Frequency-Weighted Robust Fault Reconstruction Using a Sliding Mode Observer

Transcription

1 Frequency-Weighted Robust Fault Reconstruction Using a Sliding Mode Observer C.P. an + F. Crusca # M. Aldeen * + School of Engineering, Monash University Malaysia, 2 Jalan Kolej, Bandar Sunway, 4650 Petaling, Jaya, Malaysia, tan.chee.in@eng.monash.edu.my # Deartment of Electrical and Comuter Systems Engineering, Building 72, Monash University, Melbourne, Victoria 3800, Australia. francesco.crusca@eng.monash.edu.au * Deartment of Electrical & Electronic Engineering, University of Melbourne, Parkville, Victoria, 300, Australia. aldeen@unimelb.edu.au corresonding author Keywords: Sliding mode observer, robust fault reconstruction, frequency weighting filter. Abstract: A new method for the design of robust fault reconstruction filters is introduced in this aer. he method is based on shaing the ma from disturbance inuts to the fault estimation error through frequency weighting functions and formulating the roblem as a robust observer design where the inut-outut error ma is minimized by solving a set of matrix inequalities. A numerical examle consisting of ten states, two known inuts, four measured oututs, one disturbance and three faults is considered. hrough this examle, it is shown that the frequency shaing renders far suerior erformance than existing conventional methods when roerly incororated in the observer design roblem. Page of 29 4/2/2006

2 Introduction Fault detection and isolation (FDI) has been the subject of extensive research for some time now, but esecially since early 990s [-4]. he interest in this line of research stems from its ractical alication to a variety of industries such as aerosace [5, 6], energy systems [7, 8], industrial alications [9], [0], and rocess control [], to name a few. he main function of an FDI scheme is to detect a fault when it haens, which may then be acted on in a variety of ways, such as sending alarm signals, taking rotection measures, or reconfiguring a running control scheme. he most commonly used schemes are observer based [4]. wo aroaches have been successfully extensively studied: the residual generation aroach [2] and the direct aroach [2]. In the residual generation aroach, a signal is generated and flagged when an abnormal (faulty) condition takes lace in the system. In the direct aroach the fault is directly reconstructed using unknown inut observer theory and its location and characteristics are determined online. A shortcoming of all of these aroaches is that changes in the system oerating condition and or variations in the system arameters result in estimation error and may lead to inaccurate detection of faults. o overcome this roblem, sliding mode observers have been roosed, for their ability to account for model inaccuracy. his articular class of observer based techniques was first reorted by a grou of researchers [3], [4], [5]. However, in these aroaches the sliding motion is used to detect the resence of fault only and not its location or characteristics. Page 2 of 29 4/2/2006

3 An imrovement on this aroach is reorted in [6] [7] where the nonlinear term that maintains the sliding motion is used to reconstruct the fault and therefore rovide full information about it. A major restriction of this aroach was its inability to deal with simultaneous faults, such faults that occur simultaneously in actuators and sensors, for examle. his restriction was subsequently relaxed by [8] where a sliding mode observer scheme caable of simultaneously reconstructing both sensor and actuator faults was roosed. However, in those works, there was no consideration for robustness of the fault reconstruction (from any nonlinearities or uncertainties). In [9] this work was extended by introducing a design method for the observer using Linear Matrix Inequalities (LMIs) such that the L 2 gain from the nonlinearities/uncertainties to the fault reconstruction is minimized. In this aer, we roose a further imrovement to [20] by introducing frequency shaing features in the design of sliding mode observers. he frequency characteristics of the inut disturbances are first determined. hen, these disturbances are assumed to be the outut of a filter (with the reviously mentioned frequency characteristics). he filter dynamics are then augmented with the original system, and a robust sliding mode observer [9] is designed for the augmented system. his is shown to be quite effective in enhancing the design feasibility and thus roducing far suerior results in fault detection and identification. Furthermore, the roosed frequency-shaing aroach is caable of detecting and reconstructing multile simultaneous faults on line and in real time. Page 3 of 29 4/2/2006

4 o illustrate the salient features of the new aroach, we consider a 0 th order system and inject different combinations of simultaneous faults in the state and outut measurements. We will show that for this articular system, the frequency shaing roduces exact fault reconstruction while the alternatively designed observer without frequency shaing fails to do so. 2 Problem Statement In this aer we consider a system (lant) of the following descrition: () x = A x + Bu+ M fx + Qξ y = Cx + Du+ N fy (2) where n m x is the state vector, u is the inut vector, y is the outut q k vector, f is the actuator faults vector, ξ is an unknown rocess disturbance x vector, and r f is the sensor faults vector. he matrices A, B, M, Q, C, D, N y are aroriately dimensioned matrices associated with the standard state-sace model of the lant. he term Qξ may be considered to include both rocess disturbances and also lant uncertainties [4]. We assume (as in [7]) that matrices C, N, full rank, and n > r+q M all have. he roblem addressed in this aer is to estimate the actuator and sensor fault signals in the resence of the unknown disturbances using a sliding mode observer [2] of the form: Page 4 of 29 4/2/2006

5 xˆ = Axˆ+ Bu Gley + Gnν (3) yˆ = Cxˆ+ Du where ˆx is the state vector of the observer, ŷ (4) is the outut vector of the observer, ey y ˆ y is the outut estimation error, ν is a nonlinear switching term defined by e y ν = ρ, e 0 where ρ is a ositive scalar, and G y l and Gn are gain matrices of the e y observer to be designed. Conventionally this roblem has been fully addressed and solved in [7] using an LMI based otimization aroach. Defining the oerator : ξ ef, the roblem is conventionally formulated as follows: min H where e f is the fault estimation error. In this aer we extend existing results by introducing a frequency shaing filter into the design rocedure thus allowing the design to be otimized over secific frequency ranges. he new roblem formulation is, therefore, min Ω where Ω is a H frequency shaing filter matrix with state-sace data ( A, B, C, D ) Ω Ω Ω Ω. We show that this extension results in a new LMI based roblem, which when solved yields a significantly suerior estimation outcome. his will be verified through an extensive set of comarison studies using a 0 th order faulty system, where the faults under various scenarios are estimated by both this new aroach and the conventional method [7]. Page 5 of 29 4/2/2006

6 3 Develoment of Fault Detection Method 3. State-Sace Model In the following the lant outut described by equation (2) is first artitioned into faultfree and fault-deendent arts. he fault deendent art is then assed through a dynamic filter [8], which is then augmented with the lant state equation and the frequency shaing filter to roduce a standard overall state-sace model. Let orthogonal matrix such that: r be an N r 0 r = N ; r 2 r y r y Cx + Du r = y = Cx Du 2 + N2 fy (5) r where N is invertible and C, C 2, D and D 2 are aroriate artitions of C and D. 2 Now introduce the following filter for the fault-deendent comonent of the outut, y 2 : ( ) z = A z y = A z A C x A D u A N f (6) f 2 f f 2 f 2 f 2 y where A f is chosen as a stable matrix. Let a frequency weighted filter be defined as z 2 = AΩz2 + BΩξ and its outut be defined as ξ Ω 2 k = C z + DΩξ, where ξ ; z h 2, h k. he matrices A, B, C, D Ω Ω Ω Ω are aroriately dimensioned matrices that reresent the state-sace model of the filter. Substituting these definitions into () yields: Page 6 of 29 4/2/2006

7 x = A x + Bu+ M f + QC z + QDξ x Ω 2 Ω (7) Augment equations (7), with the filter equations to obtain: n x A 0 QCΩ x B M 0 QDΩ fx r z AfC2 Af 0 z AfD 2 u 0 AfN 2 0 = + + ξ f + (8) y h z A Ω z B Ω n r h m q r k x r y C 0 0 D z r z = 0 I 0 + u r 0 z n r h 2 m (9) he system of equations (8) - (9) can now be written in the standard state sace form: x = Ax + Bu + Mf + Qξ (0) y = Cx+ Du () A sliding mode observer in the form of (3) - (4) can now be designed for this system to reconstruct the fault signals f, [2]. 3.2 Robust Fault Reconstruction Using a Sliding Mode Observer heorem : If the following conditions are satisfied (i) rank(cm)=q+r ; and (ii) the zeros of (A,M,C) (if any) are stable, then for the state-sace model described by equations (0) Page 7 of 29 4/2/2006

8 and () there exists a change of coordinates such that the trile (A,M,C) can be rewritten as: A A 0 Q 0 = = = [ ] = = 2 A, M, C 0, Q, M2 A A M Q M o (2) where A. he matrix ( n ) ( n ) is orthogonal, Q ( n ) k, M 2 q, and M o q q is invertible. Any unobservable modes of (A, A 2 ) are the invariant zeros of (A,M,C) and are stable. Full roof is given in [2]. A sliding mode observer for this system of the form described by equations (3) and (4) has been roosed in [2], where, in the coordinates of (2), the gain matrix G n is assumed to have the following structure G n L = P o ; L ( n ) o = L 0, where P o is a symmetric ositive definite matrix, and o ( n ) ( q) L. If we define the state estimation error as e xˆ x, then from equations (0), (), (3), and (4) the following error system is obtained:. e= ( A GC) e+ Gν Mf Qξ (3) l n It has been roven in [9] that the error vector e is norm bounded, and a sliding motion will take lace in finite time on the surface S { e: Ce 0} e = = if the following conditions hold: P P L Condition (i): there exists a matrix P with the structure P = 0 LP P LPL > ; + o P R ( n ) ( n ), that satisfies P( A G C) + ( A G C) P < 0 l l. Page 8 of 29 4/2/2006

9 Condition (ii): if the scalar ρ in the switching function ν satisfies the following 2 P A PQ ξ ρ + ) o 2 o 2 where μ o 2 inequality > ( P Q ξ ) + ( P M f max ( P( A GC l ) ( A GC l ) P) μ = λ Observer Design Introduce a further change of coordinates L I n = 0 L on the arameters in (2) to obtain. ( ) ( ) A + LA A + LA L + A + LA 0 + A =, M = A M 2 A 2 L A 22 2 (4) Q + LQ 0 2 Gl, C = 0 I, Q, G, G = n l Q = = P G 2 o l,2 (5) he structure of C in (5) imlies that the bottom elements of the error e are the outut estimation error e y. Partition the error system (3) in the new coordinate system as follows:. ( ( ) ( ) l ) e = ( A + LA ) e + A + LA L + A + LA G e ( Q + LQ )ξ (6) , y 2 ( ). e y A e A L A G e Q ξ M f P l,2 y 2 2 o = ν (7) Assume that a sliding motion has occurred, and hence ey. = ey = 0. hen equations (6) and (7) reduce to:. e = ( A + LA ) e ( Q + LQ )ξ (8) 2 2 Page 9 of 29 4/2/2006

10 0 = A e Q ξ M f P o ν eq (9) where ν, which is a version of ν required to achieve and maintain sliding motion. he eq signal ν is comutable online by relacing ν with ν where eq δ νδ ey = ρ e + δ y (20) and δ is a small ositive scalar that governs the degree of accuracy of ν eq [2]. Furthermore, by relacing ν with ν, there will be no singularity when δ e y = 0, which is feasible in ractice. his results in ey being driven towards S e but it will not slide erfectly on S e and will be bounded inside a small boundary layer around S e. he term δ in (20) needs to be chosen small enough so that the boundary layer is negligible (so that the fault reconstruction analysis in (8) - (9) is alicable, because it is based on the assumtion that ey. = e = 0 y ), but not too small that it causes difficulties in terms of the numerical integration methods necessary to solve the differential equations. Now, we define an estimate for the fault f signals as: ^ f (2) W P ν o eq where W W M o q ( q) with W being a design matrix [9]. Page 0 of 29 4/2/2006

11 Pre-multily both sides of equation (9) by W and define ef fˆ f as the fault estimation error to get the following fault error equation: e f = WA2e + WQ2ξ (22) Equations (8) and (22) show a state-sace system from the uncertainty vector ξ to the fault estimation error e f. In the ideal case where there is zero uncertainty, the vector e f will be zero and the fault estimate ^ f will be an exact relication of the fault f. he roblem that needs to be solved now is to minimize the effect of the uncertainty on the fault estimation error, by aroriately choosing matrices W, L and Gl. his roblem can be formulated and solved using LMIs in the form of the Bounded Real Lemma [9]. he observer design is such that the L2 gain from the uncertainty ξ to the fault estimation error e f is minimized, i.e Ω < γ. he roblem osed is therefore equivalent to the H following: Minimize γ with resect to the variables P, W, E 2 subject to the following matrix inequalities: PA + A P + PA A2 P2 PQ PQ 2 2 A2 W Q P Q2 P2 γ Ik Q2 W < 0 WA2 WQ2 γ I q (23) PA + A P γ C ( D D ) C PB E o d d d B P γ I H d o + k E H γ I o q < 0 (24) Page of 29 4/2/2006

12 P P 2 P = > 0, P = [ P P P ] 2 22 (25) where P = P, P = P with P, P, P, P. ( n ) ( n ) ( n ) ( n ) ( q) he other matrices are defined as: B = [ 0 Q] ; D = [ D ] ; H [ 0 WQ ] [ ] E = WA E. 2 2 d d 0 = ; and 2 Solution: Provided that conditions and 2 are satisfied, the LMI solver will return the values of P, W and γ, which are then used to calculate L= P P ; 2 G = γ P C ( D D ) ; ( P = P P P P ) ; l o d d o G n L = P o. he arameters γ o, D > 0 are user-defined design arameters which can be used as a means of tuning the gains. It is clear that when γ o increases, the value of γ decreases, which results in G l having a larger gain. Decreasing the gain of D has the same effect. he final ste is to erform the inverse of the first coordinate transform on G l and G n to get back to the original coordinate system. When the observer has been designed, let the transfer function from ξ to e f be G(s). herefore the transfer function from ξ to e f is Ω ( sgs ) ( ) where Ω( s) is the transfer function of the filter characteristics of ξ. In the case of the unweighted system, Ω ( s) is simly an identity matrix, which is the same as in [9]. However, for the frequency weighted case Ω ( s) is given. hus, if the frequency content of the disturbance ξ is known, then ( s) Ω can be chosen such that it has high Page 2 of 29 4/2/2006

13 singular values at those articular frequencies. his will cause the transfer function from ξ to e f, Ω ( sgs ) ( ) contribution of this aer., to have low singular values at those frequencies, which is the main It is worth mentioning that the objective of the design method roosed in this aer is to introduce additional flexibility into the design rocedure. his is achieved by lacing emhasis over a desired frequency range by adding frequency weights to shae the solution. herefore the conventional erformance otimization roblem has been greatly enhanced. he effect of this can be seen in (8) where a set of states associated with the frequency weighting filter is introduced. his is reflected in inequalities (23) - (25) and as a result the feasible domain of solutions to the affine matrix inequalities (23) - (25) is now a function of the frequency-weighting filter introduced. It should be borne in mind that according to Bode s integral relationshi [22] any erformance gain over one frequency will generally be obtained at the exense of some erformance deterioration over some other (comlementary) frequency range or ranges. herefore what is ossible is that one may trade-off erformance imrovements over one set of frequency ranges (of interest) with ossible erformance deteriorations over a comlementary set of frequency ranges (which are not of interest). his has been demonstrated by the examle given in Section 4. Page 3 of 29 4/2/2006

14 4 Design Examles A 0 th -order system with two known control inuts, one unknown external disturbance inut has been generated to test the erformance of the new aroach resented in this aer. o illustrate the fault detection caabilities, we introduced two faults in the state equations, reresenting actuator failures, and one fault in the outut measurements, reresenting sensor failure. he arameters for this test system in the notation of () - (2), which have been randomly generated by MALAB, are listed below: A = B = ; M = ; Q = ; N P 0 = 0 0 Page 4 of 29 4/2/2006

15 C = In order to be able to design a frequency-shaing observer, the frequency characteristics of the unknown inut disturbance is required to be known. While this is normally the case in ractice, for this articular Matlab generated examle, we arbitrarily select the frequency range of the external disturbance vector to be to 0.5 rad./s. Furthermore, the disturbance ξ and faults f are assumed to be norm bounded by ξ <.5 and f < Sliding Mode Observer Design In the following we design two sliding modes observers for the system described above. he design of the first is a straightforward alication of existing theory [8] (without frequency weighting), while the design of the second is based on the aroach of this aer, where frequency shaing is used. Later in section 5, detailed comarison of the two erformances is rovided. 4.. Observer design without frequency weighting Combine x and z to obtain: Page 5 of 29 4/2/2006

16 x 0 0 A x B M fx Q u z = ξ AC f 2 A f z + AD + f 2 0 AN f 2 f + y 0 (26) y C 0 x D u z = 0 I + r z 0 (27) For the observer design, the design arameters were chosen as γ o =00 and D =I 4, and A f = -. he LMI toolbox returned the following arameters in the original coordinates. 0 4 G = l G = n W = It is obvious that the value of ρ can be calculated from condition (ii), as by assumtion ξ and f are known. For the examle resented in section 4.2., ρ was chosen to be 200 (since the right-hand side of condition (ii) turned out to be 70, and this serves as a lower bound on ρ ). Page 6 of 29 4/2/2006

17 4..2 Observer design with frequency weighting he sectral density of the frequency-shaing filter is shown in Figure. Figure : Frequency resonse of shaing filter In the design of an observer, the frequency shaing filter of figure is used to shae the ath between the external disturbance and the observer outut error. For the system of equations (8),(9), the same design arameters were used as in the revious section. As a result the LMI toolbox has returned the following observer gains Page 7 of 29 4/2/2006

18 0 5 G = l G = 0 n W = For this case, in order to guarantee sliding motion on e, a value of S 4 ρ > must be chosen to satisfy condition (ii). Hence ρ was icked to be 50. By way of comarison, the sigma lots for the frequency-weighted and frequency-unweighted cases are shown in Figure 2. Insection of this figure reveals that a significant imrovement in the attenuation of the error ma from the disturbance inut, ξ, to the fault estimation error, e f, is obtained over the low frequency range of interest ( rad./s) to the study. Page 8 of 29 4/2/2006

19 Figure 2: he sigma values of the ma from the disturbance, ξ, to the fault estimation error, Dashed line is without frequency weighting, dotted line is with frequency weighting. e f 5 Simulation Studies In this section we reort on the erformance of the two observer-based fault detection schemes designed in section 4. he erformance of each of the two schemes is tested through Simulink. hree case studies are considered as outlined below. In the remainder of this aer we will refer to the frequency weighted design as our observer, and to the frequency unweighted design as the alternative design. In each of the three case studies, the system is subjected to the following conditions.. he control inuts are reresented by ste signals of magnitude, acting at and 5 seconds on channels and 2 resectively. 2. he disturbance is reresented by a sine wave of amlitude, with a frequency of 0.03 radians er second alied at time t = 0 (s) Page 9 of 29 4/2/2006

20 hese control inuts and disturbances are shown in Figure 4. Figure 3: he inuts and disturbance In the following studies, the initial condition of the states is arbitrarily set to x = o [ ] observer initial states are set to zero value. and the 5. Case Study # his initial study tests the erformance of the designed sliding mode observer. he resonses of the outut variables are shown in Figure 4. he figures demonstrate the convergence roerty of the observer, as the observer oututs converge to the true oututs of the lant after around 0.5 seconds. And once the convergence takes lace, the observer emulates exactly the behaviour of the lant, as exected. Page 20 of 29 4/2/2006

21 Figure 4: Case study frequency-weighted observer convergence roerties. Outut resonses: actual (solid), estimated (dotted) he resonses of the fault detection filter are shown in Figure 6, for the following two scenarios: (i) when the fault detection observer is designed without frequency-weighting using existing theory [9]; and (ii) and when the fault detection observer is designed with frequency-weighting, using the novel aroach roosed in this aer. he aim of this study is to comare the two cases and highlight the advantages that are aarent when frequency weighting is incororated into the observer design rocedure. his is evident in the figures where the frequency-weighted filter does not resond to any of the inuts or disturbances, while the traditional observer drifts away after aroximately 0 (s), indicating the resence of some sort of fault when in fact there is none. Page 2 of 29 4/2/2006

22 Figure 5: Resonses for case study #. 5.2 Case Study #2 In this study, we assume that the lant is in equilibrium oerating under a normal (fault free) condition. We also assume that the observer was switched on for long enough for it to track the states of the lant. he same scenario as described in Case Study # is used, however, the following faults are now introduced. At time t = 6 (s), a fault is introduced into the first of the two fault inut channels, then at time t = 2 (s), a further fault is introduced into the second of the two fault inut channels. Finally, at time t =8 (s), a sensor fault is introduced into the outut fault channel. he urose of this study is to (i) detect the occurrence of the faults; (ii) identify the seriousness of the fault by reconstructing it in its entirety; and (iii) comare the fault detection erformances of the Page 22 of 29 4/2/2006

23 roosed frequency weighted observer against an existing unweighted observer. Figure 6 shows the resonses of the traditional and the frequency weighted observer based fault detection filters. he three figures show the reconstructions of the fault signals, the first two being the actuator faults, f x, and the third being the sensor fault, f y. From these figures it can be concluded that while the alternative observer fails to reconstruct the faults, the roosed frequency weighted observer is able to reconstruct the three simultaneous faults in their entirety in real-time. It can also be concluded that the roosed fault detection filter is insensitive to external disturbances, and therefore would only react when a fault or a combination of faults occur. Figure 6: Resonses for case study #2. Page 23 of 29 4/2/2006

24 5.3 Case Study #3 In this study, we reeat the same scenario of case study 2, but with the following alterations. A fault in actuator occurs at time t = 8 (s) and is maintained thereafter. A fault in actuator 2 occurs at time t = 5 (s), and is then cleared after second. However the sensor channel is assumed to be fault-free for the entire eriod. he simulation results are shown in Figure 7. he first two figures show the reconstructions of the actuator faults by the two observers. he third figure shows the reconstruction of the sensor signal, which is fault free. Here again the figures further demonstrate the fact that while the alternative observer fails to accurately identify any of the faults, our frequency shaing aroach rovides exact detection and identification of all the faults, as well as rovides information about their magnitude, all in real time. Note that when the fault on actuator 2 is cleared at t =6 (s), the fault filter detects it by switching back to no-fault mode. It can be observed that while the frequency-weighted observer rovides excellent fault reconstructions on all three fault channels, the fault reconstructions rovided by the alternative observer tend to drift away from the true values which, after a eriod of time, will rovide false fault signals. Page 24 of 29 4/2/2006

25 Figure 7 Resonses for case study #3. 6 Conclusions A new fault detection scheme has been introduced in this aer. he scheme has been shown to be able to rovide exact information about any fault or a combination of faults when and where they occur. he main advantages of the roosed aroach are: (i) robustness can be incororated in the design through inclusion of uncertainties or arameter variation in the model of the external inut term; (ii) a reviously ineffective design method can be made effective through the introduction of frequency shaing refilters, as demonstrated in the design examle; and (iii) only one observer design is Page 25 of 29 4/2/2006

26 required to detect any number of faults, rovided that the outut measurement contains enough information about the state of the system. he new scheme has been tested on a 0 th order system with a total of three inuts and four oututs. he system is assumed to have two ossible actuators faults and one sensor fault. In addition, the system is also assumed to have an unknown disturbance. wo sliding mode fault detection filters are designed, with and without our frequency shaing aroach. he erformance of the two schemes is then tested through simulation using a number of different fault scenarios. Simulation results have shown that in all of the studied cases the alternative design failed to accurately reconstruct any of the faults, while our fault detection scheme was not only able to accurately detect the occurrence of faults, but also determine their exact locations. References [] P. M. Frank, "Analytical and qualitative model-based fault diagnosis - a survey and some new results," Euroean Journal of Control, vol. 2,. 6-28, 996. [2] P. M. Frank and X. Ding, "Survey of robust residual generation and evaluation methods in observer-based fault detection systems.," Journal of Process Control, vol. 7, , 997. [3] R. J. Patton and J. Chen, "A survey of robustness roblems in quantitative modelbased fault diagnosis," Alied Mathematics and Comuter Science, vol. 3, , 993. Page 26 of 29 4/2/2006

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29 [8] C. P. an and C. Edwards, "Sliding mode observers for reconstruction of simultaneous actuator and sensor faults," resented at 42nd IEEE International Conference on Decision and Control, 9-2 Dec. 2003, Maui, HI, USA, [9] C. P. an and C. Edwards, "Sliding mode observers for robust detection and reconstruction of actuator and sensor faults," International Journal of Robust and Nonlinear Control, vol. 3, , [20] C. P. an and C. Edwards, "Sliding mode observers for robust detection and reconstruction of actuator and sensor faults," International Journal of Robust and Nonlinear Control, , [2] C. Edwards and S. K. Surgeon, "On the develoment of discontinuous observers," International Journal of Control, vol. 59, , 994. [22] H. W. Bode, Network analysis and feedback amlifier design. Princeton, N.J.: Van Nostrand, 945. Page 29 of 29 4/2/2006