general case is more involved and it will be described in the next section. Thus consider a system of the form (3), (4), without unknown input, in which only sensor faults are taken into account
x(t) = A:c(t) + Bu(t)
(32)
yet) = Cx(t)
(33)
+ Du(t) + I(t)
The output equation can be written component-
= Ci,.x(t) + Di,.U(t) + /i(t) i = 1," ' ,p
where C i ,. (D.,.) denotes the (D).
ith
u
Supervised
system
12 7,
~
~
Obscrvcr 1
·,1
~
~
Obscrvcrp
ew
Fig. 2. Dedicated observer scheme If some of the pairs (Ci,., A) are not observable, another option is to design p Luenberger observers, the ith one being based on all measurements but the ith output !fi(t). In this way, upon occurrence of a single sensor fault, say on sensor j, all residuals will become non zero except for the j'h one. This is the generalized observer scheme which allows one to isolate single faults, but not simultaneous faults. The above two approaches correspond respectively to coding sets of the type described in table 1 and 2. Both schemes make use of full order (or full-dimensional) state observers. However, it is not generally necessary to estimate the whole state vector in order to obtain residual signals. Reconstructing the whole state imposes more stringent conditions than needed, and this motivates the use of functional observers, namely observers that estimate only l (with l < n) specific linear combinations of the state variables.
wise as
Yi(t)
-..,
(34)
4.3 Functional obseroer lor FDI
row of matrix C
In this section and in the remaining part of the paper, it will be assumed that the fault vector I (t) does not enter the output equation. This amounts to representing sensor faults I.(t) = GJI(t) (see (4)) by pseudo actuator faults. It can be achieved
If (Ci ,., A), i = 1"" ,p are observable pairs, p Luenberger state observers can be designed on the basis of equation (32), (34), one Cor each output, as depicted in Fig. 2. They have the form (26)
42
by considering f.(t) as the output of a LT! system driven by an appropriate input, and imposing that the LT! system has no direct feedthrough term (Massoumnia et al., 1989). This method for handling sensor faults is considered in all the literature on the geometric approach to fault detection and isolation. It is adopted here in order to present a unified framework encompassing results on FDI for nonlinear systems, some of which are exclusively based on geometric system theory.
existence of constant matrices P, A, B, Ll and irJ, with P, Ll and L2 different from zero, such that
AB was seen at the end of section 3.3, once coding sets are chosen, the design of residual generators for FDI amounts to determining a set of filters that solve specific fault detection problems. Each filter (residual generator) is designed for appropriate vectors of unknown inputs and faults. Hence the fundamental problem to be solved is based on a model of the form
Indeed, when such matrices exist, one can determine the required observable subsystem as follows. Let e(t) = Px(t) be the state of this subsystem, then (35) yields 9
PA - AP= BC
PE" =0 LtC-irJP=O PE,IO
(4O) (41)
(42)
and the pair (~, A) is observable.
e=p Ax+P Bu+P E"d+P EJf (43) Substituting (39) for P A and (40) for PE", (43) becomes
x(t) = Ax(t) + Bu(t) + E"d(t} + EJf(t} (35) yet) = Cx(t} (36)
e=~ + BCx+ PBu + PE,I
(44)
By (36) the second term in the right band side is nothing but By(t) 80 that (44) finally becomes
The type of filter that will be considered in order to design a residual generator is:
xr(t) = Arxr(t) + Bru(t} + Mry(t) r(t} = Crxr(t} + Dru(t} + Nry(t)
(39)
e= A( + By + PBu + PEJf
(37)
(45)
Defining 'let) = L 1 y(t), equation (41) together with the definition of {et) imply
(38)
'l = L1Cx
The problem which has already been considered in the section on the parity space method is now stated in a more formal way; it is called the fundamental problem of residual generation.
= L2e
(46)
Now system (45), (46) is a subsystem in which input d(t) does not appear anymore. Besides, (L2' A) is observable. A Luenberger observer for this system takes the form
FUndamental problem of residual generation (FPRG) Determine a filter of the form (37), (38) such that
and the output estimation error can be used as a residual
(I) When f(t} = 0 for all t, r(t} asymptotically decays to zero for any inputs u(t) and d(t}. (2) In the presence of a fault (namely when I(t) I 0 for all t ~ to), to being the fault occurrence time), r(t) is affected by the fault; it takes a non-zero value for at least some t ~ to, whatever the initial conditions of the filter and the process.
r(t) = 'let) - ~~(t)
(48)
Indeed, computing the state estimation error e( = yields
e- e
e(t) = (A - L ~)edt) + PEJf(t)
When a solution exists, the procedure to solve the FPRG is made of two steps. First one extracts from model (35), (36) an observable subsystem which has not d(t) 88 input, but for which I(t) is an input. The second step consists in designing a state observer for the observable subsystem. The output estimation error of the observer is a suitable residual. There are several ways to achieve the first step. One of them relies on the fact that a necessary and sufficient condition for the existence of a solution to the FPRG is the
r(t)
= ~e(t)
(49)
(SO)
It is clear from these equations that neither u{t) nor d(t) can affect r(t), while I(t) will do so generally since PE, is non-zero by «(2) and (~,A LL2) is observable. If PE, has full column rank, system (49), (SO) is actually input observable 11 The time argument is omitted in :z:,{,U,II,d &Dd / fOl' the sake of compact_.
43
w.r.t. J(t) (Massoumnia et al., 1989); this means that the magnitude! of a step-like fault J(t) = !1(t) can be determined uniquely from r(t), t 2! 0 when edO) = 0 10 • System (47) is a functional observer for Px(t), hence the title of this section. The set of equations (39)-(41) can be solved either by a method involVing the transformation of the state equations to Kronecker canonical form (Frank and Wunnenberg, 1989) or by an iterative algorithm based on singular value decomposition (Kinnaert, 1999). Another approach to solve the FPRG, in which the two step procedure introduced here does not appear explicitly is the method based on observer eigeDBtructure 888ignment (Chen and Patton, 1999).
be performed around different set points, there are several ways to account for the nonlinear efect. For instance, one can use nonlinear models BB explained in sections 6 and 7, or use a bank of linear models, each one corresponding to a different set point (Arte et al., 1995). Around a given set point, modelling uncertainties due to the linear approximation and/or to the error in the estimate of the model parameters can be accounted for in different ways. One can • use a nonlinear models • use a linear models and represent the higher order terms BB unstructured uncertainties (Mangoubi, 1998), (Mangoubi and Edelmayer, 2000), (Frank and Ding, 1994), • use unknown inputs to represent the efect of model uncertainties (Patton and Chen, 1993), (Chen and Patton, 1999), (Sauter and Hamelin, 1999), • use a set of models, each one corresponding to a different possible setting of the parameters (Lou et al., 1986), (Kinnaert, 1996), • use a model in the form of transfer functions with parameters in intervals (Hamelin and Sauter, 2000)
Yet another method to solve this problem is the geometric approach. Its interest is that it can be generalized to a wide class of nonlinear systems. It yields both a necessary and sufficient condition for the existence of a solution to the FPRG and a systematic design method. The results are expressed in terms of subspaces called (C,A)-unobservability subspaces. The design method also essentially consists in extracting an observable subsystem which hBB not d(t) BB input; the operations required to achieve this goal can be performed using backward stable algorithms deduced from (Van Dooren, 1981) as shown in (Alexandre and Kinnaert, 1993).
The existence of meBBurement noise, and possibly state noise to account for disturbances with a known spectrum also requires adaptation of the theory presented in the previous section. Indeed, the residual can no longer decay to zero in the presence of noise; thus a different objective has to be imposed in its design (Nikoukhah, 1994), (Mangoubi, 1998). Besides, simple comparison of the residual to a threshold becomes impracticable when noise is important, BB the efect of a fault may be buried in noise. One has to resort to statistical change detection algorithms in this C8Be (Willsky and Jones, 1976), (Basseville and Nikiforov, 1993). The problem of robustness with respect to modelling uncertainties and noise effects is thus a very broad area.
Considering arbitrary matrices A, C, Ed, EJ of ap. propriate dimensions, the FPRG h8B a solution if and only if RJ + Rd < R and Rd < P (Massoumnia et al., 1989). In the above results, it is 8B8umed that the plant model is perfectly known and that the me8Burements are exact in fault free working mode. In practice this does not hold BB discussed in the next section.
5. DEALING WITH MODELLING UNCERI'AINTIES AND MEASUREMENT NOISE
Here we have chosen to exploit nonlinear models in order to decrease the effect of modelling uncertainties. Therefore, the exterunon of the theory presented for linear systems will successively be performed for polynomial nonlinear systems (in the parity space framework), for bilinear systems and for control affine nonlinear systems (using an observer based approach).
The linear time invariant (LT!) model used in the previous sections is suitable to describe the plant behaviour around a specific set point. AB most plants are nonlinear, this LT! model corresponds to a first order approximation of the nonlinearity in a Taylor series expansion. The further away the process moves from its nominal operating point, the more important the efect due to the higher order terms becomes. If supervision must
6. PARITY SPACE APPROACH FOR POLYNOMIAL NONUNEAR SYSTEMS The presentation in this section is in line with (!Sidori et al., 2001). A model of the form (1), (2) is considered, where g(x(t), u(t), d(t), J(t» and
IOTbe only type of fault that may not affect r(t) c:orre8pOnds to inputs U80ciated to the tranamisBion zeros of the transfer matrix between I and r in this cue.
44
Finally, let us notice that the problem of generating polynomial analytical redundancy relations can actually be solved for a more general class of nonlinear systems, namely systems described by polynomial differential algebraic equations. An informal presentation of the method based on characteristic sets for this type of model is presented in (Zhang et al., 1998)
h(x(t), u(t), d(t), J(t» are polynomial functions of their arguments. Let Yj denote the lit. output: Yj = h;(x(t), u(t), d(t), J(t». By deriving this expression si times and substituting (1) for the derivative of the state, one gets equations equivalent to (8)-(11) in the linear case (a slight generalization is considered here since the order of the derivative might differ from one output to the other, which was not the case in the linear framework). These equations can be written in a compact form as: Y1;
= Hi(x, U.j,D.i,F.j )
7. OBSERVER BASED APPROACH FOR NONLINEAR SYSTEMS
(51)
Y~ is the vector made of Yj(t) and its derivatives up to order Si, U.jI D.J and F. J are defined as
Before dealing with fault detection, some prerequisites on observability for nonlinear systems are introduced.
in the linear case. The concatenation of equations (51) for j = 1"" ,p gives a set of E~_l (Si + 1) equations
7.1 Observability oJ nonlinear systems
Y = H(x, U., D., F.)
(52)
Consider the nonlinear system
where s = maxi Si and Y = [y(.l)T y(.2)T •.• y(.p)T]T. Analytical redundancy relations are obtained by eliminating x and D. from (52). This yields polynomial relations: P(Y,U.,F,) =0
(53)
Due to the polynomial structure, this expression , can be decomposed as
P(Y, u" F,) = Pc(Y, U,) - Pe(Y, U., F,) = 0 (54)
where any term in the polynomials which make the rows of Pe(Y, U" F,) is of degree at least one with respect to an entry of F,. Pe(Y, U" F,) is thus equal to zero in faultless working mode. Hence Pc(Y, U.) can be considered as a parity vector, and one can define a residual vector as
r(t) = Pc(Y(t), U,(t»
X=g(x,u)
(56)
y=h(x)
(57)
where 9 and h are smooth nonlinear functions of their arguments. Let U denote the set of admissible inputs. By definition system (56), (57) is observable if, for every pair of initial states (XO , Xl), xD =F Xl, there exists an admissible input signal u : [0, T) - U and a time instant t E [0, T] such that Y(XO,u,t):f. y(xl,u,t) where y(xO,u,t) denotes the unique output trajectory of (56), (57) for the initial condition xO and the input u. An input u : [0, T] - U that distinguishes any pair of initial states is a universal input on [0, T). System (56), (57) is said to be uniformly observable if, for every T > 0, every admissible input u : [0, T) - U is a universal input on (0, T).
Bilinear systems are not generally uniformly 01>sevable. Indeed, consider the following simple example:
(55)
Xl =UX2
To achieve fault isolation, structured residuals have to be computed. This amounts to choosing coding sets and performing the above design with a vector of unknown inputs augmented by the faults that should not affect the residual.
X2=0
y=Xl
(58) (59) (60)
Clearly, if u(t) = 0 for all t, no information on X2(t) can be deduced from the measurement of yet) and hence this state is not observable. Any piecewise-continuous non-zero input will make X2 observable. An input for which the system is not observable is called a singular input. A specific class of non singular inputs will be used to state a convergence result for bilinear systems in the next section, namely ~ed persistently exciting inputs (Kinnaert, 1999). Roughly speaking
There are several ways to perform the elimination of x and D. that leads to (53). One approach relies on Groebner basis (Comtet-Varga, 1997); another uses characteristic sets and rutt's algorithm (Ljung and Glad, 1994). A study of existence conditions for parity functions and of fault sensitivity is performed in (Staroswiecki and ComtetVarga, 2(01);
45
such inputs guarantee the observability of the system over some horizon T at any time.
and the system
7.2 Fault detection Jor bilinear systems
{et) = A(u){(t)
(68)
'1(t) = ~e(t)
(69)
is observable. Continuous-time bilinear systems described by the following equation are considered
n"
x(t) = A(u)x(t) + Bu(t) + ~)Ftx(t)
{B(t) = A(u){B(t) + pB Bu(t) + B(u)y(t)
+ Et)~(t)
n,
+ 2: pB(F/ x(t) + EI)J(t)
i=1
n,
+ "L(F! x(t) + E{)/;(t) yet) =Cx(t)
(62)
where A(u) = Ao + 2.::1 UiAi and it is assumed that the input vectors u(t), d(t) and J(t) are such that every trajectory is defined on the whole time interval (0,00(. In this case again, the design of a residual generator can be separated in two steps. First an observable subsystem which has not d(t) 88 input, but for which J(t) is an input is extracted. Next a suitable observer is designed for the resulting subsystem.
Two types of subsystems have been used in the literature: linear time invariant subsystems up to output injection (Yu and Shields, 1996), (Mechmeche et al., 1994), and bilinear subsystems up to output injection (Kinnaert, 1999), (Hammouri et al., 2(01). The approach based on a LT! subsystem leads to a LT! residual generator, which can be implemented easily. However, the class of bilinear systems for which a residual generator based on a LT! subsystem exists is smaller than the class of systems for which a residual generator based on a bilinear subsystem exists (Kinnaert, 1999). The latter option is briefly presented here.
,!here A(u) = B(u).
...to + 2.::1 AiUi
and similarly for
Defining '1B(t) by '1B(t) = Lfy(t) = LfCx(t), one gets
(71) As the dynamics of (70) is bilinear, observers for bilinear systems now have to be considered. Contrary to the linear case, observability of system (70), (71) depends on the input u(t) 88 W88 discussed in subsection 7.1. Let u(t) be a regularly persistent input for system (68), (69). For such an input, it can be proved that the following system is an exponential observer 11 (Bomard et al., 1988) for system (70), (71) when J(t) = 0 for all t:
l
(t) = A(u){B(t) + pB Bu(t) + B(u)y(t) +R(t)-lCT ('1B(t) - Lf{B(t»
(72)
k(t) = OR(t) - A(U)TR(t) - R(t)A(u) + LFLf (73)
nf nf
where eB(O) E mn:' , R(O) is a x symmetric positive definite matrix, and 6 E m+. The output reconstruction error
r(t) = '1B(t) - L~{B(t)
nf
Without loss of generality, the -dimensional state of this subsystem can be taken 88 a set of linear combinations of the entries of x(t}
exponentially decays to zero whatever d(t),{(O), {B(O) and u(t) provided the latter is regularly persistent. The reader is referred to (Kinnaert, 1999) for a discussion regarding the sensitivity of r(t) to the fault J(t).
(63) The extraction of the bilinear subsystem consists in determining the matrices pB, Ai, Bi, i = 0, ... , rn, Lf and Lf with pB, Lf and Lf different from zero such that
nf,
pB ~ _ AipB =BiC
(70)
i=1
(61)
i=1
Using an appropriate definition of the sensitivity of a residual to a fault, a precise statement of the fundamental problem of residual generation for bilinear systems can be obtained. A necessary and sufficient condition for the existence
i = 0,··· ,m (64)
pB(Et Ft)=O Lfc - Lf pB = 0
i=l,···,nd
(65) (66)
F!) i: 0
i = 1"", nJ
(67)
pB(E{
From (64) - (67), one deduces
11ThiB means that O(B(t) _ iB(t)II :S ~ e-" for all t > 0, where >. is a. positive conatant which depends on the initial con
46
of a solution to this problem can then be expressed in a very similar form as for the linear case (Hammouri et al., 2001) by introducing the notion of (C, A)-unobservability subspaces where A stands for (.40,"', Am). An algorithm to check such a geometric condition can be determined using standard algebraic tools. A similar methodology can be used to handle state affine systems, namely systems of the form:
x(t) = A(u)x(t) + Ed(X)d(t)
+ EJ(x)J(t) (74)
yet) = Cx(t)
(75)
where A(u), Ed(X) and E,(x) are matrices of which the entries are smooth functions of the components of u and x. The required concepts have been developed in (Hammouri et al., 1998), (Hammouri et al., 2(00), (De Persis and Isidori, 2(00). The main difference with the bilinear case is that symbolic computations must be performed to implement the design method.
7.3 Fault detection Jor control affine nonlinear system Systems modelled by equations of the form (76), (77) below are now considered
x = go (x) + gu(x)u + 9d(X)d + 9,(X)J
(76)
y = hex)
(77)
Vn(
r = hr(z, h(x»
= 4>n( (Vlt t12, .•• , Vn (' y, U)
Then this system is uniformly observable, and a high gain observer can be designed to estimate the state and the output q, under some additional technical conditions(De Persis and Isidori, 2001). The residual is again obtained as r = q - h(€) where €is the estimate provided by the observer.
e
(78)
8. CONCLUSION
(79)
The parity space approach and the observer based method have been presented for the design of residual generators, first on the basis of a LT! model, next from a nonlinear plant model. Although both approaches are equivalent for LT!
(:~~~~h(X») + (:~~~:h(X»)U + (gd(X») d + (g'(X») J
= 4>1 (VI, t12, y, u)
q = h l1 (v).
with a similar smoothness condition on the different functions as above, and with 90(0,0) = 0, h r (0,0) = 0, such that the output r of the cascaded system
(;) =
where the input d does not appear anymore. Should an asymptotic observer exist for this system, one could proceed as for the linear or the bilinear case to obtain a residual generator. Unfortunately, local weak observability is not sufficient to guarantee the existence of such an observer, and extra hypotheses are needed at this stage. For instance, let us assume that system (82), (83) with J = 0, can be brought by a global change of coordinates v = ~(e) into the form
~=4>2(VI,t12,V3,Y,U)
The problem to be solved is the determination of a filter of the form
i = 90(Z, y) + 9:(Z, y)u hr(z,y)
By resorting to differential geometry, and in particular to the notion of unobservability distribution (the generalization to the nonlinear framework of (C, A)-unobservability subspaces), a necessary condition for the existence of a solution to this problem can be obtained (De Persis and Isidori, 2001). Algorithms are available to check this condition, and a methodology can be deduced to extract from the original system (76),(77) a locally weakly observable 12 subsystem of the following form
VI
with the n-dimensional state x defined in a neighborhood X of the origin, inputs u, d and output y with the same dimension as before, and scalar fault J. All the entries in the vectors 90(X),9,(X),h(x) and matrices gu(X),9d(X) are smooth functions oftheir arguments and 90(0) = 0, h(O) = O. The last two conditions can always be achieved by an appropriate translation of the state and output.
r =
is affected by J, is not affected by d, and asymptotically decays to zero when J is identically equal to zero, whatever the input u.
13 Roughly speaking a system is weakly obeervable if any state can be distinguished from its neighbors, and it is locally obeervable if a IItate can be distinguished from its neighbol'1l with an input that keeps the IItate trajectory close to the initiallltate (Hermann and Krener, 1977).
(SO) (81)
47
systems, this is not proved for nonlinear systems, in general.
et al., 1977), automobile (Gertler et al., 1995), (Isermann et al., 2000) , (Nyberg, 1999), diesel engine actuator (Blanke et al., 1995), process industries (Kinnaert et al., 2000), ...
The parity space approach can be applied to polynomial nonlinear systems. It is systematic, and existing symbolic computation softwares can be used to perform the elimination of the unknown variables. However, even for models of moderate size (5 to 10 states), the parity function resulting from the elimination might contain several tens of terms. These terms might have very different orders of magnitude, which could make the evaluation of the residual ill-conditioned. Another drawback of the parity space approach is the need for the derivatives of the signals. Such derivatives are sensitive to noise. In practice they have to be obtained most often from sampled data. Obtaining suitable estimates for high order derivatives in this context might not be an easy task.
The design of diagnosis systems based on analytical models is now reaching a certain stage of maturity. However, several research directions are still widely open. One of them is the evaluation of the reliability of such diagnosis systems and the way to take into account reliability requirements at an early stage in the design. Another one is the study of the interplay between the diagnosis system and the controller in a fault tolerant control scheme.
9. REFERENCES
Alexandre, P. and M. Kinnaert (1993). Numerically reliable algorithm for the synthesis of linear fault detection and isolation filters based on the geometric approach. In: Proceeding! of the 1999 IEEE Conference on Sy!tems, Man and Cybernetics. Vo!. 5. pp. 359-
On the other hand, for the observer based approach, systematic methods relying exclusively on algebraic operations exist for bilinear systems as seen in section 7.2, and for systems with polynomial nonlinearities of higher order (Isidori et al., 2001). In the latter case however, quite stringent conditions are imposed on the system to be able to design an observer with linear error dynamics. For state and control affine systems, the observer based approach requires symbolic computations to extract from the original system a subsystem which is not affected by certain faults. Once the subsystem is found, the design of the observer is systematic in the state affine case, but not for control affine systems.
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Arte, Ph., Y. Peng and M. Kinnaert {1995}. Parity space and statistical test for fault detection in nonlinear systems : a multi- model approach. In: Proceedings of the 9rd European Control Conference. Rome. Basseville, M. and LV. Nikiforov (1993). Detection of Abrupt Changes: Theory and Applicatiom. Prentice Hall. Englewood Cliffs, N.J. Besa.n~n, G. {1999}. A nonlinear observer with disturbance attenuation and its application to fault detection. In: Proceedings of the 5th European Control Conference. Karlsruhe, Germany. Blanke, M., S. Bogb, R.B. Jorgensen and R.J. Patton (1995). Fault detection for diesel engine actuator - a benchmark for fdi. Vo!. 5. pp. 1763-1768. Bornard, G., N. Couenne and F. Celle (1988). Regularly persistent observers for bilinear systems. In: New trends in nonlinear control theory, Lecture notes in control and in/orT1l4tion science. Vol. 122. Springer-Verlag. pp. 13(}-140. Chen, J. and R.J. Patton (1999). Robust modelbased fault diagnosis for dynamic systems. Kluwer Academic Publishers. Chow, E.Y. and A.S. Willsky {1984}. Analytical redundancy and the design of robust failure detection systems. IEEE 7l-am. Automatic Control 29, 603~14. Comtet-Varga, G. (1997). Surveillance des systemes non lineaires - application awe machines asynchrones. PhD thesis. Universite des Sciences et Thchnologies de Lille.
Only perfect decoupling of the residual from the faults not to be detected has been considered here. This requirement can be relaxed by imposing that the effect on the residual, of the faults not to be detected be sufficiently small. This yields optimization problems that have been considered both for linear (Frank and Ding, 1997), (Chen and Patton, 1999) and nonlWear models (Besanc;on, 1999). Nonlinear adaptive observers have also been used to estimate fault signals instead of assuring fault decoupling (Xu and Zhang, 20(2). The nonlinear models considered in this paper are often deduced from physical laws. For certain pr~ cesses, physical descriptions might be too involved or poorly known, an a black box model like a neural network might be a suitable alternative. For examples on how to use neural networks in the design of fault detection and isolation systems the reader is referred to (Zhang et al., 2002), (Chen and Patton, 1999). Fault detection and isolation systems based on analytical models have been used for a variety of applications including aeronautics (Deckert
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De Persis, C. and A. Isidori (2000). An addendum to the discussion on the paper "fault detection and isolation for state affine systems" . Eur. J. Control 6, 290-294. De Persis, C. and A. Isidori (2001). A geometric approach to nonlinear fault detection. IEEE nuns. Automatic Control 46, 853-865. Decirert, J.C., M.N. Desai, J.J. Deyst and A.S. Willsky (1977). F-8 dfbw sensor failure identification using analytic redundancy. IEEE nuns. Automatic Control 22, 795-803. Frank, P. and J. Wunnenberg (1989). State estimation schemes for instrument fault detection. In: Fault Diagnosis in Dynamic Systems - Theory and Applications (P. Frank R. Patton and R. Clark, Eds.). Prentice Hall. London, U.K.. pp. 21-45. Frank, P.M. and X. Ding (1994). Frequency domain approach to optimally robust residual generation and evaluation for model-based fault diagnosis. Automatica 30, 798-804. Frank, P.M. and X. Ding (1997). Survey of robust residual generation and evaluation methods in observer-based fault detection systems. J. of Process ControlT, 403-424. Gertler, J . (1993). Residual generation in modeIbased fault diagnosis. Control- Theory and Advanced Technology 9(1), 25~285. Gertler, J. (1998). FaultDetection and Diagnosis in Engineering Systems. MarceI Dekker. New York. GertIer, J . and D. Singer (1990). A new structural framework for parity equation based failure detection and isolation. Automatica 26, 381388. Gertler, J. and M.M Kunwer (1993). Optimal residual decoupling for robust fault diagnosis. In: Proceedings of Tooldiag '93. Toulouse, France. GertIer, J ., M. Costin, X. Fang, Z. Kowalczuk, M. Kunwer and R. Monajemy (1995). Modelbased diagnosis for automotive engines - algorithm development and testing on a production vehicle. IEEE nuns. on Control System Technology 3, 61~9. Hamelin, F. and D. Sauter (2000) . Robust fault detection in uncertain dynamic systems. Automatica 36,1747- 1754. Hammouri, H., M. Kinnaert and E .H. El Yaagoubi (1998). Fault detection and isolation for state affine systems. Eur. J. Control 4, 2-16. Hammouri, H., P. Kabare and M. Kinnaert (2000). A differential geometric approach to fault detection and isolation for state affine systems. In: Proceedings of the 2000 IEEE CDC. Hammouri, H., P. Kabore and M. Kinnaert (2001) . A geometric approach to fault detection and isolation for bilinear systems. IEEE nuns. Automatic Control 46(9), 1451-1455.
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Hermann, R. and A.J. Krener (1977). Nonlinear controllability and observability. IEEE nuns. Automatic Control 22, 728-740. Isermann, R. (1993). Fault diagnosis of machines via parameter estimation and knowledge processing - tutorial paper. Automatica 29,815836. lsermann, R., R. Schwarz and S. Stolzl (2000) . Fault tolerant drive-by-wire systems: concepts and realizations. In: Preprints of SAFEPROCESS'2000. Budapest, Hungary. pp. lIS. Isidori, A., M. Kinnaert, V. Cocquempot, C. De Persis, P.M. Frank and D.N. Shileds (2001). Residual generation for fdi in non-linear systems. In: Control of Complex Systems (K. As-. trom, P. Albertos, M. Blanke, A. Isidori, W. Schaufelberger and R. Sanz, Eds.) . Springer. Kinnaert, M. (1996). Design of redundancy relations for failure detection and isolation by constrained optimisation. International Journal of Control 63(3), 6O~622. Kinnaert, M. (1999). Robust fault detection based on observers for bilinear systems. Automatica 35, 182~1842. Kinnaert, M., D. Vrancic, E. Denolin, D. Juricic and J . Petrovcic (2000). Model-based fault detection and isolation for a gas-liquid separation unit. Control Engineering Practice 8, 1273-1283. Ljung, L. and T. Glad (1994). On global identifiability for arbitrary model ps.rametrizations. Automatica 30(2), 265-276. Lou, X., A.S. WHlsky and G.C. Verghese (1986). Optimally robust redundancy relations for failure detection in uncertain systems. Automatica 22, 333-344. Magni, J-F. and Ph. Mouyon (1994) . On residual generation by observer and parity space approaches. IEEE 7rans. Automatic Control 39, 441-447. Mangoubi, R.S. (1998). Robust estimation and failure detection: a concue treatment. Springer. Mangoubi, R.S. and A. M. Edelmayer (2000) . Model based fault detection: the optimal past, the robust present and a few thoughts on the future. In: Preprints of SAFEPROCESS '2000. Vol. 1. pp. 64-75. Massoumnia, M. (1986). Geometric approach to the synthesis of failure detection filters. IEEE nuns. Automatic Control AC-31 , 839-846. Massoumnia, M.A. and W.E. Vander Velde (1988). Generating parity relations for detecting and identifying control system component failures. AIAA Journal of Guidance and Controlll,60-65. Massoumnia, M.A. , G.C. Verghese and A.S. Willsky (1989) . Failure detection and iden-
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Copyright (;> IFAC Fault Detection, Supervision and Safety of Technical Processes, Washi ngton, D.C., USA, 2003
[: 0
[>
Publications www.elsevier.com/locate!ifac
FAULT DIAGNOSIS BASED ON ANALYTICAL MODELS FOR LINEAR AND NONLINEAR SYSTEMS - A TUTORIAL Michel Kinnaert Dpt. of Control Engineering and System Analysis, Universite Libre de Bruxelles. CP 165/55, 50 Av. F. D. Roosevelt, B-1050 Brussels, BELGIUM
Abstract: The diagnosis systems considered in this paper rely on the inconsistency between the actual process behaviour and its expected behaviour as described by an analytical model. The inconsistency is exhibited in signals called residuals. Two methods for residual generation are presented in a tutorial way: the parity space and the observer based approaches. Linear and nonlinear models are successively considered as a basis for the design of the residual generators. Copyright © 2003 IFAC Keywords: Fault detection and isolation, linear and nonlinear systems, observer, analytical redundancy, residual generation, parity space method.
1. INTRODUCTION
it increases the weight which is a critical issue in some applications, and it is quite costly.
For the complex highly automated systems encountered in process industries, in aeronautics or power plants for instance, it is fundamental to be able to monitor the health of the installation at each time instant in order to detect incipient faults and to locate the deteriorated components. Indeed, for safety critical systems such as nuclear power plants and airplanes, the total failure of a component can cause hazard to personnel or to the population and hence it must be avoided at any cost. Moreover the early detection of a malfunction allows one to plan the required maintenance actions and decreases the number of emergency shutdowns of a process, which are often very costly.
Most industrial monitoring systems that are not based on hardware redundancy rely on the comparison of measured signals to specified thresholds. They do not exploit the correlation existing between the different measured signals. Therefore they only allow the detection of important deviations from normal operating conditions and not incipient changes. These drawbacks motivate the development of diagnosis systems based on analytical models. The main idea behind such systems is to check the consistency between the measurements of different variables taken on the supervised process, and the expected behavior of this process as described by an analytical model. As shown in figure 1, typical diagnosis systems are made of two parts: a residual generator and a decision making module. The residuals are signals that are generated by processing the measured plant inputs and outputs. In the absence of fault, they deviate from zero only due to modelling uncertainties and measurement
For safety critical systems, the present monitoring tools heavily rely on hardware redundancy. For instance three to four sensors are often used to measure the same quantity, and a voting scheme determines its most likely value. This requires additional space to accommodate the equipment,
37
noise. When a fault occurs, the deviation from zero is such that the new condition can be distinguished from the faultless working mode. The role of the decision module is to determine whether the residuals differ significantly from zero, and, from the pattern of zero and non-zero residuals, to decide which are the most likely faulty components.
Mcaaurcd in
U
11
where 9 and h are smooth 1 nonlinear functions of their arguments, x(t) is the state vector, u(t) the vector of known inputs (the manipulated inputs for instance), y(t) the vector of measured outputs, d(t) the vector of unknown inputs and f(t) the vector of faults. The dimensions of the different vectors are respectively n, rn, p, nd, and nl' In faultless working mode, f(t) is zero for all t. The unknown inputs correspond to unmeasured disturbances such as waves acting on a ship or wind gusts on an airplane for instance. When system (1), (2) is working around nominal operating conditions that are characterized by an equilibrium state, its behaviour can be described by a linear time invariant (LTI) model of the form
±(t) = Ax(t) + Bu(t) + Edd(t) + EJf(t) (3) y(t) = Cx(t) + Du(t) + Gdd(t) + GJf(t) (4) Fig. 1. Structure of a diagnosis system In this paper the main focus is on the design of residual generators for systems described by deterministic analytical models. The first part contains a tutorial presentation of both parity space and observer based approaches to the design of residual generators for linear systems. A very limited knowledge of system theory is expected from the reader for this part (sections 2 to 4). As most processes are inherently nonlinear, the residual generators based on linear models work properly provided the process is running in the operating region for which the linear approximation is valid. To relax this condition, it is necessary either to account for modelling uncertainties due to the linear approximation or to exploit the nonlinear model of the process instead of its linear approximation around an equilibrium point. After a discussion of different ways to handle modelling errors in section 5, various classes of nonlinear models are considered and the ways to generalize either the parity space method or the observer based method for residual generation are presented (see sections 6 and 7). Some open problems are pointed out in the conclusion.
2. MODEL OF THE SUPERVISED PROCESS
The fault f(t) appears as an additional input in the linear model, and hence it is called an additive fa.ult (as opposed to a multiplicative fault which consists of a. change in the entries of the matrices A,B,C, ... ) . It might correspond to a sensor bias, the jamming of an actuator or a leak for instance. Indeed, for a sensor bias on the first measurement, occurring at time to, it suffices to take El = 0, GI = (1 0 ···0 )T, and f(t) = al(t - to) where "a" is the bias magnitude and l(t) is the unit step function (l(t) = 0 for t < 0, and let) = 1 for t > 0). The jamming of the first actuator can be modelled with El = B .,1> G I = 0, and f(t) = HI - UI(t), where B.,1 denotes the first column of matrix B, UI(t), the first component of vector u(t), and HI is the value at which the actuator is stuck. A leak in an hydraulic system can be modelled by a supplementary input in a model, namely the flow of the leaking fluid. It can thus be expressed in the form of model (3},(4) when a linear model is used. Multiplicative faults are not considered in this paper. The reader is referred to (Isermann, 1993) and (Zhang et al., 1994) for an introduction to this topic.
Using physical laws, a large class of engineering systems can be modelled by differential equations of the form
±(t) = g(x(t), u(t), d(t), f(t» yet) = h(x(t), u(t), d(t), f(t»
Although it is not indicated explicitly, it is assumed that x(t), u(t), d(t), f(t) and yet) represent deviations from their value at equilibrium in (3), (4). The linear model is obtained by performing a Taylor series expansion of the functions g(x(t), u(t), d(t), J(t» and h(x(t), u(t), d(t), f(t» around the equilibrium point, and keeping the first order terms only.
(1)
1 A smooth function haa continuous partial derivative8 of any order with respect to its arguments.
(2)
38
(8) y=Cx +Du y=Cx +Du= CAx+ CBu+ Du (9) ii = CA 2x + CABu + CBu + Dil (10)
3. PARIT Y SPACE METH OD FOR LTI SYSTEMS
3.1 Introduction A fundamental notion on which diagnosis sy~ tems based on analyti cal models rely is analytic al redundancy. Analytical redund ancy relations are equations that are deduced from an analytical model and which solely involve measured variables. They must be fulfilled in the absence of fault. As a very simple example, consider the relation
between the velocity and the acceleration of a mobile device: v(t)
= a(t)
y(.)
+CBu (·-l)
(11)
f,
(5)
Y.(t) = O.x(t)
+ T;'U.(t)
(12)
Assuming that the derivatives of u(t) and y(t) can be obtaine d from the available measurements (see section 3.4 for this issue), the only unknown in (12) is the state vector x(t). When 8 is chosen sufficiently large, the left null space of matrix 0., namely the space generat ed by the row vectors w. which fulfil
(6)
may be used to detect sensor faults (the index m indicates measurement). r(t) is zero for all t in the absence of fault and it becomes nonzero upon occurrence of sensor faults. Indeed, let vm(t) = v(t) + I,,(t) denote the measured velocity, where Iv(t) is non-zero upon occurrence of a sensor fault, and consider a similar expression for the acceleration: am(t) = a(t)+ 10 (t). Substit uting these expressions in (6) yields
r(t) = j,,(t) -/a(t)
+ Du(·)
where the time argume nt has been omitted for compactness of the notatio ns. Letting Y.(t)(U .(t» denote the vector obtaine d by stacking y(t) (u(t» and its derivatives up to order 8, T,'U be the lower triangu lar block Toeplitz 3 matrix with first column [DT (CB)T ... (CA(· -l)B)T f, and O. = equatio ns (8)-(11) can [CT (CAf ... (CAa)T as form t be written in a compac
If both variables are measured perfectly, and the derivative of the velocity can be determined exactly, the quantit y
r(t) = vm(t) - am(t)
= CA·x + CAa-1 Bu + ...
w.O.
= 0,
(13)
is non-zero. Multiplying (12) on the left by one such vector yields (14) w.(y.(t ) - T;'U.( t» = 0
(14) is an analytical redund ancy relation, namely a relation that only involves known variables and derivatives of known variables. This motivates the definition of the quantit y r .(t) as
(7)
which takes at least tempor arily non-zero values when Iv(t) and/or la(t) is non-zero 2 • r(t) is said to be a residual that allows detection of faults on the velocity and acceleration sensors and (5) is an analytical redund ancy relation.
r.(t) = w.(y.(t ) - T;'U.( t»
(15)
Obviously, r.(t) is null in the absence of fault. To determ ine its value in faulty operati ng condition, the equatio n corresponding to (12) in the presence of fault is used:
3.2 Analytical redundancy relations and parity
Y.(t) = O.x(t)
+ T;'U.(t) + T,' F.(t)
(16)
junctions where T,' is the lower triangu lar block Toeplitz T matrix with first column and F.(t) is a E,)T] l) (CA(·[Gr (CE,)T ... up to order ives derivat its and vector made of I (t) yields (15) into Y.(t) for (16) 8. Substit uting
Our aim is now to determ ine how to generate analytical redund ancy relations to monitor systems described by a model of the form (3), (4). To this end let us consider the successive derivatives of the output y(t). For the sake of simplicity, the case where there are no unknown input (Ed = 0) and no fault (/(t) = 0) is first considered. From (3) and (4), one deduces
r.(t) =
3
ate 2 The patholog ical caae where both fault. compens each other is not consider ed 88 it is improba ble in practice.
the matrix [
w.r! F.(t)
~Z CA(a-l) B ...
39
(17)
As long as
w:r/ i- 0,
Table 1. Effects of the faults on the residuals - "diagonal" coding set
(18)
1
r,(t) is an indicator of the presence of a fault; it is called a residual. (15), (16) and (17) define a parity function or parity check, r,{Y,{t), U,{t», namely a function of yet), u(t) and their derivatives that is equal to zero in the absence of fault and is non-zero in the presence of a fault. The set of all parity functions involving signal derivatives up to order 8 can be obtained by linear combinations of the entries of the parity vector~ f,(t) defined as f,(t) = n,(y'(t) - 'P.U,(t»
o
o
(19)
n,
+ 'P.U,(t) + 7,d D,(t)
w,(7,' F,(t) + 7,d D,(t»
1
A diagnosis system should not only detect faults but also isolate them, namely determine which fault(s) occurred. To this end, residuals that are sensitive to certain faults and insensitive to others must be designed. Assume for instance that n I faults have to be isolated. If residuals can be designed in such a way that the ith residual is only affected by the ith fault, then fault isolation can be achieved easily. The corresponding situation is depicted in table 1 in the case of three faults. A 1 indicates that the fault in the corresponding column affects the residual of the corresponding row. Each column of the table defines a code associated to the corresponding fault. The set of fault codes forms a coding set.
(20)
where the new notations 7,d and D,(t) are defined in a similar way as 7,' and F,(t) . Substituting this expression into (15) now yields r ,et) =
o
3.3 Fault isolation
When unknown inputs are present, it is necessary to assure that the residual is not affected by such inputs. With Ed. i- 0, (16) becomes
+7,' F,(t)
o
1
time systems. In that framework, they are defined as relations between the measured inputs and outputs over a fixed time horizon, that must be fulfilled in the absence of fault. Here we have chosen to work with Continuous-time models to ease the generalization to nonlinear systems. The parity space approach to residual generation was also generalized to transfer function models, in which case the residual may depend on the whole set of prior data, and not only on the data over a fixed time horizon (Massoumnia and Vander Velde, 1988), (Gertler, 1993), (Gertler, 1998)
where n, denotes a basis 5 for the left null space of 0,. The dimension of this subspace is np = (8+ l)p - rankO,. 8 must thus be chosen sufficiently large to assure that np is positive. Any suitable row vector w, has the form w, = v,n .. where v, is an np-dimensional row vector chosen so that v,n,T/ i- o. As n, is not unique, there exists an infinite number of parity vectors f,. The space generated by these vectors is called the parity space.
Y,(t) = O,x(t)
o
o
(21)
For the coding set of table 1 to be achievable, the number of faults to be isolated cannot be larger than the number of measured outputs, as will be clear from the next paragraph. If this condition is not met, it might still be possible to perform fault isolation, but under the hypothesis that only a single fault can occur at a time. In this case, residuals may be designed so that the ith residual is affected by all faults, except the ith one (see table 2). Clearly, when simultaneous faults occur, all residuals become non-zero for this type of coding set, which prevents isolation 6 • More generally, isolation of single faults can be achieved by choosing a coding set with no identical oodes. Yet, for avoiding erroneous isolation when the reaction of one residual is not as expected upon occurrence of a fault (due to noise, model uncertainty or time history of the fault signal), some coding sets are
To assure that r,(t) is not affected by d(t), the condition (22)
n,
must be imposed on w, besides (13) and (18). A necessary condition for the existence of a row vector w, that meets the three conditions is that the number of unknown inputs ne/. be lower than the number of measured outputs p. Remark Analytical redundancy relations were presented in (Chow and Willsky, 1984) for linear discrete 4 It is assumed that there is no state variable that is exclusively controlla.ble from the fault (see (Nyberg, 1999),
An alternative approach to handle multiple faults might be to define a different code for each fault combination but this would lead to large residual aeta.
page 2(9).
11
Such a basis can be computed by performing a singular value decomposition of matrix 0 •.
&
40
Table 2. Effects of the faults on the residuals - isolation of single faults
o
1
o
4. OBSERVER BASED APPROACH FOR LTI SYSTEMS
4.1 Refresher on Luenberyer state observer
1 1
o
preferable to others as discussed in (Gertler and Singer, 1990), (Gertler and Kunwer, 1993). Residuals which are designed according to a specific coding set are called structured residuals. The method to design residuals according to a given coding set is a straightforward extension of the results presented in the previous section. Indeed, for a given residual, the fault vector f(t) can be separated into two parts: fG(t) the faults that should affect the residual, and f no. (t) the faults that should not affect the residual. The method of the previous section can then be applied to design an appropriate residual by substituting respectively fG(t) and [d(t)T fno(t)Tf for the vectors f(t) and d(t) in section 3.2 . From the existence condition stated at the end of that section, it is seen that the dimension of [d(t)T fno(t)Tf must be lower than the number of outputs for a solution to exist.
Let us consider a LTI system with no unknown input and in faultless operation. It is described by the state space model
:t(t) = Ax(t) + Bu(t)
(24)
yet) = Cx(t) + Du(t)
(25)
A state observer is a filter that receives as inputs the signals u(t) and yet) and generates an estimate of the state x(t) denoted x(t). A Luenberger state observer is described by the following differential equation: ~(t) = Ax(t)
X(O)
+ Bu(t) + L(y(t) -
•
r.(s)
~~,--
- (s + {3)-
= (A -
LC)ez(t)
(27)
ez(O) = Xo - Xo When the system is observable 8 , matrix L can be determined so that the eigenvalues of (A - LC) take fixed values. Imposing that all eigenvalues have negative real part ensures that ez(t) asymptotically decays to zero. From (25), an estimate yet) of the output is deduced as
Due to the presence of derivatives in (15), r.(t) cannot be generated as such from yet) and u(t). Only a filtered version, say r{ (t), of this signal can be obtained. As an example, using Laplace transform expressions, r{ (8) can be generated by
r J (s) -
(26)
where L is a n x p constant matrix of which the entries are chosen in such a way that the transient in the observation error ez(t) = x(t) - x(t) decays sufficiently fast to zero. The dynamics of the error is obtained by subtracting (26) from (24) and substituting (25) for yet). It yields
ez(t) 3.4 Implementation of parity checks
Cx(t) - Du(t»
= Xo
yet) = Cx(t)
(23)
+ Du(t)
(28)
The output estimation error where /3 E m.+ is chosen in such a way that the effect of noise is sufficiently filtered 7. It has been shown that the same signal r{ (t) can be generated using an observer based approach (Magni and Mouyon, 1994). Although this equivalence is well established for linear systems, it is not the case for nonlinear systems. As the parity space and the observer based methods will both be presented for the design of residual generators from nonlinear models, an introduction is given to observer based residual generation for LTI systems first.
el/(t)
= yet) -yet)
(29)
is directly linked to ez(t), since eu(t) = Cez(t).
4.2 State observer for fault detection and isolation A LTI system without unknown input is considered. In faulty working mode, the system is described by (3), (4) with Ed = 0 and Gd = 0, and the error equations (21), (29) become 8 A system is obeervable if the output trajectories corresponding to distinct initial states differ for at least some time instant.
7 A bold character has been UII6d here for the Laplace operator in order to avoid any confusion with the order of the derivatives and the order of the filter.
41
e:l:(t) = (A - LC)e:l:(t)
+ (E, -
LG I )/(t)(30)
e:l:(O) = Xo - Xo el/(t) = Ce:l:(t) + G II(t)
(31)
The transfer function between I and el/ is obtained by taking the Laplace transform of (30) , (31); it yields:
with !fi(t) substituted for yet). Computing the estimation error as in (30), (31), one deduces that the ith output estimation error is only affected by the ith fault, hence the possibility to isolate sensor faults by analyzing the pattern of zero and nonzero residuals. This type of diagnosis system is called the dedicated observer scheme (Frank and Wunnenberg, 1989).
H'e.(s) = C(sI - A + LC)-l(E,- LGf) + Gf. As long as none ofthe columns of H/e,,(s) is equal to zero, any non zero component of I(t) will affect at least temporarily el/(t). Thus after the transient due to initial conditions has vanished, ev(t) can be used to detect the occurrence of a fault in the system. Indeed, in the absence of fault ev(t) is close to zero and it becomes distinguishably different from zero when I(t) deviates significantly from zero. ev(t) thus Qualifies as a residual signal and (26), (28), (29) is the associated residual generator. Notice that el/(t) now depends on the whole time history of y(r), u(r), for r E IO,tj.
There are two approaches to achieve fault is0lation using Luenberger observers. The first one amounts to choosing the observer gain matrix L in such a way that the trajectories induced by each type of faults are confined to independent subspaces of the el/-space. By projecting ev(t) into each of these subspaces and analysing the magnitude oC each projection, one can achieve fault isolation (Massoumnia, 1986), (White and Speyer, 1987). We shall not go along this line here, as the second approach, based on a bank of observers, offers more flexibility. We present it in the particular case oC sensor faults. The general case is more involved and it will be described in the next section. Thus consider a system of the form (3), (4), without unknown input, in which only sensor faults are taken into account
x(t) = A:c(t) + Bu(t)
(32)
yet) = Cx(t)
(33)
+ Du(t) + I(t)
The output equation can be written component-
= Ci,.x(t) + Di,.U(t) + /i(t) i = 1," ' ,p
where C i ,. (D.,.) denotes the (D).
ith
u
Supervised
system
12 7,
~
~
Obscrvcr 1
·,1
~
~
Obscrvcrp
ew
Fig. 2. Dedicated observer scheme If some of the pairs (Ci,., A) are not observable, another option is to design p Luenberger observers, the ith one being based on all measurements but the ith output !fi(t). In this way, upon occurrence of a single sensor fault, say on sensor j, all residuals will become non zero except for the j'h one. This is the generalized observer scheme which allows one to isolate single faults, but not simultaneous faults. The above two approaches correspond respectively to coding sets of the type described in table 1 and 2. Both schemes make use of full order (or full-dimensional) state observers. However, it is not generally necessary to estimate the whole state vector in order to obtain residual signals. Reconstructing the whole state imposes more stringent conditions than needed, and this motivates the use of functional observers, namely observers that estimate only l (with l < n) specific linear combinations of the state variables.
wise as
Yi(t)
-..,
(34)
4.3 Functional obseroer lor FDI
row of matrix C
In this section and in the remaining part of the paper, it will be assumed that the fault vector I (t) does not enter the output equation. This amounts to representing sensor faults I.(t) = GJI(t) (see (4)) by pseudo actuator faults. It can be achieved
If (Ci ,., A), i = 1"" ,p are observable pairs, p Luenberger state observers can be designed on the basis of equation (32), (34), one Cor each output, as depicted in Fig. 2. They have the form (26)
42
by considering f.(t) as the output of a LT! system driven by an appropriate input, and imposing that the LT! system has no direct feedthrough term (Massoumnia et al., 1989). This method for handling sensor faults is considered in all the literature on the geometric approach to fault detection and isolation. It is adopted here in order to present a unified framework encompassing results on FDI for nonlinear systems, some of which are exclusively based on geometric system theory.
existence of constant matrices P, A, B, Ll and irJ, with P, Ll and L2 different from zero, such that
AB was seen at the end of section 3.3, once coding sets are chosen, the design of residual generators for FDI amounts to determining a set of filters that solve specific fault detection problems. Each filter (residual generator) is designed for appropriate vectors of unknown inputs and faults. Hence the fundamental problem to be solved is based on a model of the form
Indeed, when such matrices exist, one can determine the required observable subsystem as follows. Let e(t) = Px(t) be the state of this subsystem, then (35) yields 9
PA - AP= BC
PE" =0 LtC-irJP=O PE,IO
(4O) (41)
(42)
and the pair (~, A) is observable.
e=p Ax+P Bu+P E"d+P EJf (43) Substituting (39) for P A and (40) for PE", (43) becomes
x(t) = Ax(t) + Bu(t) + E"d(t} + EJf(t} (35) yet) = Cx(t} (36)
e=~ + BCx+ PBu + PE,I
(44)
By (36) the second term in the right band side is nothing but By(t) 80 that (44) finally becomes
The type of filter that will be considered in order to design a residual generator is:
xr(t) = Arxr(t) + Bru(t} + Mry(t) r(t} = Crxr(t} + Dru(t} + Nry(t)
(39)
e= A( + By + PBu + PEJf
(37)
(45)
Defining 'let) = L 1 y(t), equation (41) together with the definition of {et) imply
(38)
'l = L1Cx
The problem which has already been considered in the section on the parity space method is now stated in a more formal way; it is called the fundamental problem of residual generation.
= L2e
(46)
Now system (45), (46) is a subsystem in which input d(t) does not appear anymore. Besides, (L2' A) is observable. A Luenberger observer for this system takes the form
FUndamental problem of residual generation (FPRG) Determine a filter of the form (37), (38) such that
and the output estimation error can be used as a residual
(I) When f(t} = 0 for all t, r(t} asymptotically decays to zero for any inputs u(t) and d(t}. (2) In the presence of a fault (namely when I(t) I 0 for all t ~ to), to being the fault occurrence time), r(t) is affected by the fault; it takes a non-zero value for at least some t ~ to, whatever the initial conditions of the filter and the process.
r(t) = 'let) - ~~(t)
(48)
Indeed, computing the state estimation error e( = yields
e- e
e(t) = (A - L ~)edt) + PEJf(t)
When a solution exists, the procedure to solve the FPRG is made of two steps. First one extracts from model (35), (36) an observable subsystem which has not d(t) 88 input, but for which I(t) is an input. The second step consists in designing a state observer for the observable subsystem. The output estimation error of the observer is a suitable residual. There are several ways to achieve the first step. One of them relies on the fact that a necessary and sufficient condition for the existence of a solution to the FPRG is the
r(t)
= ~e(t)
(49)
(SO)
It is clear from these equations that neither u{t) nor d(t) can affect r(t), while I(t) will do so generally since PE, is non-zero by «(2) and (~,A LL2) is observable. If PE, has full column rank, system (49), (SO) is actually input observable 11 The time argument is omitted in :z:,{,U,II,d &Dd / fOl' the sake of compact_.
43
w.r.t. J(t) (Massoumnia et al., 1989); this means that the magnitude! of a step-like fault J(t) = !1(t) can be determined uniquely from r(t), t 2! 0 when edO) = 0 10 • System (47) is a functional observer for Px(t), hence the title of this section. The set of equations (39)-(41) can be solved either by a method involVing the transformation of the state equations to Kronecker canonical form (Frank and Wunnenberg, 1989) or by an iterative algorithm based on singular value decomposition (Kinnaert, 1999). Another approach to solve the FPRG, in which the two step procedure introduced here does not appear explicitly is the method based on observer eigeDBtructure 888ignment (Chen and Patton, 1999).
be performed around different set points, there are several ways to account for the nonlinear efect. For instance, one can use nonlinear models BB explained in sections 6 and 7, or use a bank of linear models, each one corresponding to a different set point (Arte et al., 1995). Around a given set point, modelling uncertainties due to the linear approximation and/or to the error in the estimate of the model parameters can be accounted for in different ways. One can • use a nonlinear models • use a linear models and represent the higher order terms BB unstructured uncertainties (Mangoubi, 1998), (Mangoubi and Edelmayer, 2000), (Frank and Ding, 1994), • use unknown inputs to represent the efect of model uncertainties (Patton and Chen, 1993), (Chen and Patton, 1999), (Sauter and Hamelin, 1999), • use a set of models, each one corresponding to a different possible setting of the parameters (Lou et al., 1986), (Kinnaert, 1996), • use a model in the form of transfer functions with parameters in intervals (Hamelin and Sauter, 2000)
Yet another method to solve this problem is the geometric approach. Its interest is that it can be generalized to a wide class of nonlinear systems. It yields both a necessary and sufficient condition for the existence of a solution to the FPRG and a systematic design method. The results are expressed in terms of subspaces called (C,A)-unobservability subspaces. The design method also essentially consists in extracting an observable subsystem which hBB not d(t) BB input; the operations required to achieve this goal can be performed using backward stable algorithms deduced from (Van Dooren, 1981) as shown in (Alexandre and Kinnaert, 1993).
The existence of meBBurement noise, and possibly state noise to account for disturbances with a known spectrum also requires adaptation of the theory presented in the previous section. Indeed, the residual can no longer decay to zero in the presence of noise; thus a different objective has to be imposed in its design (Nikoukhah, 1994), (Mangoubi, 1998). Besides, simple comparison of the residual to a threshold becomes impracticable when noise is important, BB the efect of a fault may be buried in noise. One has to resort to statistical change detection algorithms in this C8Be (Willsky and Jones, 1976), (Basseville and Nikiforov, 1993). The problem of robustness with respect to modelling uncertainties and noise effects is thus a very broad area.
Considering arbitrary matrices A, C, Ed, EJ of ap. propriate dimensions, the FPRG h8B a solution if and only if RJ + Rd < R and Rd < P (Massoumnia et al., 1989). In the above results, it is 8B8umed that the plant model is perfectly known and that the me8Burements are exact in fault free working mode. In practice this does not hold BB discussed in the next section.
5. DEALING WITH MODELLING UNCERI'AINTIES AND MEASUREMENT NOISE
Here we have chosen to exploit nonlinear models in order to decrease the effect of modelling uncertainties. Therefore, the exterunon of the theory presented for linear systems will successively be performed for polynomial nonlinear systems (in the parity space framework), for bilinear systems and for control affine nonlinear systems (using an observer based approach).
The linear time invariant (LT!) model used in the previous sections is suitable to describe the plant behaviour around a specific set point. AB most plants are nonlinear, this LT! model corresponds to a first order approximation of the nonlinearity in a Taylor series expansion. The further away the process moves from its nominal operating point, the more important the efect due to the higher order terms becomes. If supervision must
6. PARITY SPACE APPROACH FOR POLYNOMIAL NONUNEAR SYSTEMS The presentation in this section is in line with (!Sidori et al., 2001). A model of the form (1), (2) is considered, where g(x(t), u(t), d(t), J(t» and
IOTbe only type of fault that may not affect r(t) c:orre8pOnds to inputs U80ciated to the tranamisBion zeros of the transfer matrix between I and r in this cue.
44
Finally, let us notice that the problem of generating polynomial analytical redundancy relations can actually be solved for a more general class of nonlinear systems, namely systems described by polynomial differential algebraic equations. An informal presentation of the method based on characteristic sets for this type of model is presented in (Zhang et al., 1998)
h(x(t), u(t), d(t), J(t» are polynomial functions of their arguments. Let Yj denote the lit. output: Yj = h;(x(t), u(t), d(t), J(t». By deriving this expression si times and substituting (1) for the derivative of the state, one gets equations equivalent to (8)-(11) in the linear case (a slight generalization is considered here since the order of the derivative might differ from one output to the other, which was not the case in the linear framework). These equations can be written in a compact form as: Y1;
= Hi(x, U.j,D.i,F.j )
7. OBSERVER BASED APPROACH FOR NONLINEAR SYSTEMS
(51)
Y~ is the vector made of Yj(t) and its derivatives up to order Si, U.jI D.J and F. J are defined as
Before dealing with fault detection, some prerequisites on observability for nonlinear systems are introduced.
in the linear case. The concatenation of equations (51) for j = 1"" ,p gives a set of E~_l (Si + 1) equations
7.1 Observability oJ nonlinear systems
Y = H(x, U., D., F.)
(52)
Consider the nonlinear system
where s = maxi Si and Y = [y(.l)T y(.2)T •.• y(.p)T]T. Analytical redundancy relations are obtained by eliminating x and D. from (52). This yields polynomial relations: P(Y,U.,F,) =0
(53)
Due to the polynomial structure, this expression , can be decomposed as
P(Y, u" F,) = Pc(Y, U,) - Pe(Y, U., F,) = 0 (54)
where any term in the polynomials which make the rows of Pe(Y, U" F,) is of degree at least one with respect to an entry of F,. Pe(Y, U" F,) is thus equal to zero in faultless working mode. Hence Pc(Y, U.) can be considered as a parity vector, and one can define a residual vector as
r(t) = Pc(Y(t), U,(t»
X=g(x,u)
(56)
y=h(x)
(57)
where 9 and h are smooth nonlinear functions of their arguments. Let U denote the set of admissible inputs. By definition system (56), (57) is observable if, for every pair of initial states (XO , Xl), xD =F Xl, there exists an admissible input signal u : [0, T) - U and a time instant t E [0, T] such that Y(XO,u,t):f. y(xl,u,t) where y(xO,u,t) denotes the unique output trajectory of (56), (57) for the initial condition xO and the input u. An input u : [0, T] - U that distinguishes any pair of initial states is a universal input on [0, T). System (56), (57) is said to be uniformly observable if, for every T > 0, every admissible input u : [0, T) - U is a universal input on (0, T).
Bilinear systems are not generally uniformly 01>sevable. Indeed, consider the following simple example:
(55)
Xl =UX2
To achieve fault isolation, structured residuals have to be computed. This amounts to choosing coding sets and performing the above design with a vector of unknown inputs augmented by the faults that should not affect the residual.
X2=0
y=Xl
(58) (59) (60)
Clearly, if u(t) = 0 for all t, no information on X2(t) can be deduced from the measurement of yet) and hence this state is not observable. Any piecewise-continuous non-zero input will make X2 observable. An input for which the system is not observable is called a singular input. A specific class of non singular inputs will be used to state a convergence result for bilinear systems in the next section, namely ~ed persistently exciting inputs (Kinnaert, 1999). Roughly speaking
There are several ways to perform the elimination of x and D. that leads to (53). One approach relies on Groebner basis (Comtet-Varga, 1997); another uses characteristic sets and rutt's algorithm (Ljung and Glad, 1994). A study of existence conditions for parity functions and of fault sensitivity is performed in (Staroswiecki and ComtetVarga, 2(01);
45
such inputs guarantee the observability of the system over some horizon T at any time.
and the system
7.2 Fault detection Jor bilinear systems
{et) = A(u){(t)
(68)
'1(t) = ~e(t)
(69)
is observable. Continuous-time bilinear systems described by the following equation are considered
n"
x(t) = A(u)x(t) + Bu(t) + ~)Ftx(t)
{B(t) = A(u){B(t) + pB Bu(t) + B(u)y(t)
+ Et)~(t)
n,
+ 2: pB(F/ x(t) + EI)J(t)
i=1
n,
+ "L(F! x(t) + E{)/;(t) yet) =Cx(t)
(62)
where A(u) = Ao + 2.::1 UiAi and it is assumed that the input vectors u(t), d(t) and J(t) are such that every trajectory is defined on the whole time interval (0,00(. In this case again, the design of a residual generator can be separated in two steps. First an observable subsystem which has not d(t) 88 input, but for which J(t) is an input is extracted. Next a suitable observer is designed for the resulting subsystem.
Two types of subsystems have been used in the literature: linear time invariant subsystems up to output injection (Yu and Shields, 1996), (Mechmeche et al., 1994), and bilinear subsystems up to output injection (Kinnaert, 1999), (Hammouri et al., 2(01). The approach based on a LT! subsystem leads to a LT! residual generator, which can be implemented easily. However, the class of bilinear systems for which a residual generator based on a LT! subsystem exists is smaller than the class of systems for which a residual generator based on a bilinear subsystem exists (Kinnaert, 1999). The latter option is briefly presented here.
,!here A(u) = B(u).
...to + 2.::1 AiUi
and similarly for
Defining '1B(t) by '1B(t) = Lfy(t) = LfCx(t), one gets
(71) As the dynamics of (70) is bilinear, observers for bilinear systems now have to be considered. Contrary to the linear case, observability of system (70), (71) depends on the input u(t) 88 W88 discussed in subsection 7.1. Let u(t) be a regularly persistent input for system (68), (69). For such an input, it can be proved that the following system is an exponential observer 11 (Bomard et al., 1988) for system (70), (71) when J(t) = 0 for all t:
l
(t) = A(u){B(t) + pB Bu(t) + B(u)y(t) +R(t)-lCT ('1B(t) - Lf{B(t»
(72)
k(t) = OR(t) - A(U)TR(t) - R(t)A(u) + LFLf (73)
nf nf
where eB(O) E mn:' , R(O) is a x symmetric positive definite matrix, and 6 E m+. The output reconstruction error
r(t) = '1B(t) - L~{B(t)
nf
Without loss of generality, the -dimensional state of this subsystem can be taken 88 a set of linear combinations of the entries of x(t}
exponentially decays to zero whatever d(t),{(O), {B(O) and u(t) provided the latter is regularly persistent. The reader is referred to (Kinnaert, 1999) for a discussion regarding the sensitivity of r(t) to the fault J(t).
(63) The extraction of the bilinear subsystem consists in determining the matrices pB, Ai, Bi, i = 0, ... , rn, Lf and Lf with pB, Lf and Lf different from zero such that
nf,
pB ~ _ AipB =BiC
(70)
i=1
(61)
i=1
Using an appropriate definition of the sensitivity of a residual to a fault, a precise statement of the fundamental problem of residual generation for bilinear systems can be obtained. A necessary and sufficient condition for the existence
i = 0,··· ,m (64)
pB(Et Ft)=O Lfc - Lf pB = 0
i=l,···,nd
(65) (66)
F!) i: 0
i = 1"", nJ
(67)
pB(E{
From (64) - (67), one deduces
11ThiB means that O(B(t) _ iB(t)II :S ~ e-" for all t > 0, where >. is a. positive conatant which depends on the initial con
46
of a solution to this problem can then be expressed in a very similar form as for the linear case (Hammouri et al., 2001) by introducing the notion of (C, A)-unobservability subspaces where A stands for (.40,"', Am). An algorithm to check such a geometric condition can be determined using standard algebraic tools. A similar methodology can be used to handle state affine systems, namely systems of the form:
x(t) = A(u)x(t) + Ed(X)d(t)
+ EJ(x)J(t) (74)
yet) = Cx(t)
(75)
where A(u), Ed(X) and E,(x) are matrices of which the entries are smooth functions of the components of u and x. The required concepts have been developed in (Hammouri et al., 1998), (Hammouri et al., 2(00), (De Persis and Isidori, 2(00). The main difference with the bilinear case is that symbolic computations must be performed to implement the design method.
7.3 Fault detection Jor control affine nonlinear system Systems modelled by equations of the form (76), (77) below are now considered
x = go (x) + gu(x)u + 9d(X)d + 9,(X)J
(76)
y = hex)
(77)
Vn(
r = hr(z, h(x»
= 4>n( (Vlt t12, .•• , Vn (' y, U)
Then this system is uniformly observable, and a high gain observer can be designed to estimate the state and the output q, under some additional technical conditions(De Persis and Isidori, 2001). The residual is again obtained as r = q - h(€) where €is the estimate provided by the observer.
e
(78)
8. CONCLUSION
(79)
The parity space approach and the observer based method have been presented for the design of residual generators, first on the basis of a LT! model, next from a nonlinear plant model. Although both approaches are equivalent for LT!
(:~~~~h(X») + (:~~~:h(X»)U + (gd(X») d + (g'(X») J
= 4>1 (VI, t12, y, u)
q = h l1 (v).
with a similar smoothness condition on the different functions as above, and with 90(0,0) = 0, h r (0,0) = 0, such that the output r of the cascaded system
(;) =
where the input d does not appear anymore. Should an asymptotic observer exist for this system, one could proceed as for the linear or the bilinear case to obtain a residual generator. Unfortunately, local weak observability is not sufficient to guarantee the existence of such an observer, and extra hypotheses are needed at this stage. For instance, let us assume that system (82), (83) with J = 0, can be brought by a global change of coordinates v = ~(e) into the form
~=4>2(VI,t12,V3,Y,U)
The problem to be solved is the determination of a filter of the form
i = 90(Z, y) + 9:(Z, y)u hr(z,y)
By resorting to differential geometry, and in particular to the notion of unobservability distribution (the generalization to the nonlinear framework of (C, A)-unobservability subspaces), a necessary condition for the existence of a solution to this problem can be obtained (De Persis and Isidori, 2001). Algorithms are available to check this condition, and a methodology can be deduced to extract from the original system (76),(77) a locally weakly observable 12 subsystem of the following form
VI
with the n-dimensional state x defined in a neighborhood X of the origin, inputs u, d and output y with the same dimension as before, and scalar fault J. All the entries in the vectors 90(X),9,(X),h(x) and matrices gu(X),9d(X) are smooth functions oftheir arguments and 90(0) = 0, h(O) = O. The last two conditions can always be achieved by an appropriate translation of the state and output.
r =
is affected by J, is not affected by d, and asymptotically decays to zero when J is identically equal to zero, whatever the input u.
13 Roughly speaking a system is weakly obeervable if any state can be distinguished from its neighbors, and it is locally obeervable if a IItate can be distinguished from its neighbol'1l with an input that keeps the IItate trajectory close to the initiallltate (Hermann and Krener, 1977).
(SO) (81)
47
systems, this is not proved for nonlinear systems, in general.
et al., 1977), automobile (Gertler et al., 1995), (Isermann et al., 2000) , (Nyberg, 1999), diesel engine actuator (Blanke et al., 1995), process industries (Kinnaert et al., 2000), ...
The parity space approach can be applied to polynomial nonlinear systems. It is systematic, and existing symbolic computation softwares can be used to perform the elimination of the unknown variables. However, even for models of moderate size (5 to 10 states), the parity function resulting from the elimination might contain several tens of terms. These terms might have very different orders of magnitude, which could make the evaluation of the residual ill-conditioned. Another drawback of the parity space approach is the need for the derivatives of the signals. Such derivatives are sensitive to noise. In practice they have to be obtained most often from sampled data. Obtaining suitable estimates for high order derivatives in this context might not be an easy task.
The design of diagnosis systems based on analytical models is now reaching a certain stage of maturity. However, several research directions are still widely open. One of them is the evaluation of the reliability of such diagnosis systems and the way to take into account reliability requirements at an early stage in the design. Another one is the study of the interplay between the diagnosis system and the controller in a fault tolerant control scheme.
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