Disturbance Decoupled Fault Reconstruction using Sliding Mode Observers

Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 28 Disturbance Decoupled Fault Reconstruction using Sliding Mode Observers Ko Yew Ng Chee Pin Tan Rini Ameliawati Christopher Edwards School of Engineering, Monash University, Jalan Lagoon Selatan, Bandar Sunway, 4615 Selangor Darul Ehsan, Malaysia Engineering Department, Leicester University, University Road, LE1 7RH UK Abstract: The objective of a robust fault reconstruction scheme is to generate an accurate reconstructionofthe faultthatis unaffectedbydisturbances. A typicalmethodforrobust fault reconstruction is to reconstruct the faults and disturbances, which is conservative and reuires stringent conditions. This paper investigates and presents conditions that guarantee a fault reconstruction that rejects the effects of disturbances, which are less stringent than those of previous wor. A VTOL aircraft model is used to validate the wor of this paper. Keywords: Disturbance decoupling; fault reconstruction; actuator fault; sliding mode observer 1. INTRODUCTION Fault detection and isolation FDI is an important area of research activity. A fault is deemed to occur when the system being monitored is subject to an abnormal condition, such as a malfunction in the actuators or sensors. The fundamental purpose of an FDI scheme is to generate an alarm when a fault occurs and also to determine its location. An overview of wor in this area appears in Fran 199. The most commonly used FDI methods are observer-based, where the measured plant output is compared to the output of an observer, and the discrepancy is used to form a residual, which is then used to decide as to whether a fault condition is present. A useful alternative to residual generation is fault reconstruction, which not only detects and isolates the fault, but provides an estimate of the fault so that its shape andmagnitudecanbe better understoodandmore precise corrective action can be taen Edwards et al. 2; Saif & Guan 1993. However, as the fault reconstruction scheme is observer-based, it is usually designed about a model of the system. This model usually does not perfectlyrepresentthe system, as certain dynamics are either unnown or do not fit exactly into the framewor of the model. These dynamics are usually represented as a class of disturbances within the model which will corrupt the reconstruction, producing a nonzero reconstruction when there are no faults, or mas the effect of a fault, producing a zero reconstruction in the presence of faults Patton & Chen 1993. Hence, the scheme needs to be designed so that the reconstruction is robust to disturbances. Correspondence to: School of Engineering, Monash University, Jalan Lagoon Selatan, Bandar Sunway, 4615 Selangor DE, Malaysia, tan.chee.pin@eng.monash.edu.my Edwards et al. 2 used a sliding mode observer from Edwards & Spurgeon 1994 to reconstruct faults, but there was no explicit consideration of the disturbances. Tan & Edwards 23 builtonthis woranddesignedthe slidingmodeobserverusingthe LinearMatrixIneualities LMIs method in Boyd et al. 1994 to minimize the L 2 gain from the disturbances to the fault reconstruction. Saif & Guan 1993 combined the faults and disturbances to form a new fault vector and used a linear observer to reconstructthe new fault. Althoughthis methodmanages to decouple the disturbances from the fault reconstruction, very stringent conditions need to be fulfilled, and is conservative because the disturbance vector does not need to be reconstructed. Edwards & Tan 26 compared the fault reconstruction performances of Edwards et al. 2 and Saif & Guan 1993, and found that it was redundant to reconstruct the disturbance in order to generate a fault reconstructionthatis robusttodisturbance. Acounterexample was presented to prove this point, but the conditions fordisturbance decoupling were notformallyinvestigated. This paper builds on the wor in Edwards & Tan 26. Its main contribution is the investigation of conditions that guarantee the fault reconstruction is decoupled from disturbances. The conditions that guarantee disturbance decoupling are found to be less stringent than those in Saif & Guan 1993 which proves that disturbance reconstruction is redundant for disturbance decoupling. In addition, the conditions in this paper are easily testable on the original system matrices, maing it possible to immediately determine whether disturbance decoupled fault reconstruction is feasible. A VTOL system taen from the FDI literature will be used to validate the results in this paper. The paper is organized as follows: Section 2 introduces the system and sets up the coordinate transformation and framewor for the investigation of the existence conditions; Section 3 investigates the conditions 978-1-1234-789-2/8/$2. 28 IFAC 7215 1.3182/2876-5-KR-11.722

17th IFAC World Congress IFAC'8 Seoul, Korea, July 6-11, 28 such that disturbance decoupled fault reconstruction is achieved; an example to validate the conditions is given in Section 4 and finally Section 5 maes some conclusions. The notation used throughout this paper is uite standard; inparticular. represents theeuclideannorm forvectors and the induced spectral norm for matrices, and λ. denotes the spectrum of a suare matrix. For an arbitrary matrix H with ran m, denote by H the left-pseudoinverse with the property that H H = H 1 H I 1 T m where H 1 is an orthogonal matrix. The matrix H can be easily found using the Singular Value Decomposition SVD and is not necessarily uniue. For the case when H has full column ran, then H H = I m. 2. PRELIMINARIES AND PROBLEM STATEMENT Consider the following system ẋt = Axt + Mft + Qξt 1 yt = Cxt 2 where x R n, y R p, f R and ξ R h are the states, outputs,unnownfaults andunnowndisturbances respectivelywhileξ encapsulates allnonlinearities andunnowns in the system. Assume ranm =, ranq = h and ranc = p. Also assume that p >. The main goal is to reconstruct f whilst being robust to ξ. The robust fault reconstruction scheme in Tan & Edwards 23 minimized the effect of ξ on the fault reconstructionusinglinearmatrixineualities ifandonly if the following conditions are satisfied A1. rancm = ranm A2. A,M, C is minimum phase. Their method however, does not fully reject the effects of ξ. Saif & Guan 1993 managed to reject the effects of the ξ by combining f and ξ vectors and then designed a fault reconstructionscheme toreconstructthis new fault. The necessary and sufficient conditions for their scheme are B1. ranc M Q = ran M Q B2. A, M Q,C is minimum phase. Since only the reconstruction of f is reuired, therefore reconstructing the new fault is conservative. Therefore, this paper investigates the conditions that guarantee disturbance decoupled fault reconstruction with less stringent conditions than Saif & Guan 1993. Firstly, define = rancq where h and assume N. rancm = ranm N1. ran C M Q = rancm + rancq Notice that N1 implies p > +. The implication of this assumption will be discussed later in the paper. Proposition 1. IfNandN1holdthenthere exist nonsingular linear transformations x T 2 x, ξ T1 ξ such that A,M,C, Q when partitioned have the structure n-p A1 p n-p A A= 2 n-p p A 3 A 4,M=,C= C p M2 2 Q1 3,Q= Q 2 h p- where M 2 = with M Mo o,c 2 being invertible. Further partition Q 1, Q 2 to be Q1 Q 1 =,Q 2 = Q 2 p-- 4 n-p-h+ where Q 1, Q 2 are suare and invertible. Proof. From Edwards & Spurgeon 1994, since CM is full ran, there exists a change of coordinates such that A,M,C can be written as Ã1 Ã Ã = 2 Ã 3 Ã 4, M = where M 2 = M T o M2, C = T, Q = Q1 Q 2 T with Mo and T being full ran. Let T 1 R h h be an orthogonal matrix such that 5 6 Q 2 T 1 = Q22 p where ran Q 22 =. This follows since by assumption, CQ = T Q 2 is ran. Therefore, QT1 can be partitioned to have the structure of Q11 QT 1 = Q 12 n-p Q 22 p Using the partitions in 5, N1 results in 7 ran T M2 Q 22 = + 8 Since T is orthogonal, using 6 and 7 result in ran M2 Q 22 = + 9 The expanded structure of 7 according to 9 is Q 11 QT 1 = Q 12 Q 221 Q 222 n-p p- 1 where Q 221 and Q 222 are appropriate partitions of Q22. Hence, Q221 R p has full column ran which means there exists a matrix Q 221 such that Q 221 Q 221 = I. Let X 1 and X 2 be orthogonal matrices such that Q1 X 1 Q11 = n-p-h+ p-- 11, X 2 Q221 = Q2 Now define a nonsingular change of coordinates represented by the matrix T 2 where T 2 := n-p p- X 1 X 1 Q12 Q 221 X 2 Q 222 Q 221 I 12 such that the matrices Q, C in their new coordinates have the following structure 7216

17th IFAC World Congress IFAC'8 Seoul, Korea, July 6-11, 28 Q 1 n-p-h+ T 2 QT1 = Q p-- 2 X 2 where Y 1 = Q 222 Q 221 I, CT2 = TY1 13 and TY 1 is invertible. The coordinate transformation T 2 does not alter M. Euating M o = M o and C 2 = TY 1, the proof is thus complete. 2.1 A sliding mode observer for fault reconstruction A sliding mode observer from Edwards & Spurgeon 1994 for the system 1-2 is ˆxt = Aˆxt G l e y t + G n ν, ŷt = Cˆxt 14 where ˆx R n is the estimate of x and e y = ŷ y is the output estimation error. The term ν is defined by ν = ρ e y e y, e y, ρ R + 15 where ρ is an upper bound of f and ξ. The matrices G l,g n R n p are the observer gains to be designed. In the coordinates of 3 in Proposition 1, G n is assumed to have the structure L G n = C I 2 P o 16 p where P o R p p is symmetric positive definite s.p.d. and L = L o R n p p with L o R n p p. Define the state estimation error as e := ˆx x. Therefore, by combining 1-2 and 14, the state estimation error system can be expressed as ėt = A G l Cet + G n ν Mft Qξt 17 Introduce a change of coordinates such that e L := cole 1,e y = T e e where In p L T e = 18 C 2 Hence, A,M,C, Q in the new coordinates are A1 A A = 2, M =, C = I A 3 A p, Q = 4 M2 Q1 Q 2 19 where A 1 = A 1 + LA 3, A 3 = C 2 A 3, M 2 = C 2 M 2, Q 1 = Q 1 + LQ 2, Q 2 = C 2 Q 2 and G n will become G n = Po 2 Therefore, the state estimation error euation after the coordinates transformation is ė L t = A G l Ce L t + G n ν Mft Qξt 21 Lemma 2. From Tan & Edwards 23, if there exists a value of G l to satisfy PA G l C + A G l C T P < where P = P11 P 12 P12 T P 22 > with P 12 = P 121 n-p p- then if P o = P 22 P T 12P 11 P 12, and for a large enough choice of ρ, an ideal sliding motion taes place on S = {e : Ce = } in finite time. Assume that the observer 14 has been designed and a sliding motion has been achieved ė y = e y =. Then, 21 can be partitioned according to 19-2 as ė 1 t = A 1 + LA 3 e 1 t Q 1 + LQ 2 ξt 22 = C 2 A 3 e 1 t + Po ν e C 2 M 2 ft C 2 Q 2 ξt 23 where ν e is the euivalent output error injection term reuired to maintain the sliding motion and can be approximated to any degree of accuracy by replacing ν with e y ν = ρ e y + δ 24 where δ is a small positive scalar. Since e y is a measurable signal, therefore ν e can be computed online Edwards et al. 2; Edwards & Spurgeon 2 for full details. To reconstruct the fault, define a fault reconstruction signal to be ˆft := WC ν e where W := W 1 Mo 2 P o with W 1 R p. Then define the fault reconstruction error signal e f = ˆf f. From 23 and ˆf it becomes e f t = WA 3 e 1 t + WQ 2 ξt 25 It is desired that e f = ˆf = f. From 22 and 25, it is clear that ξ is an excitation signal of e f and that the design freedom is represented by L o and W 1. Hence the goal is to decouple e f from ξ by choice of L o and W 1. Define Q a to be the left h columns of Q in 3. The following theorem states the main result of this paper. Theorem 3. Suppose N and N1 hold. Then e f will be decoupled from ξ by appropriate choice of L o and W 1 if the following conditions are satisfied C1. ran CAQ a CM CQ ranm rancq = ran AQ a Q ranq C2. A, M Q,C is minimum phase. Note that C1 and C2 are easily testable conditions onto the original system matrices. In addition C1 is not as stringent as B1 but the disturbance could still be decoupled from the fault reconstruction. The following section provides a constructive proof of Theorem 1. 3. DISTURBANCE DECOUPLED FAULT RECONSTRUCTION In order to mae e f necessary condition is completely decoupled from ξ, a WQ 2 = 26 since itrepresents thedirect feedthroughcomponentinthe system formed from 22 and 25. Partition Q 2 from 3 and 13 such that Q21 p- Q 2 = = Q 22 Q 2 27 which results in WQ 2 = W 1 Q 21. Thus thenecessaryandsufficientconditiontoensure 26 holds is that W 1 Q 21 =. Lemma 4. A general solution for W 1 that will satisfy W 1 Q 21 = and hence maing WQ 2 = is given by W 1 = W 12 I p Q 21 Q 21 28 7217

17th IFAC World Congress IFAC'8 Seoul, Korea, July 6-11, 28 where Q 21 = Q 2 and W 12 R p is design freedom such that W 12 = W 121 W 122 p-- Proof. This is straightforward; substituting 28 and 27 into WQ 2 results in 26 being satisfied. Remar 1. If N1 is not satisfied, then ran C M Q < rancm + rancq. Since rancq = if N1 is not satisfied, then from M in 3, it is clear that ranq 21 < ranq 2 =:. Expanding 26 according to W gives WQ 2 = W 1 Q 21 + Mo Q 22. In order to satisfy 26, W 1 Q 21 = Mo Q 22 29 must be satisfied. To satisfy 29 by choice of W 1 reuires ranq 21 = ran Q T 21 Q T T 22 = ranq2 3 which is a contradiction. Therefore, N1 is a necessary condition in order for 26 to be satisfied. Lemma 4 has shown that N1 is a sufficient condition for 26 tobe satisfiedas itmaes thecoordinatetransformationinproposition1feasible. Therefore, N1is anecessary and sufficient condition for 26 to be satisfied. From the solution in 28, W 1 is constrained to be W 1 = W 12 I p Q 21 Q 21 = W 121 31 Partition A 1,A 3,L o from 3 and 2.1 as A A11 A A 1 = 12 32 p-- A 13 A 14, A 3 = n-p-h+ L o = p-- L11 L 12 L 21 L 22 A 34 A 35 A 36 32 Then, from W 1 in 31, WA 3 can be written as WA 3 = W 121 A 31 + Mo A 35 W 121 A 32 + Mo A 36 33 The terms A 1 + LA 3 and Q 1 + LQ 2 in 22, when expressedusingthe system partitions in 32 will produce A11 +L A 1 +LA 3 = 11 A 31 +L 12 A 12 +L 11 A 32 +L 12 A 34 A 13 +L 21 A 31 +L 22 A 14 +L 21 A 32 +L 22 A 34 Q1 L Q 1 +LQ 2 = 12 Q2 L 22 Q2 Therefore, from W 1 in 31 and L o in 32, the system 22 and 25 form the following state space system ė11 t = A ė 12 t 1 + LA 3 }{{} n-p-h+ Ā e11 t Q e 12 t 1 + LQ 2 }{{} B ξ1 t 34 ξ 2 t e11 t e f t = WA 3 35 }{{} e 12 t C Expressing 34-35 in terms of the triple Ā, B, C the state space matrices will have the form Ā1 Ā Ā= 2 B1 Ā 3 Ā, B= B 2, 4 n-p-h+ B C= C1 C2 4 36 n-p-h+ It is obvious that e 11 will be affected by ξ 1 because B 1 is full ran. However, e 12 can be decoupled from ξ by setting B 4 = L 22 = and Ā3 =. In order for e f not to be affected by e 11 and subseuently ξ, it is essential to mae C 1 =. Maing Ā3 = and C 1 = respectively reuires rana 31 = ran A13 A 31 since M o is nonsingular. and rana 31 = ran A 35 Therefore, combining the reuirements that satisfy both Ā 3 = and C 1 = reuires E1. rana 31 = ran A T 13 A T 31 A T 35 From 35, to mae C 1 = W 121 A 31 M o A 35 =. Hence, a general solution for W 121 that satisfies C 1 = is W 121 = M o A 35 A 31 + W 1211I A 31 A 31 37 where W 1211 is design freedom. Lemma 5. The condition E1 is satisfied if and only if ran CAQ a CM CQ ran CM CQ = ran AQ a Q ranq Proof. See A in the appendix. T 38 Since it has been assumed in 2 that rancm = ranm, then Condition N1 becomes ran C M Q = ranm + rancq 39 Substituting this result into 38 in Lemma 5 yields ran CAQ a CM CQ ranm rancq = ran AQ a Q ranq 4 which corresponds to Condition C1. Substitute L 22 = into 34 to get Ā3 = A 13 + L 21 A 31 and choosing L 21 = A 13 A 31 + L 211I A 31 A 31 41 where L 211 is design freedom, maes Ā3 =. As a result, Ā1 Ā Ā = 2 42 Ā 4 In order for Ā to be stable, Ā 1 and Ā4 have to be stable. Using 41, Ā 4 from 34 and 36 can be written as Ā 4 = A 14 A 13 A 31 A 32 + L 211 I A 31 A 31 A 32 43 So for Ā4 to be stable, A 14 A 13 A 31 A 32,I A 31 A 31 A 32 must be detectable. Liewise Ā1 can be written as Ā 1 =A 11 + L 11 A 31 + L 12 =A 11 + L 11 L 12 44 This implies that A 11, has to be detectable if Ā 1 is to be stable. Proposition 6. A 14 A 13 A 31 A 32,I A 31 A 31 A 32 and A 11, are detectable if A, M Q,C is minimum phase. Proof. See B in the appendix. Proposition6matches ConditionC2, guaranteeingastable sliding motion and the proof of Theorem 1 is complete. 7218

17th IFAC World Congress IFAC'8 Seoul, Korea, July 6-11, 28 Remar 3.6 of Edwards & Tan 26 provided a counter-example where B1 is not satisfied but it is still possible to reconstruct the fault robustly. In the notation of 1-2, the matrices that describe the example are A = 1 1 2 1 1 1,M = 1,Q = 1,C = 1 1 45 C1 and C2 are satisfied for the matrices in 45 above. Remar 2. Note that C1 is less conservative than B1, and hence can be applied to a wider class of systems. Condition B1 implies that Q a = empty matrix, which will satisfy C1. However, the converse is not necessarily true. Remar 3. There have been efforts to generate disturbances decoupled fault detection residuals using linear observers in Xiong & Saif 2 and Patton & Chen 2 which respectively utilize the Special Coordinate Basis and Eigenstructure Assignment. However, their methods reuired certain elements in the matrix A to be zero. Fromtheanalysis inthis paper, nosuchconditionis reuired; the only reuirement on the matrix A is that E1 is satisfied. Hence, this paper has also shown how the conditions for robust fault reconstruction using sliding modeobserver is less stringentthaniflinearobservers are used. 4. AN EXAMPLE The method proposed in this paper will be verified using a VTOL aircraft model taen from Saif & Guan 1993 where the states are the horizontal velocity, vertical velocity, pitch rate and pitch angle. The inputs are the collective pitch control and the longitudal cyclic pitch control. Assume thatthe horizontalandverticalvelocities and the pitch angle are measurable and that both inputs are faulty. Thus,.4422.1761 1 3.5446 7.5922 B = M = 5.52 4.49, C = 1 1 Suppose that the matrix A is imprecisely nown and that there exists parametric uncertainty such that ẋ = A + Ax + Bu + Mf 46 where A is the discrepancy between the nown matrix A and its actual value. Due to the nature of the state euations, any parametric uncertainty will appear in the third row of A. Writing 46 in the framewor of 1 using Ax = Qξ = 1 T.5 2 x }{{}}{{} Q ξ The coordinate transformation in Proposition 1 shows that Q 1 =,A 11 = 1.6459 and Q 2,A 12,A 13 and A 14 are all empty matrices. Also, Q 2 = 3 1 and from 27, it is easy to see that Q 21 =. Choosing W 1211 =.75 1 T and substituting into 37 and then into 31, yields W 1 =.7475.3824 T which satisfies 26. It can be shown that C1 and C2 are satisfied, hence it is possible toobtainafaultreconstructionthatis decoupled from the disturbances. Since ranq = 1 and ranm = 2 and ran C M Q = 2, then B1 is not satisfied and hence the method in Saif & Guan 1993 cannot be used. Inthe followingsimulation, the parameters associatedwith ν were chosen as ρ = 5, δ =.1. The faults were induced in both actuators and Figure 1 shows the faults and their reconstructions. It can be seen that ˆf provides accurate estimates of f that are independent of ξ. 1.8.6.4.2.2.4.6.8 Actual faults 5 1 15 2 Seconds 1.8.6.4.2.2.4.6.8 Reconstructed faults 5 1 15 2 Seconds Fig. 1. The left subfigure is f, the right subfigure is ˆf. 5. CONCLUSION This paper has investigated and presented conditions thatguarantee disturbancedecoupledfaultreconstruction, which are easilytestable ontothe originalsystem matrices. In previous wor, the effects of the disturbances on the fault reconstruction were only minimized, and conditions that guarantee disturbance decoupling were not nown. Otherworhas achieveddisturbance decouplingbyreconstructingthedisturbances togetherwiththefault, butthis reuires stringent conditions to be fulfilled and the disturbance reconstructionis notneededforanalysis. This paper has successfully investigated the conditions such that the fault reconstruction is decoupled from the disturbance, and it was found that the conditions are less stringent compared to earlier wor. A VTOL aircraft model was used to validate the proposed method. REFERENCES S.P. Boyd, L. El-Ghaoui, E. Feron, and V. Balarishnan. Linear Matrix Ineualities in Systems and Control Theory. SIAM: Philadelphia, 1994. C. Edwards and C.P. Tan. A comparison of sliding mode and unnown input observers for fault reconstruction. European Journal of Control, 12:245 26, 26. C. Edwards and S.K. Spurgeon. A sliding mode observer based FDI scheme for the ship benchmar. European Journal of Control, 6:341 356, 2. C. Edwards, S.K. Spurgeon, and R.J. Patton. Sliding mode observers for fault detection and isolation. Automatica, 36:541 553, 2. C. Edwards and S.K. Spurgeon. On the development of discontinuous observers. Int. Journal of Control, 59:1211 1229, 1994. P.M. Fran. Fault diagnosis in dynamic systems using analytical and nowledge based redundancy - a survey and some new results. Automatica, 26:459 474, 199. R.J. Patton and J. Chen. Optimal unnown input distribution matrix selection in robust fault diagnosis. Automatica, 29:837 841, 1993. R.J. Patton and J. Chen. On eigenstructure assignment for robust fault diagnosis. International Journal of Robust and Nonlinear Control, 1:1193 128, 2. M. Saif and Y. Guan. A new approach to robust fault detection and identification. IEEE Trans. Aerospaceand Electronic Systems, 29:685 695, 1993. 7219

17th IFAC World Congress IFAC'8 Seoul, Korea, July 6-11, 28 C.P. Tan and C. Edwards. Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int. Journal of Robust and Nonlinear Control, 13:443 463, 23. Y. Xiong and M. Saif. Robust fault detection and isolation via a diagnostic observer. International Journal of Robust and Nonlinear Control, 1:1175 1192, 2. Appendix A. PROOF OF LEMMA 5 From the partitions in 3-4 and 32 as well as the structure of Q a in Theorem 3, it can be shown that CAQ a = C 2 A Q 31 1 Q1 A.1 A 35 Q1 Using the partitions of M,Q from 3-4 results in rancaq a CM CQ rancm CQ = rana 31 A.2 since C 2,M o and Q 2 are full ran and invertible. Also, it can be shown that A 11 Q1 A 13 Q1 AQ a = A 31 Q1 A 33 Q1 A 35 Q1 As a result, the right hand side of 38 becomes ran AQ a Q ranq = ran A Q 13 1 A 31 Q1 A 35 Q1 A.3 A.4 Since Q 1 is invertible, 38 is euivalent to rana 31 = ran A T 13 A T 31 A T T 35 A.5 which corresponds to E1 and the proof is complete. Appendix B. PROOF OF PROPOSITION 6 From Edwards & Spurgeon 1994, the Rosenbroc system matrix associated with A, M Q,C is si A M Q E a,1 s = B.1 C and the zeros of the system are the values of s that cause its Rosenbroc matrix to lose normal ran. From 3 and 32, it can be shown that E a,1 s loses ran if and only if E a,2 s loses ran where si A14 A E a,2 s := 13 B.2 A 32 A 31 Hence, the invariant zeros of A, M Q,C are the invariant zeros of A 14,A 13,A 32,A 31. Define r = rana 31, by a singular value decomposition A 31 = R 1 R A 2 B.3 312 where R 1,R 2 are orthogonal matrices and A 312 R r r is invertible. Define A 31 = RT 2 A R1 T 312 Since by assumption rana 31 = ran A T 13 A T T 31 then the structure of A 31 in B.3 results in A 13 = A 132 R 2 where A 132 R n p h+ r. Partition A321 p---r A 32 = R 1 A 322 r B.4 B.5 Substituting B.3 - B.5 into B.2 and pre-multiply it with T z where T z = I n p h+ A 132 A 312 I p r B.6 I r then it is clear that E a,2 s loses normal ran when the following matrix pencil loses ran si A14 A E a,3 s = 132 A 312 A 322 B.7 A 321 From the Popov-Belevitch-Hautus PBH ran test Edwards & Spurgeon 1994, the values of s that cause E a,3 s to lose ran are the unobservable modes of A 14 A 132 A 312 A 322,A 321. Therefore, the invariant zeros of A 14,A 13,A 32,A 31 are the unobservable modes of A 14 A 132 A 312 A 322,A 321. Now evaluate the pair A 14 A 13 A 31 A 32,I A 31 A 31 A 32 using the new structures of A 31, A 13 and A 32 introduced in B.3 - B.5. It is easy to verify that A 14 A 13 A 31 A 32 = A 14 A 132 A 312 A 322 B.8 I A 31 A 31 A A321 32 = R 1 B.9 Therefore, if A, M Q,C is minimum phase, then A 14 A 13 A 31 A 32,I A 31 A 31 A 32 is detectable. If A, M Q,C is minimum phase, then A,C is detectable. Performing the PBH ran test on A,C and expanding in the coordinates of 3, then the detectability of A,C implies the detectability of A 1,A 3, which by expanding further in the coordinates of 32 further implies the detectability of A 11, A T 13 A T 31 A T 33 A T 35 T. However, since by assumption the condition in E1 holds, the detectability of A 11, A T 13 A T 31 A T 33 A T T 35 implies that A 11, is detectable. Hence, if A, M Q,C is minimum phase, then A 11, is detectable and the proof is complete. 722