### Sliding mode observers for fault detection and isolation 评分

C: Edwards et al. Automatica 36/2000)541-553 543 and Zak use a Lyapunov-based approach to formulate coordinates xHTx such that in the new coordinate an observer which, under appropriate assumptions, ex system hibits asymptotic state error decay in the presence of bounded nonlinearities/uncertainties in the input chan x1(1)=11x1(1)+12x2()+B1u(l) nel(Walcott Zak, 1988). The strategy of Walcott and x2(t)=,m21x1()+m2x2()+B2t(t)+D2/(t)2(6) Zak, although intuitively appealing, necessitates the use of algebraic manipulation tools to effectively solve an y(0)=x2(t), associated constrained Lyapunov problem for systems of where x1∈RP),x2∈ R! and the matrix1 has stable reasonable order. Edwards and Spurgeon(1994) propose eigenvalues. The coordinate system above will be used as an observer strategy, similar in style to that of walcott and Zak, which circumvents thc usc of symbolic manip a platform for the design of a sliding mode observer Consider a dynamical system of the form ulation and offers an explicit design algorithm. This ap proach will be adopted here for fault detection purposes 1()=71R1(t)+m12(t)+B1(1)-1243 (t),(8) Consider the nominal linear system subject to certain faults described by x2(1)=,m21x1(1)+2282(t)+B2t(t) x()=Ax()+Bu()+Df() (1) q2)ep()+v, u(t)= Cx(t)+fo(t) (2)y(t)=x2(t), where A∈Rn",B∈Rn,C∈R,D∈ IR"4 with where %/2 is a stable design matrix and the discontinu qsp<n and the matrices C and d are of full rank. The ous vector v is dcfincd by functions fi(t)and fo(t) are deemed to represent actuator and sensor faults, respectively, and are assumed to be P|D2‖ ifey≠0 bounded. It is further assumed that the states of the system are unknown and only the signals u(t) and y(t)are otherwise availablc(Notc: in the casc when p= n all the statcs will be known and the approach of Sreedhar et al. (1993)may where P2 EIPl is a Lyapunov Matrix for 22 and the be adopted scalar p is chosen so that The objective is to synthesise an observer to generate IFa tl<p 2 a stale estimate i(o and output estimate i- CA such that a sliding mode is attained in which the output error If the state estimation errors are defined as er=ii-xi and e,=&2-x then it is straightforward to show e,(t)=(t)-y( (3) is forced to zero in finite time. The particular observer e1(t)=11e1(t) structure that will be considered can be written in the e, (t)=21e1(0)+si2e,(t)+v-D2fA(t), since in this situation e, =e i(t)-=Ax(t)+ Bu(t)-Grey(t)+GnY, It is shown in Edwards and Spurgeon(1994)that the where Gr, Gn ER"P are appropriale gain matrices and nonlinear error system in(13)-(14)is quadratically stable y represents a discontinuous switched component to in and a sliding motion takes place forcing ey =0 in finite duce a sliding motion. It will be shown that, provided time. The dynamical systcm in(8)-(10) may thus bc a sliding motion can be attained, estimates of f(t) and regarded as an observer for the system in(1)-(2). It fo(t) can be computed from approximating the so-called follows that if equivalent output injection signal required to maintain 12 0 sliding motion G,- T and G=T 22 2 2. 1. A canonical form for sliding mode observers then the observer given in 8)-(10)can be written in terms of the original coordinates in the form of (4) Consider the dynamical system given in(1)and(2)and assume that 2. 2. Reconstruction of the input fault signals rank(CD) Assume that an observer has been designed as in invariant zeros of (A, D, C)must lie in C Section 2. 1 and that a sliding motion has been estab Consider initially the case when fo=0 ed. During the sliding motion, e=0 and e=o It is argued in Proposition 1 in the appendix that Therefore Eq(14)becomes under these assumptions, there exists a linear change of 0=. 21e1(t)-D2fA(t)+veq (16) 544 rds et al. /Autor 6/2000)54-553 alled equivalent output injecti the equivalent nal. The equivalent output injection represents the aver tion Veg can be calculated from(18) and consequently if age behaviour of the discontinuous component v and ( 12)Is nonsingular the fault signal represents the effort necessary to maintain the motion can be obtained from Eq. (26). Note that even if on the sliding surface(Utkin, 1992). From(13), and using (o 1.12)Is singular, inference can still be the fact that si1 is stable, it follows that e1(t)0 made about certain fault channels depending on the and therefore precise nature of the rank deficiency. An example of thi D2f() situation will be given in the next subsection. Note that 17 from the Schur expansion One way to recover the equivalent output injection signal is by the use of a low-pass filter (Utkin, 1992). Here an det(a)= det(11)det(s22-521911J12) (27) alternative approach will be employed: suppose that the and thus(c 21. 911.912) is nonsingular if and discontinuous component in(11)is replaced by the con only if dct A≠0 tinuous approximation Remarks. The approach adopted here is quite di D ( 18) from that of Hermans and Zarrop(1996)since they attempt to design an ol er which no le where 8 is a small positive scalar. It can be shown that the when a fault occurs. In general, this is a difficult problem equivalent output injection can be approximated to any since most of the research work seeks to formulate gains degree of accuracy by (18 )fo which guarantee sliding by virtue of conservative Since rank(D2)=g it follows from(17)that Lyapunov arguments 1()-lD2(DD2)-2Dnh2() (19) It is interesting to note that, in terms of the analysis of P2e,(t川+8 the equivalent output error injection signal to extract the The key point is that the signal on the right-hand side of fault signal information. the observer gain parameter the equation above can be computed on-line and de 122 plays no role. The reason for this is that that the pends only on the output estimation error e linear output error injection term is used to help in establishing a sliding motion. Formally, whilst sliding 2. 3. Detection of faults at the output takes placc, e =0 and so the tcrm. i2e has no cffcct Practically however this term does have an influence on Now consider the case when f(t=0 and consider the the approximation to sliding that is obtained and should effect of fo(t). In this situation Eq. (7) becomes be designed with respect to the usual tradeoff between performance and noise amplification y()=x2(1)+f60(1) and therefore e, =e2-fo. It follows that 3. Example: inverted pendulum e1(1)=,z1e1()+,/12f0( (21) e2()=2e1(1)+2e2(1)+(y22-2)fo(t)+V, Consider the inverted pendulum with a cart. Assume the pendulum rotates in the vertical plane and the cart is (22) to be manipulated so that the pendulum remains upright or more convenientl The cart is linked by a transmission belt to a drive wheel which is driven by a DC motor. The cquations of motion e1(t)=m11e1(1)+m12f0( are e,(l)=s21e1(t)+y2n(t)-J0(1)+210(t)+v.(24)(M+mx+Fxx+ml(0cos0-02sin) Note that fo(t) and fo(t) appear as output disturbances J0+Fe0-mlg sin 0 ml cos 0= and thus p in Eq(11)must be chosen to be sufficiently large to maintain sliding in the presence of these distur- where the particular values of the system parameters bances Arguing as before, provided a sliding motion can such as rod length and masses, etc are given in Table 1 be attained A linearization of the nonlinear equations of motion has been made about the equilibrium point 0=21e1-f0()+m2.f0()+vcq Thus for slowly varying faults, if the dynamics of the sliding motion are suficiently fast 1 Sliding mode observers exist in which only the nonlinear output 21.011 (26) error injection term is present(Utkin, 1992) C: Edwards et al. Automatica 36/2000)541-553 545 Tablc 1 Parameter values for the inverted pendulum syslem M F values 0.535 0.06 0.365 6.2 0009 9807 Units kg m. i=0=0=x=0. This results in the system triple In this particular design the scalar function p=75 and 0 the observer design is complctc 00 Remark. In a situation such as the one presented here it A 19333 1987200091 is tempting to consider designing a nonlinear observer directly about the nonlinear equations of motion. It has 36.977162589-0.1738 been demonstrated however that for certain nonlinear systems sliding mode observers of the type used here 00 designed around linear approximations to the true sys B C=0100 (30) tem, yield state replication properties equal in perfor 0.3205 mance to nonlinear observers designed directly about the 00 0 -1.0095 nonlinear equations with the advantage of a more straightforward design procedure (Walcott. Corless where it is assumed that only x and i are available for Zak, 1987). In the context of this paper the nonlinear fault detection purposes. For simplicity, a state feedback equations of motion have been used to ensure a signifi controller has been used to assign closed-loop poles at cant plant/model mismatch between the system for which the observer is dcsigned and thc onc on which its FDI -42.-44-46.-48} properties are evaluate In this particular situation, any actuator faults will occur in the input channel and hence in the notation of 3.1. Simulations of different fault conditions Section 2 the fault distribution matrix d= B Using an algorithm similar to that proposed in It can be verified that the eigenvalues of A are Edwards and Spurgeon (1994)it can be shown that in 10, 5.8702,-6.3965,-1.6347 and thus from the argu the canonical form of (8)-(10 othe system described ment in Section 2.3 the steady-state gain from fo to vea is in(30) becomes singular. In this case it can be verified that 100000-676603314960 10000 A 100000 9.8548-3.1496 0.00910 1.8437 2.0158 =0308880 (34 19052 1.9872 0|100 D which is clearly rank deficient. However, if Veg and fo i represent the ith components of the vectors 0.3205 0001 Id fo, from Eq.(26)and using the particula from (34), it is ap t that where by design 11 10. In this particular case (35) 2=diag{-11;-12;-13}, which furnishes the linear component of the observer 3.0888f0.2 with poles approximately three times faster than the closed-loop poles of the controlled plant. The symmetric It is also clear that any fault in the first output channel positivc-definitc matrix P2 has bccn sclccted as the has no direct long-term effect on veg.Furthermore, be unique solution to the Lyapunov equation cause of the structure of D, in (31)it can be verified that +(/2)P (DD2)1D=[003201 (37) 546 C: Edwards et al. Automatica 36/2000)541-553 and so from(17) scribed in Section 2. 1, since, in the former, a sliding takes vq,3≈0.3205/ (38) place on the surface in the error space given by eER: FCe=0. As a result, in this particular example, Thus the three components of the equivalent output the equivalent output injection signal would have only injection signal, properly scaled, provide estimates of one component making it difficult to distinguish between fo.3,o,2 and i, respectively, and may be used as detector faults in different channels signals 3.1.1. Simulations in the absence of measurement noise 3. 1.2. Simulations in the presence of measurement noise The following nonlinear simulation results show the The following figures are from simulations of identical responsc of thesc thrcc detector signals to diffcrcnt fault ccnarios to thosc considcrcd abovc cxccpt noisc has conditions. Fig. 1 shows the effect of a ramp in the input been added to the output signal so that the measured channel. As predicted by the theory, the third detector signal which is used by the fault detection observer is signal reproduces the fault signal whilst not aecting the corrupted other two signals From Eq. (24)it can be seen that theoretically the Fig. 2 shows the effect of a ramp in the first output derivative of the noise appears in the output error channel. As predicted by the theory, the detector signals channel and hence constitutes a large disturbance. Thus do not reproduce this fault signal (although the second arbitrarily large values of p would be needed to sustain a detector signal approximates the gradient of the fault sliding motion. However, it can readily be observed from signal). Figs. 3 and 4 show that the appropriate detector Figs. 5-8 that the effects of the deliberately introduced signal reproduces the ramp fault signals in output faults are still apparent in the detection signals and the channels 2 and 3. In both cases the detector signal 3 is underlyingshape' of the fault is preserved influenced as well The noise introduced in the simulations has zero mean Any persistent low-frequency component would be Remark. This example highlights the difference between picked up by the FDI system and, if of sufficient magni- thc approach of walcott and Zak and the observer dc- tudc, would bc intcrprctcd as a fault and conscqucntly Fault Signal: Input Channel Indicator Signal I 0.1 0.05 0 0.05 15 Tim Indicator Signal 2 Indicator Signal 3 0.1 05 0 2 0.05 0 5 10 15 15 Time Time Fig. 1. Respense of the detection signals to a fault in the input channel C. Edwards et al. Automatica 36/2000)541-553 547 Fault Signal: Output Channel 1 Indicator Signal 1 0.1 0.05 005 0.05 0.05 -0.1 Time Indicator Signal 2 Indicator Signal 3 0.1 0.05 -0.05 Tim Time Fig. 2. Respense of the detection signals to a fault in the lst output channel Fault Signal: Output Channel 2 Indicator Signal 1 0.1 0.05 005 -0.05 0.05 4.1 15 Tim Time Indicator S 2 Indicator Signal 3 0.1 0.05 0.1 15 Time Ti Fig 3. Response of the detection signals to a fault in the second output channel 548 C. Edwards et al. Automatica 36/2000)541-553 Fault Signal: Output Channel 3 Indicator Signal l 0.1 0.1 0 0 -0.05 -0.05 -0.1 15 0 10 15 Time Time Indicator Signal 2 Indicator Signal 3 0.1 4 0.05 0 2 0.05 -0.1 15 0 Time Fig. 4. Response of the detection signals to a fault in the third output channel Fault Signal: Input Channel Indicator Signal I 0.1 -0.1 0.2 Time Time Indicator signal 2 Indicator Signal 3 0.2 6 0.1 0.2 Time Iim Fig. 5. Response of the detection signals to a fault in the input channel C. Edwards et al. Automatica 36/2000)541-553 Fault Signal: Ist Output Channel Indicator Signal 1 0.2 0.05 0.1 -0.05 Indicator Signal 2 Indicator Signal 3 0.2 0 Ime Fig. 6. Response of the detection signals lo a fault in the first output channel Fault Signal: 2nd Output Channel Indicator Signal I 0 0.05 0.1 -0.1 -0.2 Time Time Indicator Signal 2 Indicator signal 3 2 -0.1 5 LIme Time Fig. 7. Response of the detection signals to a fault in the second output channel 550 C. Edwards et al. Automatica 36/2000)541-553 Fault Signal: 3rd Output Channel Indicator Signal 1 0.1 0.2 0.05 0.1 -0.1 5 ime Time Indicator Signal 2 Indicator Signal 3 0.2 0.1 4 0.I 0 -0.2 15 Time Time Fig. 8. Response of the detection signals to a fault in the third output channel would constitute afalse alarm. In practice the equiva- injection signal it has been demonstrated that certain lent output error injection signals voq, i would be passed fault signals can be faithfully reproduced through an appropriate low-pass filter to extract the (assumed) low-frequency underlying fault signal from the overlaying noise. This is perfectly in keeping with the Appendix. Proofs of observer results notion of the equivalent output error injection signal as the low-frequency component of v(Utkin, 1992) Before proving the key proposition, a lemma will be Introduced which provides a useful intermediate canonI- 3.1.3. Simulations with model uncertainty cal form Nonlinear simulations have been performed in which the parameters of the pendulum have been altered so that Lemma 1. Let(a, D, c) be a linear system with p> g and M=4.0Kg, m=0.6 Kg and =0.04". The corre rank(CD)=g. Then a change of coordinates exists so that sponding results are virtually indistinguishable from the triple in the new coordinates (A, D, C)has the following Figs. 1-4. Of course no theoretical justification for this structure has been presented in this paper. This is seen as an area of (a)the system matrix can be written as 11 A A=A 211 4. Conclusions A 212 This paper has explored the use of sliding mode ideas for the purpose of fault detection and isolation. The where the matrix A1∈R( i p) and A21∈ approach adopted here differs significantly from the which when partitioned have the structure work of Hermans and Zarrop in that the underlying intention is to ensure that sliding is maintained even in A91A9 the presence of faults. By examining an equivalent output and A211=[0 A211

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